June 25, 2019

May 24-28, 2010
Workshop on Financial Derivatives and Risk Management

Talk Titles and Abstracts

Lorenzo Bergomi (Societe Generale)
Smile Dynamics IV
Static and dynamic properties of stochastic volatility models: a structural connection

Stochastic volatility models do two jobs at once: they produce a smile and generate a dynamics for implied volatilities. For general stochastic volatility models, working at order one in the volatility of volatility we establish the structural connection between both aspects of a model.

The derivation calls for the introduction of the Skew Stickiness Ratio, a dimensionless number that quantifies the amount by which the ATM volatility moves when the spot moves, in units of the ATM skew. We derive lower and higher bounds for the SSR and relate the SSR to the decay of the ATM skew as a function of maturity, which leads to a natural partition of stochastic volatility models into two classes.

We then consider the historical joint dynamics of spot and implied volatilites, assess whether our generic results hold in practice and introduce the notion of realized skew.


Tomasz Bielecki (IIT)
Hedging of counterparty risk

Counterparty risk is one of the fundamental forms of risks underlying financial transactions. Thus, assessment and mitigation of this risk is of primary importance to financial institutions. In this talk we shall look at counterparty risk as the risk associated with certain complex financial derivative, known as CCDS (contingent CDS). We first discuss the issue of valuation of CCDS, where the corresponding price process is called the CVA (credit valuation adjustment). We shall then discuss the issue of hedging of the CCDS. We will specify our general results to the case of so called Markovian copula model. In this context, specific formulae for self-financing hedging strategies will be given, when hedging portfolio is created from so called rolling CDS contracts, and perhaps some other contracts as well. This is a joint work with Monique Jeanblanc and Stephane Crepey.


Nicholas Bingham (Imperial College)
Multivariate elliptic processes


Dorje C. Brody (Imperial College London)
Rational Term Structure Models with Geometric Lévy Martingales

In the positive interest models of Flesaker & Hughston (1996) the nominal discount bond system is represented by a one-parameter family of positive martingales. In the present paper we extend the analysis to include a variety of distributions for the martingale family, parameterised by a function ?(x) that determines the behaviour of the market risk premium. These distributions include jump and diffusion characteristics that generate various interesting properties for discount bond returns. For example, one can generate skewness and excess kurtosis in the discount bond returns by choosing the martingale family to be given by (a) exponential gamma processes, or (b) exponential variance-gamma processes. The models are “rational” in the sense that the discount bond price process is given by the ratio of a pair of sums of positive martingales. Our findings lead to semi-analytical and Fourier-inversion style solutions for the prices of European options on discount bonds, foreign exchange rates, and foreign discount bonds. The paper is motivated in part by the results of Filipovic, Tappe & Teichmann (2009), who demonstrated that the term structure density approach of Brody & Hughston (2001) admits a natural extension to general positive term-structure models driven by a class of Lévy processes. (Based on joint work with L.P. Hughston and E. Mackie.)


Rama Cont (Paris VI-VII)
Functional Ito calculus and the pricing and hedging of path-dependent derivatives

We develop a non-anticipative calculus for path-dependent functionals of a semimartingale, using a notion of pathwise functional derivative proposed by B. Dupire. The key ingredient is a functional extension of the Ito formula, which is used to derive a martingale representation formula for square integrable martingales. Regular functionals of a semimartingale S which have the local martingale property are characterized as solutions of a functional di erential equation, for which a uniqueness result is given.
This result is used to derive a universal pricing equation for the price of path-dependent derivatives with underlying asset S: this pricing equation is shown to be a functional equation whose coefficients involve the local characteristics of S. Using these results we derive a general formula for the hedging strategy of a path-dependent contingent
claim and present a numerical method for computing this hedging strategy. By contrast with methods based on Malliavin calculus, this representation is based on non-anticipative quantities which may be computed pathwise and leads to simple simulation-based estimators for computing hedging strategies for path-dependent options.


Mark Davis (Imperial College)
On SDEs with state-dependent jump measure

Virtually all textbook treatments of jump-diffusion SDEs assume that the driving processes are a Brownian motion and an independent homogeneous Poisson random measure. In many applications, for example modelling of credit-risky securities, it seems that solution-dependence of the compensator of the random measure should be allowed. The reason for not including this goes back to the 1972 book of Gihman and Skorohod, where it is shown how a problem with state-dependent compensator can be 'reduced' to an equivalent one with homogeneous random measure. There may however be good reasons for not doing this transformation: for example the homogeneous random measure may have infinite activity even if the jump rate in the original model is a.s. finite. These questions are discussed and some general results about existence and uniqueness with state-dependent jump measure are given.


