An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

Calibrated American option pricing by stochastic linear programming

Abstract

We propose an approach for computing the arbitrage-free interval for the price of an American option in discrete incomplete market models via linear programming. The main idea is built replicating strategies that use both the basic asset and some European derivatives available on the market for trading. This method goes under the name of calibrated option pricing and it has given significant results for European options. Here, we extend the analysis to American options showing that the arbitrage-free interval can be characterized in terms of martingale measures and that it gets significantly reduced with respect to the non-calibrated case.     

Option Pricing for Time-Change Exponential Levy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure （MEMM） and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.     

Pricing and Hedging of Lookback Options in Hyper-exponential Jump Diffusion Models

ABSTRACT

In this article, we consider the problem of pricing lookback options in certain exponential Lévy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model, one typically has to rely on (rather time-intense) Monte-Carlo or partial (integro)-differential equation (P(I)DE) methods. However, for Lévy processes with double exponentially distributed jumps, the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path-dependent options with jump risk via Laplace transforms. The Kyoto Economic Review, 74(9), 1–23.). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Lévy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJDs). In fact, Jeannin and Pistorius (Jeannin, M., & Pistorius, M. (2010). A transform approach to calculate prices and Greeks of barrier options driven by a class of Lévy processes. Quntitative Finance, 10(6), 629–644.) were able to derive formulae for the Laplace transformed price of certain barrier options in market models described by HEJD processes. Here, we similarly derive the Laplace transforms of floating and fixed strike lookback option prices and propose a numerical inversion scheme, which allows, like Fourier inversion methods for European vanilla options, the calculation of lookback options with different strikes in one shot. Additionally, we give semi-analytical formulae for several Greeks of the option price and discuss a method of extending the proposed method to generalized hyper-exponential (as e.g. NIG or CGMY) models by fitting a suitable HEJD process. Finally, we illustrate the theoretical findings by some numerical experiments.     

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract

The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.     

Price formation model with zero-Neumann boundary condition

ABSTRACT

This paper concerns the mathematical analysis of a mathematical model for price formation. We take a large number of rational buyers and vendors in the market who are trading the same good into consideration. Each buyer or vendor will choose his optimal strategy to buy or sell goods. Since markets seldom stabilize, our model mimics the real market behavior. We introduce three models. All of them are modifications of the original J.-M. Lasry and P. L. Lions evolution model. In the first modified model, a random term is added to mimic the randomness of trading in the real market. This reflects markets with low volatility, where it might be difficulty to buy or sell goods at specific price. In the second model, we use cumulative density function instead of density function. We give numerical simulations on these two models in order to have a general picture on the solution. In the third model, we add a term associated with the parameter R to destabilize the original Larsy–Lions model and study oscillations and wave solutions depending on different values of R. We also study existence and uniqueness of the solution. Moreover, Several plots are given to demonstrate these results corresponding to the theoretical prediction.     

Local Volatility Pricing Models for Long-Dated FX Derivatives

Abstract

We study the local volatility function in the foreign exchange (FX) market, where both domestic and foreign interest rates are stochastic. This model is suitable to price long-dated FX derivatives. We derive the local volatility function and obtain several results that can be used for the calibration of this local volatility on the FX option's market. Then, we study an extension to obtain a more general volatility model and propose a calibration method for the local volatility associated with this model.     

Determination of the Probability Distribution Measures from Market Option Prices Using the Method of Maximum Entropy in the Mean

Abstract

We consider the problem of recovering the risk-neutral probability distribution of the price of an asset, when the information available consists of the market price of derivatives of European type having the asset as underlying. The information available may or may not include the spot value of the asset as data. When we only know the true empirical law of the underlying, our method will provide a measure that is absolutely continuous with respect to the empirical law, thus making our procedure model independent. If we assume that the prices of the derivatives include risk premia and/or transaction prices, using this method it is possible to estimate those values, as well as the no-arbitrage prices. This is of interest not only when the market is not complete, but also if for some reason we do not have information about the model for the price of the underlying.     

Optimal portfolio execution under cointegrated vector autoregressive systems

Abstract

In this paper, an optimal portfolio execution problem under price model which exhibits cointegration behaviour is proposed. The proposed problem is formulated as a quadratic programming problem. With different statistical procedures and parameter estimation methods, employed on real market financial data, the four portfolios are constructed with which, computational study is performed. It is shown that the trading strategies constructed out of portfolios with cointegrated price dynamics show significant reduction in execution cost.     

Risk Minimization for a Filtering Micromovement Model of Asset Price

Abstract

The classical option hedging problems have mostly been studied under continuous-time or equally spaced discrete-time models, which ignore two important components in the actual price: random trading times and market microstructure noise. In this paper, we study optimal hedging strategies for European derivatives based on a filtering micromovement model of asset prices with the two commonly ignored characteristics. We employ the local risk-minimization criterion to develop optimal hedging strategies under full information. Then, we project the hedging strategies on the observed information to obtain hedging strategies under partial information. Furthermore, we develop a related nonlinear filtering technique under the minimal martingale measure for the computation of such hedging strategies.     

Some results on quadratic hedging with insider trading

We consider the hedging problem in an arbitrage-free incomplete financial market, where there are two kinds of investors with different levels of information about the future price evolution, described by two filtrations F and G=F∨σ(G) where G is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches.

