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1. The Arbitrage Theorem: -­‐>Derivatives cannot be priced by discounting tech, cause underlying moves and risk changes -­‐>arbitrage pricing: Replicating portfolio: should equal cost of instr. è Prices are represented by vector ������! (������) ������! ������! = … ; N=Assets ������ = … ;K=states ������! (������) ������! ������!! … ������!! ������ = … … … ; payoff Matrix (row=assets) ������!! … ������!" S=Asset prices; arbitrage free only if Q are positive If we want q as column: D_inv*S; if not: D_tran_inv Example: 1 asset rf=10% and 1 stock=150 or 100 1) Use state prices

∆������ = ������ ∆������ where ������ = ������ 0,1 Mean of Δz=0; Variance of Δz=Δt ������������ = ������������������������ + ������������������������ è Also geometric Brownian Motion Ito’s lemma: ������������ ������������ 1 ! ! ������ ! ������ ������������ ������������ = + ������������ + ������ ������ ������������ + ������������������ ������������ ������������ 2 ������������ ! ������������ Where G(S,t) is a function of S (process); term to the left in brakets is expected change in option, while to the right is the uncertainty, that also affects derive. Over short period, the BM is almost entirely determined by random component è GBM implies that stock price at T is lognormally distributed and that continuously compounded rates of return are normally distributed µ=continuous compounded rate of return between 0 and T, then: ln
!! !!

Delta changes as stock prices changes and time passes à hedge position must be rebalanced. (shorting or longing the stock depending on the option price and the position) The delta of a European call option is N(d1) and the one of a put is N(d1)-­‐1.

-­‐ Being close to ITM -­‐> c∆ ≈ 1; p∆ ≈ -­‐1; OTM ∆ ≈ 0 -­‐ ATM à ∆ tend to increase with expiration. Gamma Γ: is the rate of change in delta with respect to the underlying Formula for options on futures(up): asset: ������Δ ������ ! ������ ������ ������1 = = Implied volatility: should be constant and unique. Consistent with the ������������ ������������ ! ������������√������ observed price.Can be used to calculate price of another option on same asset. Assumptions (1) underlying is a stock; (2) no dividends (3) European-­‐ still; (4) rf and st.dev constants; (5) no imperfect. Implied volatilities of p and c must be the same if options are European and have same strike and mat.-­‐-­‐> P-­‐C parity: c-p = S0-K*e^(-r*t) American at least as valuable as European (early exercise) Volatility skewà vol of in, at or out money changes-­‐ ATM lower volatility.

With ������ ������1 =

! √!"

������ ! !

!!

= ������

-­‐> ������! = ������! ������ ! !! ! ! !!!!"√∆!
!

Furthermore evolution

! becomes������! = ������! ������ And: ������ ������! = ������! ������ !!! 2) Replicating pf Critique: no jumps implemented, interest rates .

. inverting will solve it. 3. Black-­‐Scholes: W1=-­‐90.909 and w2=1 -­‐>borrow 90.909 and buy 1 Call is a derivative on an asset and will depend on the asset More assets: risk adj. prob. Replace real world+risk premium properties. Assume stock and follows GBM: ������(������, ������; ������, ������; ������, ������; ������) Semicolon separates different types: Same for c0 (stock, opt) -­‐S&T are variables; S stochastic & T deterministic -­‐>current prices equal their -­‐µ and σ are parameters of the asset expected values*disc. -­‐E&T are parameters associated with option contract -­‐>important because normally discounted by risk premium, -­‐r is parameter associated with currency not measurable easily For a call: asset and option positively correlated The Martingale Approach: è Black and Scholes PDE Expected value equals last observed value, submartingale: ������������ 1 ! ! ������ ! ������ ������������ + ������ ������ + ������������ − ������������ = 0 growth tendency; ������������ 2 ������������ ! ������������ Discounted at risk free rate, they will be submartingales, but è Contains all variables except drift µ discounted at risk adjusted they will be martingales Final condition: call: max(S-­‐E,0) put: max(E-­‐S,0) Lattice Models: Binomial model: T into n small intervals, each size of delta, so small that up or down only. Can change by +-­‐σ√Δ; hence volatility determines movement. Pup and pdown=1 From there we can go to the GBM: first part predicatable, N(x)+N(-­‐x)=1;N(0)=0,5; N(-­‐inf)=-­‐0; N(inf) = 1

