Showing papers in "Mathematical Finance in 2012"
TL;DR: In this article, the authors consider the continuous-time market impact model with transient price impact and prove the existence and nonexistence of price manipulation in this model by means of Bochner's theorem.
Abstract: We consider the linear-impact case in the continuous-time market impact model with transient price impact proposed by Gatheral. In this model, the absence of price manipulation in the sense of Huberman and Stanzl can easily be characterized by means of Bochner’s theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure-valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko on the nonexistence of transaction-triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.
208 citations
TL;DR: In this paper, an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion is used to obtain the first and second-order terms in the short time asymptotics of European call option prices.
Abstract: Using an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion, we obtain the first- and second-order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first- and second-order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.
186 citations
TL;DR: In this article, the authors consider the problem of finding a model-free uper bound on the price of a forward-start straddle with payoff |FT2 −FT1 |.
Abstract: We consider the problem of finding a model-free uper 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|Bτ − B0|.
169 citations
TL;DR: In this paper, the authors introduced the concept of time inconsistency in efficiency and defined the induced trade-off, and demonstrated that investors behave irrationally under the precommitted optimal mean-variance portfolio policy when their wealth is above a certain threshold during the investment process.
Abstract: As the dynamic mean-variance portfolio selection formulation does not satisfy the principle of optimality of dynamic programming, phenomena of time inconsistency occur, i.e., investors may have incentives to deviate from the precommitted optimal mean-variance portfolio policy during the investment process under certain circumstances. By introducing the concept of time inconsistency in efficiency and defining the induced trade-off, we further demonstrate in this paper that investors behave irrationally under the precommitted optimal mean-variance portfolio policy when their wealth is above certain threshold during the investment process. By relaxing the self-financing restriction to allow withdrawal of money out of the market, we develop a revised mean-variance policy which dominates the precommitted optimal mean-variance portfolio policy in the sense that, while the two achieve the same mean-variance pair of the terminal wealth, the revised policy enables the investor to receive a free cash flow stream (FCFS) during the investment process. The analytical expressions of the probability of receiving FCFS and the expected value of FCFS are derived.
146 citations
TL;DR: In this paper, the authors established a one-to-one correspondence between law-invariant convex risk measures on L8 and L1, and proved that the canonical model space for the predominant class of law invariant risk measures is L1.
Abstract: In this paper, we establish a one-to-one correspondence between law-invariant convex risk measures on L8 and L1 This proves that the canonical model space for the predominant class of law-invariant convex risk measures is L1
106 citations
TL;DR: In this paper, Fama defined an efficient market as one in which prices always fully reflect available information and provided various characterizations relating to equilibrium models, profitable trading strategies, and equivalent martingale measures.
Abstract: Fama defined an efficient market as one in which prices always “fully reflect” available information. This paper formalizes this definition and provides various characterizations relating to equilibrium models, profitable trading strategies, and equivalent martingale measures. These various characterizations facilitate new insights and theorems relating to efficient markets. In particular, we overcome a well-known limitation in tests for market efficiency, i.e., the need to assume a particular equilibrium asset pricing model, called the joint-hypothesis or bad-model problem. Indeed, we show that an efficient market is completely characterized by the absence of both arbitrage opportunities and dominated securities, an insight that provides tests for efficiency that are devoid of the bad-model problem. Other theorems useful for both the testing of market efficiency and the pricing of derivatives are also provided.
83 citations
TL;DR: In this article, the authors consider practically appealing procedures for estimating intraday volatility measures of financial assets and develop a new approach that enables to approximate the values of the efficient prices at some random times.
Abstract: This paper considers practically appealing procedures for estimating intraday volatility measures of financial assets. The underlying microstructure model accommodates the inherent properties of ultra high-frequency data with the assumption of continuous efficient price processes. In this model, microstructure noise and trading times are endogenous but do not only depend on the prices. Using the (observed) last traded prices of the assets, we develop a new approach that enables to approximate the values of the efficient prices at some random times. Based on these approximated values, we build an estimator of the integrated volatility and give its asymptotic theory. We also give a consistent estimator of the integrated covariation when two assets (asynchronous by construction of the model) are observed.
