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Homework answers / question archive / The stock price today of company A is $100

The stock price today of company A is $100

Finance

The stock price today of company A is $100. Company A does not make
any dividend payments. Company A’s volatility (σ) is 0.20. Two European
call options with maturity one year from now are traded on the stock of
company A. The first call option (C1) has an exercise price of $90, while the
second call option (C2) has an exercise price of $110.
You obtain a portfolio by buying two company A stocks, selling two C1
call options, and buying one C2 call option. You use the Black-Scholes option pricing formula. The premium for the C1
call option is $16.70. The corresponding European put option (P1) with the
same maturity and an exercise price as the C1 call option has a premium of
$2.31.

What is the risk-free interest rate?
Calculate the premium for the C2 call option?

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1). Risk-free interest rate:

Using put-call parity,

C + K*e^(rt) = P + S where

C = call option premium; K = strike price; P = put option premium; S = price of underlying; r = risk-free rate; t = time to maturity

C1 = 16.70; K = 90; P = 2.31; S = 100; t = 1

16.70 + 90*e^(-r*1) = 2.31 + 100

e^-r = 0.95122, solving for r, we get r = 5.001%

2). C2 call option premium (using Black-Scholes model): s = stock volatility = 0.20 or 20%; K = 110

d1 = {ln(S/K) + (r +s^2/2)t}/(s(t^0.5))

= {ln(100/110) + (5.001% + 20%^2/2)*1}/(20%*1^0.5)

= -0.1265

d2 = d1 - (s*t^0.5) = -0.1265 - (20%*1^0.5) = -0.3265

N(d1) = 0.4497; N(d2) = 0.5503

C2 = S*N(d1) - N(d2)*K*(e^(-rt))

= 100*0.4497 - 0.5503*110*e^(-5.001%*1) = 6.0405

C2 = 6.04

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