Publications (74)57.57 Total impact

Article: Mimicking martingales
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ABSTRACT: Given the univariate marginals of a realvalued, continuoustime martingale, (respectively, a family of measures parameterised by $t \in [0,T]$ which is increasing in convex order, or a double continuum of call prices) we construct a family of purejump martingales which mimic that martingale (respectively, are consistent with the family of measures, or call prices). As an example, we construct a fake Brownian motion. Then, under a further `dispersion' assumption, we construct the martingale which (within the family of martingales which are consistent with a given set of measures) has the smallest expected total variation. We also give a pathwise inequality, which in the mathematical finance context yields a modelindependent subhedge for an exotic security with payoff equal to the total variation along a realisation of the price process.  [Show abstract] [Hide abstract]
ABSTRACT: The subject of this paper is an optimal consumption/optimal portfolio problem with transaction costs and with multiple risky assets. In our model the transaction costs take a special form in that transaction costs on purchases of one of the risky assets (the endowed asset) are infinite, and transaction costs involving the other risky assets are zero. Effectively, the endowed asset can only be sold. In general, multiasset optional consumption/optimal portfolio problems are very challenging, but the extra structure we introduce allows us to make significant progress towards an analytical solution. For an agent with CRRA utility we completely characterise the different types of optimal behaviours. These include always selling the entire holdings of the endowed asset immediately, selling the endowed asset whenever the ratio of the value of the holdings of the endowed asset to other wealth gets above a critical ratio, and selling the endowed asset only when other wealth is zero. This characterisation is in terms of solutions of a boundary crossing problem for a first order ODE. The technical contribution is to show that the problem of solving the HJB equation, which is a second order, nonlinear PDE subject to smooth fit at an unknown free boundary, can be reduced to this much simpler problem involving an explicit first order ODE. This technical contribution is at the heart of our analytical and numerical results, and allows us to prove monotonicity of the critical exercise threshold and the certainty equivalent value in the model parameters.  [Show abstract] [Hide abstract]
ABSTRACT: In this article we consider a special case of an optimal consumption/optimal portfolio problem first studied by Constantinides and Magill and by Davis and Norman, in which an agent with constant relative risk aversion seeks to maximise expected discounted utility of consumption over the infinite horizon, in a model comprising a riskfree asset and a risky asset with proportional transaction costs. The special case that we consider is that the cost of purchases of the risky asset is infinite, or equivalently the risky asset can only be sold and not bought. In this special setting new solution techniques are available, and we can make considerable progress towards an analytical solution. This means we are able to consider the comparative statics of the problem. There are some surprising conclusions, such as consumption rates are not monotone increasing in the return of the asset, nor are the certainty equivalent values of the risky positions monotone in the risk aversion.  [Show abstract] [Hide abstract]
ABSTRACT: This paper studies a variant of the contest model introduced by Seel and Strack. In the SeelStrack contest, each agent or contestant privately observes a Brownian motion, absorbed at zero, and chooses when to stop it. The winner of the contest is the contestant who stops at the highest value. The model assumes that all the processes start from a common value $x_0>0$ and the symmetric Nash equilibrium is for each agent to utilise a stopping rule which yields a randomised value for the stopped process. In the twoplayer contest this randomised value should have a uniform distribution on $[0,2x_0]$. In this paper we consider a variant of the problem whereby the starting values of the Brownian motions are independent, nonnegative, random variables that have a common law $\mu$. We consider a twoplayer contest and prove the existence and uniqueness of a symmetric Nash equilibrium for the problem. The solution is that each agent should aim for the target law $\nu$, where $\nu$ is greater than or equal to $\mu$ in convex order; $\nu$ has an atom at zero of the same size as any atom of $\mu$ at zero, and otherwise is atom free; on $(0,\infty)$ $\nu$ has a decreasing density; and the density of $\nu$ only decreases at points where the convex order constraint is binding.  [Show abstract] [Hide abstract]
ABSTRACT: In the Seel–Strack contest (J Econ Theory 148(5):2033–2048, 2013), \(n\) agents each privately observe an independent copy of a drifting Brownian motion which starts above zero and is absorbed at zero. Each agent chooses when to stop the process she observes, and the winner of the contest is the agent who stops her Brownian motion at the highest value. The objective of each agent is to maximise her probability of winning the contest. Seel and Strack derived a Nash equilibrium under a joint feasibility condition on the drift and the number of players. We consider a generalisation of the Seel–Strack contest in which the observed processes are independent copies of some timehomogeneous diffusion. This naturally leads us to consider the problem in cases where the analogue of the feasibility condition is violated. We solve the problem via a change of scale and a Lagrangian method. Unlike in the Seel–Strack problem, it turns out that the optimal strategy may involve a target distribution which has an atom, and the rule used for breaking ties becomes important.  [Show abstract] [Hide abstract]