70 citations
TL;DR: In this paper, a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals is proposed, which involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions.
Abstract: We propose a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invari- ance, subadditivity and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we refor- mulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.
64 citations
TL;DR: In this paper, the problem of forecasting volatility for the multifractal random walk model was studied and a limiting object defined in a quotient space was introduced to avoid the ill-posed problem of estimating the correlation length T of the model.
Abstract: We study the problem of forecasting volatility for the multifractal random walk model. In order to avoid the ill posed problem of estimating the correlation length T of the model, we introduce a limiting object defined in a quotient space; formally, this object is an infinite range logvolatility. For this object and the non limiting object, we obtain precise prediction formulas and we apply them to the problem of forecasting volatility and pricing options with the MRW model in the absence of a reliable estimate of the average volatility and T.
50 citations
TL;DR: In this article, the authors developed an equilibrium asset and option pricing model in a production economy under jump diffusion and provided analytical formulas for an equity premium and a more general pricing kernel that links the physical and risk-neutral densities.
Abstract: This paper develops an equilibrium asset and option pricing model in a production economy under jump diffusion. The model provides analytical formulas for an equity premium and a more general pricing kernel that links the physical and risk-neutral densities. The model explains the two empirical phenomena of the negative variance risk premium and implied volatility smirk if market crashes are expected. Model estimation with the S&P 500 index from 1985 to 2005 shows that jump size is indeed negative and the risk aversion coefficient has a reasonable value when taking the jump into account.
48 citations
TL;DR: In this paper, the authors introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level, and then defaults upon the next arrival of a Poisson process.
Abstract: It is well known that purely structural models of default cannot explain short-term credit spreads, while purely intensity-based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short-term spreads.
TL;DR: In this article, the authors study the super-replication problem in discrete time but with no assumptions on the portfolio process and show that the superreplicating cost differs from the Black-Scholes value of the claim, thus proving the existence of liquidity premium.
Abstract: We study the binomial version of the illiquid market model introduced by Ce tin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Cetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L 2 -approximating sense. Ce tin, Soner, and Touzi investigated the super-replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super-replicating cost differs from the Black-Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super-replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous-time limit. This is an independent justification of the restrictions introduced in Cetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the option's price for a binomial market.
TL;DR: In this paper, specific nonlinear transformations of the Black-Scholes implied volatility were studied to show remarkable properties of the volatility surface, and pricing formulas for European payoffs were given in terms of the implied volatility smile.
Abstract: We study specific nonlinear transformations of the Black–Scholes implied volatility to show remarkable properties of the volatility surface. No arbitrage bounds on the implied volatility skew are given. Pricing formulas for European payoffs are given in terms of the implied volatility smile.
TL;DR: In this paper, power utility maximization for exponential Levy models with portfolio constraints is studied, where utility is obtained from consumption and/or terminal wealth, and a solution in terms of the Levy triplet is constructed under minimal assumptions by solving the Bellman equation.
Abstract: We study power utility maximization for exponential Levy models with portfolio constraints, where utility is obtained from consumption and/or terminal wealth. For convex constraints, an explicit solution in terms of the Levy triplet is constructed under minimal assumptions by solving the Bellman equation. We use a novel transformation of the model to avoid technical conditions. The consequences for q-optimal martingale measures are discussed as well as extensions to nonconvex constraints.
TL;DR: In this article, the authors derived general analytic approximations for pricing European basket and rainbow options on N assets, where the underlying asset prices are assumed to follow lognormal processes, although their results can be extended to certain other price processes for the underlying.
Abstract: We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi- or single-asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi-asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single-asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi-asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi-asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi-asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.
