Analysis of Error with Malliavin Calculus: Application to Hedging

The aim of this paper is to compute the quadratic error of a discrete time-hedging strategy in a complete multidimensional model. This result extends that of Gobet and Temam (2001) and Zhang (1999) . More precisely, our basic assumption is that the asset prices satisfy the d -dimensional stochastic differential equation   dXⁱ_t = Xⁱ_t ( bⁱ ( X_t ) dt +σ _{i , j} ( X_t ) dW^j_t )  . We precisely describe the risk of this strategy with respect to n , the number of rebalancing times. The rates of convergence obtained are     for any options with Lipschitz payoff and  1/ n ^1/4  for options with irregular payoff.     

A NOTE ON ARBITRAGE AND CLOSED CONVEX CONES

We give an example of a subspace K of     such that     , where     denotes the closure with respect to convergence in probablity. On the other hand, the cone   C ≔ K − L ^∞₊  is dense in   L ^∞  with respect to the weak-star topology  σ( L ^∞, L ¹)  . This example answers a question raised by I. Evstigneev. The topic is motivated by the relation of the notion of no arbitrage and the existence of martingale measures in Mathematical Finance.     

INTENSITY-BASED VALUATION OF RESIDENTIAL MORTGAGES: AN ANALYTICALLY TRACTABLE MODEL

This paper presents an analytically tractable valuation model for residential mortgages. The random mortgage prepayment time is assumed to have an intensity process of the form h _t = h ₀( t ) +γ ( k − r _t )⁺ , where h ₀( t ) is a deterministic function of time, r _t is the short rate, and γ and k are scalar parameters. The first term models exogenous prepayment independent of interest rates (e.g., a multiple of the PSA prepayment function). The second term models refinancing due to declining interest rates and is proportional to the positive part of the distance between a constant threshold level and the current short rate. When the short rate follows a CIR diffusion, we are able to solve the model analytically and find explicit expressions for the present value of the mortgage contract, its principal-only and interest-only parts, as well as their deltas. Mortgage rates at origination are found by solving a non-linear equation. Our solution method is based on explicitly constructing an eigenfunction expansion of the pricing semigroup, a Feynman-Kac semigroup of the CIR diffusion killed at an additive functional that is a linear combination of the integral of the CIR process and an area below a constant threshold and above the process sample path (the so-called area functional). A sensitivity analysis of the term structure of mortgage rates and calibration of the model to market data are presented.     

STOCHASTIC VOLATILITY MODELS, CORRELATION, AND THE q-OPTIMAL MEASURE

The aim of this paper is to study the minimal entropy and variance-optimal martingale measures for stochastic volatility models. In particular, for a diffusion model where the asset price and volatility are correlated, we show that the problem of determining the q -optimal measure can be reduced to finding a solution to a representation equation. The minimal entropy measure and variance-optimal measure are seen as the special cases   q = 1  and   q = 2  respectively. In the case where the volatility is an autonomous diffusion we give a stochastic representation for the solution of this equation. If the correlation ρ between the traded asset and the autonomous volatility satisfies  ρ² < 1/ q   , and if certain smoothness and boundedness conditions on the parameters are satisfied, then the q -optimal measure exists. If  ρ²≥ 1/ q   , then the q -optimal measure may cease to exist beyond a certain time horizon. As an example we calculate the q -optimal measure explicitly for the Heston model.     

The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time

We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's modeling of foreign exchange markets under transaction costs. The financial market is described by a   d × d   matrix-valued stochastic process  (Π _t ) ^T _{t =0}  specifying the mutual bid and ask prices between d assets. We introduce the notion of "robust no arbitrage," which is a version of the no-arbitrage concept, robust with respect to small changes of the bid-ask spreads of  (Π _t ) ^T _{t =0}  . The main theorem states that the bid-ask process  (Π _t ) ^T _{t =0}  satisfies the robust no-arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison-Pliska and Kabanov-Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang, Morton, and Willinger and Kabanov, Rásonyi, and Stricker, pertaining to the case of general Ω. An example of a  5 × 5  -dimensional process  (Π _t )² _{t =0}  shows that, in this theorem, the robust no-arbitrage condition cannot be replaced by the so-called strict no-arbitrage condition, thus answering negatively a question raised by Kabanov, Rásonyi, and Stricker.     

Asymptotics of the price oscillations of a European call option in a tree model

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black-Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/ n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/ n in the case of usual binomial models since the second term (in     ) vanishes. The next term is of type   C ₂( n )/ n   , with   C ₂( n )  some explicit bounded function of n that has no limit when n tends to infinity.     

