# Institute for Mathematical Sciences Programs & Activities

**Inter-Faculty Workshop On Financial Mathematics**

(Saturday, 12 January 2002)

(Saturday, 12 January 2002)

**~ Abstracts ~**

**Quadratic Hedging for Interest Rates Models with
Stochastic Volatility
**

*By Dr Francesca Biagini, University of Bologna, Italy*

The quadratic hedging approaches called mean-variance hedging and local risk minimization for pricing and hedging claims in incomplete markets were originally introduced for risky assets. Here we apply these criteria to interest rate models in presence of stochastic volatility. By following the Heath-Jarrow-Morton approach, we suppose that the forward rate satisfies the following equation:

whereis a standard Brownian motion ( possibly n-dimensional)
and and represents
an additional source of randomness. In this
framework, a perfect replication is not possible even by using
an infinite number of bonds. In the terminology of Follmer and
Schweizer, the market incompleteness is a result of “insufficient
information”.

In order to find an explicit formula for the
variance-optimal and the minimal measure, first we characterize
the set of all the equivalent martingale measure for ,
where is
time of expiration of the option we wish to hedge. If , where , are times of maturity such that the volatility
matrix is invertible, the set of the equivalent martingale
measures for coincides with and we provide explicit formulas for
the densities of the variance-optimal and the minimal measure
which are independent by the chosen

Several models for the stochastic volatility are analyzed: the simplest one is when the
volatility jumps according to a totally inaccessible stopping
time and assumes values with a certain probability distribution.
The natural extension of the previous example it is the case
when the volatility is given by a Markov process and then by a
multivariate point process.

As a application of the previous
results, we set a finite number of bonds and compute the
mean-variance hedging and the risk-minimizing strategies for a
caplet on a , which reduces to evaluate the optimal strategy
for a put option on a .

Finally, we provide some numerical
simulations for the call price according to the two different
methods and compare them.

**Modeling Large Diversified Portfolios in a Jump-Diffusion
Market
**

*By Dr Liu Xiaoqing, Dept of Mathematics, National University of Singapore*

Reducing risk by diversification is the basic idea behind modern portfolio theory (MPT). The Capital Asset Pricing Model (CAPM) improves the MPT by classifying investment risks into two types: non-systematic (or diversifiable) risk and systematic (or non-diversifiable) risk. Under the CAPM model, non-systematic risk is eliminated as the number of assets grows. Only systematic risk is priced by the market and investors are compensated in expected return for bearing systematic risk. The APT and CAPM, which won their founders Nobel Prize in Economics, have long been widely investigated and applied in financial research and practice. However, it was until recently that the rationality of non-systematic risk diversification was proved on a rigorous mathematical basis and a reduced-form model for portfolios of lognormal assets was derived by N. Hofmann and E. Platen (Mathematical Finance 2000) for the first time in the literature. In this talk, the speaker will present a recent collaborative work with Prof KG Lim and Assoc Prof KC Tsui on a more realistic portfolio model that incorporates jump risks caused by unusual events on the market or by defaults of counter parties of individual assets. Three categories of portfolios are considered: namely, money (almost) equally allocated portfolios, risk minimizing portfolios, and financial indices. A stochastic dynamical system driven by only one Brownian motion and one Poisson process is derived for the asymptotic behaviours of such portfolios. We also prove that derivative contracts written on a portfolio can be priced by treating the asymptotic dynamics as the underlying if the number of assets in the portfolio is sufficiently large. Analytical and Monte Carlo VaR can be computed for the portfolios based on the derived asymptotic dynamics.

**Differential Geometry in Incomplete Markets**

By Mr Gao Yuan,
Centre for Financial Engineering, National University of
Singapore

We study the differential geometry of certain subset of the equivalent martingale measures in an incomplete market. Given a utility function, we define the entropy and the cross entropy in terms of the dual function. We construct a Riemannian geometric structure of which the distance is related to the cross entropy. We relate the geometry, entropy and cross entropy with terminal utility maximization, arbitrage-free equilibrium pricing, model risk and absolute risk aversion. A numerical algorithm for the pricing problem in the incomplete market is given as an application of the geometry.

