# Seminars 2007-2008

## 2007-2008

### Summer Term 2008

**THURSDAY 3 APRIL 2008 **

**Emmanuel Gobet (InP Grenoble ENSIMAG): Smart Expansions and Fast Calibration methods for Jump Diffusion models**** 17:30 – 18:30 Room 140, Huxley Building**

Abstract: Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and jump Poisson process. We show that the accuracy of the formula depends on the smoothness of the payoff. In practice, it is excellent. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency. As a consequence, the calibration of such model becomes very fast.

Joint work with E. Benhamou (Pricing Partners) and M. Miri (Pricing Partners and ENSIMAG)

**WEDNESDAY 30 APRIL 2008 **

**Jan Palczewski (University of Leeds and University of Warsaw): Market Selection of Self-financing Strategies in Continuous Time**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: While mathematical finance has offered deep insights in the dynamics of portfolio payoffs under the assumption of an exogenous price process, economists prefer - in a market context - equating demand and supply through an endogenous price mechanism. In our approach the market interaction of investors plays a central role. This interaction is modeled by introducing endogenous prices (driven by demand and supply) in the classical mathematical finance model. Randomness stems from exogenous asset payoff processes (dividends) and variation in traders' behavior. Our analysis focuses on the survival and extinction of investment strategies which is defined through the asymptotic outcome of the wealth dynamics. Our results identify a unique investment strategy that is asymptotically optimal in a market in which only time-invariant strategies are present. This strategy prescribes to divide wealth proportional to the average relative dividend intensity of assets. We show that any other time-invariant investment strategy interacting in the market will become extinct. The route to prove above result goes via an asymptotic analysis of a random dynamical system that, to the best of our knowledge, has not been studied before and requires the application of new techniques.

This talk is based on joint works with K. R. Schenk-Hoppe (University of Leeds).

**TUESDAY 13 MAY 2008 **

** Marc Yor (Universite Paris 6): A last passage time viewpoint for the Black Scholes formula**** 11:30 – 12:30 Room 140 , Huxley Building**** Tea served from 11:00 in the Huxley Senior Common Room (room 549)**

Abstract: For a given strike level K, the put version of the Black-Scholes formula may be interpreted as the distribution function of the last passage time at K of geometric Brownian motion.

**WEDNESDAY 14 MAY 2008 **

**Sergey Levendorskiy (University of Texas at Austin): The Wiener-Hopf factorization as a general method for valuation of American and barrier options**** One-day workshop (10-12 and 14-18), IMS Seminar Room**

**WEDNESDAY 21 MAY 2008 **

**Mark Owen (Heriot-Watt University): Multivariate utility maximization with proportional transaction costs**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: The setting is a model of variable proportional transaction costs, as found in the recent paper of Campi and Schachermayer. Trading with friction is allowed between a mixture of investment and consumption assets. The objective is to maximise expected terminal utility of the consumption assets. My talk will focus on the properties of (direct and indirect) multivariate utility functions which are sufficient for the existence of a solution to the optimal investment problem. In particular I will discuss the condition of Reasonable Asymptotic Elasticity of the direct utility function (or more precisely a growth condition on its dual function), and explain how this can be replaced by the weaker, and more natural condition of "asymptotic satiability" of the indirect utility function.

This is joint work with Luciano Campi.

**WEDNESDAY 28 MAY 2008 **

**Giacomo Scandolo (Università di Firenze): A New approach to liquidity risk (Presentation: Scandolo, 28 May 2008)**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: We present an hypotheses–free formalism for marking–to–market a portfolio in general illiquid markets. In this formalism coherent measures of risk turn out to be appropriate to measure general portfolio risk including liquidity risk. Coherent Risk maps and Value maps, defined on the space of portfolios, turn out to be convex and concave respectively, displaying two distinct faces of the diversification principle, namely the traditional “correlation” benefit and a newly observed “granularity” benefit. We show that the optimization problem implicit in the definition of the value of a portfolio is always a convex problem, ensuring straightforward industrial applicability of the method. Finally, some numerical applications are presented.

**WEDNESDAY 4 JUNE 2008 **

**Jose Manuel Corcuera (Universitat de Barcelona): Lévy term structure model: a martingale model perspective**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: We consider bond markets where the noise is a Lévy process. By beginning with the evolution of the short rates under the historical probability we consider the bonds as derivates valued under certain risk neutral probability. The completeness problem is analysed by using new representation theorems for martingales with jumps in analogy with an additive stock market.

**WEDNESDAY 11 JUNE 2008 **

**John Crosby (Glasgow University): A new class of Levy process models with almost p erfect calibration to both barrier and vanilla fx options**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: It is very well appreciated that it would be desirable to be able to fit a model, for eg fx options, to the market prices of both vanilla options and barrier options (especially the prices of Double No Touch options). Indeed, recent talks at practitioner conferences h ave highlighted the desirability of doing this – unfortunately, it is harder said than done. We offer a solution which makes the desirable possible. We describe a class of Levy-type models which are designed to be calibrated to the market prices of liquid barrier options (such as Double No Touch options) and vanilla options. The calibration problem is simplified by virtue of the fact that it is reduced to a two-stage problem and that both barrier options and vanilla options can be priced semi-analytically (up to Transform Inversion). By virtue of the fact that the model allows for jump processes (with either finite or infinite activity) and stochastic volatility, the model can generate realistic smiles and realistic future dynamics of the spot.