Giuseppe Di Graziano (Deutsche Bank London)
Target Volatility Option Pricing

In this talk we shall present three approximation methods for the pricing of Target Volatility Options (TVOs), a recent market innovation in the field of volatility derivatives. TVOs allow investors to take a joint view on the future price of a given underlying (e.g. stocks, commodities, etc) and its realized volatility. For example, a target volatility call pays at maturity the terminal value of the underlying minus the strike, floored at zero, rescaled by the ratio of a given Target Volatility (an arbitrary constant) and the realized volatility of the underlying over the life of the option. TVOs are typically used by investors and hedgers to cheapen the price of an option or to leverage their exposure to the underlying.

We present three approaches for the pricing of TVOs: a power series expansion, a Laplace transform method and approximations based on Bernstein polynomials. The three approximations have been tested numerically and results are provided.


Rudiger Frey (Leipzig)
Portfolio optimization under partial information with expert opinions (joint with R. Wunderlich and A.Gabih)

We investigates optimal portfolio strategies for utility maximizing investors in a market with partial information on the drift. The drift is modelled by a continuous-time Markov chain with finitely many states which is not directly observable. Information on the drift is obtained from the observation of stock prices. Moreover, and this is the novel feature of this paper, expert opinions are included in the analysis. This additional information is modeled by a marked point process with jump-size distribution depending on the current state of the hidden Markov chain. We derive the filtering equation for the return process and incorporate the filter into the state variables of the
optimization problem. For this reformulated completely observable problem we investigate for the case of power utility the associated Hamilton-Jacobi-Bellman equation. Since this equation contains non-linearities in a jump part we adopt a policy improvement method to obtain an approximation of the optimal strategy. Numerical results are presented.


Pavel Gapeev (London School of Economics)
Pricing and filtering in a two-dimensional dividend switching model

In our recent joint paper with Monique Jeanblanc, we have studied a model of a financial market in which the dividend rates of two risky assets change their initial values to other constant ones at the times at which certain unobservable external events occur. The asset price dynamics were described by geometric Brownian motions with random drift rates switching at exponential random times that are independent of each other and the constantly correlated driving Brownian motions. We have obtained closed form expressions for the rational values of European contingent claims through the filtering estimates of occurrence of the switching times and their conditional probability density derived given the filtration generated by the underlying asset price processes.

Building on the results described above, we consider the model in which two underlying assets are driven by dependent (compound) Poisson processes belonging to exponential families. We obtain closed form expressions for the prices in the case in which the parameters of the asset price dynamics change one constants to other at the times of occurrence of unobservable external events and derive stochastic differential equations for the filtering estimates. We also discuss the solution to the problem of pricing of perpetual American options in a one-dimensional continuous diffusion model for the asset price with switching dividend rates under partial information.


Jim Gatheral (Merrill Lynch, NY)
Implied Volatility from Local Volatility

There is a well-known simple formula for computing local volatility given implied volatility as a function of strike and expiration. Given local volatilities, implied volatilities may be computed numerically using numerical PDE techniques. However, such computations are typically too time-consuming to permit fast calibration of local volatilities to option prices. In this talk, we review various methods that have been proposed for computing implied volatility from local volatility including heat kernel-based expansions and parameter averaging. We focus in particular on the most-likely-path approximation showing by specific example that it tends to perform better in practice than competing approximations.


David Hobson (Warwick)
Model independent bounds for variance swaps

Under an assumption of continuity on the price process, and under an assumption that a continuum of calls on the underlying are traded, the work of Neuberger and Dupire gives that the price for the variance swap is equal to twice the price of a log contract. This price is model-free.

But what if we are not prepared to assume continuity? Then, given call prices a range of possible prices is consistent with no-arbitrage. In this talk we try to characterise this range.


Lane Hughston (Imperial College)
Implied Density Models for Asset Pricing

In this paper we model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the asset is driven by Brownian motion we derive an associated "master equation" for the dynamics of the conditional probability density, and express this equation in integral form. By a "model" for the conditional density process we mean a solution to the master equation along with a specification of the initial density and a specification of the volatility structure for the density. The volatility structure in particular is assumed at any given time and for each value of the argument of the density function to take the form of a functional that depends on the history of density up to that time. The choice of this functional determines the particular model for the conditional density, and in practice one specifies the functional modulo sufficient parametric freedom to allow for the input of additional option data apart from that already implicit in the specification of the initial density. The scheme is sufficiently flexible to allow for the input of various different types of data depending on the nature of the options market under consideration and the class of valuation problem being undertaken. Various specific examples are studied in detail, with exact solutions provided in some cases. (Co-authors: D. Filipovic, Ecole Polytechnique Fédérale de Lausanne, Switzerland, and A. Macrina, King's College London and Kyoto Institute of Economic Research.)


