A Novel Simple but Empirically Consistent Model for Stock Price and Option Pricing
HUADONG(HENRY) PANG∗
Quantitative Research, J.P. Morgan Chase & Co. 277 Park Ave., New York, NY, 10017 Third draft, May 16, 2009

Abstract In this paper, we propose a novel simple but empirically very consistent stochastic model for stock price dynamics and option pricing, which not only has the same analyticity as log-normal and Black-Scholes model, but can also capture and explain all the main puzzles and phenomenons arising from empirical stock and option markets which log-normal and Black-Scholes model fail to explain. In addition, this model and its parameters have clear economic interpretations. Large sample empirical calibration and tests are performed and show strong empirical consistency with our model’s assumption and implication. Immediate applications on risk management, equity and option evaluation and trading, etc are also presented. Keywords: Nonlinear model, Random walk, Stock price, Option pricing, Default risk,
Realized volatility, Local volatility, Volatility skew, EGARCH.

This paper is self-funded and self-motivated. The author is currently working as a quantitative analyst at J.P. Morgan Chase & Co. All errors belong to the author. Email: henry.na.pang@jpmchase.com or hdpang@gmail.com.

∗

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Electronic copy available at: http://ssrn.com/abstract=1374688

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1. Introduction
The well-known log-normal model for stock price was first proposed by Louis Bachelier (1870-1946) and published in his doctoral thesis at 1900. His “theory of speculation” (Theorie de la Speculation, see Bachelier(1900)) was discounted by none other than Henri Poincare, observing that “Mr. Bachelier has evidenced an original and precise mind [but] as the subject is somewhat remote from those our other candidates are in the habit of treating.” Nevertheless, his model foresaw many of the mathematical discoveries made later by Wiener and Markov, and outlined the importance of such ideas in today’s financial markets, stating that “it is evident that the present theory solves the majority of problems in the study of speculation by the calculus of probability.” Later on, in 1973, Fischer Black and Myron Scholes first used log-normal model as underlying stock price dynamics to derive the analytical formula for stock options in their famous paper, “The Pricing of Options and Corporate Liabilities” (see Black and Scholes(1973)). The now so called Black-Sholes formula is treated as foundation of option pricing and Robert C. Merton was the first to publish a paper expanding its mathematical understanding (see Merton(1973)). Due to its mathematical simplicity, the log-normal model and option pricing model based on it are now widely used at modern financial industry. Despite the success of log-normal and Black-Scholes model, lots of empirical evidence showed that the realized return of stock price diverged significantly from Brownian motion. For example, in 1988, Lo, A. and Mackinlay A.C. wrote a paper “Stock Market Prices do not follow random walk: evidence from a simple specification test” (see Lo and MacKinlay(1988)) rejected the random walk hypothesis. And the following six empirical phenomenons arising from equity and option markets have drawn much attention recently: (1) the leptokurtic feature that the realized return distribution of stocks often has a higher peak and two (asymmetric) heavier tails than those of normal distribution; (2) the default risk feature that company has positive probability to bankrupt in finite time period (e.g. Garlappi, Shu and Yan(2006)). The probability to fail in short term for many financial service, automobile, and other high leverage companies were significantly high in recent financial crisis. When the company’s stock is very stressful, the correlation between stock price and the spread of credit default swap, which measures the company’s default chance, is observed to be much higher than the one in bull market; (3) the local volatility feature that the volatility of stock highly depends on the stock

Electronic copy available at: http://ssrn.com/abstract=1374688

A Novel Simple but Empirically Consistent Model for Stock Price and Option Pricing

