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ECON $315$ HW $3.$docx

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Please feel free to work in groups of three or less. If you do so$,$ please make sure only one of you is submitting the HW$.$ Make sure you write everyone's name.

Show all your work. Only scanned pdf copies or neatly typed word documents are accepted. If you do use generative AI$,$ make sure you look into the sources from where the model has pulled the answers from because you might get wrong answers.

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

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ECON $315$ HW $3.$docx

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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$)$

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