45.4: One share of Stock A is used as the underlying asset on an exchange option

Question

45.4: 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 0.4 and pays dividends at the continuously compounded yield rate of 2%. Stock B has an annual price volatility of 0.3 and pays no dividends. The correlation between the continuously compounded returns on the two assets is 0.5. The exchange option expires in 1 year. Find the Black-Scholes price of this call option.

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^(-0.02*1)/(40*e^(-0*1))+0.5*0.469042^2*1)/(0.469042*sqrt(1))=0.277873

d2=0.277873-0.469042*sqrt(1)=-0.19117

N(d1)=0.609445

N(d2)=0.424197

Call price=42*e^(-0.02*1)*0.609445-40*e^(-0*1)*0.424197=8.122

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