-­‐ Long positions are long gamma, short are short gamma; gamma greatest when close to the money. When the stock changes by S, the delta of the option changes by ΓS It addresses Delta Hedging errors of the curvature. ITM and OTM à Γ small, ATMà large. Call ≈ 1; Put ≈ 0 (if S rises) Theta Φ: Is the rate of change of the value with respect to the passage of time: U-­‐shaped. ������������ ������������(������1)������ =− − ������������������ !!" ������(������2) Reverse skew: long term ������������ 2√������ equity options & index. High It is usually negative for the long position, if time passes and volatility for high K. Buy nothing changes, value declines. Positive e.g. for deep ITM puts for protection, as put and calls, appreciating at expiration (as a T-­‐Bill. higher probability of large ATM à lower theta price decrease. Relationship: Forward skew: high K, high volatility. OTM calls and ITM puts. Positive gamma(trader) large swings will result in profit, Commodities market. Supply low, frost in crobs… negative gamma(short) small swings in profit .

. Vega: (parameter) 4. The Greeks: Rate of change of value with respect to volatility: Rewrite the BS PDE: ������������ = ������ ������ ∗ ������(������1) 1 ! ! ������������ ������ + ������ ������ Γ + ������������Δ − ������������ = 0 2 Different because respect to parameter not variable Delta Δ: è hedging against model risk (because vol is not known !" Is the rate of change in respect to the underlying, namely ; with certainty) !" Delta of a Portfolio is the weighted sum of the deltas of the -­‐ High for ATM; low for I/OTM and short time to expiration individual positions (delta=slope). When the stock changes S, ������!"#$!% = ������!"#$% ∗ Ω higher volatility Rho ρ: the option price changes ∆S. Rate of change with respect to the interest rate: Example: S=49; K=50; rf=5% T=20weeks vol=20% !" è d1=0,0542 and N(d1)=0,522=Δ ; where V is the price of the option !" Delta of Portfolio: Higher interest rates imply higher call values !

Δ=
!!!

������! Δ!

second part randomness B-­‐S PDE can also be derived from the CAPM: ß������ = ß������ .

.

2. Stochastic processes: As S0 becomes very large in relation to K, d1 and d2 also Financial market data is not deterministic, variables likely become very large and N(d1), N(d2) tend to 1. ������ = ������! − random -­‐>follows stochastic process ������������ !!" , put tends to be 0 Randomness is now replacing states of the world If the other way around: F(t,w); w represents underlying randomness ������ = ������������ !!" − ������! Idea: stock price moves with expected price range+ An option is said to be at-­‐the-­‐money-­‐forward when K unexpected range: equals the forward price of the stock: ������! = ������������ !!" Weak randomness: normal distribution In that case the call option is approximately Wild randomness: distributions with fat tails ������ = 0,4 ∗ ������! ������√������ è We want a model consistent with weak form of market If stock pays dividend: efficiency: Markov type Brownian motion: Variable z follows Wiener process. Δz in Δt is:

������������ ������ ������������ ������

Option elasticity Ω: Risk of option relative to risk of stock:

Gamma and Vega neutral: w1=400 and w2=6000

è

Δ������ %������ℎ������������������������ ������������ ������������������ Δ������ = ������ = Δ������ %������ℎ������������������������ ������������ ������������ ������ ������ option is riskier than underlying stock Ω=

risk premium: Option return = weighted averaged of stock returns and rf ������ − ������ = ������ − ������ ������ à for a call, y ≥ µ ; put y < µ Sharpe ratio: Ω ≥ 1 for a Call ; Ω < 1 for a put (shorting the stock) The sharpe ratio for a call equals the sharpe ratio for the stockà Sr = (µ -­‐ r) /st.dev Delta Hedging For small move in the underlying, trader hedged gain/loss in .

. option offset by loss/gain in stock. 5. Basic Numerical procedures:

There are three different procedures for valuing: -­‐in form of a tree with binomial models -­‐finite difference methods -­‐Monte Carlo simulation Biniomial Trees: Can value American options -­‐In each small interval of time Δt the stock price moves up or down (with u or d), with p being the risk-­‐adj probabilities

Early exerciseà for a call, dividends is a reason. For puts, interest gained on K. It is a trade-­‐off between (1) time-­‐value of money, (2) dividends on underlying, (3) insurance -­‐ premium difference between otherwise identical calls with different strikes cannot be greater than the difference in K ������ ������������ − ������ ������������ ≤ ������������ − ������������

Now: continuous: ������������������������ !������ ������!������

The probability of an upward move is: ������ =

The variance is ������ ! Δ������, therefore: ������ ! Δ������ = ������������! + 1 − ������ ������ ! − ������ !!!! ! Cox.Ross-­‐Rubinstein: ������ =

Future: settlement daily (margin balance) Forward: pricing based on replication (price of forward equal the cost of same outcome synthetically). E.g. (1) buy the asset; (2) hold it to expiration time ! with that we can calculate the value of costs are: S(-­‐price in 0;+dividends;-­‐holding costs)

Assume: ������! < ������! < ������! ;

������! = ������������! + 1 − ������ ������! ������! − ������! ������ = ������! − ������! Then: strike price convexity ������ ������! ≤ ������������ ������! + 1 − ������ ������(������! ) ������ ������! ≤ ������������ ������! + 1 − ������ ������(������! ) -­‐ Option Premium = Intrinsic Value + Time Value .