A COUNTEREXAMPLE CONCERNING THE VARIANCE-OPTIMAL MARTINGALE MEASURE

The present note addresses an open question concerning a sufficient characterization of the variance-optimal martingale measure. Denote by S the discounted price process of an asset and suppose that   Q ^★  is an equivalent martingale measure whose density is a multiple of  1 −φ· S _T   for some S -integrable process φ. We show that   Q ^★  does not necessarily coincide with the variance-optimal martingale measure, not even if  φ· S   is a uniformly integrable   Q ^★  -martingale.     

Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

In this paper we briefly present the results obtained in our paper ( Talay and Zheng 2002a ) on the convergence rate of the approximation of quantiles of the law of one component of  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.     

EQUITY CORRELATIONS IMPLIED BY INDEX OPTIONS: ESTIMATION AND MODEL UNCERTAINTY ANALYSIS

We propose a method for constructing an arbitrage‐free multiasset pricing model which is consistent with a set of observed single‐ and multiasset derivative prices. The pricing model is constructed as a random mixture of N reference models, where the distribution of mixture weights is obtained by solving a well‐posed convex optimization problem. Application of this method to equity and index options shows that, whereas multivariate diffusion models with constant correlation fail to match the prices of index and component options simultaneously, a jump‐diffusion model with a common jump component affecting all stocks enables to do so. Furthermore, we show that even within a parametric model class, there is a wide range of correlation patterns compatible with observed prices of index options. Our method allows, as a by product, to quantify this model uncertainty with no further computational effort and propose static hedging strategies for reducing the exposure of multiasset derivatives to model uncertainty.     

Step Options

Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock‐out and knock‐in provisions and introduce a family of path‐dependent options: step options . They are parametrized by a finite knock‐out (knock‐in) rate , ρ. For a down‐and‐out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the knock‐out factor exp(-ρτ_B^‐) or max(1‐ρτ^-B,0), where &\tau;_B^‐ is the total time during the contract life that the underlying price was lower than a prespecified barrier level ( occupation time ). We derive closed‐form pricing formulas for step options with any knock‐out rate in the range $[0,∞). For any finite knock‐out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no‐regrets" alternatives to standard barrier options. As a by‐product, we derive a dynamic almost‐replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock‐out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed‐form pricing formulas are derived.     

BIVARIATE SUPPORT OF FORWARD LIBOR AND SWAP RATES

Based on a certain notion of "prolific process," we find an explicit expression for the bivariate (topological) support of the solution to a particular class of 2 × 2 stochastic differential equations that includes those of the three-period "lognormal" Libor and swap market models. This yields that in the lognormal swap market model (SMM), the support of the 1 × 1 forward Libor   L * _t   equals  [ l * _t , ∞)  for some semi-explicit  −1 ≤ l * _t ≤ 0  , sharpening a result of Davis and Mataix-Pastor (2007) that forward Libor rates (eventually) become negative with positive probability in the lognormal SMM. We classify the instances   l * _t < 0  , and explicitly calculate the threshold time at or before which   L * _t   remains positive a.s.     

A Martingale Representation Result and an Application to Incomplete Financial Markets

We establish necessary and sufficient conditions for an H¹-martingale to be representable with respect to a collection, of local martingales. M H¹( P ) is representable if and only if M is a local martingale under all p.m.'s Q which are "uniformly equivalent" to P and which make all the elements of local martingales (Theorem 1.1). We then give necessary and sufficient conditions which are easier to verify, and only involve expectations (Theorem 1.2). We go on to apply these results to the problem of pricing claims in an incomplete financial market-establishing two conjectures of Harrison and Pliska(1981).     

A UNIVERSAL OPTIMAL CONSUMPTION RATE FOR AN INSIDER

We consider a cash flow   X ^{( c )} ( t )  modeled by the stochastic equation where B (·) and     are a Brownian motion and a Poissonian random measure, respectively, and   c ( t ) ≥ 0  is the consumption/dividend rate. No assumptions are made on adaptedness of the coefficients  μ, σ, θ  , and c , and the (possibly anticipating) integrals are interpreted in the forward integral sense. We solve the problem to find the consumption rate c (·), which maximizes the expected discounted utility given by Here  δ( t ) ≥ 0  is a given measurable stochastic process representing a discounting exponent and τ is a random time with values in (0, ∞), representing a terminal/default time, while  γ≥ 0  is a known constant.     