TL;DR: In this paper, a closed-form representation of the joint transition density for European call options is provided. But the model is not suitable for the SABR stochastic volatility model, and the joint density is not robust to forward smile risk.
Abstract: Under the SABR stochastic volatility model, pricing and hedging contracts that are sensitive to forward smile risk (e.g., forward starting options, barrier options) require the joint transition density. In this paper, we address this problem by providing closed-form representations, asymptotically, of the joint transition density. Specifically, we construct an expansion of the joint density through a hierarchy of parabolic equations after applying total volatility-of-volatility scaling and a near-Gaussian coordinate transformation. We then establish an existence result to characterize the truncation error and provide explicit joint density formulas for the first three orders. Our approach inherits the same spirit of a small total volatility-of-volatility assumption as in the original SABR analysis. Our results for the joint transition density serve as a basis for managing forward smile risk. Through numerical experiments, we illustrate the accuracy of our expansion in terms of joint density, marginal density, probability mass, and implied volatilities for European call options.
TL;DR: The Dybvig-Ingersoll-Ross (DIR) theorem states that long-term yields and forward rates can never fall in arbitrage-free term structure models as discussed by the authors.
Abstract: The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term structure models, long-term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long-term rates at earlier dates can dominate long-term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.
TL;DR: In this article, the authors present some further developments in the construction and classification of new solvable one-dimensional diffusion models having transition densities, and other quantities that are fundamental to derivatives pricing, representable in analytically closed form.
Abstract: We present some further developments in the construction and classification of new solvable one-dimensional diffusion models having transition densities, and other quantities that are fundamental to derivatives pricing, representable in analytically closed form Our approach is based on so-called diffusion canonical transformations that produce a large class of multiparameter nonlinear local volatility diffusion models that are mapped onto various simpler diffusions Using an asymptotic analysis, we arrive at a rigorous boundary classification as well as a characterization with respect to probability conservation and the martingale property of the newly constructed diffusions Specifically, we analyze and classify in detail four main families of driftless regular diffusion models that arise from the underlying squared Bessel process (the Bessel family), Cox–Ingersoll–Ross process (the confluent hypergeometric family), the Ornstein-Uhlenbeck diffusion (the OU family), and the Jacobi diffusion (the hypergeometric family) We show that the Bessel family is a superset of the constant elasticity of variance model without drift The Bessel family, in turn, is nested by the confluent hypergeometric family For these two families we find further subfamilies of conservative strict supermartingales and nonconservative martingales with an exit boundary For the new classes of nonconservative regular diffusions we also derive analytically exact first exit time densities that are given in terms of generalized inverse Gaussians and extensions As for the two other new models, we show that the OU family of processes are conservative strict martingales, whereas the Jacobi family are nonconservative nonmartingales Considered as asset price diffusion models, we also show that these models demonstrate a wide range of local volatility shapes and option implied volatility surfaces that include various pronounced skew and smile patterns
TL;DR: In this paper, a bridge between different reduced-form approaches to pricing defaultable claims is built, showing how the well-known formulas by Duffie, Schroder and Skiadas and by Elliott, Jeanblanc, and Yor are related.
Abstract: In this paper, we build a bridge between different reduced-form approaches to pricing defaultable claims. In particular, we show how the well-known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure. Two processes will play a central role: the hazard process and the martingale hazard process attached to a default time. The crucial step is to understand the difference between them, which has been an open question in the literature so far. We show that pseudo-stopping times appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when t is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if t is an arbitrary random (default) time such that its Azema's supermartingale is continuous, then t avoids stopping times.
TL;DR: In this paper, the convergence rate of the quadratic tracking error was analyzed when a Delta-Gamma hedging strategy is used at N discrete times. And the fractional regularity of the payoff function played a crucial role in the choice of the trading dates, in order to achieve optimal rates of convergence.
Abstract: We analyse the convergence rate of the quadratic tracking error, when a Delta-Gamma hedging strategy is used at N discrete times. The fractional regularity of the payoff function plays a crucial role in the choice of the trading dates, in order to achieve optimal rates of convergence.
