Pricing Of Contingent Claims Research Articles

Risk measures based on weak optimal transport

In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting.

Information Theory and the Pricing of Contingent Claims: An Alternative Derivation of the Black–Scholes–Merton Formula

This paper seeks to determine the best subjective probability to use to carry out expectation values of uncertain future cash flows with the smallest number of assumptions. This results in the unique distribution that guarantees no more information is present other than the stated assumptions. The result is a novel derivation of the well-known Black–Scholes equation without the need to introduce high-level mathematical machinery. This formalism fits nicely into introductory courses of finance, where the value of any financial instrument is given by the present value of uncertain future cash flows.

A Theory of Equivalent Expectation Measures for Contingent Claim Returns

ABSTRACTThis paper introduces a dynamic change of measure approach for computing analytical solutions of expected future prices (and therefore, expected returns) of contingent claims over a finite horizon. The new approach constructs hybrid probability measures called equivalent expectation measures (EEMs) that provide the physical expectation of the claim's future price before the horizon date, and serve as pricing measures on or after the horizon date. The EEM theory can be used for empirical investigations of both the cross‐section and the term structure of returns of contingent claims, such as Treasury bonds, corporate bonds, and financial derivatives.

OPTIMAL INVESTMENT AND CONTINGENT CLAIM VALUATION WITH EXPONENTIAL DISUTILITY UNDER PROPORTIONAL TRANSACTION COSTS

We consider indifference pricing of contingent claims consisting of payment flows in a discrete-time model with proportional transaction costs and under exponential disutility. This setting covers utility maximization of terminal wealth as a special case. A dual representation is obtained for the associated disutility minimization problem, together with a dynamic procedure for solving it. This leads to efficient and convergent numerical procedures for indifference pricing, optimal trading strategies and shadow prices that apply to a wide range of payoffs, a large range of time steps and all magnitudes of transaction costs.

Analytically Pricing Formula for Contingent Claim with Polynomial Payoff under ECIR Process

Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. For a simple process, the moments are usually directly obtained from its transition probability density function (PDF). However, if the transition PDF is unavailable in simple form, the derivations of the moments and the contingent claim prices will not be accessible in closed forms. This paper provides a closed-form formula for pricing contingent claims with nonlinear payoff under a no-arbitrage principle when underlying assets follow the extended Cox–Ingersoll–Ross (ECIR) process with the symmetry properties of the Brownian motion. The formula proposed here is a consequence of successfully solving an explicit solution for a system of recurrence partial differential equations in which its solution subtly depends on the conditional moments. For the particular CIR process, we obtain simple closed-form formulas by solving the Riccati differential equation. Furthermore, we carry out a complete investigation of the convergent case for those formulas. In case such as that of the unsolvable Riccati differential equation, ECIR case, a numerical method for numerically evaluating the mentioned analytical formulas and numerical validations for the formulas are examined. The validity and efficiency of the formulas are numerically demonstrated by comparison with results from Monte Carlo simulations for various modeling parameters. Finally, the proposed formula is applied to the value contingent claims such as coupon bonds, interest rate swaps, and arrears swaps.

The Metric Generalized Inverse and Its Single-Value Selection in the Pricing of Contingent Claims in an Incomplete Financial Market

The Metric Generalized Inverse and Its Single-Value Selection in the Pricing of Contingent Claims in an Incomplete Financial Market

On the Quantitative Properties of Some Market Models Involving Fractional Derivatives

We review and discuss the properties of various models that are used to describe the behavior of stock returns and are related in a way or another to fractional pseudo-differential operators in the space variable; we compare their main features and discuss what behaviors they are able to capture. Then, we extend the discussion by showing how the pricing of contingent claims can be integrated into the framework of a model featuring a fractional derivative in both time and space, recall some recently obtained formulas in this context, and derive new ones for some commonly traded instruments and a model involving a Riesz temporal derivative and a particular case of Riesz–Feller space derivative. Finally, we provide formulas for implied volatility and first- and second-order market sensitivities in this model, discuss hedging and profit and loss policies, and compare with other fractional (Caputo) or non-fractional models.

Explainable neural network for pricing and universal static hedging of contingent claims

We present here a regress-later based Monte Carlo approach that uses neural networks for pricing multi-asset discretely-monitored contingent claims. The choice of specific architecture of the neural networks used in the proposed algorithm provides for interpretability of the model, a feature that is often desirable in the financial context. Specifically, the interpretation leads us to demonstrate that any discretely monitored contingent claim —possibly high-dimensional and path-dependent— under Markovian and no-arbitrage assumptions, can be semi-statically hedged using a portfolio of short maturity options. We also show, for Bermudan style derivatives, how the method can be used to obtain an upper and lower bound to the true price, where the lower bound is obtained by following a sub-optimal policy, while the upper bound is found by exploiting the dual formulation. Unlike other duality based upper bounds where one typically has to resort to nested simulation for constructing super-martingales, the martingales in the current approach come at no extra cost, without the need for any sub-simulations. We demonstrate through numerical examples the simplicity and efficiency of the method for both pricing and semi-static hedging of path-dependent options.