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price level and is significantly large in very stressful situation. One recent example is: at the first two weeks of September 2008, just before its filing for bankruptcy, Lehman Brother’s stock volatility is about 10 times higher than that during its 2004-2006 period. In this financial crisis, this phenomenon is observed almost everywhere; (4) the volatility clustering and EGARCH effect that for stock prices, large(small) changes tend to occur with cluster feature, in either sign, and negative changes have more impact on volatility than positive changes. In financial time series analysis, this autoregressive conditional heteroskedasticity pattern and asymmetry effect of stock price change on volatility are popularly modeled by EGARCH model (see Engle(1995)); (5) the negative volatility skew phenomenon widely observed in stock option markets. If the Black-Scholes model is correct, the implied volatility from option market should be constant. In reality, the lower strike call or put option’s implied volatility is always observed to be higher. Many studies have been conducted to modify Black-Scholes model to capture this “volatility skew” feature. The most famous of them are local volatility model, stochastic volatility model, Levy process model (including jumping process model), etc; (see Cox(1996), Cox and Ross(1976), Hull and White(1987), Heston(1993), Andersen, Benzoni and Lund(1999), Duffie, Pan and Singleton(2000), Kou(2002), etc.) (6) the phenomenon that realized volatility on average tends to be lower than implied volatility (e.g. Christensen and Hansen(2002)). If the Black-Scholes model is correct, the implied volatility from option market should be able to describe the expected future volatility of stock price. However, the implied volatility seems always over-estimate the realized volatility. This may indicate that the implied volatility is the “wrong number” from a “wrong model” to get the right option price. Each phenomenon described above is a very active research topic in modern quantitative finance and there are hundreds of related research papers devoted to explain them. Some of them are very successful in explaining part of these phenomenons either in mathematics or in economics. However, by author’s knowledge, there was no single simple model which can resolve all of them simultaneously before this paper. The current paper proposes a novel model which has the following properties: (a) It resolves and explains all the above puzzles and phenomenons. (b) It is very simple. The stock price is modeled as a simple deterministic non-linear transformation of a log-normal hidden process. This simplicity induces analytical formulas for stock price and return distribution, stock local volatility, company’s future default curve (stock price implied), and many option pricing problems, including European call

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and put, etc, whatever log-normal and Black-Scholes model can analytically solve. (c) The model and its parameters have clear intuitive economic explanation, which is consistent with corporate finance theory. The equity price is considered as a derivative product of company’s total asset. (d) The empirical calibrations and tests on large sample of liquid trading stocks (110 small capital, 391 mid-capital and 366 large capital stocks traded in current market with average annual volume over 1 million) show high consistency with model assumption and model implication. Though the empirical stock price is not random walk, the calibrated hidden process shows strong random walk characteristic.

2. The model
2.1 The model formulation We assume the risk-free interest rate r and the following log-normal hidden process under physical probability measure P : dV (t) = µV (t)dt + σV (t)dW (t), (2.1)

where W (t) is the standard Brownian motion, σ > 0. Then the stock price process S(t) is modeled by: S(t) = V (t) − Dδ+1 1min0≤s≤t V (s)>D , V (t)δ (2.2)

where 0 ≤ D < V (0) and δ = 2r/σ 2 . For notional simplicity and in order to get analytical solutions for option pricing, r, µ, σ and D are assumed to be constant. We also assume the stock pays no dividend. These assumptions, however, can be dropped to develop a more general theory. Solving the simple stochastic equation (2.1) gives the explicit dynamics of the equity price: S(t) = {V (0)e(µ− 2 σ
1 2 )t+σW (t)

−

Dδ+1 −δ(µ− 1 σ2 )t−δσW (t) 2 e }1min (µ− 1 σ 2 )s+σW (s) 2 >D 0≤s≤t V (0)e V (0)δ

(2.3)

Notice if D = 0, our model is reduced to Black-Scholes model. From now on, we concentrate on D > 0. This positive D introduces desired non-constant local volatility of equity price and negative volatility skew for option pricing as we will show later. And we have

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P (S(t) = 0) = P (min0≤s≤t V (s) ≤ D) > 0, which means there is positive probability for the company to default(or bankrupt) between time 0 and t. Another interesting thing is that the risk-free interest rate r appears in our stock price dynamics through δ under physical probability measure P . We will see later, this is a key assumption in finding risk-neutral measure for S(t) in our model. This doesn’t happen in Black-Scholes model and most option pricing models since people observed the stock price change has low correlation with interest rate change. In our model, when r increases, S(t) tends to increase since S(t) is an increasing function of δ. But it doesn’t imply a high correlation between instantaneous change of r and S(t) since the change of V (t) has much more impact on change of S(t). However, in the long run, for the same path realization of V (t), S(t) seems to be larger for bigger r by our model (note this doesn’t mean strong cause and effect relationship between stock price and interest rate policy, because the realization of V (t) is more important for S(t)’s level). Interestingly, if not totally coincidentally, S&P 500 stock index did increase a lot while Federal Reserve increased rate from 2004 to 2006 and it experienced the biggest drop of last 70 years when Federal Reserve dramatically cut interest rate during this financial crisis starting from 2007. 2.2 Motivation and parameter interpretation In corporate finance, the equity is widely considered as a call option on company’s total value. The stock shareholder enjoys the infinite upside potential of company’s value growing, but has limited possible downside loss. When the company defaults (or bankrupts), all the shareholder can lose is the money which he/she used to buy the stock. However, once the company default, the stock price will be 0 hereafter (not a bad assumption, though in real life the shareholder could still get a little). So it’s reasonable to consider equity as a down-and-out call option on company’s total value with both strike and barrier equal to the total debt. Since the stock shareholders usually don’t (or can’t) liquidate the company’s asset and instead they sell the equity to other people, it’s fair to consider this option to be European but with very long maturity. We thus model the dynamics of company’s total value per share as log-normal process V (t) by (2.1), and consider the European down-and-out call option on V (t) with both strike and barrier equal D (interpreted as the debt per share) and maturity T > t. We assume the company still alive at t (i.e. min0≤s≤t V (s) > D) and assume constant risk-free interest rate r. This exotic option price at time t is well-known as (see Merton(1973) or

