Question

1. Consider the following case of a binomial option pricing. A stock is currently trading at $50. Next period the stock price can go up to S_max or down to S_min.The call option with the exercise price of $50 is currently trading at $9.14. The risk-free rate is 7.5% and the hedge ratio is 5/7. Calculate numerical values of S_max and S_min.

Answer 1

Hedge ratio = D =5/7

Rf = 7.5%

The cost today must be equal to the payoff discounted at the risk-free rate for one period

Option price = D * current stock price - (payoff x e ^ (-risk-free rate x T)) = 9.14

9.14 = 5/7 * 50 - payoff*e^-(7.5% * 1)

=> payoff*e^-(7.5% * 1) = 26.57

=> payoff = 28.64

Cost today = $50 * 5/7 - 9.14 . . . . 1

Portfolio value (up state) = Smax *5/7 - max(Smax - $50, 0) . . . . .2

Portfolio value (down state) = Smin *5/7 - max(Smin - $50, 0) . . . . . . 3

Solving 1 and 2

Smax*5/7 - Smax + 50 = 26.57

2/7 Smax = 23.42

Smax = $81.99

Solving 1 and 3

Smin*5/7 - 0 = 26.57

Smin = $37.2

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