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\title{Financial Mathematics and Simulation\\
MATH 6740--1 -- Spring 2011 \\
Homework 3}
\date{Due Date: Monday, April 25 at 2:00 PM}
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\begin{document}
\maketitle
This homework has 195 points plus 35 bonus points available but, as always, homeworks are graded out of 100 points. Full credit will generally be awarded for a solution only if it is both correctly and efficiently presented using the techniques covered in the lecture and readings, and if the reasoning is properly explained. If you used software or simulations in solving a problem, be sure to include your code, simulation results, and/or worksheets documenting your work. If you score more
than 100 points, the extra points do count toward your homework total.
\section{Theoretical Calculations}
\subsection{Stratonovich solution method (15 points)}
One trick to solving stochastic differential equations that works in some special cases is to express the SDE in Stratonovich form. If this has a simple form, the solution can sometimes be guessed because the chain rule in Stratonovich calculus takes the familiar form.
Consider then the Ito stochastic differential equation
\begin{align*}
\difd X &= X^3 \, \difd t + X^2 \, \difd W (t), \\
X(t=0) &= \Xinit.
\end{align*}
\begin{enumerate}
\item (\textbf{5 points}) Convert the stochastic differential equation to Stratonovich form.
\item (\textbf{5 points}) Derive a solution to the stochastic differential equation by just solving the Stratonovich SDE using ordinary calculus methods.
\item (\textbf{5 points}) Verify that your solution satisfies the Ito stochastic differential equation by applying Ito's formula.
\end{enumerate}
\subsection{The Retarded Trader (25 points plus 25 bonus points)}
No, Stephen Colbert and Timothy Shriver (``Don't use the `R-word' for any reason."), I am not being politically incorrect or insulting -- I am using the adjective according to one of its acceptable technical definitions. Suppose the price $ S(t) $ of an asset obeys (generalized) geometric Brownian motion
with some given time-dependent drift $ \alpha (t) $ and volatility $ \sigma (t) S(t) $:
\begin{align*}
S(t=0) &= \Sinit, \\
\difd S(t) &= \alpha (t) S(t) \, \difd t + \sigma (t) S(t) \, \difd W (t),
\end{align*}
where $ \alpha (t) $ and $ \sigma (t) $ are deterministic functions.
Suppose that because of other activities and commitments, as well as delays
in receiving information about the market, the number of shares of an asset that a trader holds at time $ t $,
$ \Delta (t) $, is a function of the price of the asset at a deterministically retarded time $ \tau (t) $, with $ \tau (t) \leq t $:
\begin{equation*}
\Delta (t) = f (S (\tau(t))).
\end{equation*}
\begin{enumerate}
\item (\textbf{10 points})
Compute the mean of the net earnings of the trader from the asset in terms of the
model functions and parameters.
\item (\textbf{15 points}) For the special case $ \alpha \equiv 0 $, compute the variance of the net earnings of the trader from the asset in terms of the
model functions and parameters.
\item (\textbf{25 bonus points}) Compute the variance of the net earnings of the trader from the asset in terms of the
model functions and parameters for the case where $ \alpha (t) $ is a general smooth, deterministic function.
\end{enumerate}
Your answers should be completely deterministic expressions.
\subsection{Naive Higher Order Numerical Methods (20 points)}
To give some indication of the difficulties of developing higher order methods for solving stochastic differential equations,
suppose we naively try to implement an explicit trapezoidal integration rule for the stochastic differential equation:
\begin{align*}
\difd X &= a(X(t),t) \, \difd t + b (X(t),t) \, \difd W (t), \\
X(t=0) &= \Xinit
\end{align*}
The naive numerical scheme would look like:
\begin{align*}
X_0 &= \Xinit,\\
\tilde{X}_{n} &= X_n + a (X_n,t_n) \Delta t + b(X_n,t_n) \Delta W_n, \\
X_{n+1} &= X_n + \frac{1}{2} \left(a (X_n,t_n) + a (\tilde{X}_n,t_{n+1})\right) \Delta t
+ \frac{1}{2} \left(b(X_n,t_n) + b(\tilde{X}_n,t_{n+1})\right) \Delta W_n.
