Download Financial Mathematics and Applied Probability Seminars 2001-2002

Financial Mathematics and Applied Probability Seminars 2001-2002
All seminars take place at Lecture Theatre 2C, King's College London, The Strand,
London WC2R 2LS.
Tuesday 16 Dr Chenghu Ma
October, 5:30 Department of Accounting, Finance & Management, University of Essex
pm
Preferences, Levy Jumps and Option Pricing
Abstract: 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 Dr Stephen Jewson
October, 5:30 Risk Management Solutions Limited
pm
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.
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
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.
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
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.
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 Dr David Hobson
January, 5:30 Department of Mathematical Sciences, University of Bath
pm
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.
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 Dr Chris Brooks
February, 5:30 ISMA Centre for Education for Financial Markets, University of Reading
pm
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
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 Professor Phelim Boyle
March, 5:30 University of Waterloo, Canada
pm
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.
Tuesday 12 Dr Sam Howison
March, 5:30 The Mathematical Institute, University of Oxford
pm
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.
Tuesday 19 Dr Klaus Toft
March, 5:30 Goldman Sachs
pm
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.
Tuesday 26 Dr Thomas Knudsen
March, 5:30 Abbey National
pm
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.
Tuesday 14 Dr Harry Zheng
May, 5:30 pm 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.
Tuesday 21 Professor Stewart Hodges
May, 5:30 pm 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.
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 Professor Thaleia Zariphopoulou
May, 5:30 pm 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.
Tuesday 9 Professor Stathis Tompaidis
July, 5:30 pm University of Texas at Austin
Market Imperfections, Investment Optionality and Default Spreads
Abstract: This paper develops a structural model that determines default spreads on
risky debt. In contrast to previous research, the value of the debt's collateral is
endogenously determined by the borrower's investment choice, as well as by a market
demand variable that has permanent as well as temporary components. The model
also considers market imperfections that limit the borrower's ability to contract to
undertake the value-maximizing investment choice, and which may in addition limit
the borrower's ability to raise external capital. The model is calibrated with data on
office buildings and commercial mortgages, and based on our calibration, we present
numerical simulations that quantify the extent to which investment flexibility,
incentive problems and credit constraints affect default spreads.
Monday 15 Dr Matheus Grasselli
July, 5:30 pm Department of Mathematics, McMaster University, Ontario
Optimal Investment in Incomplete Markets (When the Wealth may Become
Negative)