Question

2. Consider a 1-period BOPM where a period is 1 year. The current stock price is 40. The stock could up by 20% (u=1.2) or down by 10% (0.9) and the risk-free rate is 8%. The strike price of a call option is 45. What is the hedge ratio?

3. Consider a 1-period BOPM where a period is 1 year. The current stock price is 40. The stock could up by 20% (u=1.2) or down by 10% (d=.9) and the risk-free rate is 8%. The strike price of a call option is 45. What is the fair price of this call option?

Answer 1

2) Hedge ratio will be 0.25

Calculation of hedge ratio

Hedge ratio = (Cu - Cd) / (Su - Sd)

Were,

Cu = price of call option in 1 year, if price goes up by 20%

Cd = price of call option in 1 year, if price goes down by 10%

Su = Upper stock price i.e. stock price goes up by 20%

Sd = Stock price goes down 10%

Strike price given = 45

Su = 40 * 1.2 = 48

Sd = 40 * .9 = 36

Cu = Max [(48-45),0] = 3

Cd = Max [(36-45),0] = 0

Hedge ratio = (3 - 0) / (48 - 36) = 3 / 12 = 0.25

Notes:

After 1 year, one will only exercise call option id the stock price after 1 year is higher than 45(strike price).

If stock price increases to 48 in 1 year, then the investor will exercise the option and will get the difference of stock price (48) and strike price (45), which is 3.

But in case the price drops to 36, the investor will not exercise the option, therefore will have no (0) payoff

3) Fair price of the call option is 1.69

Calculation of price of call option

Value of call option = PV (p * Cu + (1-p) * cd) = (p * Cu + (1-p) * cd) e^-rt

Where,

r = risk free rate = 8%

t = term = 1 year

p = probability of price goes up to 48 = (e^rt - d)/(u-d)

                                                            = ((e^0.08*1)-0.9)/ (1.2-0.9) = 0.6110

1-p = probability of price goes down to 36 = 1 - 0.6110 = 0.3890

Value of call option = (0.6110 * 3 + 0.3890 * 0) e^-.08*1

Value of call option = (1.833) e^-.08

Value of call option = (1.833) *(0.92312)

Value of call option = 1.69

Note:

value of e^-.08 can be calculated using =exp function in excel or in financial calculator, to calculate the value manually take value of e = 2.71828.