Question

# A stock index level is currently 2,000. Its volatility is 25%. The risk-free rate is 4%...

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?

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%

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