Financial Mathematics and Applied Probability Seminars 2001 - 2002 All seminars

This paper develops a class of equilibrium asset pricing models in continuous-time Lucas (1978) exchange economy. It distinguishes the existing literature into two parts: (a) the representative agent's preference over the life-time consumption programs is assumed to be represented by the so-called intertemporal recursive utility function formulated by Ma (2000), which generalizes that of Duffie and Epstein (1992a) by allowing non-expected utility specifications; (b) the uncertainty is Markovian with state variables driven by a Levy jump process that contains both Brownian and Poisson uncertainties. By introducing the so-called pseudo-state process, which is uniquely determined by the original state process and the representative agent's utility function, we were able to express the equilibrium price of a security as the expected net present value of its dividend streams under the original probability measure. This resolves the technical problem in identifying the risky neutral probability measure(s) in incomplete market economies as is the case in the presence of Levy jumps. The existence of equilibrium security prices as solutions to the Euler equation for the agent's optimal consumption and portfolio choices is proved. For a particular parameterisation of the economy, a closed-form formula for the European call options, as well as for other derivative securities on the aggregate equity, is derived and analyzed. Two applications are carried out. The first application involves a derivation of an equilibrium option pricing formula, and a formula for pricing all other derivative securities on the aggregate equity. This is done for a particular parametric specifications of the agent's utility function and state variables. The set-up is the same as in Merton (1976) and Naik and Lee (1990) except that we allow general homothetic recursive utility function, and we do not restrict the jump sizes to follow a log-normal distribution as assumed by the others. In such a set-up, the utility specifications become relevant for option pricing, as well as for pricing other derivative securities. The relevant option pricing formula is expressed in terms of the Laplace inverse transformation of a complex function Phi. Preference parameters and other aggregate factors of the economy affecting the option prices are explicitly reflected through the Phi-function. The derived option pricing formula is found to be attractive not only because of its mathematical simplicity, but also because of its generality. For example, it is shown that this formula generalises those of Black-Scholes, Naik-Lee, Cox-Ross and Merton in two ways: First, the magnitude of the jump may follow any distribution with finite moments; second, the utility function is recursive but is not necessarily an intertemporal additive von-Neumann Morgenstern utility function. Nevertheless, for the special cases when the market becomes complete, the equilibrium conditions will lead to the same formulation for security prices as what is implied by imposing purely the no-arbitrage conditions. As another application of the derived equilibrium asset pricing model, we re-examine Kocherlakota's (1990) observational equivalence between recursive utility functions and expected discounted utility functions in a continuous-time setting. Recursive utility function is of special interest because, in contrast to the intertemporal additive von Neumann Morgenstern utility, the recursive utility function of Epstein and Zin (1989) permits a degree of separation between risk aversion and intertemporal substitution, and agent's attitudes toward the timing of uncertainty resolution can be also captured by the recursive utility formulation. We ask: (1) how utility specifications may affect the equilibrium behaviour of security prices; and, conversely, (2) how equilibrium security prices can convey information about representative agent's preferences. For example, can we distinguish between the nonexpected recursive utility function of Epstein and Zin (1989) and the traditional expected additive utility function from the security prices? Progress has been made in the research towards some deeper understanding of these issues. Following Duffie and Epstein (1992b), Kocherlakota's observation equivalence between expected and non-expected utility functions prevails in the pure Brownian economy even when it is not i.i.d. This is because the M-function, which characterises the risk attitude towards uncertainty in continuous time, does not enter into their formulation of recursive utility. This is equivalent to assume expected utility certainty equivalent. This paper considers mixed Poisson-Brownian uncertainty. It is shown that the betweenness recursive utility function and expected utility functions are observational distinguishable even in an i.i.d. economy. Similar to Ma (1998 b) in discrete time, it is found that the prices of European call options contain the most relevant information on agent's utility function in continuous time. The observational non-equivalence finding reported here is based on the closed-form formula for pricing European call options on the aggregate equity in the presence of Levy jumps as mentioned above. Tuesday 30 October, 5:30 pm Dr Stephen Jewson Risk Management Solutions Limited Weather Derivative Pricing Abstract: This talk will introduce weather derivatives and discuss methods by which they can be priced, both now, and in a future, more liquid market. This talk will introduce weather derivatives and discuss methods by which they can be priced, both now, and in a future, more liquid market. The current weather derivative market is highly illiquid, and actuarial methods are used