Past Mathematical Finance Seminar

17 October 2008
14:15
Ernst Eberlein
Abstract
We discuss the valuation problem for a broad spectrum of derivatives, especially in Levy driven models. The key idea in this approach is to separate from the computational point of view the role of the two ingredients which are the payoff function and the driving process for the underlying quantity. Conditions under which valuation formulae based on Fourier and Laplace transforms hold in a general framework are analyzed. An interesting interplay between the properties of the payoff function and the driving process arises. We also derive the analytically extended characteristic function of the supremum and the infimum processes derived from a Levy process. Putting the different pieces together, we can price lookback and one-touch options in Levy driven models, as well as options on the minimum and maximum of several assets.
  • Mathematical Finance Seminar
13 June 2008
14:15
Dorje Brody
Abstract
A modelling framework is introduced in which there is a small agent who is more susceptible to the flow of information in the market as compared to the general market participants. In this framework market participants have access to a stream of noisy information concerning the future returns of the asset, whereas an informative trader has access to an additional information source which is also obscured by further noise, which may be correlated with the market noise. The informative trader utilises the extraneous information source to seek statistical arbitrage opportunities, in exchange with accommodating the additional risk. The information content of the market concerning the value of the impending cash flow is represented by the mutual information of the asset price and the associated cash flow. The worthiness of the additional information source is then measured in terms of the difference of mutual information between market participants and the informative trader. This difference is shown to be strictly nonnegative for all parameter values in the model, when signal-to-noise ratio is known in advance. Trading strategies making use of the additional information are considered. (Talk is based on joint work with M.H.A. Davis (Imperial) & R.L. Friedman (Imperial & Royal Bank of Scotland).
  • Mathematical Finance Seminar
6 June 2008
14:15
Jaska Cvitanic
Abstract
This talk will give a survey of results in continuous-time contract theory, and discuss open problems and plans for further research on this topic. The general question is how a ``principal" (a company, investors ...) should design a payoff for compensating an ``agent" (an executive, a portfolio manager, ...) in order to induce the best possible performance. The following frameworks are standard in contract theory: (i) the principal and the agent have same, full information; (ii) the principal cannot monitor agent's actions (iii) the principal does not know agent's type We will discuss all three of these problems. The mathematical tools used are those of stochastic control theory, stochastic maximum principle and Forward Backward Stochastic Differential Equations.
  • Mathematical Finance Seminar
30 May 2008
14:15
Lane Hughston
Abstract
We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process. An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions, and rainfall. (Co-authors: D. C. Brody, Imperial College London, and A. Macrina, King's College London and ETH Zurich. Downloadable at www.mth.kcl.ac.uk.
  • Mathematical Finance Seminar
23 May 2008
14:15
TBA
Qing Zhang
Abstract
Trading a financial asset involves a sequence of decisions to buy or sell the asset over time. A traditional trading strategy is to buy low and sell high. However, in practice, identifying these low and high levels is extremely challenging and difficult. In this talk, I will present our ongoing research on characterization of these key levels when the underlying asset price is dictated by a mean-reversion model. Our objective is to buy and sell the asset sequentially in order to maximize the overall profit. Mathematically, this amounts to determining a sequence of stopping times. We establish the associated dynamic programming equations (quasi-variational inequalities) and show that these differential equations can be converted to algebraic-like equations under certain conditions. The two threshold (buy and sell) levels can be found by solving these algebraic-like equations. We provide sufficient conditions that guarantee the optimality of our trading strategy.
  • Mathematical Finance Seminar
16 May 2008
14:15
Nicole El Karoui
Abstract
Many portfolio optimization problems are directly or indirectly concerned with the current maximum of the underlying. For example, loockback or Russian options, optimization with max-drawdown constraint , or indirectly American Put Options, optimization with floor constraints. The Azema-Yor martingales or max-martingales, introduced in 1979 to solve the Skohorod embedding problem, appear to be remarkably efficient to provide simple solution to some of these problems, written on semi-martingale with continuous running supremum.
  • Mathematical Finance Seminar
9 May 2008
14:15
Amel Bentata and Marc Yor
Abstract
14.15 - 15.00 Part I Marc Yor : The infinite horizon case. 15.00 - 15.15 A short break for questions and answers 15.15 - 16.00 Part II Amel Bentata : The finite horizon case. Roughly, the Black-Scholes formula is a distribution function of the maturity. This may be explained in terms of the last passage times at a given level of the underlying Brownian motion with drift. Conversely, starting with last passage times up to finite horizon, we obtain a 2-parameter variant of the Black-Scholes formula.
  • Mathematical Finance Seminar
25 April 2008
14:15
Karl Kunisch
Abstract
Efficient numerical solutions of several important partial-differential equation based models in mathematical finance are impeded by the fact that they contain operators which are Lipschitz continuous but not continuously differentiable. As a consequence, Newton methods are not directly applicable and, more importantly, do not provide their typical fast convergence properties. In this talk semi-smooth Newton methods are presented as a remedy to the the above-mentioned difficulties. We also discuss algorithmic issues including the primal-dual active set strategy and path following techniques.
  • Mathematical Finance Seminar
7 March 2008
13:15
Jan Obloj
Abstract
We discuss 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. Treating this market data as input, 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 Alexander Cox (University of Bath)
  • Mathematical Finance Seminar

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