**WEDNESDAY 18 JUNE 2008 **

**Sebastian Lleo (Imperial College London): Jump-Diffusion Risk Sensitive Asset Management**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: Risk-sensitive asset management theory, pioneered by Bielecki and Pliska (1999), addresses continuous-time asset allocation problems in an incomplete market, where the dynamics of securities prices is a function of some valuation factors.

Our research extends this theory by considering the possibility of random jumps in both asset prices and valuation factors levels. Because an analytical solution does not generally exist in this setting, we prove that the value function of the control problem is the unique continuous viscosity solution of the associated risk-sensitive Hamilton-Jacobi-Bellman partial integro-differential equation (RS HJB PIDE). In this presentation, we will:

- introduce risk-sensitive control and risk-sensitive asset management;

- present a jump-diffusion version of the risk-sensitive asset management problem;

- derive the Risk Sensitive Hamilton-Jacobi-Bellman Partial Integro-Differential Equation;

- show the existence of a unique optimal investment policy;

- outline a viscosity approach to solving the RS HJB PIDE.

(Joint work with Mark Davis)

### Spring Term 2008

**WEDNESDAY 16 JANUARY 2008 **

**Alexander Cox (University of Bath): Robust Pricing and Hedging of Digital Double Barrier Options**** 17:00 – 18:00 Room 140, Huxley Building**** Refreshments served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: We consider model-free pricing of digital options, which pay out depending on whether the underlying asset has crossed upper and lower levels. We make only weak assumptions about the underlying process (typically continuity), but assume that the initial prices of call options with the same maturity and all strikes are known. Under such circumstances, we are able to give upper and lower bounds on the arbitrage-free prices of the relevant options, and further, using techniques from the theory of Skorokhod embeddings, to show that these bounds are tight. Additionally, martingale inequalities are derived, which provide the trading strategies with which we are able to realise any potential arbitrages.

Joint work with Jan Obloj (Imperial College).

**WEDNESDAY 6 FEBRUARY 2008 **

**Luciano Campi (Universite Paris Dauphine): Kyle-Back equilibrium model with insider trading and credit risk**** 17:00 – 18:00 Room 140, Huxley Building**** Refreshments served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: We study, in the framework of Back [Rev. Financial Stud. 5(3), 387–409 (1992)], an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm. The market consists of a risk-neutral informed agent, noise traders, and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in the market’s view as in reduced-form credit risk models. However, we show that, in equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider’s trades are inconspicuous. Moreover, we consider the case where the traded asset is a defaultable stock and the insider knows in advance also the final value of the firm. Finally, some issues related to gradually revealing insider's information are also discussed.

This talk is based on joint works with U. Cetin (LSE) and A. Danilova (Oxford U.)

**WEDNESDAY 20 FEBRUARY 2008 **

**Michael Monoyios (University of Oxford): Optimal hedging of basis risk with partial information**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: A claim on a non-traded asset Y is to be optimally hedged using a correlated traded asset S, with both asset prices following geometric Brownian motions. The hedger is not assumed to know the asset price drifts, so strategies must be adapted to the observation filtration generated by asset returns, a partial information scenario. The asset drifts are taken to be random variables with a Gaussian prior distribution. Using a Kalman-Bucy filter the prior distribution is updated with information from asset returns over the hedging timeframe. This converts the model to a full information model with random drift parameters. The resulting stochastic control problem for the indifference price and optimal hedge under exponential utility is solved via the dual problem. A perturbation approximation for the price and hedge is derived in the limit of a small position in claims, using elementary ideas of Malliavin calculus. The optimal hedging strategy is tested over simulated asset price paths and compared with a Black-Scholes style hedge which assumes perfect correlation. The hedging error distribution indicates that optimal hedging combined with learning can indeed outperform the Black-Scholes hedge.

**WEDNESDAY 27 FEBRUARY 2008 **

**Iain J Clark (Dresdner Kleinwort): Numerical Methods for Stochastic Volatility: Fourier Methods, PDEs and Monte Carlo (Presentation: Clark, 27 Feb 2008)**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: Stochastic volatility models such as Heston (1993), and extensions thereof, are commonly used by industry practitioners, as such models allow the observed skew/smile in volatilities to be generated. We find such models particularly useful in FX options, where smiles are often closer to symmetric than in markets such as equity derivatives, due to the absence of leverage effects. The basic practical requirements of a stochastic volatility model are firstly, a method to calibrate the model parameters to European vanillas, and secondly, implementation of numerical methods for pricing exotic options on the risky asset.