TL;DR: In this paper, the authors examined the valuation of a generalized American-style option known as a game-style call option in an infinite time horizon setting and found that the value function is not convex when r > d.
Abstract: This paper examines the valuation of a generalized American-style option known as a game-style call option in an infinite time horizon setting. The specifications of this contract allow the writer to terminate the call option at any point in time for a fixed penalty amount paid directly to the holder. Valuation of a perpetual game-style put optionwasaddressedbyKyprianou(2004)inaBlack-Scholessettingonanondividend paying asset. Here, we undertake a similar analysis for the perpetual call option in the presence of dividends and find qualitatively different explicit representations for the value function depending on the relationship between the interest rate and dividend yield. Specifically, we find that the value function is not convex whenr > d. Numerical results show the impact this phenomenon has upon the vega of the option.
TL;DR: In this paper, the expected shortfall of portfolios whose value is a quadratic function of a number of risk factors, as arise from a Delta-Gamma-Theta approximation, is derived.
Abstract: Computable expressions are derived for the Expected Shortfall of portfolios whose value is a quadratic function of a number of risk factors, as arise from a Delta-Gamma-Theta approximation. The risk factors are assumed to follow an elliptical multivariate t distribution, reflecting the heavy-tailed nature of asset returns. Both an exact expression and a uniform asymptotic expansion are presented. The former involves only a single rapidly convergent integral. The latter is essentially explicit, and numerical experiments suggest that its error is negligible compared to that incurred by the Delta-Gamma-Theta approximation.
TL;DR: In this paper, the Dybvig-Ingersoll-Ross theorem was extended to the long-term limit of zero-coupon rates with respect to the maturity.
Abstract: The long-term limit of zero-coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig–Ingersoll–Ross theorem, which says that long-term spot and forward rates can never fall in an arbitrage-free model. Extensions of popular interest rate models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous-time short-rate models.
TL;DR: In this paper, the Fatou property for Schur convex lower semicontinuous functional on a general probability space is established and the existing quantile representations for coherent risk measures and law invariant deviation measures on an atomless probability space are extended for general probability spaces.
Abstract: The Fatou property for every Schur convex lower semicontinuous (l.s.c.) functional on a general probability space is established. As a result, the existing quantile representations for Schur convex l.s.c. positively homogeneous convex functionals, established on for either p= 1 or p=∞ and with the requirement of the Fatou property, are generalized for , with no requirement of the Fatou property. In particular, the existing quantile representations for law invariant coherent risk measures and law invariant deviation measures on an atomless probability space are extended for a general probability space.
TL;DR: In this paper, the authors analyzed portfolio risk and volatility in the presence of constraints on portfoliorebalancing frequency, and derived limiting results, as the rebalancing frequency increases, for the difference between discretely and continuously rebalanced portfolios.
Abstract: This paper analyzes portfolio risk and volatility in the presence of constraints on portfoliorebalancingfrequency. Thisinvestigationismotivatedbytheincremental risk charge (IRC) introduced by the Basel Committee on Banking Supervision. In contrast to the standard market risk measure based on a 10-day value-at-risk calculated at 99% confidence, the IRC considers more extreme losses and is measured over a 1year horizon. More importantly, whereas 10-day VaR is ordinarily calculated with a portfolio’s holdings held fixed, the IRC assumes a portfolio is managed dynamically to a target level of risk, with constraints on rebalancing frequency. The IRC uses discrete rebalancing intervals (e.g., monthly or quarterly) as a rough measure of potential illiquidity in underlying assets. We analyze the effect of these rebalancing intervals on the portfolio’s profit and loss distribution over a risk-measurement horizon. We derive limiting results, as the rebalancing frequency increases, for the difference between discretely and continuously rebalanced portfolios; we use these to approximate the loss distribution for the discretely rebalanced portfolio relative to the continuously rebalanced portfolio. Our analysis leads to explicit measures of the impact of discrete rebalancing under a simple model of asset dynamics.
