1. Binnenlandse Francqui Leerstoel VUB 2004-2005 1. Black Scholes and beyond André Farber Solvay Business School University of Brussels

2. August 23, 2004 VUB 01 Black Scholes and beyond |2 Forward/Futures: Review Forward contract = portfolio –asset (stock, bond, index) –borrowing Value f = value of portfolio f = S - PV(K) = S – e -rT 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 VUB 01 Black Scholes and beyond |3 Discount factors and interest rates Review: Present value of C t PV(C t ) = C t × Discount factor With annual compounding: Discount factor = 1 / (1+r) t With compounding n times per year: Discount factor = 1/(1+r/n) nt With continuous compounding: Discount factor = 1 / e rt = e -rt

4. August 23, 2004 VUB 01 Black Scholes and beyond |4 Options Standard options –Call, put –European, American Exotic options (non standard) –More complex payoff (ex: Asian) –Exercise opportunities (ex: Bermudian)

5. August 23, 2004 VUB 01 Black Scholes and beyond |5 Terminal Payoff: European call Exercise option if, at maturity: Stock price > Exercice price S T > K Call value at maturity C T = S T - K if S T > K otherwise: C T = 0 C T = MAX(0, S T - K)

6. August 23, 2004 VUB 01 Black Scholes and beyond |6 Terminal Payoff: European put Exercise option if, at maturity: Stock price < Exercice price S T < K Put value at maturity P T = K - S T if S T < K otherwise: P T = 0 P T = MAX(0, K- S T )

7. August 23, 2004 VUB 01 Black Scholes and beyond |7 The Put-Call Parity relation A relationship between European put and call prices on the same stock Compare 2 strategies: Strategy 1. Buy 1 share + 1 put At maturity T:S T K Share valueS T S T Put value(K - S T )0 Total valueKS T Put = insurance contract

8. August 23, 2004 VUB 01 Black Scholes and beyond |8 Put-Call Parity (2) Consider an alternative strategy: Strategy 2: Buy call, invest PV(K) At maturity T:S T K Call value0S T - K InvesmtKK Total valueKS T At maturity, both strategies lead to the same terminal value Stock + Put = Call + Exercise price

9. August 23, 2004 VUB 01 Black Scholes and beyond |9 Put-Call Parity (3) Two equivalent strategies should have the same cost S + P = C + PV(K) where Scurrent stock price Pcurrent put value Ccurrent call value PV(K)present value of the striking price This is the put-call parity relation Another presentation of the same relation: C = S + P - PV(K) A call is equivalent to a purchase of stock and a put financed by borrowing the PV(K)

10. August 23, 2004 VUB 01 Black Scholes and beyond |10 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

11. August 23, 2004 VUB 01 Black Scholes and beyond |11 Road map to valuation Geometric Brownian Motion dS = μSdt+σSdz continuous time continuous stock prices Binomial model uS S dS discrete time, discrete stock prices Model of stock price behavior Create synthetic option Based on Ito’s lemna to calculate df Based on elementary algebra Pricing equation PDE: p f u + (1-p) f d = f e rΔt Black Scholes formula Numerical methods

12. August 23, 2004 VUB 01 Black Scholes and beyond |12 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)

13. August 23, 2004 VUB 01 Black Scholes and beyond |13 Geometric Brownian motion illustrated

14. August 23, 2004 VUB 01 Black Scholes and beyond |14 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

15. August 23, 2004 VUB 01 Black Scholes and beyond |15 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:

16. August 23, 2004 VUB 01 Black Scholes and beyond |16 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, Π = 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

17. August 23, 2004 VUB 01 Black Scholes and beyond |17 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) Π 1- Π Π

18. August 23, 2004 VUB 01 Black Scholes and beyond |18 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

19. August 23, 2004 VUB 01 Black Scholes and beyond |19 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

20. August 23, 2004 VUB 01 Black Scholes and beyond |20 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.

21. August 23, 2004 VUB 01 Black Scholes and beyond |21 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

22. August 23, 2004 VUB 01 Black Scholes and beyond |22 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

23. August 23, 2004 VUB 01 Black Scholes and beyond |23 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

24. August 23, 2004 VUB 01 Black Scholes and beyond |24 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

25. August 23, 2004 VUB 01 Black Scholes and beyond |25 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

26. August 23, 2004 VUB 01 Black Scholes and beyond |26 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

27. August 23, 2004 VUB 01 Black Scholes and beyond |27 Convergence of Binomial Model

28. August 23, 2004 VUB 01 Black Scholes and beyond |28 Arrow securities 2 possible states: up, down 2 financial assets: one riskless bond and one stock Current price UpDown Bond1erΔterΔt erΔterΔt StockSuSdS

29. August 23, 2004 VUB 01 Black Scholes and beyond |29 Contingent claims (digital options) Consider 2 securities that pay 1€ in one state and 0€ in the other state. They are named: contingent claims, Arrow Debreu securities, states prices Current Price UpDown CC upvuvu 10 CC downvdvd 01

30. August 23, 2004 VUB 01 Black Scholes and beyond |30 Computing state prices Financial assets can be viewed as packages of financial claims. Law of one price: 1 = v u  e rΔt + v d  e rΔt S = v u  uS + v d  dS Complete markets: # securities ≥ # states Solve equations for find v u and v d

31. August 23, 2004 VUB 01 Black Scholes and beyond |31 Pricing a derivative security Using state prices: Using binomial option pricing model: State prices are equal to discounted risk-neutral probabilities

32. August 23, 2004 VUB 01 Black Scholes and beyond |32 Understanding the PDE Assume we are in a risk neutral world Expected change of the value of derivative security Change of the value with respect to time Change of the value with respect to the price of the underlying asset Change of the value with respect to volatility

33. August 23, 2004 VUB 01 Black Scholes and beyond |33 Black Scholes’ PDE and the binomial model We have: Binomial model: p f u + (1-p) f d = e r  t Use Taylor approximation: f u = f + (u-1) S f’ S + ½ (u–1)² S² f” SS + f’ t  t f d = f + (d-1) S f’ S + ½ (d–1)² S² f” SS + f’ t  t u = 1 +  √  t + ½  ²  t d = 1 –  √  t + ½  ²  t e r  t = 1 + r  t Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes BS PDE : f’ t + rS f’ S + ½  ² f” SS = r f