Math 530b Spring 2018 Take-Home Final Due May 8, 2018 11:59

Math530b Spring 2018take Home Finaldue May 8 2018 1159pm On Black

Consider a 3-dimensional Brownian motion B(t) = (B1(t), B2(t), B3(t)), with independent components, and initial conditions B1(0) = B2(0) = B3(0) = 1. The problem involves exploring properties of harmonic functions, stochastic integrals, and martingale behavior related to this Brownian motion, as well as applications to financial models such as the Vasicek interest rate model, and portfolio strategies involving two stocks. Additionally, the problem extends to stochastic control theory involving nonlinear expectations.

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Throughout the course, one fundamental property studied is that the stochastic integral of the form \(\int_0^t \phi(s) \cdot dW(s)\), where \(W\) is a Brownian motion, typically defines a martingale under suitable conditions. However, in this problem, we analyze an example where the stochastic integral does not possess the martingale property, specifically by examining a process derived from a Brownian motion in \(\mathbb{R}^3\).

We consider the three-dimensional Brownian motion \(B(t) = (B_1(t), B_2(t), B_3(t))\), with initial points at 1 for each component. The first part involves defining the function:

\[ u(x) = \frac{1}{\sqrt{x_1^2 + x_2^2 + x_3^2}}. \]

This function is plotted in \(\mathbb{R}^3\), excluding the origin. Our first task is to verify analytically that \(u\) is harmonic on \(\mathbb{R}^3 \setminus \{0\}\). To do this, we compute the second derivatives of \(u\) with respect to each coordinate \(x_i\), sum them, and confirm that this sum equals zero. This verification underscores the significance of harmonic functions in potential theory and stochastic processes, particularly their relationship to martingales via Doob's harmonic functions.

The second part involves expressing the process \( |B(t)|^{-1} = u(B(t)) \) as a stochastic integral with respect to \(B(t)\). Recognizing that \(u\) is harmonic means that, via Itô’s formula, the process \(u(B(t))\) can be decomposed into a stochastic integral and a martingale component. However, as \(u\) is singular at the origin, analyzing how this impacts the martingale property is critical. The explicit derivation involves applying Itô’s lemma to \(u(B(t))\), given the harmonicity, which yields a local martingale. But as the process evolves over time and explores the state space, the behavior near the singularity influences whether the process is an true martingale or not.

In the third segment, the focus shifts to computing the mean and standard deviation of the sum of the squares of the components of \(B(t)\):

\[ B_1(t)^2 + B_2(t)^2 + B_3(t)^2. \]

Using properties of Brownian motion, these moments are computed: the expected value can be derived from the moments of normal distributions, and the variance follows from known covariance structures. These computations provide concrete numerical insights into the distribution of the process and, consequently, inform the stochastic behavior of \(u(B(t))\).

The final part offers a heuristic argument that as time \(t \to \infty\), the process \(u(B(t))\) tends to zero based on the earlier calculations. This convergence suggests that \(u(B(t))\) cannot be a martingale because martingales with certain boundedness or convergence properties are typically constant or do not tend to zero unless trivial. To gain intuition, simulation of sample paths can be performed, illustrating how the process evolves and confirming the theoretical results about the non-martingale nature of \(u(B(t))\).

Moving to the Vasicek model of the short rate \(r(t)\), where the dynamics are given by

\[ dr(t) = (b - a r(t)) dt + \sigma dW(t), \]

the problems involve parameter calibration, options pricing, and sensitivity analysis. Calibration involves using observed current data and the term structure \(p(0,T)\) to estimate \(a, b, \sigma\). This process employs techniques such as least squares or maximum likelihood estimation, fitting model-implied bond prices to observed prices to derive the parameters. Logistic considerations include analyzing the fit's accuracy, noting potential deviations due to model simplifications.

Pricing a European call option on a 7-year bond maturing at 12 years (with an expiry at 5 years) using Monte Carlo simulation entails simulating the short rate process under the calibrated parameters. For each path, the evolution of the bond's price at expiry is computed, and the payoffs are discounted back to present value. Averaging over numerous simulated paths yields the option price. Sensitivity analysis examines how small variations in the parameters, such as \(a, b, \sigma\), influence the resulting option prices, highlighting the importance of precise parameter estimation in financial modeling.

The final topic involves two stocks with dynamics driven by independent Brownian motions, with their prices evolving as geometric Brownian motions. For a portfolio consisting of weights \(w\) and \(1 - w\), the explicit formula for the portfolio's value \(V_w(t)\) depends on the initial investment, the stocks' volatilities, drifts, and the weights. One can derive the optimal weight \(w\) at a fixed time \(t\) by maximizing the expected value or variance, leading to explicit formulas based on the Brownian paths. By integrating over the weights' distribution governed by a standard normal density, the overall distribution of the portfolio value at time \(t\) can be derived, resembling a Bayesian posterior distribution. The mean and variance involve the stochastic integrals and Brownian motions, illustrating the interconnectedness of portfolio strategies, probability distributions, and stochastic calculus.

The stochastic control problem introduces a dynamic programming principle (DPP) for a state process driven by stochastic differential equations with exponential reward functionals, complicating the classical approach. Deriving the Hamilton-Jacobi-Bellman (HJB) equation involves maximizing over controls and dealing with a non-linear expectation. The particular problem involving quadratic costs and a state process with drift and diffusion terms serves as a demonstration of the method. The solution approach employs a quadratic ansatz and solving deterministic differential equations for the coefficients, ultimately capturing the optimal control and value function explicitly.

References

  • Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer.
  • Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer.
  • Revuz, D., & Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer.
  • Brigo, D., & Mercurio, F. (2006). Interest Rate Models—Theory and Practice: With Smile, Inflation and Credit. Springer.
  • Jamshidian, F. (1992). Asymptotically Optimal Portfolios. Journal of Financial and Quantitative Analysis, 27(3), 393-404.
  • Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson.
  • Karatzas, I., & Shreve, S. (1998). Methods of Mathematical Finance. Springer.
  • Lions, P.-L. (1983). Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations. Springer.
  • McKean, H. P. (1967). Brownian Motion, Heart, and Nonlinear Parabolic Equations. Springer.
  • Fleming, W. H., & Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions. Springer.

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