Liberati, N. B. and Platen, E., "On the Efficiency of Simplified Weak Taylor Schemes for Monte Carlo Simulation in Finance", January 2004.
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Abstract:

The purpose of this paper is to study the efficiency of simplified weak schemes for stochastic differential equations. We present a numerical comparison between weak Taylor schemes and their simplified versions. In the simplified schemes discrete random variables, instead of Gaussian ones, are generated to approximate multiple stochastic integrals. We show that an implementation of simplified schemes based on random bits generators significantly increases the computational speed. The efficiency of the proposed schemes is demonstrated.

Borovkov, K. and Novikov, A., "Explicit Bounds for Approximation Rates for Boundary Crossing Probabilities for the Wiener Process", January 2004.
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Abstract:

We give explicit upper bounds for convergence rates when approximating (both one- and two-sided general curvlinear) boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries (of simpler form for which computing the possibility is feasible). In particular, we generalize and improve results obtained by Potzelberger and Wang [13] for the case when approximating boundaries are piecewise linear. Applications to barrier option pricing are discussed as well.

Liptser, R. and Novikov, A., "On Tail Distributions of Supremum and Quadratic Variation of Local Martingales", January 2004.
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Abstract:

We extend some known results on a relation between the distribution tails of the continuous local martingale supremum and its quadratic variation to the case of locally square integrable martingale with bounded jumps. The predictable and optional quadratic variations are involved in the main result.

Chiarella, C. and Ziogas, A., "McKean's Methods applied to American Call Options on Jump-Diffusion Processes", February 2004.
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Abstract:

In this paper we derive the implicit integral equation for the price of an American call option in the case where the underlying asset follows a jump-diffusion process. We extend McKean's incomplete Fourier transform approach to solve the free boundary problem under Merton's framework, with the distribution for the jump size remaining unspecified. We show how our results are consistent with those of Gukhal (2001), who derived the same integral equation using the Geske-Johnson discretisation approach. The paper also derives some results concerning the limit for the free boundary at expiry, and presents an iterative numerical algorithm for solving the linked integral equation system for the American call's price and early exercise boundary. This scheme is applied to Merton's jump-diffusion model, where the jumps are log-normally distributed.

Chiarella, C., Kucera, A. and Ziogas, A., "A Survey of the Integral Representation of American Option Prices", October 2005.
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Abstract:

This paper surveys some of the literature on American option pricing, in particular the representations of McKean (1965), Kim (1990), Jamshidian (1992) and Carr, Jarrow & Myneni (1992). In particular the paper seeks to demonstrate that the approach regarding the problem as a free boundary value problem, and its solution via incomplete Fourier transforms, is the most robust for further theoretical and applied developments involving more complex payoff structures, and higher dimensional problems such as multi-asset American options. Some comparison of different numerical solution methods is also provided.

Colwell, D., El-Hassan, N. and Kwon, O., "Hedging Diffusion Processes by Local Risk-Minimisation with Applications to Index Tracking", February 2004.
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Abstract:

The solution to the problem of hedging contingent claims by local risk-minimisation has been considered in detail in Follmer and Sondermann (1986), Follmer and Schweizer (1991) and Schweizer (1991). However, given a stochastic process Xt and tau1 <> tau2, the strategy that is locally risk-minimising for Xtau1 is in general not locally risk-minimising for Xtau2. In the case of diffusion processes, this paper considers the problem of determining a strategy that is simultaneously locally risk-minimising for Xtau for all tau. That is, a strategy that is locally risk-minimising for the entire process Xt. The necessary and sufficient conditions under which this is possible are obtained, and applied to the problem of index tracking. In particular, a close connection between the local risk-minimising and the tracking error variance minimising strategies for index tracking is established, and leads to a simple criterion for the selection of optimal set of assets from which to form a tracker portfolio, as well as a value-at-risk type measure for the set of assets used.

