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

Smax > 50 & Smin < 50

K strike price = 50

Cmax: call option pay-off when stock price is Smax

Cmax = max(Smax - 50,0) = Smax - 50

Cmin: call option pay-off when stock price is Smin

Cmin = max(Smin - 50,0) = 0

First using the concept of hedge ratio

Consider a portfolio with 5/7 long stock and 1 short call:

If stock price increases,

Value of portfolio = 5/7*Smax - Cmax = 5/7*Smax - (Smax - 50) = 50 - 2/7*Smax

If stock price decreases,

Value of portfolio = 5/7*Smin - Cmin = 5/7*Smin - 0 = 5/7*Smin

For a hedged portfolio:

50 - 2/7*Smax = 5/7*Smin

350 = 5*Smin + 2*Smax.........................equ(1)

u: up factor = Smax/50

d: down factor = Smin/50

p: probability of up movement

p = (e^(r*t) - d)/(u - d)

e: natural exponent

r: risk free rate = 7.5%

t: time to maturity = 1

p = (1.078 - Smin/50)/(Smax/50 - Smin/50) = (53.89 - Smin)/(Smax - Smin)

Value of call option = e^(-r*t)*{p*Cmax + (1-p)*Cmin}

9.14 = 0.928*{p*(Smax - 50) + (1 - p)*0}

9.14 = 0.928*{(53.89 - Smin)/(Smax - Smin)}*(Smax - 50)...............equ(2)

Solving equ1 & equ2, we get

Smin = 44.04 & Smax = 64.9