Lecture14 - value options with binomial model

1682 sowecanvaluethecallby

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SL] / (1+rf) In our case, $50 = [π × $60 + (1 ­ π) × $40] / 1.1 Solving gives π = 3/4 and 1­ π =1/4 6 Solving by Risk-Neutral Probabilities The value of the call option can thus be written as: C = π × Max [SH – E , 0] + (1 ­ π) × Max [SL – E , 0] / (1+rf) In our case, C = [(3/4) × $10 + (1/4) × $0] / 1.1 = $6.82 So we can value the call by: – calculating the risk­neutral probabilities, and – using the risk­free rate to discount the expected payoffs of the call at expiration. 7 Example 1 You bought a 100­share call contract on a firm’s stock three weeks ago. It expires five weeks from today and the strike price per share is E=$112. On the expiration date, the price of the underlying stock will either be $120 or $95. Currently the price per share is $96. You can borrow money at 10% annually. What is the value of the call contract? 8 Ex 1: Solution Using Risk-Neutral Prob. The interest rate for five weeks is (1.1)5/52 –1 = .921% Calculate the risk­neutral probabilities that satisfy risk­neutral pricing for the stock: 96 = [π×120 + (1­ π)×95]/1.00921 π = .07536 Payoff of the contract at expiration: – if the stock goes up: 100×(120­112) = $800 – if the stock goes down: 100×0 = $0 Now price the call contract using the probabilities: C = [.07536*800 + (1­.07536)*0]/1.00921 = $59.74 9 The Black-Scholes Model (Multiple The States) States) Determinants of Financial Option Values: – Current price of underlying asset (S) – Exercise price (E) – Time to expiration in years (t) – Volatility of the underlying asset per year (σ2) – Risk­free rate of return (r) 10 The Black-Scholes Value of the Call The value of a call option is: C = SN ( d1 ) − Ee Where [ ( E ) + (r + d 1 = ln S 1 2 − rt σ )t / σ t 2 2 N(d2 ) d 2 = d 1 − σ 2t and And N(d) is the probability that a standardized, normally distributed, random variable will be less than or equal to d 11 12 Example 2: Using the Black-Scholes Example Formula Formula Consider a stock with current price S=$50. Today you buy a call option on the stock with strike price E=$49, and time to expiration 199 days, so t =199/365= 0.5452 years. The risk­free rate is r = 7%, and the volatility of the stock is estimated to be σ2 = 0.09 per year. Calculate the value of the call using the formula. You should get C=$5.85.You should find that d1=.374, d2=.153, N(d1)=.646 , and N(d2)=.561 13...
View Full Document

Ask a homework question - tutors are online