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Binnenlandse Francqui Leerstoel VUB 2004-2005 1. Black Scholes and beyond

Binnenlandse Francqui Leerstoel VUB 2004-2005 1. Black Scholes and beyond. André Farber Solvay Business School University of Brussels. Forward/Futures: Review. Forward contract = portfolio asset (stock, bond, index) borrowing Value f = value of portfolio f = S - PV(K) = S – e -rT K

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Binnenlandse Francqui Leerstoel VUB 2004-2005 1. Black Scholes and beyond

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  1. Binnenlandse Francqui Leerstoel VUB 2004-20051. Black Scholes and beyond André Farber Solvay Business School University of Brussels

  2. Forward/Futures: Review • Forward contract = portfolio • asset (stock, bond, index) • borrowing • Value f = value of portfolio f = S - PV(K) = S – e-rTK Based on absence of arbitrage opportunities • 4 inputs: • Spot price (adjusted for “dividends” ) • Delivery price • Maturity • Interest rate • Expected future price not required VUB 01 Black Scholes and beyond

  3. Discount factors and interest rates • Review: Present value of Ct • PV(Ct) = Ct× 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 / ert = e-rt 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) VUB 01 Black Scholes and beyond

  5. Exercise option if, at maturity: Stock price > Exercice price ST > K Call value at maturity CT = ST - K if ST > K otherwise: CT = 0 CT = MAX(0, ST - K) Terminal Payoff: European call VUB 01 Black Scholes and beyond

  6. Exercise option if, at maturity: Stock price < Exercice price ST < K Put value at maturity PT = K - ST if ST < K otherwise: PT = 0 PT = MAX(0, K- ST ) Terminal Payoff: European put VUB 01 Black Scholes and beyond

  7. 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: ST<K ST>K Share value ST ST Put value (K - ST) 0 Total value K ST Put = insurance contract The Put-Call Parity relation VUB 01 Black Scholes and beyond

  8. Consider an alternative strategy: Strategy 2: Buy call, invest PV(K) At maturity T: ST<K ST>K Call value 0 ST - K Invesmt K K Total value K ST At maturity, both strategies lead to the same terminal value Stock + Put = Call + Exercise price Put-Call Parity (2) 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 S current stock price P current put value C current 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) 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 VUB 01 Black Scholes and beyond

  11. Road map to valuation Binomial modeluSS dS discrete time, discrete stock prices Geometric Brownian Motion dS = μSdt+σSdz continuous timecontinuous stock prices Model of stock price behavior Create synthetic option Based on elementary algebra Based on Ito’s lemna to calculate df p fu + (1-p) fd = f erΔt PDE: Pricing equation Black Scholes formula Numerical methods VUB 01 Black Scholes and beyond

  12. Modelling stock price behaviour • Consider a small time interval t: S = St+t - St • 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) VUB 01 Black Scholes and beyond

  13. Geometric Brownian motion illustrated 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 VUB 01 Black Scholes and beyond

  15. u, d and q are choosen to reproduce the drift and the volatility of the underlying process: Drift: Volatility: Cox, Ross, Rubinstein’s solution: Binomial representation of the geometric Brownian VUB 01 Black Scholes and beyond

  16. Binomial process: Example • dS = 0.15 Sdt + 0.30 S dz ( = 15%,  = 30%) • Consider a binomial representation with t = 0.5 u = 1.2363, d = 0.8089, Π= 0.6293 • Time 0 0.5 1 1.5 2 2.5 • 28,883 • 23,362 • 18,897 18,897 • 15,285 15,285 • 12,363 12,363 12,363 • 10,000 10,000 10,000 • 8,089 8,089 8,089 • 6,543 6,543 • 5,292 5,292 • 4,280 • 3,462 VUB 01 Black Scholes and beyond

  17. Time step = t Riskless interest rate = r Stock price evolution uS S dS No arbitrage: d<er t <u 1-period call option Cu = Max(0,uS-X) Cu =? Cd = Max(0,dS-X) Call Option Valuation:Single period model, no payout Π Π 1- Π 1- Π 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 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 - Bert= Cu d S - Bert = Cd Solution: Call value C =  S - B 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. VUB 01 Black Scholes and beyond

  21. 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  Cu = 9.05 dS = 91.70  Cd = 0  = 0.5216 B = 47.64 Call value= 0.5216x100 - 47.64 =4.53 Binomial valuation: Example VUB 01 Black Scholes and beyond

  22. 1-period binomial formula • Cash value =  S - B • Substitue values for  and B and simplify: • C = [ pCu + (1-p)Cd ]/ ert where p = (er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 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 ? pu + (1 - p) d = ert • Solving for p:p = (ert - d)/(u-d) • Conclusion : in binomial pricing formula, p = probability of an upward movement in a risk neutral world VUB 01 Black Scholes and beyond

  24. u²S uS S udS 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 Mutiperiod extension: European option VUB 01 Black Scholes and beyond

  25. 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 2Risk neutral probability 118.91 p²= 18.91 0.2524 109.05 9.46 100.00 100.00 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 Multiperiod valuation: Example VUB 01 Black Scholes and beyond

  26. Consider: European option on non dividend paying stock constant volatility constant interest rate Limiting case of binomial model as t0 From binomial to Black Scholes VUB 01 Black Scholes and beyond

  27. Convergence of Binomial Model VUB 01 Black Scholes and beyond

  28. Arrow securities • 2 possible  states: up, down • 2 financial assets: one riskless bond and one stock 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 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 = vu erΔt+ vd  erΔt S = vu uS+ vd dS • Complete markets: # securities ≥ # states • Solve equations for find vuand vd VUB 01 Black Scholes and beyond

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

  33. Black Scholes’ PDE and the binomial model • We have: • Binomial model: p fu + (1-p) fd = ert • Use Taylor approximation: • fu = f + (u-1) Sf’S + ½ (u–1)² S² f”SS + f’tt • fd = f + (d-1) Sf’S + ½ (d–1)² S² f”SS + f’tt • u = 1 + √t + ½ ²t • d = 1 – √t + ½ ²t • er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 VUB 01 Black Scholes and beyond

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