for pricing. A review is given of the various techniques currently used in industry, and some of their characteristics, including the roles of data cleaning, burn and simulation methods, portfolios and the incorporation of weather and seasonal forecasts. There will also be some discussion of what measures should be used for risk management in a trading organisation. There will also be some speculation about how the market might develop, and various no-arbitrage based models will be presented based on different assumptions about the future state of the market and the types of forecasts being used. The content of the talk will be a combination of actuarial science, meteorology, timeseries analysis and no-arbitrage pricing. Tuesday 13 November, 5:30 pm Professor Ragnar Norberg Department of Statistics, London School of Economics Dynamic Greeks Abstract: The sensitivity of a derivative price to changes in the parameters is given by The sensitivity of a derivative price to changes in the parameters is given by its derivatives w.r.t. the parameters the so-called Greeks. The Greeks are easily obtained when the price is a closed form expression. In any case we may determine the Greeks as solutions to differential equations derived from the differential equation of the price function by simply differentiating it w.r.t. the parameters. Existence of Greeks is the hard part. The idea extends to other dynamical entities. Some examples with numerical illustrations are given. Tuesday 20 November, 5:30 pm Dr Marek Musiela BNP Paribas A Perspective on the Pricing and Risk Management in Incomplete Markets Abstract: Incompleteness can be introduced to any complete model in a number of different ways. Typical examples are models with jumps or stochastic volatility. Alternatively one can introduce, as in Davis (2000), an additional source of uncertainty which is correlated with the source of uncertainty used in the construction of a nested complete model. I will use this model set up in my presentation. There are essentially two ways of dealing with the pricing and risk management issues in incomplete markets. As there are an infinite number of equivalent martingale measures one approach is to choose one of them by some optimality criteria. For example, one could take a martingale measure which is the closest to the historical measure, with respect to a certain distance. Once the choice is made this measure is used to compute the expectation of the discounted payoff in order to identify the price. There are numerous advantages and disadvantages of this approach. One of the fundamental disadvantages is the price dependence on the choice of numeraire. Similarity with the pricing in complete markets in many other aspects are clear advantages. Alternative approach is to derive the concept of price and risk management from the idea of optimal investment. I will follow this approach in my talk. I refer to Davis (2000) for the first results in this direction as well as for the examples of situations in which such models can be applied in practice and to Musiela and Zariphopoulou (2001) for further analysis of this model in which one trades one risky and one riskless asset but options are written on the non-traded index. The main aim of the talk is to analyse properties on the pricing and of the associated risk management of such an approach. In particular, the following issues will be discussed: 1. Consistency with the static no-arbitrage. 2. Independence on the choice of numeraire. 3. Marking to market versus marking to a portfolio. Incompleteness can be introduced to any complete model in a number of different ways. Typical examples are models with jumps or stochastic volatility. Alternatively one can introduce, as in Davis (2000), an additional source of uncertainty which is correlated with the source of uncertainty used in the construction of a nested complete model. I will use this model set up in my presentation. There are essentially two ways of dealing with the pricing and risk management issues in incomplete markets. As there are an infinite number of equivalent martingale measures one approach is to choose one of them by some optimality criteria. For example, one could take a martingale measure which is the closest to the historical measure, with respect to a certain distance. Once the choice is made this measure is used to compute the expectation of the discounted payoff in order to identify the price. There are numerous advantages and disadvantages of this approach. One of the fundamental disadvantages is the price dependence on the choice of numeraire. Similarity with the pricing in complete markets in many other aspects are clear advantages. Alternative approach is to derive the concept of price and risk management from the idea of optimal investment. I will follow this approach in my talk. I refer to Davis (2000) for the first results in this direction as well as for the examples of situations in which such models can be applied in practice and to Musiela and Zariphopoulou (2001) for further analysis of this model in which one trades one risky and one riskless asset but options are written on the non-traded index. The main aim of the talk is to analyse properties on the pricing and of the associated risk management of such an approach. In particular, the following issues will be discussed: 1. Consistency with the static no-arbitrage. 2. Independence on the choice of numeraire. 