Key words: Incomplete markets, differential geometry, utility maximization, stochastic volatility HJM model

JEL Classification: G11, G13

Mathematics Subject Classification (2000): 62P05, 91B24, 91B28

**Mean Variance versus Expected Utility in Dynamic Investment
Analysis
**

*By Dr Zhao Yonggan, Nanyang Technological University*

This paper extends Merton's continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory in which the growth optimal portfolio can be chosen as the risky fund. Given the existence of a Markovian state price density process, the CAPM holds without assuming log-normality for prices. The optimal investment policies for the case of HARA utility function are analytically derived. It is proved that only the quadratic utility exhibits global mean-variance efficiency among the family of HARA utility functions. The (global) efficient frontier for the dynamic model is linear in the space of standard deviation and expected return of the portfolio. A numerical comparison is made between the growth optimal portfolio and mean-variance analysis for the case of log-normally distributed assets. We discuss the optimal choice of target return that maximizes the probability that mean-variance analysis outperforms the expected utility approach. Finally, we discuss how to control portfolio's downside losses using a put option on the market portfolio.

**Incorporating Market Frictions into Asset Prices
**

*By Dr Mitchell Warachka, Singapore Management University*

At every point in time, a continuum of prices is introduced depending on an associated transaction size. Thus, investors face an entire curve of (exogenous) prices through time. The value of a portfolio after considering market frictions such as liquidity, transaction costs, short selling restrictions is defined. The impact of market frictions on the value of contingent claims can therefore be determined. Necessary and sufficient conditions for no arbitrage are given resulting in a generalization of the fundamental theorem of finance. Complete markets under this structure are also studied.

**High-Accuracy PDE Method for Financial Derivative Pricing
**

*By Mr Zhao Shan, Dept of Computational Science, National University of Singapore*

It is well known that many important derivatives lack a closed-form analytical solution and their estimation has to be performed by numerical procedures. Commonly used numerical methods for option pricing, such as the finite difference scheme and the binomial tree model, are quite simple, flexible, and convergent. However, the speed of convergence of these methods is usually slow. Thus, it might be expensive to achieve relevantly high accuracy in option valuations by using lower-order approximation schemes. In this talk, a brief discussion about the utility of a high-accuracy PDE scheme, the discrete singular convolution algorithm, for solving the Black-Scholes equation will be presented. The numerical results indicate that through proper implementation, higher accuracy and efficiency could be obtained by using high-accuracy methods for financial derivative pricing.

**Bayesian Risk Measures for Derivatives via Random Esscher
Transform
**

*By Dr Siu Tak Kuen, Dept of Mathematics, National University of Singapore*

We propose a model for measuring risks for derivatives which is easy to implement and satisfies a set of four coherent properties introduced in Artzner et al. (1999). We construct our model within the context of Gerber-Shiu's option-pricing framework. A new concept, namely Bayesian Esscher ``scenarios,'' which generalizes the concept of generalized ``scenarios,'' is introduced via a ``Random Esscher Transform.'' Our risk measure involves the use of the risk-neutral Bayesian Esscher ``scenario" for pricing and a family of real-world Bayesian Esscher ``scenarios" for risk measurement. Closed-form expressions for our risk measure can be obtained in some special cases.

**The Term Structure of Interest Rates as a Random Field
**

*By Mr Ricky Wong, Dept of Mathematics, National University of Singapore*

Traditional term structure models such as (Vasicek, CIR, HJM) define implicitly or explicitly that the random motion of infinite number of forwards rates as diffusions driven by a finite number of independent Brownian motions. Although this class of models allows one to fit the current yield curve, it does not permit consistency with term structure innovation. To tackle this deficiency, an infinite-dimensional process which is driven by an infinite-dimensional source of randomness was proposed in order to provide a better mathematical framework for term structure model. In my talk, I will explain some of the latest development in this area.

**Stochastic Control with Partial Observatopm and Applications
to Finance
**

*By Professor Bernt Øksendal, University of Oslo, Norway*

In many situations one does not have complete information about the state or the parameters of the system one is trying to control optimally. For example, in the classical Merton problem of optimal consumption and portfolio in a Black-Scholes market, one may not have complete information about the mean relative growth rate of the stock price process. In this case, the problem is called a stochastic control problem with PARTIAL observation. It is possible to rewrite such a problem as a stochastic control problem with COMPLETE observation, but of a system described by a stochastic PARTIAL differential equation (SPDE) instead of an ordinary stochastic differential equation (SDE). This problem can, in turn, be studied using an SPDE generalization of the stochastic maximum principle. We explain how this works and illustrate the results by some examples.