Monique Jeanblanc (Evry)
Density models for credit risk

We present a model of default times based on the conditional law of defaults. We show in particular that, in that general framework, the intensity does not contain all the needed information. In case of a single default, this model can be interpreted as an extension of the Cox model, where the barrier depends on the reference filtration. The extension of our study to several defaults can be viewed as a dynamic copula approach.


Yu Hang Kan (Columbia)
Default intensities implied by CDO spreads: inversion formula and model calibration

We propose a simple computational method for constructing an arbitrage-free CDO pricing model which matches a pre-specified set of CDO tranche spreads. The key ingredient of the method is an inversion formula for computing the aggregate default rate in a portfolio, as a function of the number of defaults, from its expected tranche notionals. This formula can be seen as an analog of the Dupire formula for portfolio credit derivatives. Together with a quadratic programming method for recovering expected tranche notionals from CDO spreads, our inversion formula leads to an efficient non-parametric method for calibrating CDO pricing models. Contrarily to the base correlation method, our method yields an arbitrage-free model.
Comparing this approach to other calibration methods, we find that model-dependent quan- tities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. On the other hand, comparing the local default intensities implied by different credit portfolio models reveals that apparently different models, such as the static Student-t copula models and the reduced-form affine jump- diffusion model, lead to similar marginal loss distributions and tranche spreads.


Thomas Kokholm (Aarhus)
A Consistent Pricing Model for Index Options and Volatility Derivatives

We propose and study a flexible modeling framework for the joint dynamics of an index and a set of forward variance swap rates written on this index, allowing volatility derivatives and options on the underlying index to be priced consistently. Our model reproduces various empirically observed properties of variance swap dynamics and allows for jumps in volatility and returns.

An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for options on the VIX as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves.

We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index. The calibration of the model is done in two steps, first by matching VIX option prices and then by matching prices of options on the underlying.


Roger Lee (Chicago)
Variation Swaps on Time-Changed Levy Processes

For a family of functions G, we define the G-variation, which generalizes power variation; G-variation swaps, which pay the G-variation of the returns on an underlying share price F; and share-weighted G-variation swaps, which pay the integral of F with respect to G-variation. For instance, the case G(x)=x^2 reduces these notions to, respectively, quadratic variation, variance swaps, and gamma swaps.

We prove that a multiple of a log contract prices a G-variation swap, and a multiple of an F log F contract prices a share-weighted G-variation swap, under arbitrary exponential Levy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the Levy driver, under integrability conditions. We solve for the multipliers, which depend only on the Levy process, not on the clock.

In the case of quadratic G and continuity of the underlying paths, each valuation multiplier is 2, recovering the standard no-jump variance and gamma swap pricing results. In the presence of jump risk, however, we show that the valuation multiplier differs from 2, in a way that relates (positively or negatively, depending on the contract) to the Levy measure's skewness.

This work, joint with Peter Carr, extends Carr-Lee-Wu's treatment of variance swaps, by generalizing from quadratic variation to G-variation; and by encompassing not only unweighted but also share-weighted payoffs.


Andrea Macrina (King's College London)
Heat Kernels for Information-Sensitive Pricing Kernels

We consider a positive propagator that is driven by time-inhomogeneous Markov processes. We multiply the propagator with a time-dependent, decreasing positive weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. Such supermartingales are suitable for the modelling of the pricing kernel in the case where it is assumed to be given by a function of time and Markov processes. This situation is encountered for example, if we assume that the pricing kernel is sensitive to partial information about economic factors, and the partial information is modelled by use of time-inhomogeneous Markov processes. We show how closed-form expressions for bond prices along with the associated interest-rate and market price of risk models can be obtained, and indicate the way towards the pricing of fixed-income derivatives within this framework. (In collaboration with Jiro Akahori, Ritsumeikan University)


Gustavo Manso (MIT)
Information Percolation

We study the "percolation" of information of common interest through a large market as agents encounter and reveal information to each other over time. We provide an explicit solution for the dynamics of the cross-sectional distribution of posterior beliefs, and calculate its rate of convergence to a common posterior. We also study how market segmentation, learning through public signals, and endogenous search intensities affect information percolation.


Aleksandar Mijatovic (Imperial College)
Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models

In this talk we describe a deterministic characterisation of the no free lunch with vanishing risk (NFLVR), the no generalised arbitrage (NGA) and the no relative arbitrage (NRA) conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other. This is joint work with Mikhail Urusov.


