Question

Let S - $57,0 -29%,r-7.5%, and 8 - 2.5% (continuously compounded). Compute the Black-Scholes vega of a $55-strike European call option with 3 months until expiration. (That is, compute the approximate change in the call price given a 1 percentage point increase in O.) O a. 0.1276 O b.0.1092 OC 0.1175 O d. 0.1862 0.0.1041

Answer 1

From the question we get the following information:

Description	Legend	Value
Underlying price	S	$ 57.00
Volatility	σ	29.0%
Risk-free Interest Rate	r	7.50%
Continuously compounded Dividend Yield	δ	2.50%
Strike Price	X	$ 55.00
Time to expiration (in Years)	t	0.25

Black-Scholes vega will be calculated in the following steps:

Step-1:- Call Option price calculation

Step-2:- Vega Calculation

Step-1:- Call Option price calculation

The formula for Call Option price (d₁) as per Black-Scholes option pricing model is

di In ++ x (r-5+) ovt

Putting value in the above formula we get the following

$d_{1}=\frac{\ln\frac{57}{55}+0.25\times (0.075 - 0.025 +\frac{29 ^{2}}{2})}{29 \sqrt{0.25}}$

   = $ 0.4050

So the Call Option price (d₁) = $ 0.4050

Step-2:- Vega Calculation

The formula for Call option Vega as per Black-Scholes Formulas for Vega is

1 Vega 1 100 x S x exp(-5 x t) Vt x X exp(- V211

Putting value in the above formula we get the following

1 Vega 1 x 57 x exp(-0.025 x 0.25) V0.25 x 100 0.40502 X exp(- 2

     = 0.1041

So, the Black Scholes Vega of a $55 - strike European call option with 3 months until expiration is 0.1041.

Answer is option e.