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:

dS(t)=μS(t)dt+σS(t)dB(t)Note-  coefficients μ  representing the drift and σ,volatility of the asset, respectively, are both constant in this model.

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:

S = Current stock price = 40 ,K = Strike Price = 42 ,

Next is, "r" the risk free rate,

risk free rate, r = mu = 0.12 ,Volatility, σ = 0.24 time to maturity, T, as we have;T= 4 months = 4/12.T = 1/3 year(360 days)

Step B:

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

The probability which we want is 1 - N(-d₂),So, we have;d₂=㏑(S/K)+(r-σ²/2)T/σ√T

Step C:

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

C is the answer

Step-by-step explanation:

Rotate 180° about the point (2, -4) -- maps the figure onto itself.

A figure is mapped onto itself when the transformation results in the original pre-image for the image.