6 Kwok(1999)):

Huadong(Henry) Pang/J.P. Morgan Chase & Co.

Cdo (V (t), T ) = V (t) N (d1 ) − (

D δ+1 D δ−1 ) N (d3 ) − De−rT N (d2 ) − ( ) N (d4 ) , V (t) V (t)

where N (x) is the cumulative density function of standard normal distribution, and
2 (t) √ ln( VD ) + (r + σ2 )T √ d1 = , d2 = d1 − σ T σ T 2 D D 2 2r d3 = d1 + √ ln( ), d4 = d2 + √ ln( ), δ = 2 . V (t) V (t) σ σ T σ T

˜ Note that when we compute this option price we used “risk-neutral” measure P such that ˜ dV (t) = rV (t)dt + σV (t)dW (t), ˜ ˜ where W (t) = W (t) + µ−r t is standard Brownian motion under P . For people familiar with σ ˜ derivative pricing theory, it’s obvious to see this P will be inherited as the risk-neutral measure for S(t) since the stock is modeled as a derivative product on V (t). Let T → +∞, we get our formula for stock price at time t: S(t) = lim Cdo (V (t), T ) = V (t) −
T →∞

Dδ+1 . V (t)δ

Notice that once V (t) hits D, then the stock price S(t) equals 0 (company defaults or bankrupts), and it should be stuck at 0 hereafter. So we model the dynamics of stock price as (2.2). As we described above, V (t) can be interpreted as company’s total value per share at time t and D can be interpreted as company’s debt per share. Notice the stock price is 2r an increasing function of δ = σ2 , which in fact controls the company leverage’s impact on stock price . When δ is high (the volatility of V (t) is low), the company’s high leverage (high V /(V − D) or high D/V ) will have less damage to company’s stock price than the low δ case. This is consistent with our economic intuition. 2.3 The simple non-linear function h(x) In our model, the stock price S(t) and hidden log-normal process V (t) are connected by the strictly increasing deterministic non-linear function h : [D, +∞) → [0, +∞) defined as Dδ+1 (2.4) h(x) = x − δ , x

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Figure 1(a) describes the shape of h(x) for D = 10 and δ = 2. h(x) is a strictly increasing concave function, approximating to identity function as x goes to +∞. So the inverse function h−1 (x) of h is well-defined.

50 45 40 a typical shape of function h(x) the 45 degree straight line

22 20 18 the simulated log−normal process V(t) the simulated stock price S(t)=h(V(t))

35 30 25 20 15 10 10 5 0 0 10 20 30 40 50 8 6 16 14 12

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(a) Shape of h(x)

(b) S(t)=h(V(t)) and log-normal process V(t)

Figure 1: Non-linear transformation h(x)

To give an initial intuition of the stock price dynamics S(t) given by our model, we draw a simulated log-normal process V (t) and corresponding h(V (t)) in Figure 1(b). When the stock price is high (so V (t) is far above D), S(t) behaves almost like a log-normal process. This is consistent with the stock price dynamics observed in bull market period. When the stock price drops a lot (so V (t) is low), the volatility of stock increases dramatically due to the leverage effect of h(x) (h (x) 1 as x is closing to D). And as the stock price is lower, the leverage effect on stock volatility is stronger (h (x) < 0). This describes the “panic effect” widely observed in bear market.

3. The distribution of S(t) and leptokurtic feature
To get the explicit formula for the distribution of S(t), we need the following standard result from probability theory (see Shepp(1979)): Lemma 3.1 Assume W (t) is a standard Brownian motion with drift η under probability measure P , W (0) = 0. For b < 0, Tb = inf {s ≥ 0; W (s) = b} is the first time W (t) hits

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b. Then for any b < 0 and a > b, we have: b − ηt b + ηt P (Tb < t) = N ( √ ) + e2ηb N ( √ ), t t
(a−2b)2 1 ηa− η2 t − a2 − 2t 2 (e 2t − e e )da, 2πt where N (x) is the cumulative density function of standard normal distribution.

