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th Lecture 17th November 2003 Options

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Basics Option contract grants the owner the right but not the obligation to take some action (see previous lectures) Call - right to purchase underlying at strike price at (european) / until (american) specified maturity Put - same as call but „sell“ instead of „buy“ Value of option is called „option‘s premium“ Options: at the money, in the money,out of the money

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Performance Profile - long call STST -C K 45 degrees V T =max(S T -K,0)

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Performance Profile - short call C KSTST

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Performance Profile - long put -P K STST V T =max(K-S T,0)

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Performance Profile - short put STST P

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Positions in the underlying instrument STST Long underlying Short underlying

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Trading strategies: underlying & call option Long underl. & short call „covered call“ Short underlying & long call „writing cov. call“

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Trading Strategies: underlying & put option Long underl. & long put Short underl. & short put

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Bull Spreads with calls STST X1X2 Short call strike X2 Long call strike X1 Payoff S T Long call Short call Total S T >X2 S T -X1 X2-S T X2-X1 X1~~
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Bull spreads with puts Short X2 Long X1 X1 X2 STST

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Bear spreads (with calls) STST X1X2 Payoff S T long call short call total S T >X2 S T -X2 X1-S T -(X2-X1) X1~~
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Bear spreads (with puts) X1X2 STST payoff S T

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Butterfly spreads (with calls) STST X1X2X3 Payoff S T long X1 long X3 short X2 Total S T

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Calendar Spreads Until now, all options had the same expiration date Calendar spreads combine options with the same strike but different expiry dates STST X Buy call & sell call

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Combinations: straddles, strips, straps... Straddle: buying a call and a put with same strike price and expiration date STST X Strips: long one call and two puts with same strike and expiration Straps: long two calls and one put with same strike and expiration

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Combinations: strangles Position in one call and one put with same expiration date but different strike prices X1X2STST

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Option Pricing - Put / Call Parity Theorem

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Boundaries of a call option STST K Call range C

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Option Pricing fundamentals - the binomial model S0S0 (1+r u )S=uS (1+r d )S=dS T=0 T=1 Probability q 1-q

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Pricing by portfolio replication Portfolio comprising n shares of underlying and B dollars in risk free debt paying interest i=r-1 replicates call at time T=1 C C u =max(0,uS- K) C d =max(0,dS- K) nuS+Br=C u ndS+Br=C d But C=nS+B, so by substituting we obtain

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Risk neutral probabilities - two period binomial case So C can be written as Important statement: traders can differ on the probabilities of underlying but still agree on the value of the call option

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Two period model as a recombining latice girder C CuCu Cd C uu C dd C ud =C du Homework: Prove using the same algorithm that: And by extrapolation to n time steps:

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Estimating u,d and r Multiplicative binomial converges to a normal in the variable ln(S T /S) The characteristics of this probability distribution are given by the first two moments: For the binomial to converge to the log-normal, then u,d and q must be chosen such that the mean and variance of the binomial model converge to the mean and variance of the log-normal To accomplish that, select:

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The Black - Scholes - Merton Framework Ito‘s lemma: Suppose stochastic variable x follows the Ito process: dx=a(x,t)dt+b(x,t)dz, z is a Wiener process, variable x has a drift rate a and variance rate b 2, then a function G follows the process: So applying this for the process of stock prices:

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Homework 2 Apply Ito‘s lemma to ln S and show that Implying that ln S T has a lognormal distribution

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Assumptions of the Black Scholes option pricing model The stock price follows a log normal distribution parametrized as in the previous slide The short selling of securities with full use of proceeds is permitted No transaction costs or taxes. All securities are perfectly divisible There are no dividends during the life of the derivative There are no riskless arbitrage opportunities Security trading is continuous The risk free rate r is constant and the same for all maturities (flat term structure)

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The Black Scholes Differential Equation Chose a portfolio of 1 short call option (f) and shares Applying Ito‘s lemma, Due to no arbitrage, Substituting we obtain:

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The fundamental PDE FPDE: Boundary conditions: European call option: t=T European put option: t=T

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Sketch of Solution for a European Call Option Transform: We get: Note: This is now a forward equation in

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Sketch of a solution for a European Call option (cont‘d) Now by substituting : And by setting: Gives: This is the famous heat diffusion equation, with the following boundary condition: Now, for a european call the final payoff at maturity is max[S-X,0] so after the first transformation: f(x,0)=max(e x -1,0) and then, And the final solution is:

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The Black Scholes solution for a European Call Where,

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Synopsis of risk neutral valuation using the equivalent martingale property of asset processes Model: deterministic r, Bond price: B t =exp(rt); stock Price: S t =S 0 exp Find the replicating strategy: 1. Probability measue Q which converts S T into a martingale (apply Girsanov) 2. Form the process E t =E Q (X t /F t ) 3. Find a previsible process φ t, such that dE t = φ t dS t Where q is the martingale measure for the discounted stock B t -1 S t Detailed derivation in Karatsas, Bingham &Kiessl, Rebonato, Baxter & Rennie

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Homework (again!!!) uh!!! Apply the fundamental PDE and the derivation of the closed form solution for finding the price of an interest rate forward an FX forward An equity forward contract good luck and be careful with the boundary conditions!

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Implied Volatility „smiles“ Strike Price Implied Vol At the money

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Risk management of options portfolios via hedge parameters See Chapter 4 from Adritti - Derivatives

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A brief dictionary of exotic options See Nelken, Chapter1

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