Andreea Minca (Paris VI)
Resilience to contagion in financial networks

Given a macroeconomic stress scenario defined in terms of the magnitude of common shocks across balance sheet, we perform an asymptotic analysis of default contagion, using analytical methods, and derive an expression for the fraction of defaulted nodes in the limit where the number of nodes is large, in terms of the empirical distribution of the in and out-degrees and the proportion of weak links in the network. We show that the size of the default cascade generated by the macroeconomic shock may exhibit a phase transition when the macroeconomic shock affecting the financial institutions reaches a certain threshold, beyond which the fraction of defaults is close to one. This result is used obtain a criterion for the resilience of a large network to macro-economic shocks The asymptotic results are shown to be in good agreement with simulations for networks whose sizes are realistic, showing the relevance of the large network limit for macro-prudential regulation.


Jan Obloj (Oxford)
On notion of arbitrage and robust pricing and hedging of variance swaps

In robust pricing and hedging one does not assume any given model but starts with market quoted prices of some options and deduces no-arbitrage bounds on a given non-traded derivative, and further specifies robust hedging strategies which enforce these bounds. In this talk, we consider the case of a weighted variance swap (e.g. a vanilla variance swap or a corridor variance swap) when prices of finite number of co-maturing call/put options are given. We analyse in some detail the arbitrage opportunities which may arise when prices are mis-specified: model independent arbitrage, weak arbitrage and weak free lunch with vanishing risk. These new notions are necessary since do not have any pre-specified probability space. Based on joint works with M. Davis and V. Raval and with A. Cox.


Goran Peskir (Manchester)
A Duality Principle for the Legendre Transform and the Valuation of Financial Contracts

We present a duality principle for the Legendre transform that yields the shortest path between the graphs of functions and embodies the underlying Nash equilibrium. A useful feature of the algorithm for the shortest path obtained in this way is that its implementation has a local character in the sense that it is applicable at any point in the domain with no reference to calculations made earlier or elsewhere. The derived results are applied to the valuation of financial contracts for Markov processes where the duality principle corresponds to the semiharmonic characterisation of the value function.


Martijn Pistorius (Imperial College)
Continuously monitored barrier options under Markov processes

In this talk we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Levy model and a local volatility jump-diffusion. We also provide a convergence proof and error estimates for this algorithm.


Marek Rutkowski (Sydney)
Market Models of Forward CDS Spreads


Thorsten Schmidt (Leipzig)
Market Models for CDOs Driven by Time-Inhomogeneous Levy Processes

This paper considers a top-down approach for CDO valuation and proposes a market model. We extend previous research on this topic in two directions: on the one side, we use as driving process for the interest rate dynamics a time-inhomogeneous Levy process, and on the other side, we do not assume that all maturities are available in the market. Only a discrete tenor structure is considered, which is in the spirit of the classical Libor market model. We create a general framework for market models based on multidimensional semimartingales. This framework is able to capture dependence between the default-free and the defaultable dynamics, as well as contagion effects. Conditions for absence of arbitrage and valuation formulas for tranches of CDOs are given.


Steve Shreve (Carnegie Mellon)
Matching Statistics of an Ito Process by a Process of Diffusion Type

Suppose we are given a multi-dimensional Ito process, which can be regarded as a model for an underlying asset price together with related stochastic processes, e.g., volatility. The drift and diffusion terms for this Ito process are permitted to be arbitrary adapted processes. We construct a weak solution to a diffusion-type equation that matches the distribution of the Ito process at each fixed time. Moreover, we show how to also match the distribution at each fixed time of statistics of the Ito process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when written on the original Ito process as when written on the mimicking process. This is joint work with Gerard Brunick.


Stuart Turnbull (Houston)
Measuring and Managing Risk in Innovative Financial Instruments

This paper discusses the difficult challenges of measuring and managing risk of innovative financial products. To measure risk requires the ability to first identify the
different dimensions of risk that an innovation introduces. The list of possible factors is long: model restrictions, illiquidity, limited ability to test models, product design,
counterparty risk and related managerial issues. For measuring some of the different dimensions of risk the implications of limited available data must be addressed. Given
the uncertainty about model valuation, how can risk managers respond? All parties within a company - senior management, traders and risk managers - have important
roles to play in assessing, measuring and managing risk of new products.


Johan Tysk (Uppsala)
Dupire's Equation for Bubbles

This is a report on a joint work with Erik Ekström. We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. Surprisingly enough, however, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.



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