. 7. Futures and Forwards:

Calculate the PF new value (CAPM with gain/losss)+hedge Forward and futures with commodities: Is used in production, is consumed Same formula, but q is now the commodity lease d. 1 ������! ������ = ������ − ln ������ ������! Slopes of forward curves: contango (dS0) &backwardation (d>r ; Ft high σ -­‐positive gamma & vega, negative theta Short straddle: selling call and put at same K-­‐> low σ -­‐profit from time (theta) but carry unlimited risk Strangles: Similar to straddle but with different K

the option at the final nodes, working back and testing for early exercise. For options with dy: r-­‐q Value = max (boundary; continuing value between two nodes) Delta: nodes at Δt1 ������! − ������! 2,16 − 6,96 = = −0,41 ������������ − ������������ 56,12 − 44,55 Gamma: nodes at Δt2 ∆1 − ∆2 ������������������������������ = 0,5 ������!! − ������!" + 0,5(������!" − ������!! ) Theta: from central nodes at 0 and 2Δt Δ������ 3,77 − 4,49 ������ℎ������������������ = = = −4,3 ������������������ ������������������������ Δ������ 0,1667 Dividends known: 1) construct tree less present value of dividends 2) add back present value at each node Control Variate Technique: ������������������������������������ ������������������������������ = ������! + (������!" − ������! ) Monte Carlo Simulation: In a risk neutral world the process for stock price is: ������������ = ������������������������ + ������������������������ We can simulate path by choosing time steps Δt: ∆������ = ������������∆������ + ������������������ ∆������ Where ε is random sample from N(0,1) In operational form (ln S):

������������ ������ = ������! + ������ ; suppose M = 0 ������������ = ������������ ������ ������!������ ������ ; ������������ = (������������ − ������)������������������ à q = avg dy Example of arbitrage: > F>Se àNow (Borrow S, buy S, sell F). Then (sell S, pay loan) >F forwards (viceversa) upside or downside Hedging with futures -­‐implement limited limited risk -­‐ Long futures hedge: you will buy asset and want to lock in bullish or bearish -­‐ Short futures “” : you will sell and want to lock in the price Suppose K2>K1 -­‐ Basis: S0-­‐F0; Basis Risk: arises for uncertainty of basis Bull spread with call: long K1, short K22; put: long k1, short K2 (investor gains sp)

è can reduce premium cost Butterfly spread: (positive theta negative gamma) Short straddle+long strangle Example: S1=55 c1=10; S2=60 c2=7; S3=65 c3=5 ������������������������ = 10 + 5 − 2 ∗ 7 = 1 short 2 middle, buy rest once. Maximum return at 60 = 5, if outside range: -­‐1 Positive theta; negative gamma. Box Spread: bull spread and bear spread. No stock price risk. 9, Exotics: Three major differences: depend on some event over the life, may have random expiration, may be written on more than one asset Path Independent: Binary: Cash-­‐or nothing, asset-­‐or-­‐nothing ������ !!!"!! = ������ !!" ������������(������2) Buy asset or nothing and sell cash or nothing is exactly like vanilla call: Ca – Cc = C Put pays M if St binomial path. Bull put and bull call, as well as bear call and bear put have the same Hit the barrier before maturity. Tree gets cut in the barrier. Price lower sensitivity. Bear call and bull put positive CF (& viceversa) than normal. Negative vega Collars: buy put and sell call with higher Kàbearish directional Asian options: -­‐ reverse collar would be bullish strategy _/-­‐ Average price-­‐>less volatile -­‐>more hedging è zero cost when premiums match Example with currency exchange, instead of securing the forward, one Lookback: floating( C St-­‐Smin; p Smax – St); fixed (c: max(Smax-­‐K;0); p: max (K-­‐Smin;0)(max or min with same K -­‐>expensive can obtain limited up and downside Zero cost dollar: same premium for different K

Path dependent:

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