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12345 6th Lecture 17th November 2003 Options

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12345 2 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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12345 3 Performance Profile - long call STST -C K 45 degrees V T =max(S T -K,0)

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12345 4 Performance Profile - short call C KSTST

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12345 5 Performance Profile - long put -P K STST V T =max(K-S T,0)

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12345 6 Performance Profile - short put STST P

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12345 7 Positions in the underlying instrument STST Long underlying Short underlying

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12345 8 Trading strategies: underlying & call option Long underl. & short call „covered call“ Short underlying & long call „writing cov. call“

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12345 9 Trading Strategies: underlying & put option Long underl. & long put Short underl. & short put

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12345 10 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<S T <X2 S T -X1 0 S T -X1 S T <X1 0 0 0

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12345 11 Bull spreads with puts Short put @ X2 Long put @ X1 X1 X2 STST

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12345 12 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<S T <X2 0 X1-S T -(S T -X1) S T <X1 0 0 0

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12345 13 Bear spreads (with puts) X1X2 STST payoff S T <X1 S T -X2 -(S T -X1) X1-X2 X1<S T <X2 S T -X2 0 S T -X2 X2<S T 0 0 0 S T Long Put Short Put Total

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12345 14 Butterfly spreads (with calls) STST X1X2X3 Payoff S T long call @ X1 long call @ X3 short call @ X2 Total S T <X1 0 0 0 0 X1<S T <X2 S T -X1 0 0 S T -X1 X2<S T <X3 S T -X1 0 -2(S T -X2) 2X2-X1 X3<S T S T -X1 S T -X3 -2(S T -X2) 2X2-X1-X3 Long two, short two

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12345 15 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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12345 16 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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12345 17 Combinations: strangles Position in one call and one put with same expiration date but different strike prices X1X2STST

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12345 18 Option Pricing - Put / Call Parity Theorem

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12345 19 Boundaries of a call option STST K Call range C

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12345 20 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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12345 21 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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12345 22 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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12345 23 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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12345 24 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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12345 25 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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12345 26 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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12345 27 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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12345 28 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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12345 29 The fundamental PDE FPDE: Boundary conditions: European call option: f=max(S-X,0) @ t=T European put option: f=max(X-S,0) @ t=T

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12345 30 Sketch of Solution for a European Call Option Transform: We get: Note: This is now a forward equation in

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12345 31 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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12345 32 The Black Scholes solution for a European Call Where,

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12345 33 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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12345 34 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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12345 35 Implied Volatility „smiles“ Strike Price Implied Vol At the money

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12345 36 Risk management of options portfolios via hedge parameters See Chapter 4 from Adritti - Derivatives

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12345 37 A brief dictionary of exotic options See Nelken, Chapter1

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