kij1

Master Thesis
Supervisor: PETER LØCHTE JØRGENSEN Author: QIAN Zhang (402847)

Pricing of principle protected notes embedded with Asian options in Denmark ---- Using a Monte Carlo Method with stochastic volatility (the Heston Model)

Aarhus School of Business and Social Science 2011

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Acknowledgements

My gratitude and appreciation goes to my supervisor Peter Lø chte Jø rgensen, for his kind and insightful discussion and guide through my process of writing. I was always impressed by his wisdom, openness and patience whenever I wrote an email or came by to his office with some confusion and difficulty. Especially on access to the information on certain Danish structured products, I have gained great help and support from him.

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Abstract

My interest came after the reading of the thesis proposal on strucured products written by Henrik, as is pointed out and suggested at the last part of this proposal, one of the main limitations of this thesis may be the choice of model. This intrigues my curiosity on pricing Asian options under assumption of stochstic volatility. At first, after the general introduction of strucutred products, the Black Scholes Model and risk neutral pricing has been explained. The following comes the disadvanges of BS model and then moves to the stochastic volatility model, among which the Heston model is highlighted and elaborated. The next part of this thesis is an emricical studying of two structured products embbeded with Asian options in Danish market and follows with a conclusion. Key words: structured products, Asian options, Black-Scholes model, stochastic volatilty, Heston model, calibration

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Table of Contents

Chapter 1 introduction .................................................................................................................. 6 1.1 structured products and its development in the world ...................................................... 6 1.2 problem statement ............................................................................................................. 9 1.3 structure of the thesis ....................................................................................................... 10 1.4 data.................................................................................................................................... 11 1.5 delimitations ..................................................................................................................... 11 Chapter 2 Risk Neutral Pricing and Black Scholes Framework .................................................... 13 2.1 binominal tree and risk neutral valuation ......................................................................... 13 2.2 continuous-time stochastic process of the underlying ..................................................... 14 2.3 Black Scholes Model .......................................................................................................... 15 2.4 Risk Neutral Valuation ....................................................................................................... 16 2.5 Black-Scholes formula for Vanilla call and put options ..................................................... 17 Chapter 3 Disadvantages of Black Scholes Model ...................................................................... 18 3.1 shortcomings of Gaussian distribution ............................................................................. 18 3.2 clustering effects and leverage effects ............................................................................. 21 3.3 The Volatility Smile ............................................................................................................ 22 Chapter 4 Assumption of Stochastic Volatility and Heston Model ............................................. 26 4.1 moving to the stochastic volatility model ......................................................................... 26 4.2 the formula of stochastic volatility model and its feature ................................................ 27 4.3 The mixing solution in pricing options .............................................................................. 29 4.4 the Heston Model ............................................................................................................. 32 4.5 advantages and disadvantages of the Heston Model ....................................................... 34 Chapter 5 Monte Carlo Method in Option Pricing ...................................................................... 36 5.1 different numerical methods ............................................................................................ 36 5.2 introduction of Monte Carlo ............................................................................................. 37

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5.3 generating random numbers ............................................................................................ 40 5.3 variance reduction techniques .......................................................................................... 42 5.5 Monte-Carlo simulation with the Heston model .............................................................. 43 Chapter 6 Calibration of parameters in Heston model and their effects ................................... 47 6.1 calibrations of parameters in Heston model..................................................................... 47 6.2 Average Relative Percentage Error and comparison with Black Scholes Model .............. 51 Chapter 7 Two specific structured products and their pricing.................................................... 52 7.1 pricing of the bonds part in PPN ....................................................................................... 52 7.2 pricing of the embedded Asian options and structured products .................................... 54 7.2.1 Råvarer Basis 2010 ..................................................................................................... 54 7.2.2 Russiske Rubler 2009-2012 ........................................................................................ 56 Chapter 8 Conclusion of the pricing ............................................................................................ 59 Appendix 1 .................................................................................................................................. 62 Appendix 2 .................................................................................................................................. 64 Appendix 3 Ito’s lemma .............................................................................................................. 65 Appendix 4 PPN with DJAIG index ............................................................................................... 66 Appendix 5 PPN with foreign exchange ...................................................................................... 67 Appendix 6: Black & Scholes Assumptions.................................................................................. 68 Appendix 7: Derivation of Black & Scholes differential equation ............................................... 69 Appendix 8: the derivation of risk-free portfolio ........................................................................ 71 Appendix 9 BScall function .......................................................................................................... 73

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Chapter 1 introduction
1.1 structured products and its development in the world
In finance, structured products, also known as a market-linked product, are generally innovative and flexible investment products based on different derivatives, such as a single security, a basket of securities, options, indices, commodities, debt issuances and/or foreign currencies, and to a lesser extent, swaps.1 It is the varieties of products that can be included in a structured product demonstrate that there is no single, uniform definition of a structured product. A common feature of a group of structured products, which are known as “principal protected notes” (PPN), is a principal guarantee function, which offers protection of principal, usually a percentage of the principal if it is held to maturity. Figure 1 elaborating one example could be found in Appendix 1. Using the leftover funds the issuer purchases the options and swaps needed to perform different investment strategies according to specific investment needs. Theoretically an investor is able to perform the strategy by himself, but the transaction costs and volume requirements of many options and swaps become obstacles to individual investors. Since the specific needs of investors cannot be met from the standardized financial products, structured products gradually becomes preferable in the financial market. The strength of structured products lies in flexibility and tailored approach to investing. With structured products, the risk exposed to the market has been reduced to certain and controllable level，meanwhile， investors still utilize the market trend with the leftover fund invested among more risky instruments. Investors on structured products may not be so interested in PPN with reduced risk. They may prefer a different performance feature. Let us look at another example. If the return on the underlying asset (RASSET) is positive - between zero and 7.5% - the investor will earn double the return (e.g. 15% if the asset returns 7.5%). If RASSET is
1

Structured Derivatives: New Tools for Investment Management A Handbook of Structuring, Pricing & Investor Applications (Published by Financial Times, London), Meraj Mattoo (1997),

7

greater than 7.5%, the investor's return will be capped at 15%. If the asset's return is negative, the investor participates one-for-one on the downside (i.e. no negative leverage). There is no principal protection. Figure 2 in Appendix 1 shows the graph of this scenario. More flexibility can be achieved by different derivatives that are embedded with structured products. For instance, a rainbow note is one that offers exposure to more than one underlying assets. A look-back is another popular feature. In such an instrument, the value of the underlying asset is based not on its final value at expiration, but on a maximum or minimum value during the life of the option. Besides, there are also another type called Asian option, in which the value of the underlying is the average performance according to the note’s term. customers diversified investments preferences. Nevertheless, there are also some potential disadvantages of structured products. They may include:3 1. Credit quality of the issuers - Although the cash flows are derived from other sources, the products themselves are legally considered to be the issuing financial institution's liabilities. 2. Lack of liquidity - structured products rarely trade after issuance, normally investors just follow and a buy-and-hold strategy. Anyone looking to sell a structured product before maturity should expect to sell it at a significant discount. 3. Highly complex pricing - the complexity of the return calculations means few investors truly understand how the structured product will perform relative to the underlying asset and how much the fair price should be. Besides some mentioned critics from both academic world and practitioners, structured products are yet sold in large quantities. Structured products originally became popular in Europe and have gained currency in the U.S where they are frequently offered as SEC-registered products, which mean they
2

2

These features contribute to satisfy

Understanding Structured Products by Katrina Lamb, CFA http://www.investopedia.com/articles/optioninvestor/07/structured_products.asp 3 Investopedia, Structured Notes: Buyer Beware!

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are accessible to retail investors in the same way that as the stocks, bonds and mutual funds are.
4

Now they are gaining popularity in the US. According to the news from

Bloomberg, Banks have sold $38.4 billion of structured notes in the U.S. this year, breaking the 2008 record as investors turn away from stocks and toward fixed-income products. 5 One reason for the popularity may lie in that, comparably, structured products generally provide the potential for higher yield than corporate debt. In addition, when the performance in the stock market is not as good as past after the financial crisis, investors are more interested in structured products. The increasing popularity combined with the critique, makes structured products quite controversial. From the critique, the complexity and ambiguity in pricing are believed to be very obvious and crucial. Meanwhile for investors, the competitiveness pricing is also ranked to be the first key criteria when selecting a structured product. This common emphasis indicates that pricing of structure products are of great interest and importance to both issuers and potential investors. Similarly, this is also the reason why the pricing of structured products are interesting from a research perspective. When it comes to the Danish market, the typical structured product issued and sold to retail investors belongs to the subclass called principal protected notes (PPN).6 From Figure 3, see Appendix 2, which shows frequency of distribution of the number of issues by main option category, Asian options are ranked as the second largest category, only after the basket options. The typical payoff structure for a principal protected note is shown in equation (1.1) where P is the principal, PR is the participation rate and OP is the option payoff Payoff= P + P *PR *OP (1.1)

A specific example of a payoff function could be a note with a call option written on a stock index, this is shown in equation (1.2) Payoff= P + P * PR * max
4

,0)

(1.2)

Understanding Structured Products by Katrina Lamb, CFA http://www.investopedia.com/articles/optioninvestor/07/structured_products.asp 5 http://www.businessweek.com/news/2010-10-15/structured-note-sales-reach-38-4-billion-to-set-u-s-record.html 6 Thesis prorosal, structured products pricing and performance evaluation, by Henrik Nørholm, 2010

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Where maturity.

the initial value of the stock is index and

is the value of the index at

The participation rate (gearing) represents the percentage at which the investor participates in the appreciation of the underlying asset.7 It is an extremely important factor for potential investors, the higher the better. It is not a surprise to see that these two types of options are so popular. The reason is simply that when the price of the option is cheap, the participation rate becomes higher. Higher participation rate makes the structured products more attractive to retail investors. If basket options include more low correlated financial products, they could be cheaper since less volatility will be involved. Asian options are relatively cheaper than vanilla options since their payoff depend on the average to the performance of the underlying assets. Even though the participation rate could be manipulated to look higher, this is not going to be deep and further explained here.

1.2 problem statement
This thesis will only focus on the pricing of PPN with Asian options embedded. In the thesis proposal on structured products written by Henrik, the total costs and the hidden costs for the structured products in Denmark, or more precisely PPN, have been calculated and analyzed. As is pointed out and suggested at the last part of the proposal, one of the main limitations of this thesis may be the choice of model in initial pricing of the embedded options and therefore better and more precise results may be achieved by using a more advanced model other than the simple Black Scholes Model, in pricing the embedded options. Therefore, in this thesis, my interest will focus on pricing Asian options under assumption of stochastic volatility. Based on the above background, my problem statement would be: 
7

Why the pricing of PPN with Asian options embedded is interesting?

