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(-d_{2})) -
(S_{0} * e^{-qt} *
N(-d_{1}))

where :

S_{0} = 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

d_{1} = (ln(S_{0} / K) + (r +
σ^{2}/2)*T) / σ√T

d_{2} = d_{1} - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d_{1} and d_{2} as below
:

- ln(S
_{0}/ K) = ln(2000 / 2020). We input the same formula into Excel, i.e. =LN(2000 / 2020) - (r + σ
^{2}/2)*t = (0.04 + (0.25^{2}/2)*0.50 - σ√t = 0.25 * √0.50

d_{1} = 0.1452

d_{2} = -0.0315

N(-d_{1}) and N(-d_{2}) are calculated in Excel
using the NORMSDIST function and inputting the value of
d_{1} and d_{2} into the function.

N(-d_{1}) = 0.4423

N(-d_{2}) = 0.5126

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

P = (K * e^{-rt} * N(-d_{2}))
- (S_{0} * e^{-qt} *
N(-d_{1})), 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(-d_{1})

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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