Assignment $2$

Assume there is no arbitrage. SHOW and explain how the risk neutral probability $p=\frac{1r-d}{u-d}$ in Page $6$ of Shreve $($additional chapter$)$ cannot have certain values i$.$e$.$ what are the possible values of $p?(15$ points$)$

Look at the following one period binomial model for stock S that follows the examples we have discussed in class:

Let risk free return be $5\%$ and strike price for the European option be $K=6.$ Find the fair price of the option and also the replication portfolio.

Show and explain how for the value of delta and B that you calculate, you are now indifferent between buying the option and only buying shares of this stock. $(35$ points$)$

Explain what we mean by martingales and Markov process. $(10$ points$)$

Graph the following equation:

where t can only be greater than or equal to zero.

a$)$ Is this function a martingale? Why$/$Why not. $(10$ points$)$

b$)$ Does this function exhibit a Markov process? Why$/$Why not? $(10$ points$)$

Consider the following equation two equations that give us stock returns$($profits$)$ for two Stocks Q and Z where

and

a$)$ Here, $x$ is a variable that captures weather behavior such that x has to be between $0$ and $5.$ Both stocks Q and Z are highly influenced by $x.$ Find maximum and minimum values for both these stocks and also find the $x$ value that achieves the maximum value for both stocks. $(10$ points$)$

b$)$ At Q starts accruing losses. Assume you do not want to lose more than $$10$ on Q $.$ How do you hedge to ensure your losses are contained? $(10$ points$)$