Thesis prorosal, structured products pricing and performance evaluation, by Henrik Nørholm, 2010

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

Why more advanced model with stochastic volatility (Heston model), instead of the constant volatility, should be implemented in pricing Asian options?



Which volatility should be used in practice and how to determine and interpret the parameters?

 

Pricing Asian options with Heston model in a Monte Carlo framework The conclusion of the pricing process under Heston assumptions

1.3 structure of the thesis
This section is constructed to give the potential audience an overview of the content that will be presented in this thesis. In Chapter 1, it starts with an introduction of structured products and its developments, especially with a focus on PPN with Asian options embedded. Problem statement, structure, data and delimitations will be mentioned. In the next chapter, risk neutral valuation and Black-Scholes framework will be explained. These are the basis and the very starting point in option pricing theory. In chapter 3, the disadvantage of the BS model will thoroughly studied and analyzed. More specifically, it includes the phenomenon of non-Gaussian distribution, the clustering and leverage effect and the volatility smile. Chapter 4 will focus on the assumption of stochasitc volatility and among different stochastic volatility models, the very popular Heston model will be expecially emphasized and ellaborated, including its parameters and critical analysis on its advantages and disadvantages. In the following Chapter, an explanation of choice of pricing method will be given. Then, the specific method named Monte Carlo Method will be explained, including its intuition, theoretical background, procedure and advantages. Furthermore, the generation of random numbers in MC method, variance reduction techniques and the discretization of the Heston model will be discussed. In chapter 6, the focus will be on the calibration of the Heston model to the market prices of vanilla options and followed by a thorough analysis on the effect of different

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parameters to the price of options. In the next chapter, two PPN embedded with Asian option in the Danish market will be priced under the assumptions of stochastic volatility (Heston model). The parameters in the Heston model will be calibrated from the market price of the vanilla call options. In Chapter 8, there will be conclusion for the pricing using Monte Carlo Method with Heston model based on the content of all previous chapters.

1.4 data
The empirical part of this thesis is to price two specific PPN in the Danish market under the assumption of stochastic volatility model (Heston model). One is Rå varer Basis 2010 from Nordea Bank Denmark A/S, and the other one is Russiske Rubler 2009-2012 from Garanti Invest Denmark. The prospectus for each product has been downloaded from its website. (Appendix 3 and Appendix 4) Information has been collected, including ISIN code, name of the product, date of issue, date of expiration, final observation date, name of issuer, issuer rating(at the time if issuance), issue organizer, currency, nominal amount issued, issue price, expiration price(if product has expired), participation rate, guaranteed price/ protection level, published annualized total costs, coupon, option type, type of underlying asset, name of underlying assets and number of underlying assets.8 In order to price the bond part, the risk-free discount factor in the Danish market has been collected from DataStream. Based on this, a Nelson-Siegel model is used to get an estimation of zero-coupon term structure. In the pricing of the embedded options, at very first, the quotes of vanilla call option with the same underlying are necessary for the calibration of the Heston model. And this information is from ivolatility.com, which is a commercial database for option quotes.

1.5 delimitations
There can be many different types of stochastic volatility model, while in this thesis the Heston model is chosen and implemented. There is possibility that other model will give
8

These are consistent with the information that has been collected in Henrik’s thesis proposal.

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different or better results and this has not been covered in the thesis.

Another limitation of this thesis is the way to achieve the calibration of Heston model. The Excel Solver has been used and it can be argued that more advanced instrument could be applied for the calibration part. The calibration risk of the Heston model can be very crucial since it has five unknown parameters; therefore, whether the Excel Solver is good enough for calibration is beyond the scope of this thesis. Furthermore, another difficulty comes from the access to data. It is not that easy to get the access to the quotes of vanilla call options, especially when they are traded over-the-counter. This makes the pricing of Asian options under the assumption of stochastic volatility more complicated and time consuming than using the simpler Black-Scholes model. Finally, in this chapter, only two specific PPN with Asian options embedded have been chosen and priced, they are just two examples to explain and show how to price under assumption of stochastic volatility and not enough to draw a conclusion based on this very limited coverage.

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Chapter 2 Risk Neutral Pricing and Black Scholes Framework
When considering the pricing of PPN with Asian options embedded, actually it can be broken down to two parts, one is the pricing for the principal protection part and the other is the pricing of Asian option. In option pricing, or more general, derivative pricing, Black Scholes Framework has built the foundation in Finance. Comprehensive derivations will not be presented here since they are not the essence of this thesis. However, the focus will mainly be on the results and concepts that are relevant to the following introduction of stochastic volatility. And the theory and explanation here is mainly from the book named Options, Futures and other derivatives by Hull (2009).

2.1 binominal tree and risk neutral valuation
The assumption of Black Scholes Model begins with the process under which the underlying asset in option pricing follows. At the very first, the valuation of options with stock is set to be a simplified situation, in which the price of the stock in the life of an option is governed by a one-step binominal tree. Then a risk-free portfolio consisting of a stock option and a certain amount of the underlying stock can be set up. According to the basic assumption in finance theory, no arbitrage opportunities should exist in the market, which indicates that this risk-free portfolio should have the same return rate as the risk-free interest rate. This enables to price the stock options in terms of stock. Furthermore, it should be pointed out that no assumption of the probabilities of up and down movements is required here, even though it could be a bit against the intuition one may have. The pricing process is very similar when the process is extended to be a multi-step binominal tree. The current value of the options is obtained by working backward from the end of the life of this option. The same as the one-step binominal tree, no-arbitrage assumption is used while assumption of the probabilities of up and down movements is not required. This leads to a very important principle in options valuation that we can assume the options move in a risk-neutral world. And it has been showed by using both numerical

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examples and algebra in Hull (2009) that no-arbitrage arguments and risk-neutral valuation is equivalent and lead to the same results in option pricing.

2.2 continuous-time stochastic process of the underlying
Any variable whose value changes over time in an uncertain way is said to follow a stochastic process, which can be classified as discrete time and continuous time. In a continuous-variable process, the underlying variable can take any value within a certain range, whereas in discrete-variable process, only certain discrete values are possible. Even though stock prices are observed to discrete values, the continuous-variable process proves to be a useful model in many purposes.9 This chapter will be based mainly on Hull (2009) and Wilmott (2008). The stochastic process usually assumes that a stock price is a geometric Brownian motion, which means the return for a holder of a stock in a small period of time is normally distributed and independent in two non-overlapping periods. This will be elaborated in a step-by-step way by following concepts: 1. A Markov process is one where only the present value of the variable is relevant for predicting the future. The past history and the way how the present value is evolved from the past are irrelevant. 2. A Wiener process dz is a process describing an evolution of a normally distributed variable. The drift of the process is 0 and the variance is 1.0 per unit time. This means that if the value of variable at time 0 is distributed with mean and standard variation , then at time T it is normally .

3. A generalize Wiener process is a process describing an evolution of a normally distributed variable with drift a and a variation rate of per unit of time, where a

and b are constants. This means that if the value of variable at time 0 is , then at time T it is normally distributed with mean and standard variation b .

4. An Itôprocess is a process where the drift and variance rate of x can be a function of both x itself and time t. The change in x in a very short period of time is, to a
9

Chapter 12, page 281, Options, Futures and other Derivatives by Hull (2009)

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good approximation, normally distributed, but its changes over a longer period of time is liable to be non-normal. 5. Itôlemma is a way of calculating the stochastic process followed by a function of a variable from the stochastic process followed by the variable itself. See Appendix 3. It plays a very important role in derivative pricing. A key point is that the Wiener process dz underlying the stochastic process for the variable is exactly the same as the Wiener process dz underlying the stochastic process for the function of the variable. Both are subject to the same source of uncertainty. Finally, generalized Brownian motion is defined as: dS = µSdt + σSdz (2.1)

The two key aspects of this model is µ and σ, µ is the expected percentage return by an investor of a stock and σ is the standard deviation of percentage return in a short period of time. Besides Marko v, geometric Brownian motion has another important property called Martingale, which means that the expected value of variable at any future time is equal to its value today. The Black-Scholes model is based on the geometric Brownian motion assumption.

2.3 Black Scholes Model
A main breakthrough in option pricing is the development of the Black-Scholes model, which has such a great influence on trades both trading and hedging options and was recognized as the Nobel Prize for economics in 1997. The building blocks of this model are delta hedging and no arbitrage, from which the model has been formed and has performed well since the idea went public. The stochastic differential equation is obtained by exploiting the correlation between the underlying assets and an option to make a perfectly risk-free portfolio. This portfolio consists one long position on option and one short position on the underlying in some quantity △, delta. Π = V(s,t) - △S (2.2)

The portfolio is created to be riskless in a short period of time by delta hedging (when

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we choose Δ =

).Generally, any reduction in

randomness is termed “hedging”. The enforcement of this delta hedging represents an insurance against any loss where the loss on one position will be perfectly offset by the gains in the other position in the portfolio. With the assumption of no arbitrage opportunity, the return of a risk-free portfolio is equal to the risk-free interest rate. dΠ = rΠdt (2.3)

The price of an option V which depends on the underlying asset S and t must satisfy the Black-Scholes equation as following: +rS =0 (2.4)

The detailed Black-Scholes assumptions and the derivation of the Black-Scholes differential equation will be presented in Appendix 6 and Appendix 7. The Black-Scholes equations for European call and put options can be derived by either solving their differential equation or by using risk-neutral valuation.