Hamada, M. amd Valdez, E., "CAPM and Option Pricing with Elliptical Disbributions", February 2004.
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Abstract:

In this paper, we offer an alternative proof of the Capital Asset Pricing Model when the returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption in modelling asset returns. The class of elliptical distributions,which includes the more familiar Normal distribution, provides flexibility in modelling the thickness of tails associated with the possibility that asset returns take extreme values with non-negligible probabilities. Within this framework, we prove a new version of Stein's lemma for elliptical distribution and use this result to derive the CAPM when returns are elliptical. We also derive a closed form solution of call option prices when the underlying is elliptically distributed. We use the probability distortion function approach based on the dual utility theory of choice under uncertainty.

Hall, A. D. and Hautsch, N., "A Continuous-Time Measurement of the Buy-Sell Pressure in a Limit Order Book Market", March 2004.
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Abstract:

In this paper, we investigate the buy and sell arrivl process in a limit order book market. Using an intensity framework allows to estimate the simultaneous buy and sell intensity and to derive a continuous-time measure for the buy-sell pressure in the market. Based on limit order book data from the Australian Stock Exchange (ASX), we show that the buy-sell pressure is particularly influenced by recent market and limit orders and the current depth in the ask and bid queue. We find evidence for the hypothesis that traders use order book information in order to infer from the price setting behavior of market participants. Furthermore, our results indicate that the buy-sell pressure is clearly predictable and is a significant determinant of trade-to-trade returns and volatility.

Chauveau, T. and Gatfaoui, H., "Pricing and Hedging Options in Incomplete Markets: Idiosyncratic Risk, Systematic Risk and Stochastic Volatility", April 2004.
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Abstract:

Starting from the European option valuation framework of Chauveau and Gatfaoui (2002), we establish the link with stochastic volatility models. And, we propose both a new vision and a general framework for valuing European options in the light of systematic and idiosyncratic risks affecting risky assets in the financial market. Therefore, we account for the well-known volatility smile in the light of the literature addressing the determinants of the smile effect among which stochastic volatility and market risk. We further discuss briefly the hedging of European options along with the local risk minimization principle. Specifically, we attempt to find a strategy which dominates the usual partial hedging technique often imposed by market's incompleteness.

Gatfaoui, H., "Idiosyncratic Risk, Systematic Risk and Stochastic Volatility: An Implementation of Merton's Credit Risk Valuation", April 2004.

Abstract:

We extend the credit risk valuation framework introduced by Gatfaoui (2003) to stochastic volatility models. We state a general setting for valuing risky debt in the light of systematic risk and idiosyncratic risk, which are known to affect each risky asset in the financial market. The option nature of corporate debt allows then to account for the well-known volatility smile along with two documented determinants, namely stochastic volatility and market risk. Under some regularity conditions, we specify diffusion functionals leading to an asymptotically (relative to time) mean reverting volatility process. The behavior of such a specification is studied along with simulation tehniques since debt is valued via a call on the firm assets value. Specifically, our examination resorts to Monte Carlo accelerators to realize related simulations. First, we consider the evolution of stochastic volatility for given parameter values. Then, we assess its impact on both risky debt and the related credit spread.

Heath, D. and Platen, E., "Local Volatility Function Models under a Benchmark Approach", April 2004.
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Abstract:

This paper studies a class of one-factor local volatility function models for stock indices under a benckmark approach. It assumes that the dynamics for a large diversified index approximates that of the growth optimal portfolio. The pricing and hedging of derivatives under the benchmark approach does not require the existence of an equivalent risk neutral martingale measure. Fair prices for index derivatives when expressed in units of the index are martingales under the real world probability measure. The real world transitin densities for the index and the underlying local volatility function can be determined from a continuum of European call option prices. As specific examples a modification of the constant elasticity of variance model and a version of the minimal market model are discussed together with a smoothed local volatility function that fits a snapshot of S&P500 index options data.