ABSTRACT: Suppose $X$ is a timehomogeneous diffusion on an interval $I^X \subseteq \mathbb R$ and let $\mu$ be a probability measure on $I^X$. Then $\tau$ is a solution of the Skorokhod embedding problem (SEP) for $\mu$ in $X$ if $\tau$ is a stopping time and $X_\tau \sim \mu$. There are wellknown conditions which determine whether there exists a solution of the SEP for $\mu$ in $X$. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution of the SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first moment. When $X$ is Brownian motion, every integrable embedding of $\mu$ is minimal. However, for a general diffusion there may be integrable embeddings which are not minimal.  [Show abstract] [Hide abstract]
ABSTRACT: We solve the Skorokhod embedding problem (SEP) for a general timehomogeneous diffusion X: given a distribution \rho, we construct a stopping time T such that the stopped process X_T has the distribution \rho? Our solution method makes use of martingale representations (in a similar way to Bass [3] who solves the SEP for Brownian motion) and draws on law uniqueness of weak solutions of SDEs. Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Then, we fully characterize the distributions that can be embedded with integrable stopping times. Finally, we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of giving a robust, modelindependent, lower bound on the price of a forward starting straddle with payoff \(F_{T_{1}}  F_{T_{0}}\), where 0<T 0<T 1. Rather than assuming a model for the underlying forward price (F t ) t≥0, we assume that call prices for maturities T 0<T 1 are given and hence that the marginal laws of the underlying are known. The primal problem is to find the model that is consistent with the observed call prices and for which the price of the forward starting straddle is minimised. The dual problem is to find the cheapest semistatic subhedge. Under an assumption on the supports of the marginal laws, but no assumption that the laws are atomfree or in any other way regular, we derive explicit expressions for the coupling which minimises the price of the option, and the form of the semistatic subhedge. 
Article: Gambling in contests with regret
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ABSTRACT: This paper discusses the gambling contest introduced in Seel & Strack (Gambling in contests, Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 375, Mar 2012.) and considers the impact of adding a penalty associated with failure to follow a winning strategy. The Seel & Strack model consists of $n$agents each of whom privately observes a transient diffusion process and chooses when to stop it. The player with the highest stopped value wins the contest, and each player's objective is to maximise their probability of winning the contest. We give a new derivation of the results of Seel & Strack based on a Lagrangian approach. Moreover, we consider an extension of the problem in which in the case when an agent is penalised when their strategy is suboptimal, in the sense that they do not win the contest, but there existed an alternative strategy which would have resulted in victory. 
Article: Fake Exponential Brownian Motion
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ABSTRACT: We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of finding a modelfree upper bound on the price of a forwardstart straddle with payoff FT2 −FT1 . The bound depends on the prices of vanilla call and put options with maturities T1 and T2, but does not rely on any modelling assumptions concerning the dynamics of the underlying. The bound can be enforced by a superreplicating strategy involving puts, calls and a forward transaction. We find an upper bound, and a model which is consistent with T1 and T2 vanilla option prices for which the modelbased price of the straddle is equal to the upper bound. This proves that the bound is best possible. For lognormal marginals we show that the upper bound is at most 30% higher than the BlackScholes price. The problem can be recast as finding the solution to a Skorokhod embedding problem with nontrivial initial law so as to maximise EB − B0. 
Article: Constructing TimeHomogeneous Generalized Diffusions Consistent with Optimal Stopping Values
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ABSTRACT: Consider a set of discounted optimal stopping problems for a oneparameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the problem value in this setting. In this article we consider an inverse problem; given the set of problem values for a family of objective functions, we aim to recover the diffusion. Under a natural assumption on the family of objective functions, we can characterize the existence and uniqueness of a diffusion for which the optimal stopping problems have the specified values. The solution of the problem relies on techniques from generalized convexity theory.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.  [Show abstract] [Hide abstract]
ABSTRACT: A variance swap is a derivative with a pathdependent payoff which allows investors to take positions on the future variability of an asset. In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths, it is well known that the variance swap payoff can be replicated exactly using a portfolio of puts and calls and a dynamic position in the asset. This fact forms the basis of the VIX contract. But what if we are in the more realistic setting where the contract is based on discrete monitoring, and the underlying asset may have jumps? We show that it is possible to derive modelindependent, noarbitrage bounds on the price of the variance swap, and corresponding sub and superreplicating strategies. Further, we characterise the optimal bounds. The form of the hedges depends crucially on the kernel used to define the variance swap.  [Show abstract] [Hide abstract]