2.4 Risk Neutral Valuation
One way to determine the derivatives price is to solve the partial differential equation (PDE), and the other way to find the price of derivatives is the so-called martingale approach or risk-neutral valuation. Risk neutral refers to an artificial random walk that has little to do with the path an asset is actually following. Both the real and risk neutral random walk have the same volatility, and the difference is in the drift rates. The risk neutral random walk has a drift that is the same as the risk-free interest rate. The martingale approach for pricing derivatives is based on the following observations: 1. The variable µ (the drift of the underlying price process of S) does not exist in the Black-Scholes equation (2.4) 2. The Black-Scholes equation is independent of all variables that are affected by risk preferences Therefore, the solution of BS equation must be the same in a risk-free world as it is in the real world. The payoff of a derivative can be valued using risk-neutral valuation by using the following procedure:

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1. assume that the expected return for the underlying asset is the risk-free interesting rate, r ( i.e. assume µ= r) 2. calculate the expected payoff from the derivative 3. discount the expected payoff at the risk-free interest rate

2.5 Black-Scholes formula for Vanilla call and put options
It is possible to obtain the Black-Scholes formula for vanilla call and put options by solving the Black Scholes differential equation explicitly.10 The BS formulas for the prices at time 0 of a European call and put option with non-dividend paying are: C= P= K Where (2.7) = (2.8) N( )-K N( )N( N( ) ) (2.5) (2.6)

The function N(x) is the cumulative probability distribution function for a standardized normal distribution. In other words, it is the probability that a variable with a standard normal distribution, ф (0, 1), will be less than x.

10

Options, Futures and other derivatives Hull(2009), page 313

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Chapter 3 Disadvantages of Black Scholes Model
Even though the Black Scholes Model has set such an important foundation in financial engineering in the past 30 years and been really recognized by both academia and practitioners, it is also well known and accepted that this model is not that accurate in capturing the features in the stock markets in reality, mainly because the idealized assumptions do rarely hold in the real world. First of all, the assumption of a normal distribution of log-returns is under challenge and critique ever since 1963 by Mandelbrot.11 Combined factors of extreme events, fat tail, high peak and the volatility clustering effects, assumption of non-Gaussian distribution is more appropriate. Secondly, the volatility smile (the volatility input to the Black-Scholes formula that generated the market price for the option) 12 is simply a violation of the constant volatility assumption. In short, in this chapter I will elaborate the disadvantages mentioned in the last paragraph with detailed information and arguments. And this will naturally point interest and necessity to investigate the Stochastic Volatility Model in the following chapters.

3.1 shortcomings of Gaussian distribution
It has been a long time in history that economists believed that prices in speculative markets, such as grain and securities markets, behave very much like random walks13, which is based on two assumptions: (1) price changes are independent random variables, and (2) the changes conform to some probability distribution.14Prior to Mandelbrot (1963) who challenged this long existing tradition, it is believed and accepted that the price changes in a speculative market is approximately Gaussian or normal. Before this assumption is going to be checked and challenged, some formula and features of Gaussian distribution will be represented. In the study of financial time-series, it is a concept to describe the actual return
11 12 13 14

Mandelbrot (1963a,b) The volatility surface,2006 Mandelbrot and the stable paretian hypothesis, page 2 by Eugene F. Fama 1965 Mandelbrot and the stable paretian hypothesis, Page 2 by Eugene F. Fama 1965

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distribution, where data or the variable turns to cluster around the mean. And the most important two parameters are the mean µ and the variance .The probability density

function for a Gaussian distribution is given by the following formula: P(x) = ( ) (3.1)

It has some notable properties including:    symmetry around its mean µ, therefore the skewness of the distribution is 0 both the mode and the median are the same as the mean µ the inflection points (points where the curve changes sign) of the curve occur one standard deviation away from the mean, i.e. at µ-σ andµ+σ  the kurtosis (describes the peakedness of a distribution) is equal to 3

Unfortunately, these properties are not suitable in capturing the probability of extreme events in the market. Take for example the famous crash in stock market of October 1987. Following the standard paradigm, the stock market returns are lognormal distributed and have an annualized volatility of 20% (which is usually believed to be 15%-60%). On October 19, 1987, the two month S&P 500 futures price went down for 29%. Under the lognormal assumption and according to the calculation from the probability density function, the probability of this event is , which is virtually

impossible.15 In history of stock market, this is not the only event that has such a tiny probability, but it actually happened. Besides the difficulty in dealing with historical extreme events, the actual return distributions in stock market have shown fatter tails and higher peak than the normal distribution. In figure 3.1, the S&P 500 log-returns are shown and compared with the normal distribution. It is quite obvious that the distribution of log-returns of S&P 500 have fatter tail and higher peak than the normal distribution. The higher peak of a distribution and the more weight on the tails are, the greater Kurtosis of the distribution is. The calculated Kurtosis for S&P 500 log-returns is 8.6764 from Eviews. Fat tails and the
15

This example is extracted from the article Recovering probability distribution from option prices, by Jackwerth and Robinstein, 1996

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high central peak are characteristics of mixture of distributions with different variances (Gatheral 2005). This is a motivation to move from assumption of constant volatility to assumption of stochastic volatility. From Table 3.1, the very high Kurtosis (much higher than 3) may indicate that the null hypothesis that the distribution of S&P 500 index is a normal distribution should be rejected. Furthermore, the skewness is -0.1297 not that close to 0. The results from Jarque-Bera test is 3579.98, which strongly indicates that the null hypothesis of normality should be rejected.

log return of S&P 500 standard normal distribution

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 3.1 comparison between the log return distribution and normal distribution

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Eviews Mean Median Maximum Minimum STD.DV Skewness Kurtosis Jarque-Bera Probability Observations -0.000011 0.000268 0.109572 -0.094 0.013427 -0.129712798 8.676365 3579.9799 0 2661

Table 3.1 S&P 500 index daily log returns 2001-2011 statistics, using statistical functions in Excel All above mentioned factors are against the assumption of a Gaussian distribution. And more drawbacks of Black Scholes model is going to be discussed following.

3.2 clustering effects and leverage effects
0.15

0.1

0.05

0

-0.05

-0.1

-0.15

Figure 3.2 the clustering effect in volatility dated back to 1991/03/15 Source: own calculation Figure 3.2 is a graph of log returns of S&P 500 in the past 10 years. We can see a trend that large moves follow large moves and small moves follow small moves, which is the so-called “volatility clustering”. This implies that actually the volatility of the

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log-returns is auto-correlated. In the model, this is a consequence of mean reversion of volatility.16 Furthermore, another feature of the volatility, to be more precise, the negative correlation between the log-returns and its volatility (named the “the leverage effect”) , has also been seen from empirical studies. This could also be explained from intuition. When the return of equity becomes negative, the reactions from the investors will be more volatile, thus the volatility will increase. Otherwise, investors will gain more confidence in the speculative market; therefore the volatility in the near future would decrease. Consequently, this is also an implication that the constant volatility assumption is far away from the reality. The EARCH model actually is able to catch this effect in the stock market.

3.3 The Volatility Smile
It is important to first understand the concept of implied volatility before we can move to discuss the volatility smile. In the Black-Scholes model, the inputs including , T, K, r, σ are needed in order to get the value of an option. Except one important parameter, others can be observed from the market directly or be specified in the option contracts. This exception is the volatility σ, which is assumed to be knows and constant. Different approaches have been suggested and exploited to assess the volatility which should be plugged in the formula. One is the approximation from historical data, and the other is by inverting the Black-Scholes formula. The latter one is called implied volatility, and it has been most used in practice. The higher the potential volatility is, the high return an option may offer. Just the same as the yield of a bond is quoted as the price for it, the implied volatility is often quoted as the price for an option. The more accurate definition of implied volatility is that it is the volatility of the underlying assets which when substituted into the Black-Scholes formula gives a theoretical price equal to the market price (Wilmott 2007).

16

The volatility surface, page 33, by Gatheral 2005

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If the assumption of constant volatility in the BS model would hold in the market, the implied volatility of the underlying one could get given a spot price with different maturities and strikes should be the same. In other words, using Black Scholes option pricing model, for options with the same expiration date, the implied volatility is expected to be the same regardless the value of the strike price. However, this is simply not the case. Reality showed different implied volatility across various strikes. This disparity is known as the volatility smile, the skew or the term structure of implied volatility.

Figure 3.3 implied volatilities across maturities backed out from the SPX options17
17

This figure is taken from Stochastic volatility models with application to volatility derivatives, by Sibel Ucar & Liis Kivila 2009

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Rubinstein( 1985) and more recently Jackwerth & Rubinstein (1996) and many other researchers have studied the phenomenon of volatility smile for equity options.18 Table 3.2 is from Rubinstein (1996) and shows that how implied volatility changes through different time-to-maturity. And this has become more widely observed after the big stock market crash in October 1987, since the volatility surface was rarely flat afterwards.19 The problem with this phenomenon is that, if Black Scholes could not account for implied volatility, it could not produce a reliable hedge ( Derman 2003).

Black-Scholes Implied and Historical Time Series Volatilities At-the-Money implied volatility 0.193 0.199 0.262 0.153 0.194 0.169 0.153 0.133 Historical Volatility: Prior Sampling Period 28 days 0.137 0.145 0.149 0.116 0.108 0.139 0.073 0.122 91 days 0.133 0.143 0.220 0.115 0.136 0.169 0.095 0.108 364 days 0.108 0.170 0.348 0.141 0.137 0.167 0.127 0.100 1092 days 0.118 0.136 0.233 0.238 0.231 0.149 0.145 0.134

Date 04/02/86 04/02/87 04/02/88 04/02/89 04/02/90 04/02/91 04/02/92 04/02/93

Table 3.2 Source: Rubinstein (1996)

In the first column, the table reports the implied Black-Scholes volatility for at-the-money option on the S&P 500 index. The option with a time-to-expiration between 135 to 225 days is chosen. We use 8 dates from April 2, 1986 and choose the date closest to April 2 for each year. The remaining four columns report the historical volatility for those dates, sampling the prior 28, 91, 364 and 1092 days, respectively.

In conclusion, BS model is a quick and dirty way to calculate option prices and widely used amongst investors. However, it has several idealistic assumptions which are clearly
18 19

Options, Futures and other derivatives Hull 2006 page 379 Options, Futures and other derivatives Hull 2006 page 379-380

25

not suitable to the real market, such as the assumption of lognormal distribution and constant volatility. The implied volatility is a contradiction of the constant volatility and proves the limitations of BS model. Therefore, the content of the following chapters will digress from the BS world to stochastic volatility world.

26

Chapter 4 Assumption of Stochastic Volatility and Heston Model
In previous chapters, the Black-Scholes Model and its drawbacks have been introduced and investigated. Especially, the phenomenon of the implied volatility smile points to a more realistic assumption of stochastic volatility model. In this chapter, the assumption of stochasitc volatility is going to be further and detailedly discussed and among different stochastic volatility models, the wildly-used Heston model will be highlighted and explained, including interpretation on its parameters and critical analysis on both advantages and disadvantages.

