Question

A stock index level is currently 2,000. Its volatility is 25%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2%. Using the Black-Scholes model:

a) Derive the value a 6-month European put option with a strike price of 2020.

b) Derive the position in the index that is needed today to hedge a long position in the put option. Assume that the option is written on 250 times the index.

c) What is today’s probability (implied by the Black-Scholes model) that the index price will be greater than 2020 in 6 months?

Answer 1

a]

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

The value of a put option is:

P = (K * e^-rt * N(-d₂)) -    (S₀ * e^-qt * N(-d₁))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

q = dividend yield

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(2000 / 2020). We input the same formula into Excel, i.e. =LN(2000 / 2020)
• (r + σ²/2)*t = (0.04 + (0.25²/2)*0.50
• σ√t = 0.25 * √0.50

d₁ = 0.1452

d₂ = -0.0315

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) = 0.4423

N(-d₂) = 0.5126

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

P = (K * e^-rt * N(-d₂))   -    (S₀ * e^-qt * N(-d₁)), which is (2020 * e^{(-0.04 * 0.50)} * 0.5126) - (2000 * e^{(-0.02 * 0.50)} * 0.4423) ==> $139.1873

Value of put option is $139.1873

b]

Delta of put option = -N(-d₁)

Delta of put option = -0.4423

As the delta is negative, to hedge a long position in the put option, a long position in the index should be taken.

Number of shares of index to buy (long) = -Delta of put option * option multiplier

Number of shares of index to buy (long) = -(-0.4423) * 250

Number of shares of index to buy (long) = 110.56

As fractional shares cannot be bought, this is rounded to 110.

To hedge a long position in the put option, 110 shares of the underlying index should be bought

c]

If the index price will be greater than 2020 in 6 months, the put option will expire in-the-money.

The probability of a put option expiring in-the-money is the delta of the put option (absolute value)

The probability of a put option expiring in-the-money is 0.4423, or 44.23%

Today’s probability (implied by the Black-Scholes model) that the index price will be greater than 2020 in 6 months is 44.23%