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.     

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.     

SPARSE CALIBRATIONS OF CONTINGENT CLAIMS

The two problems of determining the existence of arbitrage among a finite set of options and of calculating the supremum price of an option consistent with other options prices have been reduced to finding an appropriate model of bounded size in many special cases. We generalize this result to a class of arbitrage-free  m -period markets with    d  + 1   basic securities and with no prior measure. We show there are no dominating trading strategies for a given set of  l  contingent claims if and only if their bid-ask prices are asymptotically consistent with models supported by at most   ( l  +  d  + 1)( d  + 1) ^{m −1}   points, if    m  ≥ 1  . An example showing the tightness of our bound is given.     

A Nonstandard Approach to Option Pricing

Nonstandard probability theory and stochastic analysis, as developed by Loeb, Anderson, and Keisler, has the attractive feature that it allows one to exploit combinatorial aspects of a well-understood discrete theory in a continuous setting. We illustrate this with an example taken from financial economics: a nonstandard construction of the well-known Black-Scholes option pricing model allows us to view the resulting object at the same time as both (the hyperfinite version of) the binomial Cox-Ross-Rubinstein model (that is, a hyperfinite geometric random walk) and the continuous model introduced by Black and Scholes (a geometric Brownian motion). Nonstandard methods provide a means of moving freely back and forth between the discrete and continuous points of view. This enables us to give an elementary derivation of the Black-Scholes option pricing formula from the corresponding formula for the binomial model. We also devise an intuitive but rigorous method for constructing self-financing hedge portfolios for various contingent claims, again using the explicit constructions available in the hyperfinite binomial model, to give the portfolio appropriate to the Black-Scholes model. Thus, nonstandard analysis provides a rigorous basis for the economists' intuitive notion that the Black-Scholes model contains a built-in version of the Cox-Ross-Rubinstein model.     

Hedging under model misspecification: All risk factors are equal,but some are more equal than others …

It is often difficult to distinguish among different option pricing models that consider stochastic volatility and/or jumps based on a cross‐section of European option prices. This can result in model misspecification. We analyze the hedging error induced by model misspecification and show that it can be economically significant in the cases of a delta hedge, a minimum‐variance hedge, and a delta‐vega hedge. Furthermore, we explain the surprisingly good performance of a simple ad‐hoc Black‐Scholes hedge. We compare realized hedging errors (an incorrect hedge model is applied) and anticipated hedging errors (the hedge model is the true one) and find that there are substantial differences between the two distributions, particularly depending on whether stochastic volatility is included in the hedge model. Therefore, hedging errors can be useful for identifying model misspecification. Furthermore, model risk has severe implications for risk measurement and can lead to a significant misestimation, specifically underestimation, of the risk to which a hedged position is exposed. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Option Pricing in Discrete‐Time Incomplete Market Models

Various aspects of pricing of contingent claims in discrete time for incomplete market models are studied. Formulas for prices with proportional transaction costs are obtained. Some results concerning pricing with concave transaction costs are shown. Pricing by the expected utility of terminal wealth is also considered.     

THE TERM STRUCTURE OF INTEREST RATES AS A GAUSSIAN RANDOM FIELD

A simple model of the term structure of interest rates is introduced in which the family of instantaneous forward rates evolves as a continuous Gaussian random field. A necessary and sufficient condition for the associated family of discounted zero-coupon bond prices to be martingales is given, permitting the consistent pricing of interest rate contingent claims. Examples of the pricing of interest-rate caps and the situation when the Gaussian random field may be viewed as a deterministic time change of the standard Brownian sheet are discussed.     

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.     

Toward A Convergence Theory For Continuous Stochastic Securities Market Models1

This article reviews some recently developed approximation schemes for financial markets with continuous trading. Two methods for approximating continuous-time stochastic securities market models whose exogenously given prices have continuous sample paths are described and compared One method approximates both the paths and the information structure; the other is an approximation in distribution with a Markovian structure. In both cases, the approximating models have a finite state space, discrete time, and possess the same “structural” properties (e.g., “no arbitrage” and “completeness”) as the continuous model. the latter characteristic is an important criterion for judging the merits of the approximations. Taking advantage of the “structure-preserving” characteristic, one can formulate a convergence theory for frictionless markets with continuous trading. the theory provides convergence results for objects such as contingent claim prices, replicating portfolio strategies (hedging policies), optimal consumption policies, and cumulative financial gains (i.e., stochastic integrals), which are constructed along the approximation. the convergence theory enables one to combine the intuitive appeal of discrete models and the analytic tractability of continuous models to provide new insight into the theory of modern financial markets. We survey the current state of such a convergence theory and illustrate the results with some examples of well-known continuous securities market models.     

HEDGING WITH ENERGY

In the setting of diffusion models for price evolution, we suggest an easily implementable approximate evaluation formula for measuring the errors in option pricing and hedging due to volatility misspecification. The main tool we use in this paper is a (suitably modified) classical inequality for the L ² norm of the solution, and the derivatives of the solution, of a partial differential equation (the so-called "energy" inequality). This result allows us to give bounds on the errors implied by the use of approximate models for option valuation and hedging and can be used to justify formally some "folk" belief about the robustness of the Black and Scholes model. Surprisingly enough, the result can also be applied to improve pricing and hedging with an approximate model. When statistical or a priori information is available on the "true" volatility, the error measure given by the energy inequality can be minimized w.r.t. the parameters of the approximating model. The method suggested in this paper can help in conjugating statistical estimation of the volatility function derived from flexible but computationally cumbersome statistical models, with the use of analytically tractable approximate models calibrated using error estimates.     

When Does Convergence of Asset Price Processes Imply Convergence of Option Prices?