4.1 moving to the stochastic volatility model
Due to its inconsistentce with reality and exixtence of implied volatility smile, Black-Scholes Framework leaves much room for improvement by loosing its idealized assumptions, which has been shown in Appendix 5. It is hereafter that different researchers have contributed various models to deal with its drawbacks according to their individual and specific focus. One category of these models is those with the assumptions of a stochastic interest rate other than a constant interest rate( Merton (1973) and Amin&Jarrow (1992)). Another type to retain the property that the asset price changes continuously, but assume a process other than geometric Brownian motion. 20 One alternative to Black-Scholes is the constant elasticiy variance (CEV) model. This is a diffusion model where the risk-neutral process for a stock price S is : dS=(r-q)S dt+ When =1, the CEV model is the geometirc Brownian motions model we have alrealdy seen. When 1, the volatility increases as the stock price increases. This
21

creates a probability distribution with a heavy right tali and a less heave left tail.

Yet another alternative is to assume a process where all the asset price changes that take
20 21

Options Futures and Other Derivatives, by Hull(2009) page 584 Options Futures and Other Derivatives, by Hull(2009) page 584

27

place are jumps. A model where stock prices change continuously and are overlaid with jumps is known as a mixed-jump-diffusion model( Merton(1973), Bates(1991), and Pan(2002)). A model where all stock price changes are jumps is known as a pure jump model, one example is the variance-gamma model( Madan Carr and Chang(1998)).22 These models have been developed to fit the volatiliy smiles that are observed in practice. The CEV model is more suitable for catching the valotility smile similar to that observed for equity options. While the jump-diffusion model is for currency options.
23

Finally, the assumption of constant valotility in BS model has been loosen. This leads to models with the assumption of stochastic volatility. These include models by Hull&White(1987), Scott(1987), Wiggins(1987), Stein&Stein(1991) and Heston(1993) etc. The list of these models can continue to be so long as different models and some combined ones come out through the time. Therefore, it could be a real challenge to pick up the only propriet one. According to analysis from Bakshi, Cao and Chen(1997), it can be concluded that among all the effort and devotion, the most important improvement of the Black-Scholes model is by introduing stochastic volatility. Other correction measurement, such as adding jumps and assumption of stochastic interest rate are not as significant as the assumption of stochastic volatility. Besides its better fit to the implied volatility smile, stochastic volatility models assume realistic dynamics for the underlying.
24

They can be viewed as a rising from Brownian

motion surbodinated to a random clock. This clock time, often refered as trading time, may be identified with the volume of trades or the frequency of trading(Clark 1973); the idea is that as trading activity fluctuates, so does volatility.

4.2 the formula of stochastic volatility model and its feature
In a stochastic volatility model, the volatility is changing randomly satisfying some stochastic differential equation(SDE) or some discrete random process. For example, if the underlying follows a geometric Brownian mothion at time t and the volatility follows a

22 23 24

Options Futures and Other Derivatives, by Hull(2009) page 584 Options Futures and Other Derivatives, by Hull(2009) page 584 The Volatility Surface, by Gatheral 2005

28

stochastic process too, then the following SDEs are supposed to be the generalized form(Wilmott (2000)): d d with { Where the coefficients in the equation are     the time variation of volatility involves an additional source of randomness, besides Furthermore, the two Brownian motions are correlated as { and is the correlation coefficient. is the variance fro stock price returns (4.1) (4.2)

The stochastic process (4.1) followed by the stock price is equivalent to the one assumped in the derivation of Black and Scholes (1973). This ensures that the standard time-dependent volatility version of the Black-Scholes formula may be retrieved in the limit →0. In practical applications, this is a key requirement of a stochastic volatility option pricing model as practitioners’ intuition for the behavior of option prices is invariably expressed within the framework of the Black-Scholes formula.25 While the stochastic process (4.2) followed by the variance is very general. Both not been specified here in the formula and no assumptions have been made about the functional forms of The two Wiener process d (.) and (.).

are correlated with the correlation rate Rho

,

which has to satisfy the condition: -1⦤ ⦤1. Rho

is the correlation coefficient

between the process of the underlying asset and the process of the volatility. When
25

The Volatility Surface, by Gatheral (2006), page 4

29

=0, this means the two random walks are independent of each other. When =1, this means the two random walks are perfectly correlated. These are particular cases, in most of the cases, is just either a positive or a negative number within this range.

There is only one source of randomness in the Black-Scholes model, which is the changes in stock price. In contrast, in the stochastic volatility model, one more randomness is presented, which is the changes in volatility. This has a great influence on hedging. It is already known and shown that in the Black-Scoles case, the risk can be hedged by setting up a risk-free portfolio consisting of the underlying asset and an option written on the underlying. However, this does not work in the SV models anymore, because of the extra randomness of volatility. In the stochastic volatility case, random changes in volatility needs to be hedged with the help of another asset, which is denoted as depending on

volatility. Therefore, a risk-free portfolio contains of the option being price, denoted as V, a certain quantity of the underlying stock and a certain quantity of another asset , and the

market price of volatility risk can be derived from this risk-free portfolio.Ф(S,v,t) is denoted as the market price of volatility risk, which means the extra return per unit of volatility risk and so in analogy with the Capital Asset Pricing Model.
26

The logic behind

the derivation of this is quite similar to that of BS formular, using delta-hedging. This can be found in Appendix 8. Hence, the risk-neutral drift can be defined as: (4.3) as the risk-neutral drift is derived, the risk-neutral SDE for v can be obtained: dv= (4.4)

From now on, it is assumed that the pricing is going on in the risk-neutral world.

4.3 The mixing solution in pricing options
The intuition of the mixing idea is that option prices under stochastic volatility are a weighted sum of constant volatility prices. For example, the result for put and call options is simply a weighted sum of Black-Scholes prices.27 The superiority of the idea is obvious
26 27

The Volatility Surface, by Gatheral 2006, page 7 A.L.LewisMxing chapter 4

30

that since Black-Scholes has won such popularity and recognition among practitioners and academia, the technique of breaking down the more complex model to BS framework is easy to understand and implement. Hull and White (1987) first demonstrated the mixing idea in the special case of no correlation between the stock price and volatility changes. Then, it was extended to the case of a correlated process for put and call options by Romano and Touzi (1997). 28 And in Lewis (2002), the theorems were further extended to deal with more generalized payoff functions.In general, the procedure of the mixing idea consists of following three steps:  separate the stock price evolution into two independent degrees of freedom, one

independent of the volatility process   Integrate out the variance and the mean inserting the integrated variables into the Black-Scholes model

Let’s then consider a quite general risk neutral SV model of the following form:

d =(r-q) dt + d with r, q and correlation d = dt constant.

(4.5) (4.6)

And take a simple case for example, a vanilla call option striking at K and maturing at T, so the price C( C( E[max( ] (4.7)

The main idea is for each path of volatiltiy v, the model is just simply like a Black-Scholes model. Then the SV price is the average over all the volatility paths. The BS explicit solution is : (4.8) Moving to continuous time will allow fro mixing solution to be incorporated. The explicit solution for BS mode can then be rewritten in continuous time in following:

28

A.L.LewisMxing chapter 4

31

(4.9) Similar to the decomposition before, with d

Then, the two source of randomness will be separated as described in the procedure. First, the “effective spot” is introduced as following: Plugging in the decomposition re-arrange, can be reformulated as : ds + (4.11) ds + ) (4.10) and formula (4.10) and

So far, the two independent Brownian motion dZ and dW have been separated. Further more, the “effective spot” will also be defined as: ds Then the distribution of the log-spot mixture of a normal distribution: N (0, 1) With independent of N (0, 1) (4.13) (4.12) at maturity is given by a mean-variance

This is similar with the Black-Scholes model where: N (0, 1) (4.14)

Therefore, due to the mixing solution, the price of a call option with assumption of stochastic volatility can be expressed as the expectation of the BS call value. And the effective values depend on only one source of randomness ( dW ) the volatility process. So the pricing problem has been reduced to take expectations over the volatility process alone. Summing up: C( With E[ , T)] (4.15)

32

-

ds + ds

)

(4.16) (4.17)

And the variance evolves as: d Since the exact joint distribution of (4.18) unknown, and therefore a good

and typical way to deal with this problem is simulation. And this will be further and detailed explained in the next chapter.

4.4 the Heston Model
It is already known that the assumption of stochastic volatility can help to better catch the features of the implied volatility surface. However, among the unexhausive stochastic volatility models, a choice of an appropriate model seems to be a very complicated task. As is shown in Equation (4.2), by choosing different functions for models can be obtained. Some of the examples can be : Hull and White (1987) Scott (1987) Stein and Stein (1991) Heston (1993) d d d d =k( (4.19) (4.20) (4.21) (4.22) and , different SV

Among the first authors to tackle this issue, Hull and White proposed in 1987 a simple extention of Black-Scholes model. In this model, the variance is also decribed as geometric Brownian motion and the correlation between the stock price and the variance is zero.29 Hull and White explanied explicitly a cause of the volatility skew: the leverage effect , even though they have not provided any analytic formula in the correlated case.
30

Scott provided a model in which the instantaneous volatility parameter for stock prices follows a random mean-reverting process, an Ornstein-Uhlenbeck process.
29 30 31

31

The

The pricing of options on assets with stochastic volatility, by Hull, J. & White, (1987) Encyclopedia of Quantitative Finance Option Pricing when the Variance Changes Randomly: Theory, Estimation and an Application, by Scott (1987)

33

common feature of these two models is that the variance is a exponential Brownian motion , and therefore the variance is strictly positive. Stein and Stein tried to use analytic techniques to derive a explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein-Ublenbeck process. Besides this, they also studied the relationship between stochastic volatility and the nature of “fat tail” in stock price distributions. The disadvantage of this model is that the variance could be negative. Last but not the least, in literature, the Heston (1993) model has been recognized as the most widely used model for describing stochastic volatility. The volatility is related to a suqare root process, which was first introduced by Cox, Ingersoll and Ross (1985).32 The big advantage of this model is that it offers a closed form solution for European call options when the volatility process is correlated with the spot asset. And from call-and-put parity, a closed form solution for European put options can also be obtained. In addition, this model keeps the variance moving around the mean level. Hereafter, the Heston model is going to be further explained. In this model, the underlying asset price S follows a log-normal distribution process (4.9) , and the volatility follows a mean-reversion process with some diffusion (4.10). This mean-reverting property of volatility is consistent with the empirical observations. d and with d {d d d (4.23) (4.24)

the Heston Model parameters are:    is the variance of the stock price returns is the long run mean of variance

32

The Volatility Surface, by Gatheral (2006), page 15

34

 

is the volatility of volatility The two Brownian motion d are correlated with

Comparing with the more general equation of SDE in (4.2), by selecting certain functions for = =1 With the condition 2 > , the Heston model can be achieved: (4.25) (4.26) , the variance is always positive and will not

reach zero. (Mikhailov & Nogel 2003). It is known that in the Heston model, the two Wiener process d are correlated

by . This can be further explained in the Cholesky decomposition as follows: (4.27) Here d are correlated, however d are not correlated. The d can

be therefore expressed as: d So the Heston model process can be rewritten as: d d d ] (4.29) (4.30) ] (4.28)

It is assumed that this model moves in a risk-neutral world, so it holds that the drift is equal to r-q, where q is the dividend yield. This decomposition is of great importance in determining the mixing solution for the Heston model, which has been explained in the previous section.

