Question

Peter has just sold a European call option on 10,000 shares of a stock. The exercise price is $50; the stock price is $50; the continuously compounded interest rate is 5% per annum; the volatility is 20% per annum; and the time to maturity is 3 months. (a) Use the Black-Scholes-Merton model to compute the price of the European call option. (b) Find the value of a European put option with the same exercise price and expiration as the call option above. (c) What position should Peter take in the stock for delta neutrality? (d) Suppose that Peter does set up a delta neutral position as soon as the option has been sold and the stock price jumps to $55 within the first hour of trading. What trade is necessary to maintain delta neutrality?

Answer 1

a] and b]

We use Black-Scholes Model to calculate the value of the call and put options.

The value of a call and put option are:

C = (S₀ * N(d₁)) - (Ke^-rT * N(d₂))

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 expiry 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 :

· lln(S₀ / K) = ln(50 / 50). We input the same formula into Excel, i.e. =LN(50/50)

· (r + σ²/2)*T = (0.05 + (0.20²/2)*0.25

· σ√T = 0.20 * √0.25

d1 = 0.1750

d2 = 0.0.750

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

N(d₁) = 0.5695

N(-d₁) = 0.5299

N(d₂) = 0.4305

N(-d₂) = 0.4701

Now, we calculate the values of the call and put options as below:

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (50 * 0.5695) - (50 * e^{(-0.05 * 0.25)})*(0.5299)    ==> $2.3075

P = (K * e^-rT)*N(-d₂) - (S₀)*N(-d₁), which is (50 * e^{(-0.05 * 0.25)})*(0.4701) - (50 * (0.4305) ==> $1.6864

Value of call option is $2.3075

Value of put option is $1.6864

c]

The delta of a short call option = -N(d₁)

Delta of a short call option on 10,000 shares = (-0.5695) * 10,000 = -5,694.60

The delta of the underlying stock is always equal to 1.

To make the position delta neutral, the underlying stock must be bought.

Number of shares to buy = 5,694.60

As fractional shares cannot be bought, this is rounded off to 5,695

d]

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

We calculate d₁ as below :

· ln(S₀ / K) = ln(55 / 50). We input the same formula into Excel, i.e. =LN(55/50)

· (r + σ2/2)*T = (0.05 + (0.202/2)*0.25

· σ√T = 0.20 * √0.25

d₁ = 0.1281

N(d₁) is calculated in Excel using the NORMSDIST function and inputting the value of d1 into the function.

N(d₁) = 0.8704

The delta of a short call option = -N(d₁)  

Delta of a short call option on 10,000 shares = (-0.8704) * 10,000 = -8,703.62

The delta of the underlying stock is always equal to 1.

Overall delta of position =  -8,703.62 + 5,695 = -3,008.62

To maintain delta neutrality, the underlying stock must be bought.

Number of shares to buy = 3,008.62

As fractional shares cannot be bought, this is rounded off to 3,009