In particular, for LRM, we show that for a large class of additional non trivial r.v.s G both investors will pursue the same locally risk minimizing portfolio strategy and the cost process of the ordinary agent is just the projection on F of that of the insider. For the MVH approach, we study also some general stochastic volatility model, including Hull and White, Heston and Stein and Stein models. In this more specific setting and for r.v.s G which are measurable with respect to the filtration generated by the volatility process, we obtain an expression for the insider optimal strategy in terms of the ordinary agent optimal strategy plus a process admitting a simple feedback-type representation.     

A General Benchmark Model for Stochastic Jump Sizes

Abstract

Under few technical assumptions and allowing for the absence of an equivalent martingale measure, we show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process. We investigate the structure of market prices of risk as markets become approximately complete and consider the limits of traded securities, characterizing explicitly the growth optimal portfolio and investigating arbitrage and diversification in such markets.     

Random times at which insiders can have free lunches

We consider models of time continuous financial markets with a regular trader and an insider who are able to invest into one risky asset. The insider's additional knowledge consists in his ability to stop at a random time which is inaccessible to the regular trader, such as the last passage of a certain level before maturity by some stock price process, or the time at which the stock price reaches its maximum during the trading interval. We show that under very mild assumptions on the coefficients of the diffusion process describing these price processes the information drift caused by the additional knowledge of the insider cannot be eliminated by an equivalent change of probability measure. As a consequence, all our models allow the insider to have free lunches with vanishing risk, or even to exercise arbitrage.     

Option Valuation with a Discrete-Time Double Markovian Regime-Switching Model

Abstract

This article develops an option valuation model in the context of a discrete-time double Markovian regime-switching (DMRS) model with innovations having a generic distribution. The DMRS model is more flexible than the traditional Markovian regime-switching model in the sense that the drift and the volatility of the price dynamics of the underlying risky asset are modulated by two observable, discrete-time and finite-state Markov chains, so that they are not perfectly correlated. The states of each of the chains represent states of proxies of (macro)economic factors. Here we consider the situation that one (macro)economic factor is caused by the other (macro)economic factor. The market model is incomplete, and so there is more than one equivalent martingale measure. We employ a discrete-time version of the regime-switching Esscher transform to determine an equivalent martingale measure for valuation. Different parametric distributions for the innovations of the price dynamics of the underlying risky asset are considered. Simulation experiments are conducted to illustrate the implementation of the model and to document the impacts of the macroeconomic factors described by the chains on the option prices under various different parametric models for the innovations.     

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Abstract

In this paper, we study the stochastic alpha beta rho with mean reversion model (SABR-MR). We first compare the SABR model with the SABR-MR model in terms of future volatility to point out the fundamental difference in the models’ dynamics. We then derive an efficient probabilistic approximation for the SABR-MR model to price European options. Similar to the method derived in Kennedy, J. E., Mitra, S., & Pham, D. (2012). On the approximation of the SABR model: A probabilistic approach. Applied Mathematical Finance, 19(6), 553–586., we focus on capturing the terminal distribution of the underlying asset (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a wide range of parameters that cover the long-dated option and different market conditions.     

Further Results on Global Convergence and Stability of Globally Projected Dynamical Systems

Several risk management and exotic option pricing models have been proposed in the literature which may price European options correctly. A prerequisite of these models is the interpolation of the market implied volatilities or the European option price function. However, the no-arbitrage principle places shape restrictions on the option price function. In this paper, an interpolation method is developed to preserve the shape of the option price function. The interpolation is optimal in terms of minimizing the distance between the implied risk-neutral density and the prior approximation function in L ²-norm, which is important when only a few observations are available. We reformulate the problem into a system of semismooth equations so that it can be solved efficiently.     

A Model with Interacting Assets Driven by Poisson Processes

Abstract

An extension with noise given by Poisson processes of a model of financial market with several assets that are interacting, i.e., influencing each other (even in the absence of noise) is given. We present explicit formulae for the stock price process as well as for the prices of European multi-asset contingent claims based on a residual risk minimization approach. We also provide an explicit hedging formula.     

Optimal Market Making in the Foreign Exchange Market

Abstract

This paper is concerned with optimal market making in the foreign exchange market. The market maker's holdings in the different currencies are modelled as stochastic processes that are influenced by both the stochastic exchange rates and the stochastic customer buy and sell orders. The market maker can control their own bid and ask price quotes and, additionally, can buy and sell at other market participants' quotes. The resulting stochastic control problem consists of a controlled diffusion problem for the optimal quotes and a singular control problem for optimal trades at other market participants' quotes. A Markov chain approximation is used to derive optimal strategies.     

Closed Form Approximations for Spread Options

Abstract

Market mechanisms are increasingly being used as a tool for allocating somewhat scarce but unpriced rights and resources, and the European Emission Trading Scheme is an example. By means of dynamic optimization in the contest of firms covered by such environmental regulations, this article generates endogenously the price dynamics of emission permits under asymmetric information, allowing inter-temporal banking and borrowing. In the market, there are a finite number of firms and each firm's pollution emission follows an exogenously given stochastic process. We prove the discounted permit price is a martingale with respect to the relevant filtration. The model is solved numerically. Finally, a closed-form pricing formula for European-style options is derived.     

Real-World Scenarios With Negative Interest Rates Based on the LIBOR Market Model

ABSTRACT

In this article, we present a methodology to simulate the evolution of interest rates under real-world probability measure. More precisely, using the multidimensional Shifted Lognormal LIBOR market model and a specification of the market price of risk vector process, we explain how to perform simulations of the real-world forward rates in the future, using the Euler?Maruyama scheme with a predictor?corrector strategy. The proposed methodology allows for the presence of negative interest rates as currently observed in the markets.