Question

The price of a share follows the geometric Brownian motion with parameters µ = 0.2 and σ = 0.18. Presently, the share’s price is £38. Consider a call option having one year until its expiration time and having a strike price of £40. The continuously compounded interest rate is 5%.

(a) What is the risk-neutral price C of this call option?

(b) Suppose now that you are the seller of this option. At time t = 0 you get £C from the buyer of the option, where C is the risk-neutral price of the option. You then have to design a hedging strategy which would allow you to meet your financial obligation in one year’s time. Your portfolio should consist of two investments: you are allowed to buy the underlying shares and to deposit money in the bank.

(i) The price of the share evolves according to a geometric Brownian motion. State the formulae you will need to compute the number of shares in the portfolio and the capital deposited in the bank at any time t, 0 6 t 6 1.

Answer 1

a:

Risk-Neutral price of call option

$C=\frac{π\times c^{+}+(1-π)\times c^{-}}{1+r}$

where π is the probability, r is the risk-free rate

First, we need to determine the probability of up movement with help of the risk-neutral formula

$π=\frac{1+r-d}{u-d}$

where is the probability, r is the risk-free rate, u is the underlying price of the up move.

Substituting the values in the equation:

$π=\frac{1+5\%-0.18}{0.2-0.18} =\frac{0.87}{0.02}$

$π=43.5$

Substituting the values on the equation of risk-Neutral price of call option:

$c=\frac{43.5\times40+1-43.5\times38}{1+0.05}$

$c=\frac{125}{1.05}$

 $c=119.04$

Therefore, the risk-neutral price of the call option is $119.04.

(b)(i)

To compute the number of shares in a portfolio and the capital deposited in the bank at any time t, 0 ≤ t ≤ 1 in geometric Brownian motion, given equations are followed-

$∂V∂t+12σ2S2∂V∂S2+rS∂V∂S−rV=0$

C(S,t), of this equation is equal to the cost of constructing an option from the specified stock. Here, r and σ are constant, this is:

$C(S,t)=SN(d1)−Le−r(T−t)N(d2)$

L is the option's ‘strike price’

T is its time to maturity

N is the cumulative probability distribution function for a normal random variable.

Now-

$d1=log(S/L)+(r+12σ2)(T−t)σ(T−t)$

$d2=log(S/L)+(r−12σ2)(T−t)σ(T−t)=d1−σ(T−t)$