\end{align*}
Here $ t_n = n \Delta t $, the $ \Delta W_n $ are random variables representing the Wiener process increment
$ W (t_{n+1}) - W (t_n) $, the $ X_n $ are numerical approximations to $ X(n\Delta t) $, and
$ \tilde{X}_n $ are intermediary values (essentially Euler method approximations for $ X ((n+1) \Delta t) $,
which are then refined by the second step, which estimates the increment $ X((n+1) \Delta t) - X(n \Delta t) $
by averaging the coefficient value at the beginning of the time step and its rough estimate at the end of the
time step). For deterministic differential equations ($b =0 $), this explicit trapezoidal method is a second
order accurate numerical scheme.
Show that the explicit trapezoidal scheme described above is neither weakly nor strongly consistent for simulating the
stochastic differential equation. A careful formal argument can earn you full credit, and you can earn up to 15 bonus
points for making your argument mathematically rigorous.
\section{Applications}
\subsection{Accumulated Wealth in Vasicek Interest Rate Model (20 points plus 20 bonus points)}
Suppose the effective interest rate of a certain bond fund can be modeled by the Vasicek interest rate model:
\begin{subequations}
\begin{align}
\difd R (t) &= (\alpha - \beta R(t)) \, \difd t + \sigma \,\difd W (t), \\
R(t=0) &= \Rinit, \nonumber
\end{align}
where $ \alpha $, $ \beta $, and $ \sigma $ are positive constants, and $ \Rinit $ is the (nonnegative) initial value of the interest rate. If the interest is immediately reinvested in the fund, then the total value $ X(t) $ of the bond fund resulting from an initial investment of $ \Xinit $ (assuming no further investment or withdrawals) is given by:
\begin{align}
\difd X(t) &= R (t) X(t) \, \difd t, \\
X(t=0) & = \Xinit. \nonumber
\end{align}
\label{eq:vasicek}
\end{subequations}
(This neglects fluctuations in the price of the bond fund shares, even though that's quite important.)
\begin{enumerate}
\item (\textbf{10 points}) Compute the probability distribution of the value of the bond fund $ X(T) $ at the end of the investment period $ T $.
\item (\textbf{10 points}) Compute the mean and variance of the value of the bond fund $ X(T) $ at the end of the investment period $ T $.
\item (\textbf{20 bonus points})
Note that these equations can be also interpreted as a physical model: $ R(t) $ could be a fluctuating strain rate in a turbulent flow, with $ X(t) $ the trajectory of a particle in this flow. Feel free to use this physical rather than financial interpretation instead in your solution above. Actually a more realistic physical model of this type would be formulated in terms of a strain matrix $ \cR (t) $ and vectorial particle position $ \bX (t) $ in a $ d=2 $ or $ 3 $ dimensional flow. Unfortunately, this multidimensional extension doesn't seem to allow as explicit a solution. Try anyway:
\begin{enumerate}
\item (\textbf{10 bonus points}) Formulate a two- or three-dimensional version of the model (\ref{eq:vasicek}) , with physically reasonable structure. (In particular, most turbulent flows are incompressible so you should define your model accordingly). Provide at least rough physical intepretation for the parameters in your model.
\item (\textbf{10 bonus points}) Try to solve your multidimensional model in an analogous manner to the one-dimensional model
(\ref{eq:vasicek}). What is the obstacle to obtaining as explicit a solution?
\end{enumerate}
\end{enumerate}
\subsection{Accumulated Interest in Cox-Ingersoll-Ross Interest Rate Model (70 points)}
Suppose the effective interest rate of a certain bond fund can be modeled by the Cox-Ingersoll-Ross model:
\begin{subequations}
\begin{align}
\difd R (t) &= (\alpha - \beta R (t))\, \difd t + \sigma \sqrt{ R(t)} \, \difd W(t), \\
R (t=0) &= \Rinit, \nonumber
\end{align}
where $ \alpha $, $ \beta $, and $ \sigma $ are positive constants, and $ \Rinit $ is the (nonnegative) initial value of the interest rate. The interest earned is not reinvested, so the total amount of interest $ I(t) $, earned up to time $ t $ is given by
\begin{align}
\difd I (t) &= R(t) P \, \difd t \\
I(t=0) &= 0, \nonumber
\end{align}
\label{eq:cir}
\end{subequations}
where $ P $ is the deterministic constant representing the total face value of the bonds held. We are again neglecting fluctuations in the principal due to revaluation of the bond fund shares, even though that is a very significant factor.
\begin{enumerate}
\item (\textbf{10 points})
Calculate the mean and variance of the total interest $ I(T) $ earned after holding the bond fund shares for a time $ T $.
\item (\textbf{10 points}) Calculate the autocorrelation function of the interest rate $ R(t) $.
