Problem 1.

For this question, you will use SDE methods to model a chemical reaction. Suppose there exists some chemical which undergoes the reaction $X+X \rightarrow P . X$ is our chemical of interest, and $P$ is some downstream product, which does not interfere with our primary reaction in any way 1 . Assume a reaction rate $\alpha=0.01$, and an initial concentration $X(0)=100$
(a) Let $X(t)$ be your chemical concentration at time $t$. Write out a stochastic differential equation of the form $d X=f\left(X, t, W_{t}\right) d t+g\left(X, t, W_{t}\right) d W_{t}$, and give a couple sentences to justify each term in your equations.
(b) Use an appropriate change of variables to transform your equation into an equation of the form $d Z=(a Z+b) d t+(c Z+d) d W_{t}$.
(c) Rearrange your equation and use integrating factors to find an explicit expression for $Z$, and hence $X_{t}$. Your final answer is liable to contain integral terms on the RHS, so long as your $R H S$ contains only $t$ and $W_{t}$ terms, and does not contain any $X$ or $Z$ terms, everything is fine.
(d) Is the SDE model you have produced a good representation of reality for all values of $X$ ? Are there and values of $X$ for which your SDE approximation break down, or start giving meaningless results?

Problem 2.

Me and my friends do not have the money for gambling on the stock market, so instead we make a bet. I bet my friend that the price of vegetable stock will go over $\$ 3.50$by the end of the year. If the price goes over, they must buy me one box of veggie stock. If they price stays under, I must buy them one box of veggie stock. Our bet takes place January first$\left(t_{0}=0\right)$, and comes due at the end of December$(T=1)$. The value of stock obeys the stochastic differential equation: $$d S=\mu S d t+\sigma S d W_{t} .$$ Where$\mu=0.06$is the annual rate of inflation and$\sigma=0.2$is the volatility in stock price. (a) Assuming the current value of stock is$S_{0}=3.3$, give an explicit formula for$S_{t}$in terms of$t$and$W_{t}$(you do not need to deduce this formula from first principles, you are free to quote derivations given in class). (b) Given the above formula what is my expected payoff in December. Rather than calculate a probability density function for$S$, it may be easier to calculate the probability density function for$W_{t}$, and then integrate$E(V)=\int V(w) p(w) d w$. (c) For what value of$S_{0}\$ would the above be be a fair bet.
(d) The above bet can be thought of as a stock market derivative, as described by the Black-Scholes equation. Write our the corresponding Black-Scholes equation, with suitable boundary conditions. You are not required to solve this equation.

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