Let us consider an option on a dividend paying stock with dividend

$\backslash$

delta

$,$

$$volatility

$\backslash$

sigma

$,$

risk

$-$

free rate r over a period

$\backslash$

Delta t

$.$

a

$)$

$$Show that the risk

$-$

neutral

$($

martingale

$)$

$$probability q can be written as:

q

$=$

e

$($

r

$$

$\backslash$

delta

$)$

$\backslash$

Delta t

$$

$$e

$$

$\backslash$

sigma

$$

$\backslash$

Delta t

e

$\backslash$

sigma

$$

$\backslash$

Delta t

$$

$$e

$$

$\backslash$

sigma

$$

$\backslash$

Delta t

$[$

$2$

$]$

b

$)$

$$In a risk

$-$

neutral world, a future contract is treated as stock paying a continuous

dividend yield equal to the risk

$-$

free rate r

$.$

$$Deduce from question a

$)$

$,$

$$the risk

$-$

neutral

$($

martingale

$)$

$$probability q expression for a future contract.

$[$

$1$

$]$

The current price of a eight

$-$

month future is

$5$

$8$

$,$

$$the risk

$-$

free rate is

$5$

$\%$

$,$

$$the strike price

is

$6$

$0$

$$and the volatility is

$2$

$4$

$\%$

$$per annum.

c

$)$

$$Use a two

$-$

step binomial tree to display the price process of the eight

$-$

month future

contract

$.$

$$Consider u

$=$

$$e

$\backslash$

sigma

$$

$\backslash$

Delta t

$,$

$$d

$=$

$$e

$$

$\backslash$

sigma

$$

$\backslash$

Delta t

$.$

$[$

$4$

$]$

d

$)$

$$Use the two

$-$

step binomial tree defined in c

$)$

$$to calculate the value an eight

$-$

month

American put option on the future contract.