Calculate the Delta, Gamma and Vega of an option given the current stock price, strike price, time to expiration, risk-free rate, and the volatility.?

To calculate the Delta, Gamma, and Vega of an option, we need to use the Black-Scholes-Merton (BSM) option pricing model. The BSM formula for a European call option (C) and a European put option (P) are given as follows:

C = S_tN(d₁) − Ke^− rTN(d₂)

P = Ke^− rTN( − d₂) − S_tN( − d₁)

where

- S_t: Current stock price

- K: Strike price

- T: Time to expiration

- r: Risk-free rate

- σ: Volatility

- N(x): Cumulative distribution function of the standard normal distribution

- d₁ and d₂ are given by:

$$d_1 = \frac{\ln{\frac{S_t}{K}} + (r+\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}$$

$$d_2 = d_1 - \sigma\sqrt{T}$$

Now, let’s compute the Greeks:

1. Delta (Δ): Delta is the rate of change of the option price with respect to the stock price.

For call: $\Delta_C = \frac{\partial C}{\partial S_t} = N(d_1)$

For put: $\Delta_P = \frac{\partial P}{\partial S_t} = N(d_1) - 1$

2. Gamma (Γ): Gamma is the rate of change of Delta with respect to the stock price. Both call and put options have the same Gamma.

$$\Gamma = \frac{\partial \Delta}{\partial S_t} = \frac{N'(d_1)}{S_t \sigma\sqrt{T}}$$

Here, N′(x) denotes the probability density function of the standard normal distribution:

$$N'(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$$

3. Vega (ν): Vega is the rate of change of the option price with respect to the volatility.

For call:
$$\nu_C = \frac{\partial C}{\partial \sigma} = S_tN'(d_1)\sqrt{T}$$

For put:
$$\nu_P = \frac{\partial P}{\partial \sigma} = S_tN'(d_1)\sqrt{T}$$

Both call and put options have the same Vega.

Using the parameters given in the question, we can plug them into the BSM model and then compute the required Greeks. Note that actual numerical examples are necessary to get specific values for Greeks.