Breymann, W., Kelly, L. and Platen, E., "Intraday Empirical Analysis and Modeling of Diversified World Stock Indices", May 2004.
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Abstract:

This paper proposes an approach to the intraday analysis of diversified world stock accumulation indices. The growth optimal portfolio (GOP) is used as reference unit or benchmark in a continuous financial market model. Diversified portfolios, covering the world stock market, are constructed and shown to approximate the GOP, providing the basis for a range of financial applications. The normalized GOP is modeled as a time transformed square root process of dimension four. Its dynamics are empirically verified for several world stock indices. Furthermore, the evolution of the transformed time is modeled as the integral over a rapidly evolving mean-reverting market activity process with deterministic volatility. The empirical findings suggest a rather simple and robust model for a world stock index that reflects the historical evolution, by using only a few readily observable parameters.

Chiarella, C., El-Hassan, N. and Kucera, A.,"Evaluation of Point Barrier Options in a Path Integral Framework", March 2005.
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Abstract:

The pricing of point barrier or discretely monitored barrier options is a difficult problem. In general, there is no known closed form solution for pricing such options. In this paper we develop a path integral approach to the evaluation of barrier options. This leads to a backward recursion functional equation linking the pricing functions at successive barrier points. We solve this functional equation by expanding the pricing functions in Fourier-Hermite series. The backward recursion functional equation then becomes the backward recurrence relation for the coefficients in the Fourier-Hermite expansion of the pricing functions. We thus obtain a very efficient and accurate method for generating the pricing function at any barrier point. We perform a number of numerical experiments with the method in order to gain some understanding of the nature of convergence. We present results for various volatility values and different numbers of basis functions in the Fourier-Hermite expansion. Comparisons will be given between pricing of point barriers in the path integral framework and by use of finite difference methods. Date: March 10, 2005.

Corrado, C. and Truong, C., "Forecasting Stock Index Volatility: The Incremental Information in the Intraday High-Low Price Range", June 2004.
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Abstract:

We compare the incremental information content of implied volatility and intraday high-low range volatility in the context of conditional volatilityforecasts for three major market indexes: the S&P 100, the S&P 500, and the Nasdaq 100. Evidence obtained from out-of-sample volatility forecasts indicates that neither implied volatility nor intraday high-low range volatility subsumes entirely the incremental information contained in the other. Our findings suggest that intraday high-low range volatility can usefully augment conditional volatility forecasts for these market indexes.

Heath, D. and Platen, E., "Understanding the Implied Volatility Surface for Options on a Diversified Index", June 2004.
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Abstract:

This paper describes a two-factor model for a diversifed index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model,which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts.

Platen, E., "Diversified Portfolios with Jumps in a Benchmark Framework", June 2004.
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Abstract:

This paper considers diversifed portfolios in a sequence of jump di®usion market models. Conditions for the approximation of the growth optimal portfolio (GOP) by diversi¯ed portfolios are provided. Under realistic assumptions, it is shown that diversi¯ed portfolios approximate the GOP without requiring any major model speci¯cations. This provides a basis for systematic use of diversi¯ed stock indices as proxies for the GOP in derivative pricing, risk management and portfolio optimization.

Miller, S. and Platen, E., "Two-Factor Model for Low Interest Rate Regimes", August 2004.
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Abstract:

This paper derives a two factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate proces that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is niether possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew.

Novikov, A. and Shiryaev, A., "On an Effective Solution of the Optimal Stopping Problem for Random Walks", August 2004.
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Abstract:

We find a solution of the optimal stopping problem for the case when a reward function is an integer function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, … , T} converges with an exponential rate as T approaches infinity to the limit under the assumption that jumps of the random walk are exponentially bounded.

Chiarella, C. and Nikitopoulos Sklibosios, C., "A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework", September 2004.
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Abstract:

This paper considers a class of term structure models that is a parameterisation of the Shirakawa (1991) extension of the Heath, Jarrow and Morton (1992) model to the case of jump-diffusions. We consider specific forward rate volatility structures that incorporate state dependent Wiener volatility functions and time dependent Poisson volatility functions. Within this framework, we discuss the Markovianisation issue, and obtain the corresponding affine term structure of interest rates. As a result we are able to obtain a broad tractable class of jump-diffusion term structure models. We relate our approach to the existing class of jump-diffusion term structure models whose starting point is a jump-diffusion process for the spot rate. In particular we obtain natural jump-diffusion versions of the Hull and White (1990, 1994) one-factor and two-factor models and the Ritchken and Sankarasubramanian (1995) model within the HJM framework. We also give some numerical simulations to gauge the effect of the jump-component on yield curves and the implications of various volatility specifications for the spot rate distributions.