3. Marking to market versus marking to a portfolio. Friday 30 November, 5:45 pm Professor Dilip Madan Robert H. Smith School of Business, University of Maryland at College Park Pricing and Hedging in Incomplete Markets Abstract: We present a new approach for positioning, pricing and hedging in incomplete markets, which bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether to undertake a particular position involves specifying a set of probability measures and associated floors which expected payoffs must exceed in order that the hedged and financed investment be acceptable. By assuming that the liquid assets are priced so that each portfolio of them has negative expected return under at least one measure, we derive a counterpart to the first fundamental theorem of asset pricing. We also derive a counterpart to the second fundamental theorem, which leads to unique derivative We present a new approach for positioning, pricing and hedging in incomplete markets, which bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether to undertake a particular position involves specifying a set of probability measures and associated floors which expected payoffs must exceed in order that the hedged and financed investment be acceptable. By assuming that the liquid assets are priced so that each portfolio of them has negative expected return under at least one measure, we derive a counterpart to the first fundamental theorem of asset pricing. We also derive a counterpart to the second fundamental theorem, which leads to unique derivative security pricing and hedging even though markets are incomplete. For products that are not spanned by the liquid assets of the economy, we show how our methodology provides more realistic bid-ask spreads. Tuesday 11 December, 5:30 pm Professor Terry Lyons The Mathematical Institute, University of Oxford Integration on High Dimensional Path Spaces Abstract: It is well known that there is a mathematical equivalence between "solving" parabolic partial differential equations and "the integration" of certain functionals on Wiener Space. Monte Carlo simulation of stochastic differential equations is a naive approach based on this underlying principle. It is well known that there is a mathematical equivalence between "solving" parabolic partial differential equations and "the integration" of certain functionals on Wiener Space. Monte Carlo simulation of stochastic differential equations is a naive approach based on this underlying principle. In one dimension, it is well known that Gaussian quadrature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener Space. In the process we develop high order numerical schemes valid for high dimensional SDEs and semi-elliptic PDEs. Tuesday 29 January, 5:30 pm Dr David Hobson Department of Mathematical Sciences, University of Bath Coupling and Option Price Comparisons in a Jump Diffusion Model Abstract: In this paper we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities comparing the prices of contingent claims under various martingale measures which have been proposed in the literature as candidate pricing measures. In this paper we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities comparing the prices of contingent claims under various martingale measures which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities. Tuesday 26 February, 5:30 pm Dr Chris Brooks ISMA Centre for Education for Financial Markets, University of Reading Autoregressive Conditional Kurtosis Abstract: This paper proposes a new model for autoregressive conditional heteroscedasticity and kurtosis. Via a time-varying degrees of freedom parameter, the conditional variance and conditional kurtosis are permitted to evolve separately. The model uses only the standard Student's t density and consequently can be estimated simply using maximum likelihood. The method is applied to a set of four daily financial asset return series comprising US and UK stocks and bonds, and significant evidence in favour of the presence of autoregressive conditional kurtosis is observed. Various extensions to the basic model are examined, and show that conditional kurtosis appears to be positively but not significantly related to returns, and that the response of kurtosis to good and bad news is not significantly asymmetric. A This paper proposes a new model for autoregressive conditional heteroscedasticity and kurtosis. Via a time-varying degrees of freedom parameter, the conditional variance and conditional kurtosis are permitted to evolve separately. The model uses only the standard Student's t density and consequently can be estimated simply using maximum likelihood. The method is applied to a set of four daily financial asset return series comprising US and UK stocks and bonds, and significant evidence in favour of the presence of autoregressive conditional kurtosis is observed. Various extensions to the basic model are examined, and show that conditional kurtosis appears to be positively but not significantly related to returns, and that the response of kurtosis to good and bad news is not significantly asymmetric. A multivariate model for conditional heteroscedasticity and conditional kurtosis, which can provide useful information on the co-movements between the higher moments of series, is also proposed. Tuesday 5 March, 5:30 pm Professor Phelim Boyle University of Waterloo, Canada Asset Allocation using Quasi Monte Carlo Abstract: The asset allocation decision is an important one for investment managers of pension plans, mutual funds and other financial institutions. In recent years many institutions have increasingly passed this decision down to the individual investor. We examine the extent to which modern finance provides useful tools to assist the investor in this decision. Some recent advances which lead to new approaches to this problem use Monte Carlo methods. In particular we will discuss how one of these approaches can be implemented and discuss some preliminary work that uses a quasi Monte Carlo approach. The asset allocation decision is an important one for investment managers of pension