Calibration involves pricing a collection of vanilla options given a set of model parameters, and optimisation over these parameters until the volatility fit to the market is within a suitable tolerance. We therefore need an efficient method for pricing European options under stochastic volatility, in conjunction with a least squares optimisation routine. In Black-Scholes, the analytic expression for European call prices is well known. For Heston and other stochastic volatility models, no such simple solution exists in terms of special functions. However, an expression for the characteristic function of the asset price distribution can be symbolically computed (we shall give an overview of the relevant literature). We therefore, instead of taking the product of the probability den sity function with the payout function and performing a numerical integration, take the product of the characteristic function with the Fourier transform of the payout function. A simple (and fast) one-dimensional numerical integration then recovers the price of the European option. However this technique only works for pricing non-path dependent options, and is therefore of great use for efficient ca libration but useless for pricing path-dependent exotic options. The final part of the seminar will describe how path dependent options can be priced consistently using two-dimensional numerical methods such as the ADI method for solving the option pricing PDE, and Monte Carlo simulation.

**WEDNESDAY 5 MARCH 2008 **

**David Hobson (University of Warwick): Liquidation strategies fo r perfectly divisible option portfolios**** 17:00 – 18:00 Room 140, Huxley Building**** Tea served from 16:30 in the Huxley Senior Common Room (room 549)**

Abstract: We consider the exercise of a number of American options in an incomplete market by a risk averse agent. In contrast to much of the literature in which that options must be sold as a single block, in this paper we are interested in the case where the options are infinitely divisible. The contribution is to solve this problem explicitly (at least in the case of exponential utility and infinite maturity) and we show that, except at the initial time when it may be advantageous to exercise a positive fraction of holdings, it is never optimal for the holder to exercise a tranche of options. Instead the process of option sales is continuous; however, it is singular with respect to calendar time. Exercise takes place when the stock price reaches a convex boundary which we identify.

### Autumn Term 2007

**WEDNESDAY 10 OCTOBER 2007 **

**Rudiger Frey (Leipzig): Constructing Credit Ri sk Models Via Nonlinear Filte ring**** 17:00 - 18:00, Room 140, Huxley Building**

Abstract: We use nonlinear filtering techniques for the construction and the analysis of reduced-form portfolio credit risk models. We start from an artificial fundamental model where default times are conditionally independent and driven by a finite-state Markov chain X. Prices of traded instruments are determined by so-called informed market participants who observe the default history and some abstract signal Z, modelled as observations of X in additive Gaussian noise. Assuming absence of arbitrage, prices are thus given as conditional expectation of the payoff with respect to the information available to the informed market participants. It is shown that in this setup the pricing of credit derivatives leads to a nonlinear nonlinear filtering problem. Asset price (credit spread) dynamics are derived in analogy with the innovations approach to nonlinear filtering; it is shown that these dynamics exhibit a number of very desirable features. Finally we discuss the problem of pricing and hedging derivatives from the viewpoint of a `normal' investor who is restricted to observing the default history and prices of traded securities.

**WEDNESDAY 17 OCTOBER 2007 **

**Dmitry Kramkov (Carnegie Mellon): A model for a large investor trading market indifference prices**** 16:00 - 17:00, Room 140, Huxley Building**

Abstract: We develop a continuous-time model for a large economic agent where she is trading with market makers at their utility indifference prices. We start with the case of simple strategies, where trades occur at a finite number of times, and then use them to define general investment policies. This passage from discrete to continuous-time trading constitutes our main result and relies on stability theory of stochastic differential equations. The presentation is based on a joint project with Peter Bank.

**WEDNESDAY 17 OCTOBER 2007 **

**Xunyu Zhou (Oxford): Thou Shalt Buy and Hold**** 17:00 - 18:00, Room 140, Huxley Building**

Abstract: An investor holding a stock needs to decide when to sell the stock. It Is tempting to think that he should sell at the maximum price over a given investment horizon -- which is what investors always dream of. Unfortunately this is not possible. A close yet realistic goal is to sell the stock at the time when the expected relative error between the current price and the maximum price is minimized. This problem is thoroughly investigated for a geometric Brownian motion model, and it is shown that when the stock is good enough -- which we quantify explicitly -- the optimal strategy is to sell at the end of the horizon. Moreover, the resulting expected relative error is at most 25% - a universal number that is independent of the investment opportunities. This result justifies the conventional wisdom that one should buy and hold a good stock.

**WEDNESDAY 21 NOVEMBER 2007 **

**Alex Mijatovic (Imperial): Local time and the pricing of complex-barrier options**** 17:30 - 18:30, Room 140, Huxley Building**

Abstract: A complex-barrier option is a derivative security which delivers the terminal value of the payoff at expiry T if neither of the continuous time-dependent barriers b, B: [0,T] -> R (satisfying b(t) < B(t) for all t) have been hit during the time interval [0,T]. In this talk we describe a decomposition of the complex-barrier option price into the corresponding European option price minus the barrier premium for a wide class of linear diffusions, possibly discontinuous payoff functions and twice differentiable barrier functions b, B. We show that the barrier premium can be expressed as an integral of the option's delta at the barriers and that the pair of functions describing the deltas at the barriers solves a system of Volterra integral equations of the first kind.