Question

PROBLEM №4

Using the BSM model, estimate value of a 3-month call option and 3-month put option on a share of ETF if:

an exercise price, X: $250

sport price, S0: 260

the annual risk-free rate is 5 %

historical standard deviation of share’s returns: 12%.

Implied standard deviation: 13.5%

Please show how you calculated the following parameters: d1, d2, N(d1). N(d2), N(-d1), N(-d2)

Answer 1

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. Here, we use implied standard deviation, and not historical standard deviation.

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(260 / 250). We input the same formula into Excel, i.e. = LN(260/250)
• (r + σ²/2)*T = (0.05 + (0.135²/2)*0.25
• σ√T = 0.135 * √0.25

d₁ = 0.8000

d₂ = 0.7325

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) = 0.7881

N(d₂) = 0.7681

N(-d₁) = 0.2119

N(-d₂) = 0.2319

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (260 * 0.7881) - (250 * e^{(-0.05 * 0.25)})*(0.7681)    ==> $15.2858

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁), which is (250 * e^{(-0.05 * 0.25)})*(0.2319) - (260 * (0.2119) ==> $2.1803

Value of call option is $15.2858

Value of put option is $2.1803