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6th Lecture 17th November 2003

6th Lecture 17th November 2003. Options. 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

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6th Lecture 17th November 2003

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  1. 6th Lecture17th November 2003 Options

  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

  3. Performance Profile - long call K 45 degrees ST -C VT=max(ST-K,0)

  4. Performance Profile - short call C K ST

  5. Performance Profile - long put K ST -P VT=max(K-ST,0)

  6. Performance Profile - short put P ST

  7. Positions in the underlying instrument Long underlying ST Short underlying

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

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

  10. Bull Spreads with calls Long call strike X1 X1 X2 ST Short call strike X2 Payoff ST Long call Short call Total ST>X2 ST-X1 X2-ST X2-X1 X1<ST<X2 ST-X1 0 ST-X1 ST<X1 0 0 0

  11. Bull spreads with puts Short put @ X2 ST X1 X2 Long put @ X1

  12. Bear spreads (with calls) X1 X2 ST Payoff ST long call short call total ST>X2 ST-X2 X1-ST -(X2-X1) X1<ST<X2 0 X1-ST -(ST-X1) ST<X1 0 0 0

  13. Bear spreads (with puts) ST X1 X2 payoff ST Long Put Short Put Total ST<X1 ST-X2 -(ST-X1) X1-X2 X1<ST<X2 ST-X2 0 ST-X2 X2<ST 0 0 0

  14. Butterfly spreads (with calls) Long two, short two Long two, short two ST X1 X2 X3 Payoff ST long call @ X1 long call @ X3 short call @ X2 Total ST<X1 0 0 0 0 X1<ST<X2 ST-X1 0 0 ST-X1 X2<ST<X3 ST-X1 0 -2(ST-X2) 2X2-X1 X3<ST ST-X1 ST-X3 -2(ST-X2) 2X2-X1-X3

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

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

  17. Combinations: strangles • Position in one call and one put with same expiration date but different strike prices X1 X2 ST

  18. Option Pricing - Put / Call Parity Theorem

  19. Boundaries of a call option C Call range K ST

  20. Option Pricing fundamentals - the binomial model Probability (1+ru)S=uS q S0 1-q (1+rd)S=dS T=0 T=1

  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 Cu=max(0,uS-K) nuS+Br=Cu ndS+Br=Cd C Cd=max(0,dS-K) But C=nS+B, so by substituting we obtain

  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

  23. Two period model as a recombining latice girder Cuu Homework: Prove using the same algorithm that: Cu Cud=Cdu C And by extrapolation to n time steps: Cd Cdd

  24. Estimating u,d and r • Multiplicative binomial converges to a normal in the variable ln(ST/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:

  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 b2, then a function G follows the process: So applying this for the process of stock prices:

  26. Homework 2 • Apply Ito‘s lemma to ln S and show that Implying that ln ST has a lognormal distribution

  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)

  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:

  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

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

  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(ex-1,0) and then, And the final solution is:

  32. The Black Scholes solution for a European Call Where,

  33. Synopsis of risk neutral valuation using the equivalent martingale property of asset processes • Model: deterministic r, Bond price: Bt=exp(rt); stock Price: St=S0exp Find the replicating strategy: 1. Probability measue Q which converts ST into a martingale (apply Girsanov) 2. Form the process Et=EQ(Xt/Ft) 3. Find a previsible process φt, such that dEt= φtdSt Where q is the martingale measure for the discounted stock Bt-1St Detailed derivation in Karatsas, Bingham &Kiessl, Rebonato, Baxter & Rennie

  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!

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

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

  37. A brief dictionary of exotic options • See Nelken, Chapter1

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