We consider weak convergence of a sequence of asset price models (Sⁿ) to a limiting asset price model S . A typical case for this situation is the convergence of a sequence of binomial models to the Black–Scholes model, as studied by Cox, Ross, and Rubinstein. We put emphasis on two different aspects of this convergence: first we consider convergence with respect to the given "physical" probability measures (P^n) and second with respect to the "risk‐neutral" measures (Q^n) for the asset price processes (Sⁿ) . (In the case of nonuniqueness of the risk-neutral measures the question of the "good choice" of (Qⁿ) also arises.) In particular we investigate under which conditions the weak convergence of (Pⁿ) to P implies the weak convergence of (Qⁿ) to Q and thus the convergence of prices of derivative securities.
The main theorem of the present paper exhibits an intimate relation of this question with contiguity properties of the sequences of measures (Pⁿ) with respect to (Qⁿ) , which in turn is closely connected to asymptotic arbitrage properties of the sequence (Sⁿ) of security price processes. We illustrate these results with general homogeneous binomial and some special trinomial models.     

PRICING OF AMERICAN PATH-DEPENDENT CONTINGENT CLAIMS

We consider the problem of pricing path-dependent contingent claims. Classically, this problem can be cast into the Black-Scholes valuation framework through inclusion of the path-dependent variables into the state space. This leads to solving a degenerate advection-diffusion partial differential equation (PDE). We first estabilish necessary and sufficient conditions under which degenerate diffusions can be reduced to lower-dimensional nondegenerate diffusions. We apply these results to path-dependent options. Then, we describe a new numerical technique, called forward shooting grid (FSG) method, that efficiently copes with degenerate diffusion PDEs. Finally, we show that the FSG method is unconditionally stable and convergent. the FSG method is the first capable of dealing with the early exercise condition of American options. Several numerical examples are presented and discussed. ²     

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.     

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.     

Dividend‐Rollover Effect and the Ad Hoc Black‐Scholes Model

One of the most widely used option‐valuation models among practitioners is the ad hoc Black‐Scholes (AHBS) model. The main contribution of this study is methodological. We carefully consider three dividend strategies (No dividend, Implied‐forward dividend, and Actual dividend) for the AHBS model to investigate their effect on pricing errors. We suggest a new dividend strategy, implied‐forward dividend, which incorporates expectational information on dividends embedded in option prices. We demonstrate that our implied‐forward dividend strategy produces more consistent estimates between in‐sample market and model option prices. More importantly our new implied‐forward dividend strategy makes more accurate out‐of‐sample forecasts for one‐day or one‐week ahead prices. Second, we document that both a “Return‐volatility” Smile and a “Return‐pricing Error” Smile exist. From these return characteristics, we make two conclusions: (1) the return dependency of implied volatility is an important explanatory variable and should be controlled to reduce the pricing error of an AHBS model, and (2) it is important for the hedging horizon to be based on return size, that is, the larger the contemporaneous return, the more frequent an option issuer must rebalance the option's hedge. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:742‐772, 2012     

On the Rate of Convergence of Discrete‐Time Contingent Claims

This paper characterizes the rate of convergence of discrete‐time multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of all‐or‐nothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discrete‐time solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuous‐time limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results.     

A Closer Look at Barrier Exchange Options

A barrier exchange option is an exchange option that is knocked out the first time the prices of two underlying assets become equal. Lindset, S., & Persson, S.‐A. (2006) present a simple dynamic replication argument to show that, in the absence of arbitrage, the current value of the barrier exchange option is equal to the difference in the current prices of the underlying assets and that this pricing formula applies irrespective of whether the option is European or American. In this study, we take a closer look at barrier exchange options and show, despite the simplicity of the pricing formula presented by Lindset, S., & Persson, S.‐A. (2006), that the barrier exchange option in fact involves a surprising array of key concepts associated with the pricing of derivative securities including: put–call parity, barrier in–out parity, static vs. dynamic replication, martingale pricing, continuous vs. discontinuous price processes, and numeraires. We provide valuable intuition behind the pricing formula which explains its apparent simplicity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:29–43, 2013     

Improving lattice schemes through bias reduction

Lattice schemes for option pricing, such as tree or grid/partial differential equation (p.d.e.) methods, are usually designed as a discrete version of an underlying continuous model of stock prices. The parameters of such schemes are chosen so that the discrete version “best” matches the continuous one. Only in the limit does the lattice option price model converge to the continuous one. Otherwise, a discretization bias remains. A simple modification of lattice schemes which reduces the discretization bias is proposed. The modification can, in theory, be applied to any lattice scheme. The main idea is to adjust the lattice parameters in such a way that the option price bias, not the stock price bias, is minimized. European options are used, for which the option price bias can be evaluated precisely, as a template to modify and improve American option methods. A numerical study is provided. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:733–757, 2006     

OPTIMAL CONTINUOUS-TIME HEDGING WITH LEPTOKURTIC RETURNS

We examine the behavior of optimal mean–variance hedging strategies at high rebalancing frequencies in a model where stock prices follow a discretely sampled exponential Lévy process and one hedges a European call option to maturity. Using elementary methods we show that all the attributes of a discretely rebalanced optimal hedge, i.e., the mean value, the hedge ratio, and the expected squared hedging error, converge pointwise in the state space as the rebalancing interval goes to zero. The limiting formulae represent 1-D and 2-D generalized Fourier transforms, which can be evaluated much faster than backward recursion schemes, with the same degree of accuracy. In the special case of a compound Poisson process we demonstrate that the convergence results hold true if instead of using an infinitely divisible distribution from the outset one models log returns by multinomial approximations thereof. This result represents an important extension of Cox, Ross, and Rubinstein to markets with leptokurtic returns.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

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.