4.5 advantages and disadvantages of the Heston Model
Both academia and practioners have recognized the significance of Heston Model, nevertheless, it is not a model without any drawbacks. A brief summery of its advantages and disadvantages will be represents in the following.

35

Main advantages:   A close form solustion for European call and put can be acquired. The Heston model better describes the property of a non-Gaussion distribution of stock returns with higher peak and fat tails.   The implied volatility surface of option prices in the market can be fitted. The leverage effect in the stock market can also be captured by choosing a negative correlation coefficient. Main disadvantages:  These parameters can not be easily observed or estimated, and therefore the calibration of them is very crucial.  The price of derivatives is very sensitive to the parameters; hence the estimation of parameters would have big influence on the fitness of the model.33  The Heston model fails to capture the skew at short maturity as the one given in the market. An improvement of this is by adding jumps.

33

Heston’s stochastic volatility model implementation calibration and some extensions, by Mikhailov& Nogel 2003

36

Chapter 5 Monte Carlo Method in Option Pricing
In the last chapter, the advantages and features of SV models, particularly the Heston model, have been illustrated and explained. Thereby, in this chapter, the implementation of this model is going to be explored and discussed. At first, an explanation of choice of pricing method will be given. Then, the specific method named Monte Carlo Method will be thoroughly described, including its intuition, theoretical background, procedure and advantages. Furthermore, the generation of random numbers in MC method and variance reduction techniques will also be a necessary part to understand the MC method. Last but not the least, Monte Carlo simulation with the Heston model will be explored, and especially the discretization of the Heston model would be emphasized. Most content of this chapter is based on the textbooks named Monte-Carlo Methods in Financial Engineering written by Glasserman (2003) and Introduces Quantitative Finance by Paul Wilmott (2007).

5.1 different numerical methods
In chapter 2, the explicit solution for European put and call options in Black-Scholes model has been given. This is a case when the payoff of the option is not complicated so that it is possible to find the explicit solution to the partial differential equation. However, if the payoff turns to be more complex, the explicit solution could be hard to achieve, and therefore different numerical methods are necessary to price options. In general, there are the three most typical and widely used numerical methods: the binominal tree, Finite Difference Method and Monte Carlo simulation. And the appropriate choice of model depends on the type of the option and the feature of the contract. The binominal tree and Finite Difference Method work well when two dimensions are incorporated. While in the case of three dimensions, for instance, when the stochastic volatility is incorporated, it is complicated to apply these tree-featured models. This has also been supported by argument in Glasserman (2003) and in which it is clearly stated that Monte Carlo Method is attractive in evaluating integrals in high

37

dimensions.34 Another reason for choosing Monte Carlo Method is that it is more acceptable and easier to apply for pricing and hedging path dependent derivatives since valuing a derivative security by MC typically involves simulating paths of stochastic processes, that are used to describe the evolution of underlying assets prices, interest rates, model parameters and other factors relevant to the security in question.35 In the next section, MC Method will be further elaborated.

5.2 introduction of Monte Carlo
Even though the theory of Monte Carlo was established decades ago, not until computers had been invented and widely implemented did it gain such popularity and recognition due to its feature of much repetition and time consuming. In principle, Monte Carlo method is based on the analogy between probability and volume. The definition of probability is that, associating an event with a set of outcomes and the probability is equal to the number of outcomes in the event divided by the number of total possible outcomes. MC method uses this intuition in a reversed way by calculating the fraction of a set and interpreting the fraction as an approximation for the probability.36 The theoretical background of MC method is the law of large numbers which says that estimator converges to the correct volume when the number of simulation increases. Let X be a random variable with E[X] = < , let be independent = , then

realizations from the distribution of X. Define the sample mean the law of large numbers states that =

(5.1)

In words, when enough independent realizations from the distribution of X have been simulated, then the sample mean estimator converges to the true mean . It also means that any problem that can be broken down to the calculation of an
34 35 36

Monte-Carlo Methods in Financial Engineering, by Glassermen(2003), page 17 Monte-Carlo Methods in Financial Engineering, by Glassermen(2003), page 17 Monte-Carlo Methods in Financial Engineering, by Glassermen(2003), page 17

38

expectation of a random variable can be solved by using MC method. Even though the law of large numbers has ensured the convergence, how good this estimate could actually be is still a concern for practitioners of MC simulation. Fortunately, the central limit theorem has provided some insight on this. The following proof is taken from Domingo A. Tavella (2002).

Assume n independent and identically distributed (IID) random variable { }. Each is a normalized (discounted) evaluation of the payoff

function as a result of the ith Monte Carlo cycle. Consider the sample mean:

(5.2)
Take expectation

E[ ]= E[
Since the expectation of each

=

(5.3)

is the same as the expectation of the population,

(5.4)
This gives

E[

(5.5)

This shows that the expectation of the sample is an unbiased estimator of the expectation of the population. How good is this estimator? To answer this, first the variance of is given:

= var ( =
This gives us:

(5.6) (5.7)

(5.8)
It means that the standard error of the mean estimator is inversely proportional to the square root of the number of samples. In a simple case, it shows that in order to decrease the standard error to 50% of its previous level, the number of simulations

39

should increase to 4 times of its previous level. This is a central feature of the Monte Carlo method. normally distributed.
37The

central limit theorem tells that the mean estimator is

E[
Then for =

, Var[

=

, i=1,2,...N,

(5.9)

(
This also means that

-

N(0,1)

(5.10)

P( (

-

)

(x)

(5.11)

Where (x) is the distribution function for the standard normal distribution.

After setting the theoretical background, now it is moving to the common steps when applying MC method. Here an estimate for the value of an option is shown as an example:38  Simulate the risk-neutral random walk for the underlying asset, starting at today’s value of the asset, over the required horizon. This time period starts today and continues until the expiration of the option. This simulation gives one realization of the underlying price path     Calculate the option payoff for this realization Repeat the first two steps and obtain different realizations over the time horizon Calculate the average payoff for all these realizations Discount the average payoff back at the risk-free rate to get a final estimate for the value of the option Now that the procedure of MC has been given, its simplicity and clarity may to some extent indicate that why it has gained increasing popularity in financial engineering. Besides this intuition, the benefits of using MC simulations can include:39

37 38 39

Monte-Carlo Methods in Financial Engineering, by Glasserman (2003), page 16 Introduces Quantitative Finance by Paul Wilmott (2007), page 582 Introduces Quantitative Finance by Paul Wilmott (2007), page 585

40



The mathematics that is used in perform MC simulation is very easy to understand. As stated above, it can be reducible to the calculation of an expectation.



Different kinds of software are available to apply MC simulation, i.e. SAS, VBA, , or at least the spreadsheet functions can be used for most of the time.

 

The more simulations are run, the more accurate the estimate will be. With the powerful calculation ability of modern computers, the effort in getting a relative good answer is low.



Complex path dependency can often be easily incorporated in MC simulation, since each path has to be individually simulated.



Correlation can be easily modeled by Cholesky decomposition.

5.3 generating random numbers
Random numbers are the core of MC simulations, which influence the individual shape of each path, the estimator and its error. Therefore, it is of great importance to improve and ensure the quality of random numbers that are generated and going to be used in Monte Carlo simulations. However, true and perfect randomness is more like a theoretical construct and it is difficult or to be more precisely, impossible to achieve in practice. Nowadays, computers are used to generate numbers that are very close to random by using certain deterministic algorithms, hence the realizations generated in this way is often referred to as pseudo random numbers. Fortunately, these algorithms have been already built in most programming languages.40 A series of different methods have been explored to design random numbers generators and there is elegant theory that has been applied to the problem of random numbers generation. This is not going to be investigated in depth in this thesis. Instead, some crucial considerations in the construction of a random number generator will be pointed out and followed by a comparison of several typical models. The following are those considerations:41
40 41

Monte-Carlo Methods in Financial Engineering, by Glasserman (2003), page 54 Monte-Carlo Methods in Financial Engineering, by Glasserman (2003), page 56

41

1. Period length. Other things being equal, generators with longer periods are preferred, i.e. generators products more distinct values before repeating, since the longer the period is, the more closely the values can be an approximation of a uniform distribution. 2. Reproducibility. The inputs in one simulation can also be used in simulations afterwards or in other different simulations. 3. Speed. A random number generator usually has to be repeated for thousands or even millions of times in a simulation, and therefore speed is a crucial judgment. 4. Portability. This means an algorithm for generating random numbers should provide the same realizations on all computing platforms. 5. Randomness. The most significant measurement of a random number generator and this can be checked by some statistic tests. Next, four different methods of generating random numbers will be represented and analyzed. The first very common way is the NORMSINV (RAND ()) function in Excel, which can fit random numbers into a normal distribution. This function is easy to understand and implement, however, the speed of it is really not satisfactory. An improved method is the Box-Muller methods, which relies on two uniformly distributed variables that are turned into normal variates.
42

If

and they are independent, then: cos(2 sin(2 Then and (5.12) (5.13)

are independent random variables. So if a uniform

distribution has been simulated, a standard normal distribution can be easily obtained. Similarly, other normal distributions can be simulated based on the standard normal distribution: X
42

+ Z and Z N (0, 1)

(5.14)

Introduces Quantitative Finance, by Paul Wilmott, page 587

42

This is very popular way to generate normal variates since it is easy to implement. Yet, an improvement on this method is the Marsaglia-Bray method since it reduces the computing time and is said to be twice as fast as the Box-Muller way. And the algorithm is: , X= , (5.15) (5.16) (5.17)