(3.1)

P (Wt ∈ da, Tb > t) = √

(3.2)

By Lemma 3.1 and (2.3), we immediately get: Theorem 3.2 The probability density function of S(t) modeled by (2.1) and (2.2) under physical probability measure P is β − γt β + γt P (S(t) = 0) = N ( √ ) + e2βγ N ( √ ), (3.3) t t g(x)2 (g(x)−2β)2 γ2t 1 P (S(t) ∈ dx) = √ eγg(x)− 2 (e− 2t − e− 2t )dx, for x > 0, σ 2πt[(1 + δ)h−1 (x) − δx] (3.4) h−1 (x) µ 1 D σ 1 where β = σ ln( h−1 (S(0)) ), γ = σ − 2 and g(x) = σ ln( h−1 (S(0)) ).

140

0.035 0.03

the distribution on positive region for S(5) with P(S(5)=0)=0.14% the log−normal distribution with same mean and variance
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distribution histogram of realized return of S(t) normal distribution with same mean and deviation

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(a) distribution of S(t)

(b) leptokurtic effect

Figure 2: The distribution of S(t) and the leptokurtic effect Figure 2(a) shows the distribution density function (solid curve) of S(t), with t = 5, r = 0.01, µ = 0.01, σ = 0.1, D = 20 and S(0) = 35. The log-normal density function (dashed curve) with the same mean and variance is also drawn for comparison. The distribution of S(t) has an obvious left fat and right thin tail comparing to log-normal. There is 0.14% probability for S(t) to be 0, which makes the left tail even fatter.

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As we described in section 2.3, the return process of S(t) is not stationary, and it’s also not ergodic. So we can’t use the realized return in one realized S’s path to estimate the S’s true return distribution at fixed time t. If we do draw the histogram of realized daily return for one simulated S’s path by our model, we get the shape as in Figure 2(b). There is an obvious leptokurtic feature: a higher peak and two (asymmetric) heavier tails than those of normal distribution. This is consistent with what we observed in empirical stock market. Note this histogram highly depends on the realization of S’s path and has no “stationary” or “invariant” feature. In the modern financial industry, many people tend to use the realized historical stock return to estimate the stock’s future return distribution or to estimate the correlation between two stocks, which could be significantly wrong, especially in a stressful period when S(t) is very un-stationary and un-ergodic. Our model in fact provides a more accurate method for this distribution and correlation estimation problem by instead looking at the stationary and ergodic return process of V (t). The following is a simple but powerful application in dynamic hedging problem: Theorem 3.3 Assume constant interest rate is r, for i = 1, 2, stock Si (t) has hidden process Vi (t) such that, dVi (t) = µi Vi (t)dt + σi Vi (t)dWi (t) Si (t) = Vi (t) − δ Di i +1 1min0≤s≤t Vi (s)>D Vi (t)δi

2 where δi = 2r/σi . W1 (t) and W2 (t) are two standard Brownian motions with correlation ρ. Assume S1 and S2 are still alive(still positive) at time t, if we use the change of stock S2 to hedge the change of stock S1 , the optimal hedge ratio at time t is:

αhedge (t) = −

σ1 [V1 (t) + δ1 V11 δ1 ] (t) σ2 [V2 (t) +
D 2 δ2 V22 δ2 ] (t) δ +1

D

δ1 +1

ρ,

(3.5)

where Vi (t) = h−1 (Si (t)) and hi (x) = x − i

Di i xδi

δ +1

.

The proof is trivial by solving variance minimization problem minα V ar(dS1 (t)+αdS2 (t)) and using first order condition.

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4. Local volatility, volatility clustering and EGARCH effect
When S(t) > 0, we use (2.1) and apply Ito’s lemma to S(t) = V (t) − DV δ to get the following local volatility representation of our model under physical probability P : dS(t) = [µ+(δµ−δr −r)( D h−1 (S(t)) )δ+1 ]h−1 (S(t))dt+[1+δ( D h−1 (S(t)) )δ+1 ]h−1 (S(t))σdW (t), δ+1 (4.1) where r, σ, δ, D and h−1 (x) are the same as in section 2. Notice when S(t) is close to 0 (then h−1 (S(t)) is close to D), the variance of dS(t) is approximating to (1 + δ)2 D2 σ 2 dt. So the process S(t) in (4.1) can hit 0, and we let S(t) = 0 after the first time S(t) hits 0. So if S(t) > 0, the local volatility of S(t)’s return at time t is σlocal (t) = σ[1 + δ( h−1 (S(t)) D )δ+1 ] . h−1 (S(t)) S(t) (4.2)