plans, mutual funds and other financial institutions. In recent years many institutions have increasingly passed this decision down to the individual investor. We examine the extent to which modern finance provides useful tools to assist the investor in this decision. Some recent advances which lead to new approaches to this problem use Monte Carlo methods. In particular we will discuss how one of these approaches can be implemented and discuss some preliminary work that uses a quasi Monte Carlo approach. Tuesday 12 March, 5:30 pm Dr Sam Howison The Mathematical Institute, University of Oxford Asymptotic Techniques in Derivatives Pricing Abstract: This talk will be concerned with a number of uses of asymptotic techniques in derivatives pricing, including transactions costs analysis of vega hedging and pricing of volatility derivatives. This talk will be concerned with a number of uses of asymptotic techniques in derivatives pricing, including transactions costs analysis of vega hedging and pricing of volatility derivatives. Tuesday 19 March, 5:30 pm Dr Klaus Toft Goldman Sachs How Firms should Hedge Abstract: Substantial academic research has explained why firms should hedge, but little work has addressed how firms should hedge. We assume that firms face costly states of nature and derive optimal hedging strategies using vanilla derivatives (e.g., forwards and options) and custom "exotic" derivative contracts for a valuemaximizing firm that faces both hedgable (price) and unhedgable (quantity) risks. Optimal hedges depend critically on price and quantity volatilities, the correlation between price and quantity, and profit margin. A close relationship exists between the optimal number of forward contracts and the optimal custom hedge: At the forward price of the traded good, the optimal forward hedge and the optimal exotic hedge have identical "deltas". At prices different from the forward price, the exotic contract fine-tunes the firm's exposure by including a non-linear payoff component. We also determine the benefits from choosing customized exotic derivatives over vanilla contracts for different types of firms. Customized exotic derivatives are typically better than vanilla contracts when correlations between prices and quantities are large in magnitude and when quantity risks are substantially greater than price risks. Substantial academic research has explained why firms should hedge, but little work has addressed how firms should hedge. We assume that firms face costly states of nature and derive optimal hedging strategies using vanilla derivatives (e.g., forwards and options) and custom "exotic" derivative contracts for a valuemaximizing firm that faces both hedgable (price) and unhedgable (quantity) risks. Optimal hedges depend critically on price and quantity volatilities, the correlation between price and quantity, and profit margin. A close relationship exists between the optimal number of forward contracts and the optimal custom hedge: At the forward price of the traded good, the optimal forward hedge and the optimal exotic hedge have identical "deltas". At prices different from the forward price, the exotic contract fine-tunes the firm's exposure by including a non-linear payoff component. We also determine the benefits from choosing customized exotic derivatives over vanilla contracts for different types of firms. Customized exotic derivatives are typically better than vanilla contracts when correlations between prices and quantities are large in magnitude and when quantity risks are substantially greater than price risks. Tuesday 26 March, 5:30 pm Dr Thomas Knudsen Abbey National Calibration and Vega of Exotic Derivative Pricing Models Abstract: Most interest rate models and several models for pricing equity and FX options postulate a stochastic evolution of the state variables where the equations for the evolution are parametrised by a number of unobservable parameters. These parameters are then determined by calibrating the model to a set of liquid market instruments. Specific models include HJM/BGM and Hull-White for interest rate derivatives and the Heston model in equity/FX. Traditionally the vega of these models is the price sensitivity with respect to certain of these parameters. This vega is standard output from the models. However, these model vegas do not depend on the calibration. So the way market moves are transformed into model moves does not influence the model vega. This suggests that simply hedging model vega is insufficient. Furthermore, calculation of Value at Risk (VaR) requires time series of the (unobservable) parameters. Hence both for hedging and VaR, it is desirable to be able to calculate the sensitivity of the model price with respect to the market instruments. This seminar shows how it is possible to transform model vega into market vega and further explores the properties of the different ways of hedging vega. Most interest rate models and several models for pricing equity and FX options postulate a stochastic evolution of the state variables where the equations for the evolution are parametrised by a number of unobservable parameters. These parameters are then determined by calibrating the model to a set of liquid market instruments. Specific models include HJM/BGM and Hull-White for interest rate derivatives and the Heston model in equity/FX. Traditionally the vega of these models is the price sensitivity with respect to certain of these parameters. This vega is standard output from the models. However, these model vegas do not depend on the calibration. So the way market moves are transformed into model moves does not influence the model vega. This suggests that simply hedging model vega is insufficient. Furthermore, calculation of Value at Risk (VaR) requires