1. Define the stochastic integral X(t) := J： e^^udW(u) for t > 0.

1. Compute the mean E[X(t)] and the variance Var[X(t)].

2. Write the stochastic differential equation dX(t) and derive informally d[X, X](t). Integrate to obtain [X,X](T).

3. Compute the quadratic variation [X, X](T) by using a partition n = {to, ti, ...,t_n} of (0,T]. Show that

• Compute ^{E e}" (l^W(知i) - ^W(切)12 - ^(tj+i - ^tj)).

• Compute ^Var £；二： ^e2^j|W(tj+i) ^{- W(t}j^{)|2 - (t}j+i ^{- t}j)).

• Obtain [X, X](T).

2. Read the chapter "Derivations and Applications of Greek Letters: Review and Integration“ of Handbook of Quantitative Finance and Risk Management. This is available online through Cornell Library.

3. The goal of this exercise is to study more in depth the greeks, i.e., the sensitivities of the Black & Scholes call price c(So,T, K, r, a) with respect to the parameters So,T, K, r, a (initial price, maturity, strike, interest rate, volatility).

Generate at least

• two plots of the price of the call option for varying parameters

• two plots of the price of the put option for varying parameters

• four plots of some of the Greeks (of your choice) for varying parameters

For example, a — c(S：,T, K, r, a) gives the variation of the call price with the volatility.

You can consider different cases: in the money call option (Ke^-rT < So), out of the money call option (Ke^-rT < So), at the money call option (Ke^-rT = So).

You can also consider second order variations, for example the variation of the vega function with respect to the strike as in the Figure 1.

4. Derive the option delta for the Black and Scholes model, i.e. 5(So)= 急c(So,T, K, r, a). Draw the plot of Delta.