Past Mathematical and Computational Finance Internal Seminar

In this paper we employ the quantile formulation to solve the SP/A portfolio choice model in continuous time. We show that the original version of the SP/A model proposed by Lopes is ill-posed in the continuous-time setting. We then generalise the SP/A model to one where a utility function is included, while the probability weighting

(distortion) function is still present. The feasibility and well-posedness of the model are addressed and an explicit solution is derived. Finally, we study how the aspiration level and the probability weighting function affect the optimal solution

Virtually all decisions taken by living beings, from financial investments to life history, mate choice or anti-predator responses involve uncertainties and inter-temporal trade offs. Thus, hypothesis and formal models from these different fields often have heuristic value across disciplines. I will present theories and experiments about temporal discounting and risky choice originating in behavioural research on birds. Among other topics, I will address empirical observations showing risk aversion for gains and risk proneness for losses, exploring parallels and differences between Prospect Theory, Risk Sensitivity Theory and Scalar Utility Theory.

We solve the problem of an agent with prospect theory preferences who seeks to liquidate a portfolio of (divisible) claims.

Our methodology enables us to consider different formulations of prospect preferences in the literature (piecewise exponential or piecewise power) and various price processes. We find that these differences in specification matter - for instance, with piecewise power functions, the agent may liquidate at a loss relative to break-even, albeit the likelihood of liquidating at a gain is much higher than liquidating at a loss. This is consistent with the disposition effect documented in empirical and experimental studies. We find the agent does not choose to partially liquidate a position, but rather, if liquidation occurs, the entire position is sold. This is in contrast to partial liquidation when agents have standard concave utilities.

I consider the problem of maximising the final utility of a portfolio which is constrained to satisfy the draw-down condition, i.e. the current value of the portfolio can not drop below a pre-specified funciton of its running maximum. It turns out that martingale techniques yield an explicit and rather elegant solution. The so- called Azema-Yor processes appear naturally and I take some time to introduce this class and discuss some of their remarkable properties.

In particular, I show how they can be characterised as (unique,

strong) solutions to SDEs called the Bachelier Eq and the Draw-Down Eq.

The talk is based (in particular) on a joint work with L. Carraro, N.

El Karoui and A. Meziou.

I shall discuss a number of problems to do with approximating the value function of an American Put option in the Black-Scholes model. This is essentially a variant of the oxygen-consumption problem, a parabolic free boundary (obstacle) problem which is closely related to the Stefan problem. Having reviewed the short-time behaviour from the perspective of ray theory, I shall focus on constructing approximations in the case when there is a discretely paid dividend yield.

A new portfolio choice model in continuous time is formulated and solved, where the quantile function of the terminal cash flow, instead of the cash flow itself, is taken as the decision variable. This formulation covers and leads to solutions to many existing and new models including expected utility maximisation, mean-variance, goal reaching, VaR and CVaR, Yaari's dual model, Lopes' SP/A model, and behavioural model under prospect theory.

We identify a natural way of ordering functions, which we call the interval dominance order, and show that this concept is useful in the theory of monotone comparative statics and also in statistical decision theory. This ordering on functions is weaker than the standard one based on the single crossing property (Milgrom and Shannon, 1994) and so our monotone comparative statics results apply in some settings where the single crossing property does not hold. For example, they are useful when examining the comparative statics of optimal stopping time problems. We also show that certain basic results in statistical decision theory which are important in economics - specifically, the complete class theorem of Karlin and Rubin (1956) and the results connected with Lehmann's (1988) concept of informativeness – generalize to payoff functions that obey the interval dominance order.

We study an equilibrium model for a defaultable bond in the asymmetric dynamic information setting. The market consists of noise traders, an insider and a risk neutral market maker. Under the assumption that the insider observes the firm value continuously in time we study the optimal strategies for the insider and the optimal pricing rules for the market maker. We show that there exists an equilibrium where the insider’s trades are inconspicuous. In this equilibrium the insider drives the total demand to a certain level at the default time. The solution follows from answering the following purely mathematical question which is of interest in its own: Suppose Z and B are two independent Brownian motions with B(0)=0 and Z(0) is a positive random variable. Let T be the first time that Z hits 0. Does there exists a semimartingale X such that

1) it is a solution to the SDE

dX(t) = dB(t) + g(t,X(t),Z(t))dt

with X(0) = 1, for some appropriate function g,

2) T is the first hitting time of 0 for X, and

3) X is a Brownian motion in its own filtration?

We consider the hedging of a claim on a non-traded asset using a correlated traded asset, when the agent does not know the true values of the asset drifts, a partial information scenario. The drifts are taken to be random variables with a Gaussian prior distribution. This is updated via a linear filter. The result is a full information model with random drifts. The utility infdifference price and hedge is characterised via the dual problem, for an exponential utility function. An approximation for the price and hedge is derived, valid for small positions in the claim. The effectiveness of this hedging strategy is examined via simulation experiments, and is shown to yield improved results over the Black-Scholes strategy which assumes perfect correlation.

In a financial market, the appreciate rates are very difficult to estimate precisely, and in general only some confidence interval will be estimated. This paper is devoted to the portfolio selection with the appreciation rates being in a certain closed convex set rather than some precise point. We study the problem in both expected utility framework and mean-variance framework, and robust solutions are given explicitly in both frameworks.

(based on joint work with Helen Haworth, William Shaw, and Ben Hambly)

The simulation of multi-name credit derivatives raises significant challenges, among others from the perspective of dependence modelling, calibration, and computational complexity. Structural models are based on the evolution of firm values, often modelled by market and idiosyncratic factors, to create a rich correlation structure. In addition to this, we will allow for contagious effects, to account for the important scenarios where the default of a number of companies has a time-decaying impact on the credit quality of others. If any further evidence for the importance of this was needed, the recent developments in the credit markets have furnished it. We will give illustrations for small n-th-to-default baskets, and propose extensions to large basket credit derivatives by exploring the limit for an increasing number of firms

This talk will be about the mathematics and computer science behind my "Smoking Adjoints: fast Monte Carlo Greeks" article with Paul Glasserman in Risk magazine. At a high level, the adjoint approach is simply a very efficient way of implementing pathwise sensitivity analysis. At a low level, reverse mode automatic differentiation enables one to differentiate a "black-box" to get the sensitivity of a single output to multiple inputs at a cost no more than 4 times greater than the cost of evaluating the black-box, regardless of the number of inputs

The setting is a lognormal basis risk model. We study the optimal hedging of a claim on a non-traded asset using a correlated traded asset in a partial information framework, in which trading strategies are required to be adapted to the filtration generated by the asset prices. Assuming continuous observations, we take the assets' volatilites and the correlation as known, but the drift parameters are not known with certainty.

We assume the drifts are random variables with a Gaussian prior distribution, derived from data prior to the hedging timeframe. This distribution is updated via a Kalman-Bucy filter. The result is a basis risk model with random drift parameters.

Using exponsntial utility, the optimal hedging problem is attacked via the dual to the primal problem, leading to a representation for the hedging strategy in terms of derivatives of the indifference price. This representation contains additional terms reflecting uncertainty in the assets' drifts, compared with the classical full information model.

An analytic approximation for the indifference price and hedge is developed, for small positions in the claim, using elementary ideas of Malliavin calculus. This is used to simulate the hedging of the claim over many histories, and the terminal hedging error distribution is computed to determine if learning can counteract the effect of drift parameter uncertainty.