Chiarella, C., He, X. and Hommes, C., "A Dynamic Analysis of Moving Average Rules", May 2005 (update).
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Abstract:

The use of various moving average (MA) rules remains popular with financial market practitioners. These rules have recently become the focus of a number empirical studies, but there have been very few studies of financial market models where some agents employ technical trading rules of the type used in practice. In this paper we propose a dynamic financial market model in which demand for traded assets has both a fundamentalist and a chartist component. The chartist demand is governed by the difference between current price and a (long-run) MA. Both types of traders are boundedly rational in the sense that, based on a fitness measure such as realized capital gains, traders switch from a strategy with low fitness to the one with high fitness. We characterize the stability and bifurcation properties of the underlying deterministic model via the reaction coefficient of the fundamentalists, the extrapolation rate of the chartists and the lag length used for the MA. By increasing the intensity of choice to switching strategies, we then examine various rational routes to randomness for different MA rules. The price dynamics of the moving average rule are also examined and one of our main findings is that an increase of the window length of the MA rule can destabilize an otherwise stable system, leading to more complicated, even chaotic behaviour. The analysis of the corresponding stochastic model is able to explain various market price phenomena, including temporary bubbles, sudden market crashes, price resistance and price switching between different leves.

Chiarella, C., Dieci, R. and Gardini, L., "Asset Price and Wealth Dynamics in a Financial Market with Heterogeneous Agents", October 2004.
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Abstract:

This paper considers a discrete-time model of a financial market with one risky asset and one risk-free asset, where the asset price and wealth dynamics are determined by the interaction of two groups of agents, fundamentalits and chartists. In each period each group allocates its wealth between the risky asset and the safe asset according to myopic expected utility maximization, but the two groups have heterogeneous beliefs about the price change over the next period: the chartists are trend extrapolators, while the fundamentalists expect that the price will return to the fundamental. We assume that investors have CRRA utility, so that their optimal demand for the risky asset depends on wealth. A market maker is assumed to adjust the price at the end of each trading period, on the basis of the excess demand and according to particular stabilization policies. The model results in a three-dimensional nonlinear discrete-time dynamical system, with growing price and wealth processes, but it is reduced to a stationary system in terms of asset returns and wealth shares of the two groups. It is shown that the long-run market dynamics are highly dependent on the parameters which characterize agents' behavior (in particular the risk aversion coefficient and the chartist extrapolation parameter) as well as on the initial condition (in particular the initial wealth shares of fundamentalists and chartists). It is also shown that the for wide ranges of the parameters a (locally) stable fundamental steady state may coexist with a stable "nonfundamental" steady state, where price grows faster than the fundamental and only chartists survive in the long-run. In such cases, the role played by the initial condition is analysed by means of numerical investigations and graphical representation of the basins of attraction. Other dynamic scenarios include limit cycles, periodic orbits or more complex attractors, where in general both types of agents survive in the long run, with time varying wealth fractions.

Chiarella, C., Schlögl, E. and Nikitopoulos-Sklibosios, C., "A Markovian Defaultable Term Structure Model with State Dependent Volatilities", October 2004.
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Abstract:

The defaultable forward rate is modeled as a jump diffusion process within the Schonbucher (2000, 2003) general Heath, jarrow and Morton (1992) framework where jumps in the defaultable term structure f^d(t, T) cause jumps and defaults to the defaultable bond prices P^d(t, T). Within this framework, we investigate an appropriate forward rate volatility structure that results in Markovian defaultable spot rate dynamics. In particular, we consider state dependent Wiener volatility functions and time dependent Poisson volatility functions. The corresponding term structures of interest rates are expressed as finite dimensional affine realisations in terms of benchmark defaultable forward rates. In addition, we extend this model to incorporate stochastic spreads by allowing jump intensities to follow a square-root diffusion process. In that case the dynamics become non-Markovian and to restore path independence we propose either an approximate Markovian scheme or, alternatively, constant Poisson volatility functions. We also conduct some numerical simulations to gauge the effect of the stochastic intensity and the distributional implications of various volatility specifications.

