The price of a certain security follows a geometric brownian motion with drift parameter µ = 0.12 and the volatility parameter σ = 0.24.

(a) if the current price of the security is $40, find the probability that a call option, having four months until expiration and with a strike price of k = 42 will be exercised.

(b) in addition to the above information as in part (a) if the interest rate is 8%, find the risk-neutral arbitrage free valuation of the call option.

Brownian Motion- The usual model for the time-evolution of an asset price S(t) is given by the geometric Brownian motion.

Now the geometric Brownian motion is represented by the following stochastic differential equation:

To solve the problem now we have the been Data provided:

μ= 0.12,

σ=0.24,

Step-by-step explanation:

Step A:

we have, the variables of Black Scholes Model, by putting the values of variables available, we get:

Next is, "r" the risk free rate,

Step B:

We now need to calculate the parameter d₂ of the Black Scholes Model. .

Step C:

As step C is done on excel for further calculations so, do use it if you are solving it on computer.

answer: b.48 for

step-by-step explanation: