When a continuous-time diffusion is observed only at discrete
dates, not necessarily close together, the likelihood function of the
observations is in most cases not explicitly computable. Researchers have
relied on simulations of sample paths in between the observation points, or
numerical solutions of partial differential equations, to obtain estimates
of the function to be maximized. By contrast, we construct a sequence of
fully explicit functions which we show converge under very general
conditions, including non-ergodicity, to the true (but unknown) likelihood
function of the discretely-sampled diffusion. We document that the rate of
convergence of the sequence is extremely fast for a number of examples
relevant in finance. We then show that maximizing the sequence instead of
the true function results in an estimator which converges to the true
maximum-likelihood estimator and shares its asymptotic properties of
consistency, asymptotic normality and efficiency. Applications to the
valuation of derivative securities are also discussed.
A class of models for analysis of observational series is described. In the
context of finance the models have the interpretation ob being of the stochastic
volatility type. However, they are very generally applicable and turbulence is
another area of application. There are striking similarities, as well as
important differences, between the main stylised features of empirical data from
finance and turbulence (and there is now an emerging research field termed
'econophysics'). These features are discussed and related to models and data
stets on stocks, exchange rates, term sturctures of interest rates and
turbulent fluids. A discussion of (real or apparent) scaling phenomena in
finance and turbulence is also given.
Given a nonlinear forward rate model, we investigate when the (inherently
infinite dimensional) forward rate process evolves on a finite dimensional
submanifold in the space of forward rate curves. As an application we give
necessary and sufficient conditions for the existence of an underlying
finite dimensional factor model. In particular we give conditions for when
the short rate (in a forward rate model) is a markov process.
As an application we study when it is possible to of fit a a class of
short rate models to an arbitrary initial bond price curve.
When out of the money a convertible bond is much like an ordinary bond, and
must have a value that is consistent with the credit quality of the issuer.
On the other hand when it is in the money there is substantial conversion
option value and it is inconsistent with standard Black-Scholes pricing to
discount with any kind of credit spread. This leads to an apparent paradox,
which has only recently been studied in any systematic way, by Tsiveriotis
and Fernandez (J Fixed Income, 1998). In this paper we use a model that
explicitly includes possible default events, and then pricing can be done by
the usual no-arbitrage arguments. We will discuss the relation between the
three sources of volatility (stock price, interest rate and credit spread)
in the model.
This lecture will summarize some recent research on
the modeling of default times, and the valuation
of securities that are sensitive to credit risk,
such as credit derivatives and corporate or sovereign bonds.
One portion of the talk will focus on the implications of
incomplete information regarding the
assets and liabilities of the issuer. First
passage of the asset-liability ratio to
a default boundary will be used to illustrate
a model of corporate credit spreads that incorporates
incomplete accounting information.
Another portion of the talk will concentrate
on models of correlation across issuers in
default timing. The valuation of first-to-default swaps and
of collateralized loan obligations will
be used for illustration. Alternative computational methods
are considered.
Portions of this presentation are based on joint work with
David Lando and with Ken Singleton. Related
working papers can be downloaded at http://www.stanford.edu/~duffie/.
Empirical analysis of data from financial markets, such as stock prices,
interest rates or foreign exchange rates, shows that generalized hyperbolic
(GH) distributions allow a more realistic description of returns than the
classical normal distribution. GH distributions contain as subclasses
hyperbolic as well as normal inverse Gaussian (NIG) distributions. We use GH
distributions as the basic ingredient to model price processes. An option
pricing formula is derived, which generalizes that from the hyperbolic model
developed in Eberlein and Keller (1995). We compare the new approach with the
classical normal one from various points of view. General issues of risk
management are discussed, in particular we show the increase in accuracy of VaR
estimates. The new concept can also be used in interest rate theory, where term
structure models driven by general L\'evy processes are considered. This talk
is partly based on joint work with Karsten Prause and Sebastian Raible.
There are currently two paradigms for term structure modelling: modelling
the spot rate, and modelling the term structure of forward rates. Each
has advantages and disadvantages: For spot rate modelling the question of
model choice is unclear, while for most HJM models computations are
difficult. We present a new class of term structure models essentially as
general as either of the above and for which differences between models
are easy to understand and, for a class of interesting models,
computations are easy.
We consider a financial market with primitive assets and derivatives on
these primitive assets. The derivative assets are non-redundant in the
market, in the sense that the market is complete, only with
their existence. In such a framework, we derive an
equilibrium restriction on the admissible prices of derivative assets. The
equilibrium imposes a monotonicity principle that restricts the set of
probability measures that qualify as candidate equivalent martingale
measures. This restriction is preference free and applies whenever the
utility functions belong to the general class of Von-Neumann Morgenstern
functions. We provide numerical examples that demonstrate the
applicability of the restriction for the computation of option prices in a
discrete time as well as in a continuous time framework. We also provide
some interesting convergence results and some empirical validation of our
approach.
The talk is based on joint papers with G. Last and Ch. Stricker. We
consider a general semimartingale model of a currency market with
transaction costs. Our aim is to prove a hedging theorem describing the
set of initial endowments allowing to hedge a contingent claim in various
currencies by a self-financing portfolio. One of the major problem here is
a ``financially acceptable" definition of the class admissible strategies.
Our choice is are the strategies having value processes bounded from below
(in the sense of the partial ordering induced by the solvency cone) by the
price process times negative constants. The techniques used a version of
the bipolar theorem in $L^0$.
This paper describes the implementation of a simulation algorithm
for lognormal forward rate "market" models of the Brace, Gatarek and
Musiela (1997) and Jamshidian (1997) types. Each simulated rate is a
lognormal martingale under an equivalent martingale measure corresponding
to a different numeraire asset;these measures are linked by relationships
which follow from no-arbitrage condition. Brownian increments are
generated under one chosen measure and the increments under the remaining
measures are obtained by transformation. The modelled rates are lognormal
martingales under the corresponding measures, which converge to their
continuous counterparts as the time step of the discretization goes to
zero. Choosing a measure for simulation is the same as choosing a
numeraire, and care must be taken to evaluate all payoffs in terms of the
chosen numeraire.
The talk describes a stock or currency market model, where asset
prices are formed by ratios of squared Bessel processes. These asset
prices have leptokurtic return distributions. Their stochastic
volatilities show properties that coincide closely with those typically
observed in developed financial markets. The volatility of indices is
negatively correlated with the asset itself, which models the well-known
leverage effect. Some empirical evedence will be presented that
supports the model.
We take a number of different perturbations of the standard
Black-Scholes world in which liquidity effects are represented
by a delay in the application of new portfolio choices. This
reflects the fact that you cannot reduce your holding of a particular
share until someone is prepared to buy them from you. We investigate
what the `cost of liquidity' is in these examples, and try to
clarify what investment strategies should be used.
We consider an incomplete market setting, where the
prices of the underlying risky assets satisfy dynamic equations
with coefficients driven by an unobservable random process.
Given a liability to be hedged, the problem is that of determining,
for various initial capitals, a self financing strategy that
minimizes a risk criterion related to shortfall (shortfall risk,
shortfall probability).
We approach this problem from the point of view of stochastic
control under partial information, taking into account a rebalancing
of the strategy only at discrete time points. The resulting strategy
uses all the information as it becomes successively available by an
agent who observes the prices of the underlying assets.
The paper documents a persistent and thus far largely overlooked
empirical regularity in the yield curve: the tendency for long term
forward rates, and correspondingly, zero coupon rates, to slope
downwards. We present evidence from a number of different estimates of
the US Treasury yield curve and also for the UK nominal and
index-linked markets. The explanation for this feature of the yield
curve has to do with the effect of interest rate volatility. When long
term zero coupon rates have non-zero volatility, we show - in the
context of the affine class of term structure models - that the long
term forward rate curve will be downward sloping whenever the
volatility of the corresponding long term is sufficiently high.
We present some new results on enlargements of filtrations and their
applications in financial mathematics. Basically, the goal is to
understand how typical finance problems change if an ordinary investor
in a financial market gets some additional information at the beginning
of his trading period. The results include new martingale representation
theorems and links between logarithmic utility maximization and relative
entropy.
