Question

Suppose the current stock price is $100. The continuously compounded interest rate is 4%. Assume that 1 year down the road, the stock price will be either $140 or $60. No other possibilities (I know this is an extreme assumption). There are no dividend payments.

Please compute the delta and fair value of (i) a 1-year call option on the stock with a strike price of $110; (ii) a 1-year put option on the stock with a strike price of $90.

Answer 1

a. Delta and Fair Value of (i)

Su = Upward Factor = Upward Price / Spot Price = 140 / 100 = 1.40

Sd = Downward Factor = Downward Price / Spot Price = 60 / 100 = 0.60

Risk Neutral Probability = (e^rt - Sd) / (Su - Sd)

Risk Neutral Probability = (e^0.04 - 0.60) / (1.40 - 0.60)

Risk Neutral Probability = (1.040811 - 0.60) / (1.40 - 0.60)

Risk Neutral Probability = 0.440811 / 0.80

Risk Neutral Probability = 0.551013

Call payoff at 140 = Spot Price at year end - Strike Price = 140 - 110 = 30

Call payoff at 60 = 60 - 110 = $0 (There will no negative Payoff's)

Delta = Difference in Call Payoff at Exercise Date / Difference in Spot price at year end

Delta = 30 / (140-60)

Delta = 0.375

Fair Value of Call Option = Risk Neutral Probability * Call Payoff * e^-rt

Fair Value of Call Option = 0.551013 * 30 * e^-0.04

Fair Value of Call Option = 0.551013 * 30 * 1/1.040811

Fair Value of Call Option = $15.88

b Delta and Fair Value of (ii)

Su = Upward Factor = Upward Price / Spot Price = 140 / 100 = 1.40

Sd = Downward Factor = Downward Price / Spot Price = 60 / 100 = 0.60

Risk Neutral Probability = (e^rt - Sd) / (Su - Sd)

Risk Neutral Probability = (e^0.04 - 0.60) / (1.40 - 0.60)

Risk Neutral Probability = (1.040811 - 0.60) / (1.40 - 0.60)

Risk Neutral Probability = 0.440811 / 0.80

Risk Neutral Probability = 0.551013

Put payoff at 140 = Strike Price - Spot Price at year end= $0 (there will be no negative payoss's)

Put payoff at 60 = Strike Price - Spot Price at year end= $90 - 60 = $30 (There will no negative Payoff's)

Delta = Difference in Put Payoff at Exercise Date / Difference in Spot price at year end

Delta = 0-30 / (140-60)

Delta = -0.375

Fair Value of Put Option = (1 - Risk Neutral Probability) * Put Payoff * e^-rt

Fair Value of Call Option = (1 - 0.551013) * 30 * e^-0.04

Fair Value of Call Option = 0.448987 * 30 * 1/1.040811

Fair Value of Call Option = $12.94

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