It’s very interesting to compare our model with other local volatility models, like the most famous constant elasticity of variance(CEV) model: dS(t) = κS(t)dt + νS α (t)dWt , (4.3)

where κ, ν are constants and α > 0. In CEV model, the variance of dS(t) is ν 2 S 2α (t)dt and the local volatility of stock return is thus σCEV = νS α−1 (t). To be consistent with the phenomenon that stock volatility is higher as price is lower, we must have α < 1 in CEV model. Figure 3(a) shows the variance of dS(t) to stock price S(t), and figure 3(b) shows the local volatility to stock price for both models. In CEV model, when stock price is very small, the variance of dS(t) converges to 0, while our model’s variance of dS(t) is still above a significant level. On the local volatility side, both models show significant growing as stock price is low. However, CEV’s local volatility converges to 0 when stock price is very large and the one in our model converges to a positive constant level. In reality, even when the stock price is very low, the magnitude of stock price change doesn’t converge to 0 and it’s very unrealistic for stock’s local volatility converges to 0 when stock price is very high. Our model is much more consistent with these empirical phenomenons than other local volatility models. The volatility clustering and EGARCH effect are kind of obvious from our local volatility representation (4.2). When the stock price moves into a high level region, the local volatility is almost constant and low, we are very likely to see a group of small changes clustering there. When the stock price drops into a low level region, the local volatility of stock will enter

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50 40 the local volatility of our model to stock price the local volatility of CEV to stock price

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(a) variance of dS(t) to stock price

(b) local volatility to stock price

Figure 3: Comparison of our model with CEV model into a high region, which expects a bundle of big changes during that period. By (4.2) and figure 3(b), starting in the same stock level, when the stock price drops, the impact on local volatility is bigger than when the stock price increases at the same amount/percentage, as the local volatility is a convex function to stock price in our model. (To prove the convexity of function in (4.2) to stock price S(t) is very easy, we leave it to readers.)

5. Risk-neutral measure and analytical option pricing formula
˜ Let Z(t) = exp − 1 ( µ−r )2 t − µ−r W (t) , then by Girsanov theorem, W (t) = W (t)+ µ−r t 2 σ σ σ ˜ defined by dP = ZdP . The ˜ is a Brownian motion under the new probability measure P ˜ dynamics equation of V (t) under P is: ˜ dV (t) = rV (t)dt + σV (t)dW (t). ˜ Let S(t) = V (t) − 0. By Ito’s formula,
Dδ+1 , V (t)δ

˜ ˜ and T = inf {s ≥ 0; S(s) = 0} be the first time of S(t) hitting

˜ ˜ ˜ d{e−rt S(t)} = e−rt {−rS(t)dt + dS(t)} = e−rt {−r[V (t) − = e−rt Dδ+1 δ 1 δ(δ + 1) 2 ]dt + dV (t) − Dδ+1 [− dV (t) + σ dt]} δ δ+1 V (t) V (t) 2 V (t)δ

1 δσDδ+1 ˜ Dδ+1 (δ + 1) (r − δσ 2 )dt + e−rt [σV (t) + ]dW (t) V (t)δ 2 V (t)δ =e
−rt

δDδ+1 ˜ (1 + )σV (t)dW (t), V (t)δ+1

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˜ ˜ where we used δ = 2r/σ 2 . So e−rt S(t) is a martingale under measure P . By optional stop˜ ping time theorem, e−rt S(t) = e−r(t∧T) S(t ∧ T) is also a martingale, where t ∧ T = min(t, T). ˜ P is thus the desired risk-neutral measure for stock price S(t). Note that δ = 2r/σ 2 is a key assumption in our model to find this risk-neutral measure and thus guarantee no arbitrage in option pricing. Now under this risk neutral measure, we price the European call option with strike K > 0, maturity T (European put option can be priced according to put-call parity). As˜ sume S(0) > 0, then under risk-neutral measure P , the call option price at time 0 is ˜ C(S(0), K, T ) = E[e−rT (S(T ) − K)+ ] =e
−rT