Nevertheless, the Box-Muller and Marsaglia-Bray methods suffer their drawbacks in the respect of randomness, since the extreme situations at the tails are neglected due to an asymmetrical distribution of the lower and upper bounds.43 In conclusion, the Marsaglia-Bray method is going to be chosen and simulations in this thesis. used in the

5.3 variance reduction techniques
As has been explained before, in order to get better accuracy in MC simulation, more simulations should be run, however, this will at the same time increase the computational time. Therefore, other ways to speed up convergence and diminish computational time is crucial to increase the efficiency of MC simulation. In the following, two specific techniques of reducing the variance of simulation estimates will be explained: one is the antithetic variables and the other is control variate technique. The antithetic variables method The method of antithetic variables attempts to reduce variance by introducing negative dependence between pairs of replications.44 This method works because in this way the random numbers generated are symmetric, like the normal variables. To be more specific, an unusually large or small output computed from the first path may be balanced by the value computed from the antithetic path, resulting in a reduction in variance.45
43 44 45

Jackel 2002, page 101 Monte-Carlo Methods in Financial Engineering, by Glasserman, page 219 Monte-Carlo Methods in Financial Engineering, by Glasserman, page 219

43

The method calculates two estimates for the option value using only one set of random numbers, the steps are as following:46  use one set of normal random numbers ф to generate one estimates of the option

  

use the set -ф of random numbers to generate another estimate set repeat the procedure above many times

The control variates technique The control variates technique is applicable when there are two similar derivatives A and B. Here derivative A is the derivative under consideration while derivative B has an explicit solution available . The same set of random numbers is used in

simulations for these two derivatives. The idea is that the error in the estimate for derivative A should be the same as the error in the estimate for derivative B, which is quantifiable, since the value is available from the explicit formula.

The steps of this method are as following:47  use a set of realizations of random numbers ф derivative A  in parallel, use the same set random numbers ф derivative B  a better estimate for derivative A can be calculated by: to estimate the value for to estimate the value for

In the empirical part of this thesis, the antithetic variables method has been implemented in pricing of Asian options.

5.5 Monte-Carlo simulation with the Heston model
So far in this chapter, the MC simulations has been introduced and explained. In section 4.3, the idea and general formula of the mixing solution and its application in the stochastic volatility models has been given and analyzed. Next, how the Heston model
46 47

Options, futures and other derivatives, by Hull 2009, page 447 Options, futures and other derivatives, by Hull 2009, page 447

44

is incorporated into the MC simulations by using the mixing solution is going to be explained. As is shown in formula (4.16) and (4.17), both the effective stock price effective variance and the

include the integral part. While in Monte-Carlo simulations,

this will be resolved by discretizing the time interval. Here in this thesis, the choice is a simple Euler scheme. As in formula (4.18), the variance evolves as: d And the Euler scheme for this is: , (5.18)

△t+b(

△

ф

(5.19)

In particular, in the case of the Heston model, it becomes:

△t+

△

5
, some negative is negative and large.

One critical problem with the Euler scheme is that with the values for variance V may be generated when the outcome of

So far, a lot of literature has been devoted to avoid this problem as well as to simulate the Heston model as accurately as possible. In short, there two easy solutions to handle this drawback: one is to adopt the so-called absorbing assumption, which means whenever v means whenever v ; the other is to adopt the reflecting assumption, which
48

Unfortunately, these tow solutions will both slow down the speed of convergence. Additionally, according to Gatheral (2006), the Euler discretization requires a huge number of time steps to achieve convergence. This is why in Gatheral (2006) the Milstein scheme used by Kahl et al. (2005) is suggested. However, the Milstein scheme is criticized since it is harder to implement and takes more time per replication compared with the Euler discretization. 49 Furthermore, the Euler discretization is

48 49

Gatheral 2006 Broadie & Kaya 2004

45

possible to converge to the true process when the length of time steps is made smaller. In conclusion, in this thesis the Euler scheme is applied in pricing using the MC simulations.

Euler Discretization As in formula (4.16) and (4.17): ds + ds So, the two integrals that are necessary to simulate are: ) (4.16) (4.17)

The dynamics are given by d , =0, and d = , =0

So discretizing the time integral in N points and mesh Δt = T / N , we get for i=0, 1,…, N-1: = + With = + ф

sampled as usual from a standard normal distribution.

As for the variance process, a simple Euler scheme is applied:

△t+

△

5

The algorithm for simulating  initialize current value for



for i=1 to N generate ф from N(0,1) = + vΔt + ф

46

Next i  Set -

(v-

ф

And finally to simulate the option price itself: Suppose that a vanilla call option has to be priced, and BScall is the function in VBA to calculate the call option price in Black-Scholes framework. The full version of the function is in Appendix 9.    choose a large number initialize the current sum of Black-Scholes price sum=0 for j=1 to generate ( update sum as sum = sum + BS call ( Next j  finally the option price = sum/ according to previous algorithm

47

Chapter 6 Calibration of parameters in Heston model and their effects
In the previous chapter, the MC simulation and its application with the Heston model have been elaborated. In option pricing, the unknown parameters should be first estimated appropriately since the prices can be very sensitive to even small changes in these parameters. Hence, the core of this chapter is how to obtain the right estimations for the parameters. Furthermore, the effect of different parameters, especially volatility on volatility and the correlation coefficient Rho, on the price of options will also be analyzed.

6.1 calibrations of parameters in Heston model
There are five parameters that need to be estimated in Heston models50:      The long-run volatility, or the mean volatility The speed of mean reversion level The volatility for the stock price The volatility on volatility The correlation coefficient ; ; can be interpreted as representing the degree of ; ; ;

The mean reversion level “volatility

”. This is the fact that has been observed in the equity market

as explained before; precisely, large/small movements are more likely to be followed by large/small price variations. (Moodley 2005) The volatility on volatility affects the kurtosis (peak) of the distribution. When is

equal to zero, it means the volatility is deterministic and therefore the log-returns are normally distributed. Increasing will increase kurtosis. Higher means that

the volatility is more volatile and thus there is a greater chance of extreme movements. (Moodley 2005) The correlation coefficient implies the heaviness of the tails. If coefficient ρ is positive, it means the volatility will increase when the asset price/return increases. This will
50

The volatility surface, by Gatheral 2006

48

spread the right tail and squeeze the left tail of the distribution creating a fat right-tailed distribution. If the coefficient ρ is negative, it means the volatility will decrease when the asset price/return increases. And this will spread the left tail and squeeze the right tail of the distribution and therefore making a fat left-tailed distribution. Due to the fact that equity returns and its volatility are negatively correlated, the coefficient ρ, therefore, affects the skewness of the distribution. (Moodley 2005) As stated before, when pricing options under the assumptions of stochastic volatility model, the correctness of the parameters are of great influence and therefore the method for estimation is crucial. Unfortunately, researches have shown that the implied parameters (those parameters product the correct options prices) and their time-series estimate counterparts are different. Therefore, the empirical estimates for the parameters cannot be used (Bakshi, Chao & Chen 1997). A typical way is to find those parameters which give the market price of the options, also known as the inverse problem. The inverse problem can be solved by minimizing the error difference between model prices and market prices, which is known as the non-linear least-squared optimization method. Take a call option for example, the purpose is to minimize the squared difference between the market price quoted for this option and the price calculated from the Heston model. The formula is as follows: Min( ) SqErr( ) = min ( ) where respectively. The

, i=1,2,,,N

(6.1)

are the prices for the call option from the model and the market

is a set of realizations for the parameters in the Heston model,

and N is the number of options that is used for calibration. One condition that has to be satisfied is 2k >0, so the volatility process would not reach zero, which

is also known as Feller constraint.( Cox, Ingersoll & Ross 1985). Other conditions on these parameters are as follows:   The long term volatility is constrained to 0< 0

49

  

The volatility for stock prices returns The

is positive 1 1 0

The correlation coefficient is constrained to -1<

The calibration is finished in Excel by using the built-in solver function. It uses a

Generalized Reduced Gradient (GRG) method and hence is a local optimizer
(Mikhailov & Nogel 2003). In the standard Solver, nonlinear problems are solved with GRG method as implemented in Lasdon and Waren’s GRG2 code. This method and specific implementation have been proven in use over many years as one of the most robust and reliable approaches to solving difficult nonlinear problem.51 However, GRG method is subject to its intrinsic limitations on its ability to find the globally optimal solutions. The calibration results are extremely sensitive to the initial estimates of the parameters. This optimizer only provides reasonable results when the initial estimates that are used in Solver are quite close to the optimal parameter set (www.solver.com). It is quite obvious that nonlinear problems are intrinsically more difficult to solve than linear problems, and therefore fewer guarantees about what the solver can do. The limitation of a nonlinear problem is that there may be more than one feasible region, or set of similar values for the decision variables, where all the constraints are satisfied. Within each feasible region, there may be more than one “peak” (if maximizing) or “valley” (if minimizing), and there is no general way to determine which “peak” is tallest and which “valley” is deepest. There may be also false peaks and valleys known as “ points”. Because of these possibilities, nonlinear

optimization methods can make few guarantees about finding the “true” optimal solution.

51

A Study of Generalized Reduced Gradient Method with Different Search Directions, by Hong-Tau Lee, Sheu-Hua Chen, He-Yau Kang (2004)

50

Figure 6.1 local and global solutions in GRG method52 Therefore, when dealing with a nonlinear problem, it is a good idea to run the solver starting from several different sets of initial values for the decision variables. Since the solver follows a path from the starting values (guided by direction and curvature of the objective function and constraints) to the final solution values, it will normally stop at a peak or valley that is closest to the starting points values applied. By starting from more than one points, ideally chosen based on knowledge and experience of the problems, the chances that one will find the best possible “optimal solutions” will increase. In the next chapter, two PPN products from the Danish market will be chosen and illustrated. At first, the pricing of the zero coupon bond will be explained and studied. Next the pricing of the embedded Asian options will begin with the calibration of the parameters in the Heston model using the collected data from market.

52

The Graph is taken out of slides from the website

http://www.nvc.vt.edu/rmajor/bit5724/Chapter_14.pdf

51

6.2 Average Relative Percentage Error and comparison with Black Scholes Model
Having calibrated the parameters for the Heston model, statistic measures are calculated to check the forecasting ability of this model. A very common used measure is the average relative percentage error and it formula is as follows:53 ARPE= In this thesis, only the vanilla call options prices are used in the calibration, so the number of options used is simply 1. Therefore, the formula is actually as: ARPE= Where the model price is the price calculated by using the parameters calibrated for the Heston model. In order to see how the Heston model is superior to the Black Scholes model, the ARPE of the BS model will also be calculated and made a comparison with that of Heston model. The same as stated before, the result will be presented in the next chapter when two structured products embedded with Asian options in the Danish market are priced under the assumption of stochastic volatility.

