Théorie Financière Financial Options Professeur André Farber

August 23, 2004 Tfin Options (1) |2 Options Objectives for this session: –1. Define options (calls and puts) –2. Analyze terminal payoff –3. Define basic strategies –4. Binomial option pricing model –5. Black Scholes formula

August 23, 2004 Tfin Options (1) |3 Definitions A call (put) contract gives to the owner –the right : –to buy (sell) –an underlying asset (stocks, bonds, portfolios,..) –on or before some future date (maturity) on : "European" option before: "American" option at a price set in advance (the exercise price or striking price) Buyer pays a premium to the seller (writer)

August 23, 2004 Tfin Options (1) |4 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)

August 23, 2004 Tfin Options (1) |5 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 )

August 23, 2004 Tfin Options (1) |6 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

August 23, 2004 Tfin Options (1) |7 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

August 23, 2004 Tfin Options (1) |8 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)

August 23, 2004 Tfin Options (1) |9 Valuing option contracts The intuition behind the option pricing formulas can be introduced in a two-state option model (binomial model). Let S be the current price of a non-dividend paying stock. Suppose that, over a period of time (say 6 months), the stock price can either increase (to uS, u>1) or decrease (to dS, d<1). Consider a K = 100 call with 1-period to maturity. uS = 125 C u = 25 S = 100 C dS = 80 C d = 0

August 23, 2004 Tfin Options (1) |10 Key idea underlying option pricing models It is possible to create a synthetic call that replicates the future value of the call option as follow:  Buy Delta shares  Borrow B at the riskless rate r (5% per annum) Choose Delta and B so that the future value of this portfolio is equal to the value of the call option.  Delta uS - (1+r  t) B = C u Delta 125 – B = 25  Delta dS - (1+r  t) B = C d Delta 80 – B = 0 (  t is the length of the time period (in years) e.g. : 6-month means  t=0.5)

August 23, 2004 Tfin Options (1) |11 No arbitrage condition In a perfect capital market, the value of the call should then be equal to the value of its synthetic reproduction, otherwise arbitrage would be possible: C = Delta  S - B This is the Black Scholes formula We now have 2 equations with 2 unknowns to solve.  Eq1  -  Eq2   Delta  ( ) = 25  Delta = Replace Delta by its value in  Eq2   B = Call value: C = Delta S - B =   C = 12.20

August 23, 2004 Tfin Options (1) |12 A closed form solution for the 1-period binomial model C = [p  C u + (1-p)  C d ]/(1+r  t) with p =(1+r  t - d)/(u-d) p is the probability of a stock price increase in a "risk neutral world" where the expected return is equal to the risk free rate. In a risk neutral world : p  uS + (1-p)  dS = 1+r  t p  C u + (1-p)  C d is the expected value of the call option one period later assuming risk neutrality The current value is obtained by discounting this expected value (in a risk neutral world) at the risk-free rate.

August 23, 2004 Tfin Options (1) |13 Risk neutral pricing illustrated In our example,the possible returns are: + 25% if stock up - 20% if stock down In a risk-neutral world, the expected return for 6-month is 5%  0.5= 2.5% The risk-neutral probability should satisfy the equation: p  (+0.25%) + (1-p)  (-0.20%) = 2.5%  p = 0.50 The call value is then: C = 0.50  25 / = 12.20

August 23, 2004 Tfin Options (1) |14 Multi-period model: European option For European option, follow same procedure (1) Calculate, at maturity, - the different possible stock prices; - the corresponding values of the call option - the risk neutral probabilities (2) Calculate the expected call value in a neutral world (3) Discount at the risk-free rate

August 23, 2004 Tfin Options (1) |15 An example: valuing a 1-year call option Same data as before: S=100, K=100, r=5%, u =1.25, d=0.80 Call maturity = 1 year (2-period) Stock price evolution Risk-neutral proba.Call value t=0 t=1t= p² = p(1-p) = (1-p)² = Current call value : C = 0.25  56.25/ (1.025)² = 13.38

August 23, 2004 Tfin Options (1) |16 Volatility The value a call option, is a function of the following variables: 1. The current stock price S 2. The exercise price K 3. The time to expiration date T 4. The risk-free interest rate r 5. The volatility of the underlying asset σ Note:In the binomial model, u and d capture the volatility (the standard deviation of the return) of the underlying stock Technically, u and d are given by the following formulas:

August 23, 2004 Tfin Options (1) |17 Option values are increasing functions of volatility The value of a call or of a put option is in increasing function of volatility (for all other variable unchanged) Intuition: a larger volatility increases possibles gains without affecting loss (since the value of an option is never negative) Check: previous 1-period binomial example for different volatilities VolatilityudCP (S=100, K=100, r=5%,  t=0.5)

August 23, 2004 Tfin Options (1) |18 Black-Scholes formula For European call on non dividend paying stocks The limiting case of the binomial model for  t very small C = S N(d 1 ) - PV(K) N(d 2 )  Delta B In BS: PV(K) present value of K (discounted at the risk-free rate) Delta = N(d 1 ) N(): cumulative probability of the standardized normal distribution B = PV(K) N(d 2 )

August 23, 2004 Tfin Options (1) |19 Black-Scholes : numerical example 2 determinants of call value: “Moneyness” : S/PV(K) “Cumulative volatility” : Example: S = 100, K = 100, Maturity T = 4, Volatility σ = 30% r = 6% “Moneyness”= 100/(100/ ) = 100/79.2= Cumulative volatility = 30% x  4 = 60% d 1 = ln(1.2625)/0.6 + (0.5)(0.60) =0.688  N(d 1 ) = d 2 = ln(1.2625)/0.6 - (0.5)(0.60) =0.089  N(d 2 ) = C = (100) (0.754) – (79.20) (0.535) = 33.05

August 23, 2004 Tfin Options (1) |20 Cumulative normal distribution This table shows values for N(x) for x  0. For x<0, N(x) = 1 – N(-x) Examples: N(1.22) = 0.889, N(-0.60) = 1 – N(0.60) = 1 – = In Excell, use Normsdist() function to obtain N(x)

August 23, 2004 Tfin Options (1) |21 Black-Scholes illustrated