Derivatives Introduction to option pricing André Farber Solvay Business School University of Brussels

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 Pricing options |3 Options Standard options –Call, put –European, American Exotic options (non standard) –More complex payoff (ex: Asian) –Exercise opportunities (ex: Bermudian)

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 Pricing options |6 Creation of synthetic option Geometric Brownian motion –requires advanced calculus (Ito’s lemna) Binomial –based on elementary algebra

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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)

July 16, 2015 Derivatives 07 Pricing options |9 Geometric Brownian motion illustrated

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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:

July 16, 2015 Derivatives 07 Pricing options |12 Binomial process: Example dS = 0.15 S dt S dz (   = 15%,  = 30%) Consider a binomial representation with  t = 0.5 u = , d = , q = Time ,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

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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.

July 16, 2015 Derivatives 07 Pricing options |17 Binomial valuation: Example Data S = 100 Interest rate (cc) = 5% Volatility  = 30% Strike price X = 100, Maturity =1 month (  t = ) u = d = uS =  C u = 9.05 dS =  C d = 0  = B = Call value= x =4.53

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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 = u = d = p = Risk neutral probability p²= p(1-p)= (1-p)²= Risk neutral expected value = 4.77 Call value = 4.77 e -.05(.1667) = 4.73

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 Pricing options |23 Convergence of Binomial Model

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 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 = d 2 =  1= N(d 1 ) = N(d 2 ) = European call : 100    = 8.60 European put : 100   ( )  ( ) = 3.72

July 16, 2015 Derivatives 07 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

July 16, 2015 Derivatives 07 Pricing options |27 Black-Scholes illustrated