Derivatives Introduction to option pricing Andr Farber Solvay
Derivatives Introduction to option pricing André Farber Solvay Business School University of Brussels Derivatives 07 Pricing options
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 11 February 2022 Derivatives 07 Pricing options 2
Options • Standard options – Call, put – European, American • Exotic options (non standard) – More complex payoff (ex: Asian) – Exercise opportunities (ex: Bermudian) 11 February 2022 Derivatives 07 Pricing options 3
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 February 2022 Derivatives 07 Pricing options 4
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 11 February 2022 Derivatives 07 Pricing options 5
Creation of synthetic option • Geometric Brownian motion – requires advanced calculus (Ito’s lemna) • Binomial – based on elementary algebra 11 February 2022 Derivatives 07 Pricing options 6
Options: the family tree Black Merton Scholes (1973) Analytical models Numerical models European American Option Binomial B&S Trinomial Merton Finite difference Monte Carlo 11 February 2022 Analytical approximation models Term structure models American Options on Bonds & Interest Rates Derivatives 07 Pricing options Analytical Numerical 7
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) 11 February 2022 Derivatives 07 Pricing options 8
Geometric Brownian motion illustrated 11 February 2022 Derivatives 07 Pricing options 9
Geometric Brownian motion model • S/S = t + z • S = S t + S z • = S t + S t • If t "small" (continuous model) • d. S = S dt + S dz 11 February 2022 Derivatives 07 Pricing options 10
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: 11 February 2022 Derivatives 07 Pricing options 11
Binomial process: Example • • d. S = 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 0 10, 000 11 February 2022 0. 5 12, 363 8, 089 1 15, 285 10, 000 6, 543 1. 5 18, 897 12, 363 8, 089 5, 292 2 23, 362 15, 285 10, 000 6, 543 4, 280 2. 5 28, 883 18, 897 12, 363 8, 089 5, 292 3, 462 Derivatives 07 Pricing options 12
Call Option Valuation: Single period model, no payout • • • Time step = t Riskless interest rate = r Stock price evolution q • • u. S • 1 -period call option q • • S Cu = Max(0, u. S-X) Cu =? 1 -q • Cd = Max(0, d. S-X) d. S No arbitrage: d<er t <u 11 February 2022 Derivatives 07 Pricing options 13
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 11 February 2022 Derivatives 07 Pricing options 14
Synthetic call option • Buy shares • Borrow B at the interest rate r period • Choose and B to reproduce payoff of call option u S - B er t = Cu d S - B er t = Cd Solution: Call value C = S - B 11 February 2022 Derivatives 07 Pricing options 15
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. 11 February 2022 Derivatives 07 Pricing options 16
Binomial valuation: Example • • • Data S = 100 Interest rate (cc) = 5% Volatility = 30% Strike price X = 100, Maturity =1 month ( t = 0. 0833) 11 February 2022 • • u = 1. 0905 d = 0. 9170 u. S = 109. 05 Cu = 9. 05 d. S = 91. 70 Cd = 0. 5216 B = 47. 64 Call value= 0. 5216 x 100 - 47. 64 =4. 53 Derivatives 07 Pricing options 17
1 -period binomial formula • Cash value = S - B • Substitue values for and B and simplify: • C = [ p. Cu + (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 11 February 2022 Derivatives 07 Pricing options 18
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 : r Expected return = risk free interest rate r What are the probabilities of u and d in such a world ? p u + (1 - p) d = er t - d)/(u-d) r Solving for p: p = (e • Conclusion : in binomial pricing formula, p = probability of an upward movement in a risk neutral world 11 February 2022 Derivatives 07 Pricing options 19
Mutiperiod extension: European option • (European and American options) u²S u. S S ud. S d²S Recursive method • 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 11 February 2022 Derivatives 07 Pricing options 20
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. 00 4. 73 100. 00 2 p(1 -p)= 0. 5000 84. 10 0. 00 (1 -p)²= 0. 2476 91. 70 0. 00 Risk neutral expected value = 4. 77 Call value = 4. 77 e-. 05(. 1667) = 4. 73 11 February 2022 Derivatives 07 Pricing options 21
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 11 February 2022 Derivatives 07 Pricing options 22
Convergence of Binomial Model 11 February 2022 Derivatives 07 Pricing options 23
Black Scholes formula • • • European call option: C = S N(d 1) - K e-r(T-t) N(d 2) • • European put option: P= K e-r(T-t) N(-d 2) - S N(-d 1) • or use Put-Call Parity N(x) = cumulative probability distribution function for a standardized normal variable 11 February 2022 Derivatives 07 Pricing options 24
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) 11 February 2022 • • • 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) Derivatives 07 Pricing options = 3. 72 25
Black Scholes differential equation: Assumptions • S follows a geometric Brownian motion: d. S = µS dt + S dz • Volatility constant • No dividend payment (until maturity of option) • Continuous market • • 11 February 2022 Perfect capital markets Short sales possible No transaction costs, no taxes Constant interest rate Derivatives 07 Pricing options 26
Black-Scholes illustrated 11 February 2022 Derivatives 07 Pricing options 27
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