Show that the initial value of the call option in Example 16.1 is $4.434.

a. Confirm that the spread in option values is C_u = C_d = $6.984.

b. Confirm that the spread in stock values is uS₀ = dS₀ = $15.

c. Confirm that the hedge ratio is .4656 shares purchased for each call written.

d. Demonstrate that the value in one period of a portfolio comprising .4656 shares and one call written is riskless.

e. Calculate the present value of this payoff.

f. Solve for the option value.

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EXAMPLE 16.1 Binomial Option Pricing 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 C₁. From this point, the call can rise to an expiration-date value of Cuu = $11 (since at this point the stock price is u Xux So = $121) or fall to a final value of Cud = 0 (since at this point the stock price is u X d X So = $104.50, which is less than the $110 exercise price). Therefore, the hedge ratio at this point is $11-0 Cuu Cua uus。 - udSo $121 $104.5 H = = = 2 3 Thus, the following portfolio will be worth $209 at option expiration regardless of the ultimate stock price: