(Solved): 37. a. Return to Example 16.1. Use the binomial model to value a one-year European put option with ...

37. a. Return to Example 16.1. Use the binomial model to value a one-year European put option with exercise price $\$110$ on the stock in that example. b. Show that your solution for the put price satisfies put-call parity. (LO 16-2) B?nomial Opt?on Pr?c?ng Suppose that the risk-free interest rate is $5\%$ per six-month period and we wish to value a call option with exercise price $\$110$ on the stock described in the two-period price tree just above. We start by finding the value of $C_u$. From this point, the call can rise to an expiration-date value of $C_{uu}=\$11$ (because at this point the stock price is $u\times u\times$ $S_0=\$121$ ) or fall to a final value of $C_{ud}=0$ (because at this point the stock price is $u\times$ $d\times S_0=\$104.50$, which is less than the $\$110$ exercise price). Therefore, the hedge ratio at this point is $H=\frac{C_{uv}?C_{ud}}{uu{S}_0?ud{S}_0}=\frac{\$11?0}{\$121?\$104.5}=\frac{2}{3}$ page 516 Thus, the following portfolio will be worth $\$209$ at option expiration regardless of the ultimate stock price: The portfolio must have a current market value equal to the present value of $\$209$ : $2\times \$110?3C_u=\$209/1.05=\$199.047$ Solve to find that $C_U=\$6.984$. Next we find the value of $C_d$. It is easy to see that this value must be zero. If we reach this point (corresponding to a stock price of \$95), the stock price at option expiration will be either $\$104.50$ or $\$90.25$; in both cases, the option will expire out of the money. (More formally, we could note that with $C_{ud}=C_{dd}=0$, the hedge ratio is zero, and a portfolio of zero shares will replicate the payoff of the call!) Finally, we solve for $C$ by using the values of $C_U$ and $C_d$. The following Concept Check leads you through the calculations that show the option value to be $\$4.434$.