Question

Use below-given dataset:
Stock Price	40
Exercise Price	35
Time to Maturity in years	0.25
Risk Free Rate	6%
Volatility	40%
A) Calculate Call and Put Price. (2.5+2.5=5)
Greeks are given below:
	Call	Put
Delta	0.8003	-0.1997
Gamma	0.0350	0.0350
Theta	-0.0165	-0.0108
Vega	0.0559	0.0559
Rho	0.0638	-0.0224
B) Interpret the GREEKS in detail. (3*5=15)

Answer 1

A]

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 :

• ln(S₀ / K) = ln(40 / 35). We input the same formula into Excel, i.e. =LN(40 /35)
• (r + σ²/2)*T = (0.06 + (0.40²/2)*0.25
• σ√T = 0.40 * √0.25

d₁ = 0.8427

d₂ = 0.6427

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) = 0.8003

N(d₂) = 0.7398

N(-d₁) = 0.1997

N(-d₂) = 0.2602

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

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (40 * 0.8003) - (35 * e^{(-0.06 * 0.25)})*(0.7398)    ==> $6.5049

P = (K * e^-rT)*N(-d₂) - (S₀)*N(-d₁), which is (35 * e^{(-0.06 * 0.25)})*(0.2602) - (40 * (0.1997) ==> $0.9838

Value of call option is $6.5049

Value of put option is $0.9838

B]

Delta

Delta is the sensitivity of the option premium to changes in the price of the underlying stock.

For each $1 increase in the stock price, the call premium will increase by $0.8003 and the put premium will decrease by $0.1997.

Gamma

Gamma is the sensitivity of the delta to changes in the price of the underlying stock.

For each $1 increase in the stock price, the call delta will increase by 0.0350 d for each $1 decrease in the stock price, the put delta will increase by 0.0350

Theta

Theta is the decay (decrease) in option premium with the passage of time.

With the passage of each trading day, the call premium will decrease by $0.0165 and the put premium will decrease by $0.0108.

Vega

Vega is the sensitivity of the option premium to changes in the volatility.

For each 1% increase in volatility, the call and put premium will increase by $0.0559.

Rho

Rho is the sensitivity of the option premium to changes in the risk free rate.

For each 1% change in risk free rate , the call premium will increase by $0.0638 and the put premium will decrease by $0.0224