TL;DR: In this paper, a modification of Leland's strategy is proposed to ensure that the approximation error vanishes in the limit of the number of revisions, where the transaction costs rate does not depend on the frequency of revisions.
Abstract: In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black-Scholes formula with a suitably adjusted volatility for a periodical revision of the portfolio whose terminal value approximates the pay-off. Unfortunately, if the transaction costs rate does not depend on the number of revisions, the approximation error does not converge to zero as the frequency of revisions tends to infinity. In the present paper, we suggest a modification of Leland's strategy ensuring that the approximation error vanishes in the limit.
TL;DR: In this article, the authors considered the optimal decision to sell or buy a stock in a given period with reference to the ultimate average of the stock price and provided a partial differential equation approach to characterize the free boundary (or equivalently, the optimal selling region).
Abstract: We are concerned with the optimal decision to sell or buy a stock in a given period with reference to the ultimate average of the stock price. More precisely, we aim to determine an optimal selling (buying) time to maximize (minimize) the expectation of the ratio of the selling (buying) price to the ultimate average price over the period. This is an optimal stopping time problem which can be formulated as a variational inequality problem. The problem gives rise to a free boundary that corresponds to the optimal selling (buying) strategy. We provide a partial differential equation approach to characterize the free boundary (or equivalently, the optimal selling (buying) region). It turns out that the optimal selling strategy is bang-bang, which is the same as that obtained by Shiryaev, Xu, and Zhou taking the ultimate maximum of the stock price as benchmark, whereas the optimal buying strategy can be a feedback one subject to the type of averaging and parameter values. Moreover, by a thorough characterization of free boundary, we reveal that the bang-bang optimal selling strategy heavily depends on the assumption that no time-vesting restrictions are imposed. If a time-vested stock is considered, then the optimal selling strategy can also be a feedback one. In terms of a similar analysis developed by the present paper, the same phenomenon can be proved when taking the ultimate maximum as benchmark.
TL;DR: In this paper, a new measure of skewness is proposed, which can be linked to prospective satisficing risk measures and tail risk measures such as Value-at-Risk.
Abstract: This paper presents a new measure of skewness, skewness-aware deviation, that can be linked to prospective satisficing risk measures and tail risk measures such as Value-at-Risk. We show that this measure of skewness arises naturally also when one thinks of maximizing the certainty equivalent for an investor with a negative exponential utility function, thus bringing together the mean-risk, expected utility, and prospective satisficing measures frameworks for an important class of investor preferences. We generalize the idea of variance and covariance in the new skewness-aware asset pricing and allocation framework. We show via computational experiments that the proposed approach results in improved and intuitively appealing asset allocation when returns follow real-world or simulated skewed distributions. We also suggest a skewness-aware equivalent of the classical Capital Asset Pricing Model beta, and study its consistency with the observed behavior of the stocks traded at the NYSE between 1963 and 2006.
TL;DR: In this article, a simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution, and algebraic geometry techniques are employed to show that this model is well posed for default dependence: it represents any given marginal distribution for single firms and pairwise correlation matrix.
Abstract: A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for default dependence: it represents any given marginal distribution for single firms and pairwise correlation matrix. These techniques also provide a calibration algorithm based on maximum likelihood estimation. Finally, the model is compared with standard normal copula model in terms of tails of the loss distribution and implied correlation smile.
TL;DR: In this article, the existence of a dynamic mortgage rate process is proved by constructing a solution using a newly proposed level set method, which resolves the circular dependence of the mortgage rate.
Abstract: The mortgage rate is a major factor in the refinancing decision. The refinancing behavior influences cash flow and, therefore, mortgage price. The prices of mortgage instruments drives the mortgage rates. We consider a problem of the existence of a dynamic mortgage rate process which resolves this circular dependence. The existence is proved by constructing a solution using a newly proposed level set method.