金融代写|MATH3975 Financial Derivatives

Compute the value of a stock-or-nothing option in the Black-Scholes framework, i.e., the solution $V=V(S, \tau)$ to the Black-Scholes PDE
$$
-\frac{\partial V}{\partial \tau}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+(r-q) S \frac{\partial V}{\partial S}-r V=0
$$
( $S>0, \tau \in(0, T])$ with initial condition $V(S, 0)$ which is equal to $S$ if $S>S_0$ (where $S_0$ is a predeterminted value) and zero otherwise.

Compute the value of a power option $\mathrm{n}$ the Black-Scholes framework, i.e., the solution $V=V(S, \tau)$ to the Black-Scholes PDE
$$
-\frac{\partial V}{\partial \tau}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+(r-q) S \frac{\partial V}{\partial S}-r V=0
$$
$(S>0, \tau \in(0, T])$ with initial condition $V(S, 0)=\max \left(S^n-E, 0\right)$ where $n \in \mathbb{N}$ and $E>0$ are given constants.
HINT: Make a transformation $V(S, \tau)=W(Y, \tau)$, where $Y=S^n$ and derive the PDE for $W(Y, \tau)$. You can solve this equation for $W$ more easily by comparing with to the classical Black-Scholes equation and using the Black-Scholes formula.

Source: beanactuary.com, acturial exams. You are using the Vasicek one-factor interest rate model with the short rate process calibrated as
$$
d r=0.6(b-r) d t+\sigma d w
$$
For $t \leq T$ let $P(r, t, T)$ be the price at time $t$ of a zero-coupon bond that pays 1 USD at time $T$, if the short-rate at time $t$ is $r$. The price of each zero-coupon bond in the Vasicek model follow an Itô process
$$
\frac{d P(r, t, T)}{P(r, t, T)}=\alpha(r, t, T) d t-q(r, t, T) d w
$$
for $t \leq T$. You are given that $\alpha(0.04,0,2)=0.04139761$. Find $\alpha(0.05,1,4)$.
HINT: Recall the definition of the market price of risk and the fact that it is taken to be constant in Vasicek model

Consider the Cox-Ingersoll-Ross model of interest rates and the PDE for the bond prices $P=P(r, \tau)$
$$
-\frac{\partial P}{\partial \tau}+(\kappa(\theta-r)-\lambda \sigma r) \frac{\partial P}{\partial r}+\frac{\sigma^2 r}{2} \frac{\partial^2 P}{\partial r^2}-r P=0
$$
for $r>0, \tau \in(0, T]$ and initial condition $P(r, 0)=1$. We are looking for a solution in the form $P(r, \tau)=A(\tau) e^{-B(\tau) r}$. Derive the system of ordinary differential equations for the functions $A, B$ and their initial conditions. Using the solution to this system (you do not need to derive it; you can use the formulae given in the lectures), for selected parameters plot the term structures of interest rates showing their three possible shapes – increasing, humped and decreasing.

Chooser options are a type of exotic option that, at some pre-specified time $T_1$ in the future, can be converted into either a put or call option with expiry $T_2>T_1$ and strike $K$. We consider these options in the Black-Sholes setting. See the website http://demonstrations.wolfram.com/ChooserOptions/ and the interactive demonstrations there (you will need to download a free player). Then, solve the following problems.

• Explain the put-call parity argument (“It can be shown using general put-call parity considerations…”) and derive the price of a chooser option. You can check your solution for example here: http://ww . haas . berkeley . edu/groups/finance/WP/ rpf220.pdf, pp. 56-57 (but you need to provide a more detailed explanation).
• Compute its delta and gamma and plot their graphs as functions of the stock price for some selected times. Note the behaviour for $t$ approaching $T_1$, as mentioned also on the website.