1. Options and Speculative Markets 2004-2005 Introduction to option pricing André Farber Solvay Business School University of Brussels

2. August 23, 2004 OMS 06 Pricing options |2 Forward/Futures: Review Forward contract = portfolio –asset (stock, bond, index) –borrowing Value f = value of portfolio f = S - PV(K) Based on absence of arbitrage opportunities 4 inputs: Spot price (adjusted for “dividends” ) Delivery price Maturity Interest rate Expected future price not required

3. August 23, 2004 OMS 06 Pricing options |3 Options Standard options –Call, put –European, American Exotic options (non standard) –More complex payoff (ex: Asian) –Exercise opportunities (ex: Bermudian)

4. August 23, 2004 OMS 06 Pricing options |4 Option Valuation Models: Key ingredients Model of the behavior of spot price  new variable: volatility Technique: create a synthetic option No arbitrage Value determination –closed form solution (Black Merton Scholes) –numerical technique

5. August 23, 2004 OMS 06 Pricing options |5 Model of the behavior of spot price Geometric Brownian motion –continuous time, continuous stock prices Binomial –discrete time, discrete stock prices –approximation of geometric Brownian motion

6. August 23, 2004 OMS 06 Pricing options |6 Creation of synthetic option Geometric Brownian motion –requires advanced calculus (Ito’s lemna) Binomial –based on elementary algebra

7. August 23, 2004 OMS 06 Pricing options |7 Options: the family tree Black Merton Scholes (1973) Analytical models Numerical models Analytical approximation models Term structure models B & S Merton Binomial Trinomial Finite difference Monte Carlo European Option European American Option American Option Options on Bonds & Interest Rates Analytical Numerical

8. August 23, 2004 OMS 06 Pricing options |8 Modelling stock price behaviour Consider a small time interval  t:  S = S t+  t - S t 2 components of  S: –drift : E(  S) =  S  t [  = expected return (per year)] –volatility:  S/S = E(  S/S) + random variable (rv) Expected value E(rv) = 0 Variance proportional to  t –Var(rv) =  ²  t  Standard deviation =   t –rv = Normal (0,   t) –=   Normal (0,  t) –=    z  z : Normal (0,  t) –=      t  : Normal(0,1)  z independent of past values (Markov process)

9. August 23, 2004 OMS 06 Pricing options |9 Geometric Brownian motion illustrated

10. August 23, 2004 OMS 06 Pricing options |10 Geometric Brownian motion model  S/S =   t +   z  S =  S  t +  S  z =  S  t +  S   t If  t "small" (continuous model) dS =  S dt +  S dz

11. August 23, 2004 OMS 06 Pricing options |11 Binomial representation of the geometric Brownian u, d and q are choosen to reproduce the drift and the volatility of the underlying process: Drift: Volatility: Cox, Ross, Rubinstein’s solution:

12. August 23, 2004 OMS 06 Pricing options |12 Binomial process: Example dS = 0.15 S dt + 0.30 S dz (   = 15%,  = 30%) Consider a binomial representation with  t = 0.5 u = 1.2363, d = 0.8089, q = 0.6293 Time 00.511.522.5 28,883 23,362 18,89718,897 15,28515,285 12,36312,36312,363 10,00010,00010,000 8,0898,0898,089 6,5436,543 5,2925,292 4,280 3,462

13. August 23, 2004 OMS 06 Pricing options |13 Call Option Valuation:Single period model, no payout Time step =  t Riskless interest rate = r Stock price evolution uS S dS No arbitrage: d<e r  t <u 1-period call option C u = Max(0,uS-X) C u =? C d = Max(0,dS-X) q 1-q q

14. August 23, 2004 OMS 06 Pricing options |14 Option valuation: Basic idea Basic idea underlying the analysis of derivative securities Can be decomposed into basic components  possibility of creating a synthetic identical security by combining: - Underlying asset - Borrowing / lending  Value of derivative = value of components

15. August 23, 2004 OMS 06 Pricing options |15 Synthetic call option Buy  shares Borrow B at the interest rate r per period Choose  and B to reproduce payoff of call option  u S - B e r  t = C u  d S - B e r  t = C d Solution: Call value C =  S - B

16. August 23, 2004 OMS 06 Pricing options |16 Call value: Another interpretation Call value C =  S - B In this formula: + : long position (buy, invest) - : short position (sell borrow) B =  S - C Interpretation: Buying  shares and selling one call is equivalent to a riskless investment.