This talk will explore the features of dynamic term structure
models that are empirically important for explaining the
joint distribution of yields on short- and long-term
interest rate swaps. We begin by presenting a classification
scheme for all affine term stucture models that highlights
the key similarities and differences between many of the
extant models in the literature. Then simulated method
of moment estimates and tests of over-identifying restrictions
will be presented for several special cases. The empirical
evidence suggests that extant models largely fail to
explain key features of historical term structure movements,
but that relaxing the restrictions in extant models leads
to models that pass several goodness-of-fit tests for
the sample period considered.
Market volatility is often observed to vary slowly, or cluster over a
period of a few days, in comparison with the rapid tick-by-tick
fluctuation of asset prices. We discuss how this is described by
mean-reverting stochastic volatility models, and how an asymptotic
analysis of the derivative pricing problem that exploits the
clustering phenomenon identifies three key model-independent
parameters that are needed. These are easily estimated from the
observed implied volatility surface and used directly to correct the
Black-Scholes prices of, for example, American, barrier and Asian
contracts, to account for randomly changing volatility. We investigate
the stability of the theory with S&P 500 data. Joint work with
Jean-Pierre Fouque and George Papanicolaou.
The first part of the talk is on the utility gradient (or martingale)
approach for computing portfolio and consumption plans that maximize
stochastic differential utility (SDU), a continuous-time version of
recursive utility. We characterize the first order conditions of optimality
as a system of forward-backward SDE's, which, in the Markovian case, reduces
to a system of PDE's and forward only SDE's that is amenable to numerical
computation. Another contribution is a proof of existence, uniqueness, and
basic properties for a parametric class of homothetic SDU that can be
thought of as a continuous-time version of the CES Kreps-Porteus utilities
studied by Epstein and Zin. For this class, we derive closed-form solutions
in terms of a single backward SDE (without imposing a Markovian structure),
including concrete examples involving the type of ``affine'' state price
dynamics that are familiar from the term structure literature.
The second part of the talk shows an equivalence between optimal portfolio
selection or competitive equilibrium models with utilities incorporating
linear habit formation and corresponding models without habit formation. The
equivalence is expressed through an explicit transformation of consumption
plans, utilities, endowments, state prices, wealth processes, and security
prices, that can be used to mechanically translate known solutions not
involving habit formation to corresponding solutions with habit formation.
For example, given this isomorphism, the Constantinides (JPE, 1990) solution
becomes a corollary of a familiar additive utility model. More generally,
the solutions discussed in the first part of the talk can be mechanically
transformed to solutions of corresponding problems with utility that
combines recursivity with habit formation.
In their classical paper "A Re-examination of Traditional Hypotheses about the
Term Structure of Interest Rates" (JF 1981) Cox-Ingersoll-Ross study the impact
of the different Expectation Hypotheses in the framework of bond
prices $P(t,T)$ driven by an $n$-dimensional Wiener process, where the short
rate is exogenously given. They derive the following relations
$$
P(t,T)_{LEH} > P(t,T)_{UEH} > P(t,T)_{RTM}
$$
where LEH = Local, UEH = Unbiased, RTM = Return-to-Maturity Expectation
Hypothesis (cp. also Ingersoll: Theory of Financial Decision Making,
Chapter 18).
We study the impact of the different Expectation Hypotheses on the dynamics of
the forward rates in a HJM-framework. In our study forward rates and, in
particular, short rates are endogenous, and depend on the assumed hypotheses.
Among other results we show that the above relationship is just the other way
round, i.e.
$$
P(t,T)_{LEH} < P(t,T)_{UEH} < P(t,T)_{RTM}.
$$
This general result, which seems counter intuitive at first sight, is
examplified in some concrete term structure models. Also the CIR
conclusions on risk premiums have to be revised. We also study the
implications on term premiums, i.e. the difference between forward rates
and (expected) future spot rates.
The talk is based on a joint paper with Y. Kabanov. We consider a
discrete-time model of a currency market with transaction costs.
First we investigate absence of arbitrage. Then we give the dual
description of initial endowments which allow
to hedge a contingent claim in various currencies by a
self-financing porfolio without assuming the existence of an
equivalent martingale measure.
I will discuss a class of valuation models with diffusion stock prices
whose coefficients are affected by correlated stochastic factors
representing the nontraded assets. Using a nonlinear tranformation one
obtains reduced form solutions for the value function and the optimal
policies which can in turn be used in a variety of problems in portfolio
management and derivative pricing in markets with frictions.
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