Question

. Use the Black-Scholes model to find the price for a call option with the following inputs: (1) current stock price is $45, (2) exercise price is $50, (3) time to expiration is 3 months, (4) annualized risk-free rate is 3%, and (5) variance of stock return is 0.50.

. Using the information from question above, find the value of a put with a $50 exercise price.

Answer #1

Std dev = variance^(1/2)=0.5^(1/5)=87.06%

As per Black Scholes Model | ||||||

Value of call option = S*N(d1)-N(d2)*K*e^(-r*t) | ||||||

Where | ||||||

S = Current price = | 45 | |||||

t = time to expiry = | 0.25 | |||||

K = Strike price = | 50 | |||||

r = Risk free rate = | 3.0% | |||||

q = Dividend Yield = | 0% | |||||

σ = Std dev = | 87% | |||||

d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2) | ||||||

d1 = (ln(45/50)+(0.03-0+0.8706^2/2)*0.25)/(0.8706*0.25^(1/2)) | ||||||

d1 = -0.007162 | ||||||

d2 = d1-σ*t^(1/2) | ||||||

d2 =-0.007162-0.8706*0.25^(1/2) | ||||||

d2 = -0.442462 | ||||||

N(d1) = Cumulative standard normal dist. of d1 | ||||||

N(d1) =0.497143 | ||||||

N(d2) = Cumulative standard normal dist. of d2 | ||||||

N(d2) =0.329077 | ||||||

Value of call= 45*0.497143-0.329077*50*e^(-0.03*0.25) | ||||||

Value of call= 6.04 |

As per put call parity | ||||||

Call price + PV of exercise price = Spot price + Put price | ||||||

6.04+50/(1+0.03)^0.25=45+Put value | ||||||

Put value = 10.67 |

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