Dδ+1 ˜ E[(V (T ) − − K)+ 1min0≤s≤T V (s)>D ], δ V (T )
˜

where V (s) = V (0)e(r− 2 )s+σW (s) . By Lemma 3.1 and after easy simplification, we get the analytical formula for call option price: Theorem 5.1 Assume stock price S(t) is modeled by (2.1)-(2.2) and is still alive at 0 (S(0) > 0), then the price of European call option with strike K and maturity T at time 0 is C(S(0), K, T ) = h−1 (S(0))[N (d1 ) − e−λ1 N (d2 )] − [h−1 (S(0)) − S(0)][N (d3 ) − eλ1 N (d4 )] −e−rT K[N (d5 ) − eλ2 N (d6 )], (5.1)

σ2

where N (x) is the cumulative density function of standard normal distribution, h−1 (x) is the δ+1 2r inverse function of h(x) = x − Dxδ , δ = σ2 and
(S(0)) ln( hh−1 (K) ) + (r + √ d1 = σ T
−1

σ2 )T 2

2 h−1 (S(0)) , d2 = d1 − √ ln( ), D σ T

d3 = d1 − (

√ √ 2r 2r + σ) T , d4 = d2 − ( + σ) T , σ σ √ √ d5 = d1 − σ T , d6 = d2 − σ T ,

h−1 (S(0)) h−1 (S(0)) λ1 = (1 + δ) ln( ), λ2 = (1 − δ) ln( ). D D Not surprisingly, as D → 0, our model for stock price converges to log-normal model and (5.1) converges to Black-Scholes formula.

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6. Volatility skew and realized volatility
Our model intrinsically carries the volatility skew for option pricing. It comes from the left fat tail of stock price distribution as shown in figure 2(a). The positive probability to default feature in our model can make the left tail to be fat in any extent we want, which dramatically increases the ability of fitting very steep volatility skew curve. Figure 4(a) shows the model volatility skew for different σ and D parameters, with S(0) = 20, r = 0.01 and maturity T = 5.

2 1.8 1.6 implied volatility of call option 1.4 1.2 implied volatility

1.4

sigma=0.4, D=20 sigma=0.2, D=20 sigma=0.4, D=40 sigma=0.2, D=40

the implied volatility for 5 year call options 1.2 1 0.8

0
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1.2 1 0.8 0.6 0.4 0.2 0 200 400 600 date 800 1000 1200 1400 simulated realized volatility in next 5 years

(a) volatility skew for different parameters

(b) realized volatility and implied volatility

Figure 4: Volatility skew and realized volatility In real life, on average, the realized volatility seems always below the implied volatility. Our model can explain this phenomenon very easily: the realized volatility is computed conditional on the stock didn’t default during that period, while the implied volatility captures the future stock volatility with default possibility. Obviously, the latter one should have a bigger number. Let r = 0.01, σ = 0.4, D = 20, S(0) = 20, figure 4(b) shows the implied volatility curve for call options with maturity T = 5 and the realized volatility in next 5 years by simulation. On average, the realized volatility is obviously below the implied volatility from option market at time 0.

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7. Empirical test and applications
7.1 The empirical test Let’s take AIG(American International Group) as an empirical test example. The data we used is the AIG historical dividend and splitting adjusted close stock prices from 3/3/2003 to 9/30/2008, downloaded from http://finance.yahoo.com. The risk-free interest rate r is set to be 1%, and we get the calibrated σ = 0.0298, D = 72.7037 and µ = −0.0066. Figure 5(a) is the AIG stock price(solid curve) and its 20-day window realized volatil-

5 AIG stock price S(t) AIG stock S(t)‘s 20−day window realized volatility

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(a) AIG stock price and 20-day window realized volatility

(b) calibrated AIG hidden process V(t) and 20-day window realized volatility

4 3 2 QQ plot curve for AIG stock price return QQ plot curve for return of AIG hidden process V(t) 45 degree straight line

3 AIG stock 20−day window realized volatility AIG stock model local volatility 2.5

2 1 0 −1 1 −2 0.5 −3 −4 −4 0 Jan03 1.5

−3

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(c) QQ-plot with normal distribution for AIG stock price and the hidden process V (t)