\item (\textbf{10 points}) Generalize your answers to the case where the initial interest rate $ \Rinit $ is itself a random variable, which you may assume is independent of the noise driving the interest rate at future times. (This would correspond to contemplating a bond investment at a future time (which here is $0 $, but that doesn't really matter), when the starting interest rate would be currently unknown.)
\item (\textbf{10 points}) Simulate the equations (\ref{eq:cir}) numerically using the Euler-Marayama method and plot a realization of the exact solution together with a strong numerical approximation.
\item (\textbf{10 points})
Examine the strong order of accuracy of the Euler-Marayama method for this equation (such as by adapting \texttt{emsstrong.m}) and comment on your findings.
\item (\textbf{10 points})
Develop a numerical approximation for the correlation function of the interest rate $ R(t) $ using your numerical simulations (not the exact solution).
\item (\textbf{10 points}) Examine how the accuracy of this approximation of the correlation function depends on the time step, and comment on your findings.
\end{enumerate}
The model (\ref{eq:cir}) also has a sort of physical interpretation: take $ I(t) $ as the spatial position of a particle (moving in one dimension) and $ R(t) $ as its velocity. These equations are appropriate for a microscale particle,
nondimensionalized to unit mass, moving through a fluid and experiencing three forces:
an external constant applied (electrical, gravitational, etc.) field ($\alpha $), friction ($-\beta R$), and random forces due to interactions (such as collisions)
with the fluid molecules ($\sigma \sqrt{ R} \frac{\difd W (t)}{\difd t} $). Well, not quite -- this corresponds to a fluid whose temperature $ k_B T = \sigma^2 R $ depends on the velocity of the particle, which is nonsensical -- but by simply dropping the $ \sqrt{R (t)} $ factor (in other words going to the Vasicek interest rate model) we have a good model for a constant temperature fluid.
\section{Mathematical Problems}
\subsection{The Collision of Two Beautiful Mathematical Structures (\textbf{15 points})}
We define the complex Wiener process by
\begin{equation*}
\Wcomp (t) = \frac{W_1 (t) + \mathi W_2 (t)}{\sqrt{2}}
\end{equation*}
where $ W_1 (t) $ and $ W_2 (t) $ are independent Wiener processes. Show that if $ f $ is a complex analytic function,
then $ f(\Wcomp (t)) $ is a martingale.
\subsection{The Futility of Finance for Honest Traders (30 points plus 15 bonus points)}
Consider two assets in an efficient market, for which their prices can be modelled:
\begin{equation*}
S_1(t) = \exp \left[\int_{0}^t \mu (\tp) \, \difd \tp\right] M_1 (t), \qquad \qquad
S_2(t) = \exp \left[\int_{0}^t \mu (\tp) \, \difd \tp\right] M_2 (t)
\end{equation*}
where $ M_1 (t) $ and $ M_2 (t) $ are martingales with respect to the filtration generated by these asset prices. Note that we do not assume that $ M_1 $ and $ M_2 $ are independent (their price fluctuations could have a common cause).
The function $ \mu (t) $ is a deterministic growth rate, which is typically taken as constant but we can account for seasonality and other effects by allowing it to depend on time. Note that we are assuming it is the same for both assets (as might be the case if they were in the same sector). Suppose that an investor starts with a certain amount of money $ c $ that is fully invested in some combination of the two assets. At certain prescribed times $ t_1 < t_2 < \ldots t_n $, which need not be equally spaced, the investor rebalances his portfolio between the two assets, meaning he reduces his investment in one of the assets by a certain amount and increases his investment in the other asset by the same amount. (That is, he remains fully invested, and incurs no transaction costs.) The strategy the investor uses for the amount of money transferred between the assets can depend on any information about the asset prices up to the time of the trade
\begin{enumerate}
\item (\textbf{20 points}) Show that no matter what trading strategy the investor uses, the expected value of his portfolio after time $ t $ will be
$ c \exp \left[\int_{0}^t \mu (\tp) \, \difd \tp\right] $. \label{part:marttrade}
\item (\textbf{10 points}) Show that if the investor is allowed to look into the future (from, for example, insider information), he can improve his expected portfolio value above this ``fair value."
\item (\textbf{15 bonus points}) Extend the result from part~\ref{part:marttrade} to the case where the times of trading $ \{t_j\}_{j=1}^n $ are no longer prescribed ahead of time, but allowed to depend on the information the investor has about the asset prices. That is, the investor's decision to trade at a certain moment of time can only depend on the information he has up to that moment of time.
\end{enumerate}
\end{document}