53

The formula is from Cont & Kokholm 2009

52

Chapter 7 Two specific structured products and their pricing
In order to explain and test how the stochastic volatility model can be used in the pricing of structured products embedded with Asian options, two specific PPN in the Danish market have been chosen. One is Rå varer Basis 2010 from Nordea Bank Denmark A/S, and the other one is Russiske Rubler 2009-2012 from Garanti Invest Denmark. For these two products, detailed information has been collected, including ISIN code, name of the product, date of issue, date of expiration, final observation date, name of issuer, issuer rating(at the time if issuance), issue organizer, currency, nominal amount issued, issue price, expiration price(if product has expired), participation rate, guaranteed price/ protection level, published annualized total costs, coupon, option type, type of underlying asset, name of underlying assets and number of underlying assets.54 The ExcelSheet containing this information can be found in ExcelSheet named “Information on PPN”.

7.1 pricing of the bonds part in PPN
The pricing of PPN embedded with Asian options can be broken down into two part, the first one is the pricing of the zero coupon bonds, and the other part is the pricing of the embedded Asian options. So first, the pricing of zero coupon bonds will be handled. The general formula for bond pricing is: B(r,t) = (7.1)

Where, c is the coupon payments, r is the interest rate, t is time to maturity and F is the face value of the bond. Here since there is no coupon payment during the life time of the PPN, the pricing process of the bond is actually reduced to the term structured of the discount rate. Even though various models exist for estimating zero-coupon yield curve, the most adopted methods are either the Nelson & Siegel (1987) method or its extended version suggested by Svensson (1994). In the Nelson & Siegel mode, the forward rate function can be written as:
54

These are consistent with the information that has been collected in Henrik’s thesis.

53

; (7.2) Where f(m) denotes the instantaneous forward rate f(t, t+m) with time to settlement for a given trade date t. And b = ( is a vector of parameters. The forward rate , the second is an exponential

consists of three components. The first is a constant, term

monotonically decreasing (or increasing, if

toward zero as a function of the time to settlement, and the third is a term which generates a hump-shape ( or a U-shape, if settlement, And when the time to maturity approaches infinity, the forward rate approaches the constant , and when the time to maturity approaches zero, the forward rate approaches the constant . as a function of the time to

The spot rate can be derived by integrating the forward rate according to

let i(m) denote the spot rate i(t, t+m) with time to maturity m, for a given trade time t, it is given by : ; (7.3) Where the discount function is given by ; (7.4) The next step is using a bond pricing formula for finding the bond price over its life time.55 Using the Excel Solver, the estimation table we can achieve is as follows:
;

55

All the formula and explanation is extracted from “Seminarium in Analytical Finance II”, Institutionen för matematik och fysik

54

Parameters name Long-run levels of interest rates Short-run component Medium-term component Decay parameter 1 b0 b1 b2 t1

Råvarer Basis 2010 0.0023 0.02 0.09 0.25

Russiske Rubler 2009-2012 0.0024 0.01 0.09 0.25

Table 1 the realizations of parameter using Excel Solver And from the formula above, the spot rate and discount factor can also be achieved: time to maturity spot rate discount factor 4 0.905% 0.9626 3 1.07% 0.9684

Table 2 the spot rate and discount factors Therefore, the price for bond part of Rå varer Basis 2010 is: B (RB 2010) = 100* 0.9626 = 96.26 The price for bond part of Russiske Rubler 2009-2012 is: B (RR 09-12) = 100* 0.9684 = 96.84

7.2 pricing of the embedded Asian options and structured products
As been explained in the previous chapter, the first step of the option pricing part should be the calibration of the Heston model. After the calibration, the pricing of Asian options can be done in VBA in Excel.
7.2.1 Råvarer Basis 2010

The embedded option is an Asian option, and the underlying is DJ AIG index, which is an index for commodity worldwide in the Dow Jones index family. The issue date is 16 June 2006, and the maturity date is 16 June 2010. The determination dates include 01 December 2009, 04 January 2010, 01 February 2010, 01 March 2010, 06 April 2010, 04 May 2010 and 01 June 2010. The payoff of the option can be written as MAX is the official closing price of the DJAIG on 16 June 2006. And . is the

55

arithmetic means of the official closing price of the DJAIG on each of the determination dates. In order to price the Asian option in VBA, at first the calibration of the parameters in the Heston model should be implemented. Here the Excel Solver is used in the calibration of parameter in Heston model. Heston parameters 0.14 0.23 vbar vzero rho ρ -0.35 lambda λ 1.46 eta η 0.25

Table 7.3 the calibration of the Heston model for Rå varer Basis 2010 These are the realization of parameters that will be used in the pricing process. Form the sensitivity test of these parameters, it has been found that among these five parameters, rho and vbar have more significant influence. The ExcelSheet named “Asian option pricing DJP” contains all the tables of these. And following is the ARPE for this calibration: ARPE ARPE 1 (strike from 33 to 38) 14% 2 (strike from 33 to 43) 21%

Table 7.4 the ARPE for the calibration for Rå varer Basis 2010 It can be easily noticed that with strike price ranging from 33 to 43, the ARPE is much higher than that with strike price ranging from 33 to 38. This phenomenon is due to the fact that with the very high strike price, the market quotes for the vanilla call options become small. Therefore, in the far end, the relative percentage error is relative bigger. In order to compare with the Black-Scholes model which means under the assumption of constant volatility, the ARPE is as follows: ARPE 1 ARPE 2 (strike from 33 to 38) (strike from 33 to 43) 21% 42%

Table 7.5 the ARPE for BS model for Rå varer Basis 2010 Form the tables above, it can be seen that the ARPE for Heston model is lower than the Black Scholes model, which shows the superiority of the stochastic volatility model.

56

This result is consistent with the analysis that has been extended in the previous chapters. Using the parameters calibrated in Excel Solver, the price for the Asian option can be achieved with some programming in VBA. This is done in the ExcelSheet named “Asian option pricing DJP”. And the price for this Asian option under the assumption of Heston model is 0.097. Record the payoff of a PPN (chapter 1) is: Payoff= P + P *PR *OP Therefore, the price of a PPN is Price= (P+P*PR*OP)* exp(-r*T) = Price(bond)+ P*PR*OP*exp(-r*T) The participation rate is guaranteed to be on less than 70%. Hereafter the fair price is calculated base on the lower bound and the result is 103.07. Comparing with the issue price 104, it is lower and this result is consitent with the research that has been done by Henrik. If the same product is price under the BS model, the result is 103.42.
7.2.2 Russiske Rubler 2009-2012

(1.1)

(1.2)

The embedded option is also an Asian option, and the underlying is foreign exchange RUB/EUR, which means the exchange rate between RUB and EUR expressed as the price in EUR of one RUB. The issue date is 23 June 2009, and the maturity date is 25 June 2012. The initial observation dates means 4 June, 2009, 5 June 2009 and 8 June 2009. The final observation dates include 21 May 2012, 28 May 2012, 04 June 2012. The payoff of the option can be written as MAX . equals the arithmetic average of the fixings p of RUB/EUR on each of the three initial observation dates divided by the reduction factor. The reduction factor has been calculated by the calculation agent on 3 June 2009 and is equal to 1.3. And is the

arithmetic average of the fixings of RUB/EUR on each of the three final observation dates.

57

In order to price the Asian option in VBA, at first the calibration of the parameters in the Heston model should be implemented. Here the Excel Solver is used in the calibration of parameter in Heston model. Heston parameters 0.25 0.29 vbar vzero rho ρ -0.27 lambda λ 0.39 eta η 0.88

Table 7.3 the calibration of the Heston model for Russiske Rubler 2009-2012 These are the realization of parameters that will be used in the pricing process. Form the sensitivity test of these parameters, it has been found that among these five parameters, vzero and vbar have more significant influence. The ExcelSheet named “Asian option pricing RUBEUR” contains all the tables of these. And following is the ARPE for this calibration: ARPE 1 (strike from 38 to 48) 7%

Table 7.4 the ARPE for the calibration for Russiske Rubler 2009-2012 In order to compare with the Black-Scholes model which means under the assumption of constant volatility, the ARPE is as follows: ARPE 1 (strike from 38 to 48) 38%

Table 7.5 the ARPE for BS model for Russiske Rubler 2009-2012 Form the tables above, it can be seen that the ARPE for Heston model is lower than the Black Scholes model, which shows the superiority of the stochastic volatility model. This result is consistent with the analysis that has been extended in the previous chapters. Using the parameters calibrated in Excel Solver, the price for the Asian option can be achieved with some programming in VBA. This is done in the ExcelSheet named “Asian option pricing RUBEUR”. And the price for this Asian option under the assumption of Heston model is 0.095. Record the payoff of a PPN (chapter 1) is: Payoff= P + P *PR *OP (1.1)

58

Therefore, the price of a PPN is Price= (P+P*PR*OP)* exp(-r*T) = Price(bond)+ P*PR*OP*exp(-r*T) The participation rate is set to be 100%. Hereafter the fair price is calculated base on the gearing lever and the result is 104.8. Comparing with the issue price 105, it is lowe and this result is consitent with the research that has been done by Henrik. If the same product is priced under the BS model, the result is 108.37. (1.2)

59

Chapter 8 Conclusion of the pricing
The financial markets have changed and developed for the past decades. It provides more investment choices for potential investors, thanks to the variety and diversity in financial products. On this ground structured products have become popular for its limited downward risk and its potential of higher return. The embedded option contains both vanilla and exotic options. Usually, Fischer Black & Myron Scholes model offers a closed form analytical formula to price options, but it is under criticize for a number of reasons. Besides other reasons, it assumes that the log-return follows the normal distribution and the variance of the price of the underlying is both constant and known. However, this rigorous assumption is rejected by empirical studies and has been blamed for its simplicity.

