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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 RiskNeutral 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 RiskNeutral 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 BlackScholes 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 BlackScholes 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 BlackScholes
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
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 Spring '13

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