David Hobson

The University of Warwick, Coventry, England, United Kingdom

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Publications (64)40.84 Total impact

  • Han Feng, David Hobson
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    ABSTRACT: This paper studies a variant of the contest model introduced by Seel and Strack. In the Seel-Strack 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 two-player 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, non-negative, random variables that have a common law $\mu$. We consider a two-player 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.
    05/2014;
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    ABSTRACT: We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous 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.
    06/2013;
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    David Hobson, Martin Klimmek
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    ABSTRACT: In this article we consider the problem of giving a robust, model-independent, 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 $(F_t)_{t \geq 0}$, 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 which 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 semi-static subhedge. Under an assumption on the supports of the marginal laws, but no assumption that the laws are atom-free 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 semi-static subhedge.
    04/2013;
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    Han Feng, David Hobson
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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.
    Mathematical Finance 01/2013; · 1.25 Impact Factor
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    David G. Hobson, Anthony Neuberger
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    ABSTRACT: We consider the problem of finding a model-free upper bound on the price of a forward-start 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 super-replicating 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 model-based 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 Black-Scholes price. The problem can be recast as finding the solution to a Skorokhod embedding problem with non-trivial initial law so as to maximise E|B − B0|.
    Mathematical Finance 01/2012; · 1.25 Impact Factor
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    Vicky Henderson, David Hobson
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    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.
    Mathematical Finance 06/2011; 21(3):365 - 382. · 1.25 Impact Factor
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    David Hobson, Martin Klimmek
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    ABSTRACT: A variance swap is a derivative with a path-dependent 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 model-independent, no-arbitrage bounds on the price of the variance swap, and corresponding sub- and super-replicating strategies. Further, we characterise the optimal bounds. The form of the hedges depends crucially on the kernel used to define the variance swap.
    Finance and Stochastics 04/2011; · 1.21 Impact Factor
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    ABSTRACT: Given a centred distribution, can one find a time-homogeneous martingale diffusion starting at zero which has the given law at time 1? We answer the question affirmatively if generalized diffusions are allowed.
    Probability Theory and Related Fields 03/2011; · 1.39 Impact Factor
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    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 Black-Scholes 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.
    International Journal of Theoretical and Applied Finance 01/2011;
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    David Hobson, Martin Klimmek
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    ABSTRACT: Consider a set of discounted optimal stopping problems for a one-parameter 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.
    Stochastics An International Journal of Probability and Stochastic Processes 01/2011; · 0.51 Impact Factor
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    David Hobson, Martin Klimmek
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    ABSTRACT: The Az\'{e}ma-Yor 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}ma-Yor 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}ma-Yor and Perkins stopping times to converge. In the case of the Az\'{e}ma-Yor 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).
    The Annals of Applied Probability 12/2010; 23(5). · 1.37 Impact Factor
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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, risk-neutral 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 Black–Scholes model (but if we know the prices of more than one call then together they will typically be inconsistent with the Black–Scholes 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 1–1 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.
    10/2010: pages 267-318;
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    ABSTRACT: In this article we study a decoupled forward backward stochastic differential equation (FBSDE) and the associated system of partial integro-differential obstacle problems, in a flexible Markovian set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper. This financial motivation is developed in the first part of the paper, which provides a synthetic view of the theory of pricing and hedging financial derivatives, using backward stochastic differential equations (BSDEs) as main tool. In the second part of the paper, we establish the well-posedness of reflected BSDEs with jumps coming out of the pricing and hedging problems exposed in the first part. We first provide a construction of a Markovian model made of a jump-diffusion – like component X interacting with a continuous-time Markov chain – like component N. The jump process N defines the so-called regime of the coefficients of X, whence the name of jump-diffusion with regimes for this model. Motivated by optimal stopping and optimal stopping game problems (pricing equations of American or game contingent claims), we introduce the related reflected and doubly reflected Markovian BSDEs, showing that they are well-posed in the sense that they have unique solutions, which depend continuously on their input data. As an aside, we establish the Markov property of the model. In the third part of the paper we derive the related variational inequality approach. We first introduce the systems of partial integro-differential variational inequalities formally associated to the reflected BSDEs, and we state suitable definitions of viscosity solutions for these problems, accounting for jumps and/or systems of equations. We then show that the state-processes (first components Y ) of the solutions to the reflected BSDEs can be characterized in terms of the value functions of related optimal stopping or game problems, given as viscosity solutions with polynomial growth to related integro-differential obstacle problems. We further establish a comparison principle for semi-continuous viscosity solutions to these problems, which implies in particular the uniqueness of the viscosity solutions. This comparison principle is subsequently used for proving the convergence of stable, monotone and consistent approximation schemes to the value functions. Finally in the last part of the paper we provide various extensions of the results needed for applications in finance to pricing problems involving discrete dividends on a financial derivative or on the underlying asset, as well as various forms of discrete path-dependence.