time series of the (unobservable) parameters. Hence both for hedging and VaR, it is desirable to be able to calculate the sensitivity of the model price with respect to the market instruments. This seminar shows how it is possible to transform model vega into market vega and further explores the properties of the different ways of hedging vega. Tuesday 14 May, 5:30 pm Dr Harry Zheng Department of Mathematics, Imperial College Risk Minimizing Hedging of Liabilities Abstract: Duration is a useful tool in hedging liabilities. Macaulay duration is a wellknown one but is criticized for its stringent assumptions. Many other durations have been suggested to accommodate more general type of shifts. Unfortunately, hedging is achieved only against assumed type of rate change. In this talk we introduce a risk (maximum loss) minimizing measure which is applicable to any pattern of changes. We discuss different hedging strategies and compare their performances with US Tbond and STRIPS data. We use a multi-stage linear programming model to select the optimal portfolio that minimizes the hedging loss and transaction cost. We also extend the measure to credit-risky and option-embedded bonds, the uncertainty of timing and amount of cash flows of these securities is characterized by risk-neutral survival probabilities derived from the market. Duration is a useful tool in hedging liabilities. Macaulay duration is a wellknown one but is criticized for its stringent assumptions. Many other durations have been suggested to accommodate more general type of shifts. Unfortunately, hedging is achieved only against assumed type of rate change. In this talk we introduce a risk (maximum loss) minimizing measure which is applicable to any pattern of changes. We discuss different hedging strategies and compare their performances with US Tbond and STRIPS data. We use a multi-stage linear programming model to select the optimal portfolio that minimizes the hedging loss and transaction cost. We also extend the measure to credit-risky and option-embedded bonds, the uncertainty of timing and amount of cash flows of these securities is characterized by risk-neutral survival probabilities derived from the market. Tuesday 21 May, 5:30 pm Professor Stewart Hodges FORC, University of Warwick The Relation between Implied and Realised Probability Density Functions Abstract: A number of financial regulators [see Neuhaus (1995), Bahra (1996, 1997), McManus (1999) and Shiratsuka (2001)] have suggested that risk neutral densities (RND) associated with options markets could provide useful indicators of future market turbulence. Critical to this assumption is that such RNDs should provide an unbiased forecast of realised probability density functions. To date, this assumption has not been fully examined. A number of financial regulators [see Neuhaus (1995), Bahra (1996, 1997), McManus (1999) and Shiratsuka (2001)] have suggested that risk neutral densities (RND) associated with options markets could provide useful indicators of future market turbulence. Critical to this assumption is that such RNDs should provide an unbiased forecast of realised probability density functions. To date, this assumption has not been fully examined. In this research, we test the ability of RNDs for options on the S&P 500 and the British Pound / US Dollar to predict future probability densities. We consider four approaches to estimate the RNDs, which are consistent with approaches proposed and used by financial regulators. We also provide a number of new testing procedures to assess the efficiency and unbiasness of the forecasts. These tests provide more power than the usual Komolgorov/Smirnov tests. Using non-overlapping quarterly data from the mid 1980s to 2001, we find that we can reject the hypothesis that the RNDs for both the S&P 500 and British Pounds are unbiased forecasts. Even with a limited number of observations, the tests are powerful enough to allow rejection. However, when an adjustment for the risk premium is made, the results become more controversial. Depending on the nature of the adjustment, we can or cannot reject the accuracy of the adjusted implied densities. When a power utility adjustment is made, like Bliss and Panigirtzolou (2001), we are unable to reject the hypothesis that RNDs are unbiased forecasts of realised densities. Overall, our results tend to support the conclusions of Shiratsuka (2001), that unadjusted RNDs should not be used by financial regulators as financial indicators, and that such use could prove counterproductive; actually increasing future market turbulence rather than alleviating it. Moreover, we observe that return distributions simulated on the basis of historical volatility processes of some GARCH-type exhibit better forecasting performance than unadjusted implied RNDs. These findings seem to suggest that also in relative terms (unadjusted) implied densities do not constitute an efficient forecast of realised probability density. Tuesday 28 May, 5:30 pm Professor Thaleia Zariphopoulou University of Texas at Austin Indifference Prices and Valuation in Incomplete Markets Abstract: A new pricing algorithm will be presented for an incomplete market framework. The valuation method incorporates risk preferences and provides a coherent way to identify and price the risk components emerging from the market frictions. Results on the risk monitoring policies and the related pricing measures will be presented. A new pricing algorithm will be presented for an incomplete market framework. The valuation method incorporates risk preferences and provides a coherent way to identify and price the risk components emerging from the market frictions. Results on the risk monitoring policies and the related pricing measures will be presented.