Question

You hold one share of IBM stock. The current stock price is $56. You are worried that the stock price may decline. Your objective is to get at least $50 from the stock at the end of one year. Assume the volatility of IBM stock (sigma) is 30% and the risk-free rate is 8%. (i) How will you hedge using put option on IBM stock to make sure that you receive at least $50 at the end of the year? (ii) Suppose put option on IBM stock is not available. How can you create a portfolio using only IBM stocks and a risk-free bond that has exactly same payoff as the portfolio in (a) at the end of the year? [Hint: You need to use Black-Scholes formula for put option. Calculate d1 and d2 (show your calculations). Then, use N(d1) =0.79 and N(d2)=0.69].

Answer 1

i]

To hedge, we buy a put option with strike price of $50

We use Black-Scholes Model to calculate the value of the put option today

The value of a put option is:

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁)

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

t is the time to maturity in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(56 / 50). We input the same formula into Excel, i.e. =LN (56 / 50)
• (r + σ²/2)*T = (0.08 + (0.30²/2)*1
• σ√T = 0.30 * √1

d₁ = 0.7944

d₂ = 0.4944

N(-d₁),and N(-d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(-d₁) = 0.2135

N(-d₂) = 0.3105

Alternatively, N(-d₁) can be calculated as 1 - N(d₁) and N(-d₂) can be calculated as 1 - N(d₂)

Now, we calculate the values of the put option as below:

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁), which is (50 * e^{(-0.08 * 1)})*(0.3105) - (56 * (0.3105) ==> $2.377

Value of put option is $2.377

Put option on IBM stock with strike price of $50 can be bought for $2.377. This will ensure that the stock can be sold for at least $50 at the end of one year

ii]

The portfolio can be replicated by shorting IBM stock, and using the proceeds of the short sale to buy a risk-free bond which pays $50 at maturity in 1 year. This is called a synthetic put option. The payoff replicates the payoff of the put option