He, X. and Westerhoff, F. H., "Commodity Markets, Price Limiters and Speculative Price Dynamics", October 2004.
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Abstract:

We develop a behavioral commodity market model with consumers, producers and heterogeneous speculators to characterize the nature of commodity price fluctuations and to explore the efectiveness of price stabilization schemes. Within our model, nonlinear interactions between market participants can create either bull or bear markets, or irregular price fluctuations between bulland bear markets. Both the imposition of a bottoming price level (to support producers) or a topping price level (to protect consumers) can reduce market price volatility. However, simple policy rules, such as price limiters, may have unexpected consequences in a complex environment: a minimum price level decreases the average price while a maximum price limit increases the average price. In addition, price limiters influence the price dynamics in an intricate way and may cause volatility clustering.

Bohl, M. T. and Siklos, P., "Empirical Evidence on Feedback Trading in Mature and Emerging Stock Markets", October 2004.
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Abstract:

We investigate the hypothesis that some participants in mature and emerging capital markets engage in feedback trading. The analysis is based on the Shiller-Sentana-Wadhwani noise trader model. It has the attractive property that it yields testable implications about the presence of positive and negative feedback traders in stock markets. This theoretical framework, together with an asymmetric GARCH-type model, allows us to draw conclusions about whether differences exist between mature and emerging capital markets in terms of the degree of feedback trading. The empirical results show that positive and negative feedback trading strategies exist in both types of markets but are more pronounced in emerging stock markets than in their mature counterparts. Hence, non-fundamental trading strategies seems to play a more important role in emerging relative to mature stock markets.

Platen, E., "A Benchmark Approach to Finance", October 2004.
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Abstract:

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling and risk measurement. It is based on the natural assumption that investors prefer more or less, in the sense that the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account of his or her home currency. If the market portfolio is diversified or monetary authorities aim to maximize the growth rates of the portfolios of their market participants through corresponding interest policies, then the market portfolio is the growth optimal portfolio (GOP). In this setup the capital asset pricing model follows without the use of expected utility functions or equilibrium assumptions. The expected increase of the discounted value of GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real world probability measure as pricing measure. An equivalent risk neutral martingale measure does not exist under the derived minimal market model.

Christensen, M. and Platen, E., A General Benchmark Model for Stochastic Jump Sizes, November 2004.

Abstract:

This paper extends the benchmark framework of Platen (2002) by introducing a sequence of incomplete markets, having uncertainty driven by a Wiener process and a marked point process. By introducing an idealized market, in which all relevant economical variables are observed, but may not all be traded, a generalized growth optimal portfolio (GOP) is obtained and calculated explicitly. The problem of determining the GOP is solved in a general setting which extends existing treatments and provides a clear link to the market prices of risk.  The connection between traded securities, arbitrage and market incompleteness is analyzed. This provides a framework for analyzing the degree of incompleteness associated with jump processes, a problem well-known from insurance and credit risk modeling. By staying under the empirical measure, the resulting benchmark model has potential advantages for various applications in finance and insurance.

Platen, E., West, J. and Breymann, W., "An Intraday Empirical Analysis of Electricity Price Behaviour", November 2004
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Abstract:

This paper proposes an approach to the intraday analysis of the dynamics of electricity prices.  The Growth Optimal Portfolio (GOP) is used as a reference unit in a continuous financial electricity price model.  A diversified global portfolio in the form of a market capitalisation weighted index aproximates the GOP.  The GOP, measured in units of electricity, is normalised and then modeled as a time transformed square root process of dimension four.  The dynamics of the resulting process is empirically verified.  Intraday spot electricity prices from the US and Australian markets are used for this analysis.  The empirical findings identify a simple but realistic model for examining the volatile behavious of electricity prices.  The proposed model reflects the historical price evolution reasonably well by using a only a few robust but readily observable parameters.  The evolution of the tranformed times is modeled via a rapidly evolving market activity.  A periodic, ergodic process with deterministic volatility is used to model market activity.