ABSTRACT: Given a centred distribution, can one find a timehomogeneous martingale diffusion starting at zero which has the given law at time 1? We answer the question affirmatively if generalized diffusions are allowed.  [Show abstract] [Hide abstract]
ABSTRACT: We pursue an inverse approach to utility theory and consumption & investment problems. Instead of specifying an agent's utility function and deriving her actions, we assume we observe her actions (i.e. her consumption and investment strategies) and ask if it is possible to derive a utility function for which the observed behaviour is optimal. We work in continuous time both in a deterministic and stochastic setting. In the deterministic setup, we find that there are infinitely many utility functions generating a given consumption pattern. In the stochastic setting of the BlackScholes complete market it turns out that the consumption and investment strategies have to satisfy a consistency condition (PDE) if they are to come from a classical utility maximisation problem. We show further that important characteristics of the agent such as her attitude towards risk (e.g. DARA) can be deduced directly from her consumption/investment choices. 
Article: Lecture Notes in Mathematics
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ABSTRACT: This set of lecture notes is concerned with the following pair of ideas and concepts: 1. The Skorokhod Embedding problem (SEP) is, given a stochastic process X=(X t )t≥0 and a measure μ on the state space of X, to find a stopping time τ such that the stopped process X τ has law μ. Most often we take the process X to be Brownian motion, and μ to be a centred probability measure. 2. The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, riskneutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model? If we know a single call price, then we can calibrate the volatility of the BlackScholes model (but if we know the prices of more than one call then together they will typically be inconsistent with the BlackScholes model). At the other extreme, if we know the prices of call options for all strikes and maturities, then we can find a unique martingale diffusion consistent with those prices. If we know call prices of all strikes for a single maturity, then we know the marginal distribution of the asset price, but there may be many martingales with the same marginal at a single fixed time. Any martingale with the given marginal is a candidate price process. On the other hand, after a time change it becomes a Brownian motion with a given distribution at a random time. Hence there is a 11 correspondence between candidate price processes which are consistent with observed prices, and solutions of the Skorokhod embedding problem. These notes are about this correspondence, and the idea that extremal solutions of the Skorokhod embedding problem lead to robust, model independent prices and hedges for exotic options.  [Show abstract] [Hide abstract]
ABSTRACT: The Az\'{e}maYor solution (resp., the Perkins solution) of the Skorokhod embedding problem has the property that it maximizes (resp., minimizes) the law of the maximum of the stopped process. We show that these constructions have a wider property in that they also maximize (and minimize) expected values for a more general class of bivariate functions $F(W_{\tau},S_{\tau})$ depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions $g$, they also maximize and minimize $\mathbb {E}[\int_0^{\tau}g(S_t)\,dt]$ amongst embeddings of $\mu$, although, perhaps surprisingly, we show that for increasing $g$ the Az\'{e}maYor embedding minimizes this quantity, and the Perkins embedding maximizes it. For $g(s)=s^{2}$ we show how these results are useful in calculating model independent bounds on the prices of variance swaps. Along the way we also consider whether $\mu_n$ converges weakly to $\mu$ is a sufficient condition for the associated Az\'{e}maYor and Perkins stopping times to converge. In the case of the Az\'{e}maYor embedding, if the potentials at zero also converge, then the stopping times converge almost surely, but for the Perkins embedding this need not be the case. However, under a further condition on the convergence of atoms at zero, the Perkins stopping times converge in probability (and hence converge almost surely down a subsequence).  [Show abstract] [Hide abstract]
ABSTRACT: These lecture notes cover a major part of the crash course on financial modeling with jump processes given by the author in Bologna on May 21–22, 2009. After a brief introduction, we discuss three aspects of exponential Lévy models: absence of arbitrage, including more recent results on the absence of arbitrage in multidimensional models, properties of implied volatility, and modern approaches to hedging in these models. Lévy processes–exponential Lévy models–absence of arbitrage–Esscher transform–implied volatility–smile modeling–quadratic hedging
Publication Stats
2k  Citations  
57.57  Total Impact Points  
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Institutions

20072014

The University of Warwick
 Department of Statistics
Coventry, England, United Kingdom 
King's College London
Londinium, England, United Kingdom


19972007

University of Bath
 Department of Mathematical Sciences
Bath, England, United Kingdom