(d) AIG 20-day window realized volatility and model local volatility

Figure 5: The calibration of our model for AIG stock

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ity(dashed curve) between 3/3/2003 and 9/30/2008. Starting from September 2007, the realized volatility of AIG increased dramatically, and became extremely high at September 2008, when Lehman Brothers filed for bankruptcy, AIG had the liquidity problem and its credit rating was downgraded. The stock price of AIG was obviously not a log-normal process during this period. Figure 5(b) is the hidden V (t) process(solid curve) computed from V (t) = h−1 (S(t)) with calibrated σ and D. V (t)’s 20-day realized volatility is also drawn(dashed curve). The V (t) process shows amazing log-normal process feature with sampling volatility 0.0288. The calibrated model volatility σ = 0.0298 is not rejected by χ2 test at 5%. And V (t) is not rejected for random walk hypothesis using variance ratio test V R(q) (see Lo and MacKinlay(1988)) for q = 4, 8, 16 at 5% and for q = 2 at 4%. Figure 5(c) is the Quantile-Quantile(QQ) plot with normal distribution for AIG stock return(solid curve) and hidden V (t)’s return(dashed curve), which shows a significant improvement on the normality for V (t)’s return than the one of original stock price S(t). Figure 5(d) is the AIG 20-day window realized volatility and the model local volatility computed by formula (4.2). The model stock local volatility shows amazing consistency with realized one. We did an exhaustible empirical calibration and tests on all liquid (with average annual volume over 1 million) 110 small-cap, 391 mid-cap and 366 large-cap company stocks. The data we used is dividend and splitting adjusted close stock prices from 3/3/2003 to 3/20/2009, downloaded from http://finance.yahoo.com. We set risk-free interest rate r = 1% and a hard constraint that V (t)’s sampling volatility doesn’t reject V (t)’s model volatility σ at 5% by χ2 test. We then calibrated σ and D for each company by minimizing the distance between its V (t) return’s QQ-plot curve with normal distribution and the 45 degree straight line(we neglected the low 2.5% and high 2.5% quantiles). µ was estimated by the sampling drift of calibrated V (t) and is not a main parameter for calibration. The random walk hypothesis for V (t) is not rejected for 72% of small-cap, 71% of mid-cap, 62% of large-cap by variance ratio test V R(q) for q = 2, 4, 8, 16 at 5%. If we lower the significance level from 5% to 1%, the hypothesis for 86% of small-cap, 88% of mid-cap and 85% of large-cap is not rejected. Notice in our empirical tests we assumed D is constant during last 6 years, which could not be a good assumption for many companies. The risk-free interest rate set to be constant 1% could also be problematic as the Federal Reserve changed the rate significantly in past 6 years. Even so, thanks to the accuracy of our model, there are still more than 2/3 of liquid stock prices showing high empirical consistency with our model. The empirical test of our model in fact answers the following question: “Can stock price be modeled through random walk?”. The short answer may be: “Yes, but it need go through a simple non-linear transformation.”

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7.2 Applications Due to the simple mathematical setup, the same analyticity with log-normal and BlackSholes model and high consistency with empirical stock and option markets, our model has wide applications in almost every area in equities. Besides the obvious application on option pricing and the dynamic hedge ratio mentioned in Theorem 3.3, we now exhibit several more immediate applications of our model (we neglect the detailed mathematics derivation, which is very easy if not trivial): (1) The Value at Risk(VaR) calculation. The traditional historical VaR method or other model-building VaR method on stock price could be significantly wrong especially in very stressful situation, because the return distribution of stock is neither stationary nor ergodic. Our model suggests that we could compute the tail distribution of stock return through its hidden process, which is both stationary and ergodic and the mapping between stock price and its hidden process is just a simple monotone non-linear function. This new VaR method can be easily generalized to stock portfolios and stock indexes. (2) The pair trading and statistic arbitrage strategy. As mentioned in section 3, the correlation between stock returns is not stable. It’s much better to consider the relation of stocks in stationary hidden process space. Assume we have two stocks whose hidden processes are highly correlated, but one of them has much higher default probability (very high D could cause this), there could be a statistical arbitrage opportunity by buying the low default-risk stock and selling the high default-risk one. We do the dynamic optimal hedging using Theorem 3.3 whenever their stock prices are above the initial trading prices. As the hidden processes drops, the stock price of the high default risk company will have a bigger drop than the other, by the leverage effect of non-linear transformation h(x). If there is a big increase on hidden process, even using the static hedging, the effect on both stock price is still comparable since the nonlinear function h(x) is concave and approximating to identity function as the variable is large. Not mentioning we have extra bonus of the high chance that high-default risk one does default before the other one. We have a very limited downside risk, but have very high upside potential and probability. A recent example is the stocks of Lehman Brothers and Goldman Sachs in 2008. (3) The exploration of mean reversion pattern. Possibly the greatest and oldest dream of stock investors is to detect when is the bottom and when is the peak of stock price. To realize

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this dream, the stock price need show some mean-reversion pattern. However, even the stock does have the desired pattern, it could still be very hard to be captured due to the stock’s high non-stationarity. As usual, the much more stationary and ergodic hidden process V (t) can obviously help us, also thanks to the monotone feature of non-linear transformation h(x). (4) The trading signal from stock and option market. Due to the high consistency of our model on historical stock prices and the phenomenon from option markets, it is now feasible to calibrate our model using historical stock prices and currently traded option prices together. The market-model option price difference and calibrated stock distribution could indicate rich-cheap signals in either option products or stock itself.

