One share of Stock A is used as the underlying asset on an exchange option, for

Question

One share of Stock A is used as the underlying asset on an exchange option, for which the benchmark asset is four shares of a Stock B. Currently, Stock A trades for $42 per share, and Stock B trades for $10 per share. Stock A has an annual price volatility of 0.3 and pays no dividends. Stock B has an annual price volatility of 0.5 and pays dividends at an annual continuously compounded yield of 0.04. The correlation between the continuously compounded returns on the two assets is 0.4. The exchange option expires in 1 year. Find the Black-Scholes price of this call option[answer:$9.3716]

Explanation / Answer

Underlying asset is 1 share of Stock A & strike asset is 4 shares of Stock B. Hence, we have S = 42and K = 10*4=40. Volatility of ln (S/K) is

sqrt(standrd devation of A^2+standard devaition of B^2-2*correlation of A &B*standard deviation of A*standard devaition of B)

sqrt(0.3^2+0.5^2-2*0.4*0.3*0.5)=0.469042

d1=(ln(42*e^(-0*1)/(40*e^(-0.04*1))+0.5*0.469042^2*1)/(0.469042*sqrt(1))=0.393771

d2=0.393771-0.469042*sqrt(1)=-0.07527

N(d1)=0.653125

N(d2)=0.47

Call price=42*e^(-0*1)*0.653125-40*e^(-0.04*1)*0.47=9.3716

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