ON THE CONSISTENCY OF REGRESSION‐BASED MONTE CARLO METHODS FOR PRICING BERMUDAN OPTIONS IN CASE OF ESTIMATED FINANCIAL MODELS

In many applications of regression‐based Monte Carlo methods for pricing, American options in discrete time parameters of the underlying financial model have to be estimated from observed data. In this paper suitably defined nonparametric regression‐based Monte Carlo methods are applied to paths of financial models where the parameters converge toward true values of the parameters. For various Black–Scholes, GARCH, and Levy models it is shown that in this case the price estimated from the approximate model converges to the true price.     

MONTE CARLO METHODS FOR THE VALUATION OF MULTIPLE-EXERCISE OPTIONS

We discuss Monte Carlo methods for valuing options with multiple-exercise features in discrete time. By extending the recently developed duality ideas for American option pricing, we show how to obtain estimates on the prices of such options using Monte Carlo techniques. We prove convergence of our approach and estimate the error. The methods are applied to options in the energy and interest rate derivative markets.     

From Discrete to Continuous Financial Models: New Convergence Results For Option Pricing

In this paper we develop a new notion of convergence for discussing the relationship between discrete and continuous financial models, D ²-convergence. This is stronger than weak convergence, the commonly used mode of convergence in the finance literature. We show that D ²-convergence, unlike weak convergence, yields a number of important convergence preservation results, including the convergence of contingent claims, derivative asset prices and hedge portfolios in the discrete Cox-Ross-Rubinstein option pricing models to their continuous counterparts in the Black-Scholes model. Our results show that D ²-convergence is characterized by a natural lifting condition from nonstandard analysis (NSA), and we demonstrate how this condition can be reformulated in standard terms, i.e., in language that only involves notions from standard analysis. From a practical point of view, our approach suggests procedures for constructing good (i.e., convergent) approximate discrete claims, prices, hedge portfolios, etc. This paper builds on earlier work by the authors, who introduced methods from NSA to study problems arising in the theory of option pricing.     

LATTICE OPTION PRICING BY MULTIDIMENSIONAL INTERPOLATION

This paper proposes a method for pricing high-dimensional American options based on modern methods of multidimensional interpolation. The method allows using sparse grids and thus mitigates the curse of dimensionality. A framework of the pricing algorithm and the corresponding interpolation methods are discussed, and a theorem is demonstrated, which suggests that the pricing method is less vulnerable to the curse of dimensionality. The method is illustrated by an application to rainbow options and compared to least squares Monte Carlo and other benchmarks.     

RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS

We study the risk indifference pricing principle in incomplete markets: The (seller's)  risk indifference price        is the initial payment that makes the  risk  involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to find a formula for       and similarly for      . In particular, we prove that  where    p _low   and    p _up   are the lower and upper hedging prices, respectively.     

PSEUDODIFFUSIONS AND QUADRATIC TERM STRUCTURE MODELS

The non-Gaussianity of processes observed in financial markets and the relatively good performance of Gaussian models can be reconciled by replacing the Brownian motion with Lévy processes whose Lévy densities decay as  exp(−λ| x |)  or faster, where  λ > 0  is large. This leads to asymptotic pricing models. The leading term, P ₀, is the price in the Gaussian model with the same instantaneous drift and variance. The first correction term depends on the instantaneous moments of order up to 3, that is, the skewness is taken into account, the next term depends on moments of order 4 (kurtosis) as well, etc. In empirical studies, the asymptotic formula can be applied without explicit specification of the underlying process: it suffices to assume that the instantaneous moments of order greater than 2 are small w.r.t. moments of order 1 and 2, and use empirical data on moments of order up to 3 or 4. As an application, the bond-pricing problem in the non-Gaussian quadratic term structure model is solved. For pricing of options near expiry, a different set of asymptotic formulas is developed; they require more detailed specification of the process, especially of its jump part. The leading terms of these formulas depend on the jump part of the process only, so that they can be used in empirical studies to identify the jump characteristics of the process.     

Pricing Multiasset Cross‐Currency Options

This study develops a general pricing method for multiasset cross‐currency options, whose underlying asset consists of multiple different assets, and the evaluation currency is different from the ones used in the most liquid market of each asset; the examples include cross‐currency options, cross‐currency basket options, and cross‐currency average options. Moreover, in practice, fast calibration is necessary in the option markets relevant for the underlying assets and the currency, which is also achieved in this study. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 34:1–19, 2014