Appendix. A simple generalization by an alternative method
In this appendix, we will give a simple generalized version to our model through an interesting mathematics way of thinking. We assume the risk-free interest rate r and a hidden process V(t) is log-normal process under some probability measure P(and assume the physical probability measure is change of measure to P): dV (t) = νV (t)dt + σV (t)dW (t), where ν is constant and W (t) is standard Brownian motion under P . Now we ask the following math question: “What’s the C 2 function h(x) such that e−rt h(V (t)) is a martingale?”. By Ito’s lemma, we have the following second order CauchyEuler equation for h(x): 1 −rh(x) + νh (x)x + σ 2 h (x)x2 = 0. 2 The general solution to this ODE is: h(x) = C1 xm1 + C2 xm2 , where m1 =
1− 2ν + 2 σ (A.1)

(A.2)

(A.3) .

( 2ν −1)2 + 8r 2 2 σ σ

2

and m2 =

1− 2ν − 2 σ ( 2ν −1)2 + 8r 2 2 σ σ

2

We model the stock price dynamics S(t) as following: S(t) = V (t)m1 − Dm1 −m2 V (t)m2 1min0≤s≤t V (s)>D , (A.4)

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Huadong(Henry) Pang/J.P. Morgan Chase & Co.

where D ≥ 0 is a constant and V (0) > D. By optional stopping time theorem, P is the risk-neutral measure for S(t). m2 ˜ ˜ Let V (t) = V (t)m1 , D = Dm1 , δ = − m1 , then we can rewrite (A.4) as:

˜ Dδ+1 ˜ S(t) = V (t) − 1min0≤s≤t V (s)>D . ˜ ˜ ˜ V (t)δ

(A.5)

˜ Now we have the same nonlinear function form as (2.4). V (t) is interpreted as the com˜ pany total asset per share. D is interpreted as the company debt per share. Notice when ν = r, this will be reduced to model (2.2), and when ν = r and D = 0, this will be reduced to Black-Scholes model. (A.5) can also be easily obtained by considering S(t) as a Euro˜ pean down and out call option with infinite maturity on V (t), with both barrier and strike ˜ ˜ equal D, calculated under measure P . There is no arbitrage since V (t) is in fact not tradable. By lemma 3.1 and simple calculation, we get the following analytical formula for European call option under this generalized model: Theorem A.1 Assume stock is still alive at 0 (S(0) > 0), then the price of European call option with strike K and maturity T at time 0 is: C(S(0), K, T ) = h−1 (S(0))m1 [N (d1 ) − eλ1 N (d2 )] − (h−1 (S(0))m1 − S(0))[N (d3 ) − eλ2 N (d4 )] −e−rT K[N (d5 ) − eλ3 N (d6 )], (A.6)

where N (x) is the cumulative density function of standard normal distribution, h−1 (x) is the inverse function of h(x) = xm1 − Dm1 −m2 xm2 , and
(S(0)) ln( hh−1 (K) ) + (σm1 + η)T 2 h−1 (S(0)) √ d1 = ), , d2 = d1 − √ ln( D T σ T √ √ d3 = d1 − σ T (m1 − m2 ), d4 = d2 − σ T (m1 − m2 ), √ √ d5 = d1 − σ T m1 , d6 = d2 − σ T m1 , 1 σ
−1

1 2 h−1 (S(0)) ν )(σm1 + η), η = − σ, λ1 = − ln( σ 2 σ D 2 h−1 (S(0)) 2 h−1 (S(0)) λ2 = − ln( )(σm2 + η), λ3 = − ln( )η. σ D σ D Not surprisingly, if ν = r, this formula will be reduced to (5.1), and if ν = r and D → 0 this formula will be reduced to Black-Scholes formula.

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References
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[15] Merton, R.C. 1973. Theory of rational option pricing. Bell Journal of Ecnomics and Management Sciences, 4, 141-183. [16] Shepp, L.A. 1979. The joint density of the maximum and its location for a Wiener process with drift. J. Appl. Prob. 16, 423-427.