17. August 23, 2004 OMS 06 Pricing options |17 Binomial valuation: Example Data S = 100 Interest rate (cc) = 5% Volatility  = 30% Strike price X = 100, Maturity =1 month (  t = 0.0833) u = 1.0905 d = 0.9170 uS = 109.05  C u = 9.05 dS = 91.70  C d = 0  = 0.5216 B = 47.64 Call value= 0.5216x100 - 47.64 =4.53

18. August 23, 2004 OMS 06 Pricing options |18 1-period binomial formula Cash value =  S - B Substitue values for  and B and simplify: C = [ pC u + (1-p)C d ]/ e r  t where p = (e r  t - d)/(u-d) As 0< p<1, p can be interpreted as a probability p is the “risk-neutral probability”: the probability such that the expected return on any asset is equal to the riskless interest rate

19. August 23, 2004 OMS 06 Pricing options |19 Risk neutral valuation There is no risk premium in the formula  attitude toward risk of investors are irrelevant for valuing the option  Valuation can be achieved by assuming a risk neutral world In a risk neutral world :  Expected return = risk free interest rate  What are the probabilities of u and d in such a world ? p u + (1 - p) d = e r  t  Solving for p:p = (e r  t - d)/(u-d) Conclusion : in binomial pricing formula, p = probability of an upward movement in a risk neutral world

20. August 23, 2004 OMS 06 Pricing options |20 Mutiperiod extension: European option u²S uS SudS dS d²S Recursive method (European and American options )  Value option at maturity  Work backward through the tree. Apply 1-period binomial formula at each node Risk neutral discounting (European options only )  Value option at maturity  Discount expected future value (risk neutral) at the riskfree interest rate

21. August 23, 2004 OMS 06 Pricing options |21 Multiperiod valuation: Example Data S = 100 Interest rate (cc) = 5% Volatility  = 30% European call option: Strike price X = 100, Maturity =2 months Binomial model: 2 steps Time step  t = 0.0833 u = 1.0905 d = 0.9170 p = 0.5024 0 1 2 Risk neutral probability 118.91 p²= 18.91 0.2524 109.05 9.46 100.00100.00 2p(1-p)= 4.73 0.00 0.5000 91.70 0.00 84.10 (1-p)²= 0.00 0.2476 Risk neutral expected value = 4.77 Call value = 4.77 e -.05(.1667) = 4.73

22. August 23, 2004 OMS 06 Pricing options |22 From binomial to Black Scholes Consider: European option on non dividend paying stock constant volatility constant interest rate Limiting case of binomial model as  t  0

23. August 23, 2004 OMS 06 Pricing options |23 Convergence of Binomial Model

24. August 23, 2004 OMS 06 Pricing options |24 Black Scholes formula European call option: C = S  N(d 1 ) - K e -r(T-t)  N(d 2 ) N(x) = cumulative probability distribution function for a standardized normal variable European put option: P= K e -r(T-t)  N(-d 2 ) - S  N(-d 1 ) or use Put-Call Parity

25. August 23, 2004 OMS 06 Pricing options |25 Black Scholes: Example Stock price S = 100 Exercise price = 100 (at the money option) Maturity = 1 year (T-t = 1) Interest rate (continuous) = 5% Volatility = 0.15 Reminder: N(-x) = 1 - N(x) d 1 = 0.4083 d 2 = 0.4083 - 0.15  1= 0.2583 N(d 1 ) = 0.6585 N(d 2 ) = 0.6019 European call : 100  0.6585 - 100  0.95123  0.6019 = 8.60 European put : 100  0.95123  (1-0.6019) - 100  (1-0.6585) = 3.72

26. August 23, 2004 OMS 06 Pricing options |26 Black Scholes differential equation: Assumptions S follows a geometric Brownian motion:dS = µS dt +  S dz Volatility  constant No dividend payment (until maturity of option) Continuous market Perfect capital markets Short sales possible No transaction costs, no taxes Constant interest rate

27. August 23, 2004 OMS 06 Pricing options |27 Black-Scholes illustrated