From the graph that has been showed in previous chapter, the distribution of the log-return shows a tendency of a higher probability of returns around the mean. It also shows fatter tails and a more skewed (negative skew) than the normal distribution. All of these distribution features are thought to explain smiles and smirks. If the assumption of normal distribution really holds, the implied volatility should always be constant across moneyness and maturity. However, the volatility smile across moneyness and maturity turns out to deviate from the assumption of a constant volatility under Black Scholes framework. Instead it shows a continuous falling tendency (except for very short maturity it showed an increasing tendency) across moneyness. In addition, the volatility in the log-return shows a tendency to cluster. These findings indicate that the Black Scholes model is not adequate and it is obvious that a more realistic and flexible model becomes necessary in order to better catch the observed market option prices.

An assumption of stochastic volatility is more realistic and better describe the process. In the past two decades, efforts on constructing such a model has been made by several theorists, such as Hull & White(1987), Scott(1987), Wiggins(1987), Stein &

60

Stein(1991). Fortunately, Heston(1993)n contributed to a mean reverting model that could account for the above mentioned problems, and specifically allowing for a volatility that could vary randomly. Even though many models exist including assumptions of stochastic interest rates and jump-diffusion, it has been concluded that to introduce stochastic volatility other than constant volatility is by far the most significant improvement. According to literature, the Heston (1993) model was the most accepted and popular choice for pricing options under assumption of stochastic volatility. The reason for its popularity is that it is able to capture a mean-reverting volatility and allows the correlation between the processes of underlying assets and its variance. Besides this, the model can be easily implemented in practice in Visual Basics.

Calibration of the parameters in Heston model is obtained by minimizing the squared difference between the theoretical price and market price of vanilla call option with the same underlying assets. Then in order to assess the validity and reliability of Heston model, the average relative percentage error has been calculated and analyzed and compared with that of Black Scholes model. From sensitivity analysis of the Heston model it shows that both the starting volatility and long-term volatility had significant impacts on the pricing. Volatility of volatility affects the kurtosis of the return distribution. A higher volatility of volatility results in higher peak which is consistent with empirical evidences on return distribution of assets. A negative coefficient of correlation means the volatility will decrease when the asset price/return increases. And this spreads the left tail and squeezes the right tail of the distribution and therefore making a fat left-tailed distribution. Conclusively, the Heston model performs much better than the Black Scholes model.

The purpose of this thesis was to price PPN with Asian options embedded under the assumption of stochastic volatility. Therefore, the Heston (1993) square root model is applied. The estimated model prices could not be compared to market-traded prices as

61

prices for these Asian options are not quoted. The sensitivity test is conducted to make sure that it is known that how the price of the option will change accordingly when the parameters changed slightly. In general, it can be concluded that Asian option prices can be very sensitive to changes in the parameters in Heston model. Therefore, calibration risk can be very crucial in the pricing process and should be taken into serious consideration.

As mentioned early, it is very difficult to find market-quoted Asian option quotes; however, the theoretical price of the PPN embedded with Asian option can be compared with its market price. The market prices for Rå varer Basis 2010 and Russiske Rubler 2009-2012 are DKK 104 and DKK 105, respectively. It has been calculated that the theoretical prices using the Heston model are DKK 103.11 and DKK 104.81. The price difference shows that the Heston model price of the product is almost close to but still lower than the market price, and this result is also consistent with the findings in Henrik’s proposal. The positive aspects of the Heston (1993) model show that the model can be used in practice with certain considerations. On the other hand there remain some disadvantages and open questions. The estimation of the parameters may be not stable and suffer from the calibration risk. And for shout maturities, good results are not obtained and thereby jumps-diffusion is recommended to be added (Gatheral 2006). In addition, for some Asian options, it is difficult to find the equivalent vanilla call options and make the calibration of the parameters in Heston model impossible to be implemented.

Generally, in literature this is a common belief shared amongst theorists; there is no stochastic volatility model that is believed to fit the market-traded option prices perfectly, and if a better model should be applied, one must go beyond stochastic volatility models (Gatheral 2006).

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Appendix 1

Figure 1 On the issue date you pay the face amount of $1,000. This note is here is fully principal protected, meaning that you will get your $1,000 back at maturity no matter what happens to the underlying asset. This is accomplished with the zero-coupon bond increasing from its original issue discount to face value. For the performance component, the underlying, priced as a European call option, will have intrinsic value at maturity if the underlying asset's value on that date is higher than its value when issued. You earn that return on a one-for-one basis. If not, the option expires worthless and you get nothing in excess of your $1,000 return of principal.

63

Figure 256

56

http://www.investopedia.com/articles/optioninvestor/07/structured_products.asp

64

Appendix 2

Figure 3: The figure shows the frequency distribution of the number of issues by main option category57

57 57

Henrik Nørholm, thesis prorosal, structured products pricing and performance evaluation

65

Appendix 3 Ito’s lemma

Ito’s lemma provides a tool for handling stochastic differentials by determining the SDE that corresponds to f( t) given a SDE for . This is extremely useful in financial math.

Given a stochastic Ito’s process for X d =a(

Where a (

are two functions depending on only

and time t, and the term and t, the

is a Wiener process. Given that f( following equation can be derived:

t) is a twice-differentiable function of

The above applies to a one-dimentional process.

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Appendix 4 PPN with DJAIG index

See the PDF file named “PPN with DJAIG index”

67

Appendix 5 PPN with foreign exchange

See the PDF file named “PPN with foreign exchange”

68

Appendix 6: Black & Scholes Assumptions
The assumptions used to derive the Black Scholes Merton differential equation are as follows:

1. The stock price follows the process dS=

with the drift µ and constant

σ. In a less strict situation, the underlying follows a lognormal random walk. 2. The short selling of securities with full use of proceeds is permitted. Delta hedging is done continuously. 3. There are no transactions costs or taxes. All securities are perfectly divisible. 4. There are no dividends during the life of the derivative. 5. There are no riskless arbitrage opportunities. 6. Security trading is continuous. 7. The risk-free rate of interest, r, is constant and the same for all maturities. Note: some of the assumptions can be relaxed. For instance, v and r can be known functions of t. Also interest rates can be allowed stochastic provided that the stock price distribution at maturity of the option is still lognormal.

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Appendix 7: Derivation of Black & Scholes differential equation
The derivation of the Black Scholes is given in the following: The value of the option is given as V(S, t; σ, µ; E, T; r) but is for simplicity denoted as V(S, t). If assumed that V(S, t) is known at time t a portfolio with value Π of one long position and a short position in some quantity Δ (delta) of the underlying:

Π = V(S, t) – ΔS

(1.1)

Where the first term shows the value of the option and second term is the short position in the underlying asset. The change of the value in the portfolio from time t to t + dt partly comes from the change in the value of the underlying and partly from change in the value of the option. The change can be written as:

dΠ = d V(S, t) – Δ dS

(1.2)

From the assumption that the underlying follows a GBM then by Ito’s lemma we have:

dV=

dS +

dt

Thus the portfolio (from (1.1)) changes by:

dΠ=

dS +

dt– Δ dS

(1.3)

Next, all the deterministic terms and the risky terms are separated:

dΠ= (

(

(1.4)

The idea in the Black Scholes argumentation is now to eliminate the random terms that represent the risk in the portfolio; i.e. the last term of (1.4). In theory (and almost in practice) this can be done by choosing the quantity Δ:

(

⇔ Δ=

(1.5)

This elimination of risk, by exploiting correlation between two instruments (the option and its underlying), is called delta hedging. The quantity , a function of the

continuously changing variables S and t, therefore also changes from one time step to another. For this reason it is necessary to continuously rebalance the position of the

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underlying asset to remain delta hedged. The portfolio held is then risk-free:

dΠ= (

(1.6)

If the change in portfolio in (1.6) is completely riskless then it must be, by the no arbitrage principle, that it yields the same amount if the equivalent amount of cash was invested in a risk-free bearing account:

dΠ = rΠdt

(1.7)

The economic argument is now that if the return on the portfolio were higher than the risk-free rate then by borrowing money from the bank, paying interest at the rate r, invest in the risk-free option/stock portfolio and make a profit from this. The reverse action is performed if the return on the portfolio was lesser. In either case an arbitrage opportunity exist. By substituting (1.1), (1.5) and (1.6) into (1.7) following is obtained and rearranged to (1.8)

(

= r (V-S S

)dt (1.8)

– rV =0

This equation is known as the Black Scholes Pricing Differential Equation (PDE). The price of any option, depended on S and t, will thus satisfy this equation otherwise, as mentioned before, arbitrages will arise.

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Appendix 8: the derivation of risk-free portfolio
Here we have two sources of randomness-the asset price, which can be hedged with the asset, and the volatility, which also has to be hedged in order to set up a riskless portfolio. However, since there is no trading of volatility the variance process cannot be used in creating the replicating portfolio. Thus, a second derivative, which also depends on s, v and t is introduced. Then it is possible to use the portfolio replicating arguments which is similar in the Black-Scholes formula. Now the portfolio П consisting of the derivative f(s,ν,t), a quantity and a quantity of another derivative (s,ν,t) depends on volatility

of the underlying asset S and can be written as : П= f(s,ν,t)(s,ν,t)(1)

a change in the portfolio value over dt is then given by the following equation: d П= df(s,ν,t)(s,ν,t)(2)

the two-dimensional case of Ito’s lemma is used in obtaining the PDE d f(s,ν,t)= analogously, the PDE for d (s,ν,t)= + (s,ν,t) can also be derived: + (4) (3)

plugging (3) and (4) into (2) and rearranging dП=( + +( + + (5) + dt

The objective now is to eliminate the uncertainty related to the portfolio so it becomes riskless. Thus, the quantities should be properly set, so that the Wiener

process of both the underlying and the volatility will be eliminated.

By inserting the weights the portfolio evolution over time must be risk-free, which

72

means that the portfolio must yield the risk-free rate dП=rПdt=r(f(s,ν,t)(s,ν,t)(6)

73

Appendix 9 BScall function

The following is the codes for BScall function based on the closed-form solution for a vanilla call option:

Function Payoff(S As Double, x As Double, callOpt As Integer) As Double

Dim Value As Double

Value = S - x If callOpt = 1 Then 'is a Call-option If Value > 0 Then Payoff = Value Else Payoff = 0 End If Else ' Put-option If Value < 0 Then Payoff = -Value Else Payoff = 0 End If End If End Function

74

Function BSCall(S As Double, x As Double, t As Double, r As Double, v As Double, div As Double) As Double

Dim vol As Double Dim nvx As Double Dim d1 As Double Dim d2 As Double Dim modifiedS As Double

modifiedS = S * Exp(-div * t) If t