    10/2010: pages 63-203;
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    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
    10/2010: pages 319-359;
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    ABSTRACT: In this first chapter, we show that a CDO tranche payoff can be perfectly replicated with a self-financed strategy based on the underlying credit default swaps. This extends to any payoff which depends only upon default arrivals, such as basket default swaps. Clearly, the replication result is model dependent and relies on two critical assumptions. First, we preclude the possibility of simultaneous defaults. The other assumption is that credit default swap premiums are adapted to the filtration of default times which therefore can be seen as the relevant information set on economic grounds. Our framework corresponds to a pure contagion model, where the arrivals of defaults lead to jumps in the credit spreads of survived names, the magnitude of which depending upon the names in question, and the whole history of defaults up to the current time. These jumps can be related to the derivatives of the joint survival function of default times. The dynamics of replicating prices of CDO tranches follows the same way. In other words, we only deal with default risks and not with spread risks. Unsurprisingly, the possibility of perfect hedging is associated with a martingale representation theorem under the filtration of default times. Subsequently, we exhibit a new probability measure under which the short term credit spreads (up to some scaling factor due to positive recovery rates) are the intensities associated with the corresponding default times. For ease of presentation, we introduced first some instantaneous default swaps as a convenient basis of hedging instruments. Eventually, we can exhibit a replicating strategy of a CDO tranche payoff with respect to actually traded credit default swaps, for instance, with the same maturity as the CDOtranche. Letus note that no Markovian assumption is required for the existence of such a replicating strategy. However, the practical implementation of actual hedging strategies requires some extra assumptions. We assume that all pre-default intensities are equal and only depend upon the current number of defaults. We also assume that all recovery rates are constant across names and time. In that framework, it can be shown that the aggregate loss process is a homogeneous Markov chain, more precisely a pure death process. Thanks to these restrictions, the model involves as many unknown parameters as the number of underlying names. Such Markovian model is also known as a local intensity model, the simplest form of aggregate loss models. As in local volatility models in the equity derivatives world, there is a perfect match of unknown parameters from a complete set of CDO tranches quotes. Numerical implementation can be achieved through a binomial tree, well-known to finance people, or by means of Markov chain techniques. We provide some examples and show that the market quotes of CDOs are associated with pronounced contagion effects. We can therefore explain the dynamics of the amount of hedging CDS and relate them to deltas computed by market practitioners. The figures are hopefully roughly the same, the discrepancies being mainly explained by contagion effects leading to an increase of dependence between default times after some defaults.
    06/2010: pages 1-61;
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    ABSTRACT: This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.
    06/2010: pages 205-266;
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    David Hobson
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    ABSTRACT: Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.
    Finance and Stochastics 01/2010; 14(1):129-152. · 1.21 Impact Factor
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    ABSTRACT: We solve explicitly the following problem: for a given probability measure mu, we specify a generalised martingale diffusion X which, stopped at an independent exponential time T, is distributed according to mu. The process X is specified via its speed measure m. We present three proofs. First we show how the result can be derived from the solution of Bertoin and Le Jan (1992) to the Skorokhod embedding problem. Secondly, we give a proof exploiting applications of Krein's spectral theory of strings to the study of linear diffusions. Finally, we present a novel direct probabilistic proof based on a coupling argument. Comment: 15 pages, 2 figures
    12/2009;
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    Vicky Henderson, David Hobson
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    ABSTRACT: In this paper we model the behavior of a risk averse agent who seeks to maximize ex-pected utility and who has a timing option over when to sell an indivisible asset. Our first contribution is to show that, contrary to intuition, optimal behavior for such a risk averse agent can include risk increasing gambles. In a discrete time setting, except for a few degenerate cases, there is always a wealth range at which it is optimal to gamble. A different perspective yields that the presence of gambling opportunities can increase the equilibrium price of an indivisible asset. In continuous time, the smoothing effect of the choice of selling time, means that it is only optimal to gamble for certain parameter ranges.
    05/2009;
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    Erik Ekstrom, David Hobson
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    ABSTRACT: It is well-known how to determine the price of perpetual American options if the underlying stock price is a time-homogeneous diffusion. In the present paper we consider the inverse problem, i.e. given prices of perpetual American options for different strikes we show how to construct a time-homogeneous model for the stock price which reproduces the given option prices.
    The Annals of Applied Probability 01/2009; · 1.37 Impact Factor

Publication Stats

1k Citations
40.84 Total Impact Points

Institutions

  • 2002–2013
    • The University of Warwick
      • • Department of Statistics
      • • Warwick Business School (WBS)
      Coventry, England, United Kingdom
  • 2012
    • Warwick Business School
      Warwick, England, United Kingdom
  • 2007
    • King's College London
      Londinium, England, United Kingdom
  • 1997–2007
    • University of Bath
      • Department of Mathematical Sciences
      Bath, England, United Kingdom
  • 2004
    • Princeton University
      • Bendheim Center for Finance
      Princeton, New Jersey, United States