Chiarella, C., He, X. and Wang, D., "A Behavioural Asset Pricing Model with a Time-Varying Second Moment", November 2004 (Updated February 2005)
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Abstract:

We develop a simple behavioural asset pricing model with fundamentalists and chartists to study price behaviour in financial markets when chartists estimate both conditional mean and variance by using a weighted averaging process. Through a stability, bifurcation, and normal form analysis, the market impact of the weighting process is examined. It is found that the weighting process leads to different price dynamics when the fundamental price becomes unstable, depending on whether the chartists act as either trend followers or contrarians. It is also found that a time varying second moment of the chartists impacts differently on the stability of the bifurcated price dynamics, but has no impact on the stability of the fundamental price. Near the flip bifurcation boundary, the bifurcated period-two price dynamics are stable for all time varying second moments, but near the Hopf bifurcation, the bifurcated (quasi)periodic cycle is stable (unstable) when the time varying second moment value is high (low). Different routes to complicated price dynamics are also observed. The analysis provides an analytical foundation for the statistical analysis ofthe corresponding stochastic version of this type of behavioural model.

Chiarella, C., He, X. and Wang, D., "Statistical Properties of a Heterogeneous Asset Price Model with Time-Varying Second Moment", November 2004
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Abstract:

Stability and bifurcation analysis of deterministic systems has been widely used in modeling financial markets. However, the impact of such dynamic phenomena on various statistical properties of the corresponding stochastic model, including skewness and excess kurtosis, various autocorrelation (AC) patterns of under and over reactions, and volatility clustering characterised by the long-range dependence of ACs, is not clear and has been very little studied. This paper aims to study this issue. Through a simple behavioural asset pricing model with fundamentalists and chartists, we examine the statistical properties of the model and their connection to the dynamics of the underlying deterministic model. In particular, our analysis leads to some insights into the type of mechanism that may be generating some of the stylised facts, such as fat tails, skewness, high kurtosis and long memory, observed in high frequency financial data.

Platen, E., "Capital Asset Pricing for Markets with Intensity Based Jumps", December 2004
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Abstract:

This paper proposes a uni¯ed framework for portfolio optimiza- tion, derivative pricing, modeling and risk measurement in ¯nancial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that investors prefer more for less, in the sense that for two given portfolios with the same variance of its increments, the one with the higher expected increment is preferred. If one additionally assumes that the market together with its monetary authority acts to maximize the long term growth of the market portfolio, then this portfolio exhibits a very particular dynamics. In a market without jumps the resulting dynamics equals that of the growth optimal portfolio (GOP). Conditions are formulated under which the well-known capital asset pricing model is generalized for markets with intensity based jumps. Furthermore, the Markowitz e±cient frontier and the Sharpe ratio are recovered in this continuous time setting. In this paper the numeraire for derivative pricing is chosen to be the GOP. Primary security account prices, when expressed in units of the GOP, turn out to be supermartingales. In the proposed framework an equivalent risk neutral martingale measure need not exist. Fair derivative prices are obtained as conditional expectations of future payo® structures under the real world probability measure. The concept of fair pricing is shown to generalize the classical risk neutral and the actuarial net present value pricing methodologies.

To, T., "A Note on the Bias of using Futures Rates as a Proxy for the Instantaneous Forward Rate", December 2004
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Abstract:

The note shows that there is a non-negligible bias in using the futures rates as a proxy for the instantaneous forward rates in the estimation of forward rate models. It is therefore desirable to derive the evolution of observable rates, then use the distributional properties of this evolution to do the estimation. In a general case where these properties are hard to obtained, a filtering technique is required..