The Contributions of Stephen A. Ross to Financial Economics

Stephen A. Ross was one of the most influential scholars in the field of financial economics in the late twentieth century. Ross's work was central to several novel domains of economic inquiry. His contributions included the arbitrage pricing theory (APT), the risk-neutral pricing of contingent claims, the binomial option pricing model, a theory of the term structure of interest rates, a seminal contribution to the economic theory of agency, and insights about conditioning biases in ex post performance measurement. In this article, we discuss his seminal papers and the broad scope of his curiosity within the arc of a remarkably productive and influential career that spanned five decades and yet ended sooner than most who knew him expected.

Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact

AbstractIn this paper, we construct the utility‐based optimal hedging strategy for a European‐type option in the Almgren‐Chriss model with temporary price impact. The main mathematical challenge of this work stems from the degeneracy of the second order terms and the quadratic growth of the first‐order terms in the associated Hamilton‐Jacobi‐Bellman equation, which makes it difficult to establish sufficient regularity of the value function needed to construct the optimal strategy in a feedback form. By combining the analytic and probabilistic tools for describing the value function and the optimal strategy, we establish the feedback representation of the latter. We use this representation to derive an explicit asymptotic expansion of the utility indifference price of the option, which allows us to quantify the price impact in options' market via the price impact coefficient in the underlying market.

Numerical Method for a System of PIDEs Arising in American Contingent Claims under FMLS Model with Jump Diffusion and Regime-Switching Process

This paper investigates a numerical method for solving fractional partial integro-differential equations (FPIDEs) arising in American Contingent Claims, which follow finite moment log-stable process (FMLS) with jump diffusion and regime switching. Mathematically, the prices of American Contingent Claims satisfy a system of d problems with free-boundary values, where d is the number of regimes of the market. In addition, an optimal exercise boundary is needed to setup with each regime. Therefore, a fully implicit scheme based on the penalty term is arranged. In the end, numerical examples are carried out to verify the obtained theoretical results, and the impacts of state variables in our model on the optimal exercise boundary of American Contingent Claims are analyzed.

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.

Too-big-to-fail: The value of government guarantee

Following the 2008 financial crisis and the government bailout of troubled companies, Too-Big-to-Fail became a standard expression to name a free protection of Wall Street by tax-payers' money. What should have been the fair cost of this protection? We offer a novel approach to estimate the value of the implicit government guarantee by combining the contingent claim pricing with the likelihood of the government intervention. We find that the cost of this implicit protection can go beyond tens of billions of dollars with an average of about $13 million per company, per year, and it rises to about $24 million if the government is assumed to intervene with certainty. Moreover, we show that the spread of the funding costs of the large banks over the small banks is strongly inversely associated with the value of the implicit government guarantee, especially after the crisis. For the Chinese companies, we use the contingent claims approach to measure the government guarantee and find that the value of the guarantee for State-owned Enterprises (SOEs) is inversely related to their bond yield spreads.

A Theory of Equivalent Expectation Measures for Expected Prices of Contingent Claims

This paper introduces a dynamic change of measure approach for computing the analytical solutions of expected future prices (and therefore, expected returns) of contingent claims over a finite horizon. The new approach constructs hybrid probability measures called the "equivalent expectation measures"(EEMs), which provide the physical expectation of the claim's future price until before the horizon date, and serve as pricing measures on or after the horizon date. The EEM theory can be used for empirical investigations of both the cross-section and the term structure of returns of contingent claims, such as Treasury bonds, corporate bonds, and financial derivatives.

Pricing Path-Dependent Payoff Using Quadrature on Wiener Space

Solution to stochastic differential equation that defines the dynamics of asset prices is interpreted as a function of underlying Brownian motion when permissible, and the expectation values of various functionals of the solution – therefore functionals of Brownian motion (Wiener functionals) – are estimated from numerical approximations to high-dimensional Gaussian integrals using quadrature. The simple method is exemplified for a few standard asset price dynamics used in practice for structured and contingent claim pricing, such as local volatility, Heston stochastic volatility, SABR, and LIBOR market model (LMM/BGM). The quadrature in these concrete implementations is performed in m-dimensional space where m is between 10 and 30 depending on the specific model or a payoff structure. The methodology appears to be a broadly viable alternative to Monte Carlo for short- to medium-dated financial products, with potential uses outside those examined in this paper.

Neural Network for Pricing and Universal Static Hedging of Contingent Claims

We present here a regress later based Monte Carlo approach that uses neural networks for pricing high-dimensional contingent claims. The choice of specific architecture of the neural networks used in the proposed algorithm provides for interpretability of the model, a feature that is often desirable in the financial context. Specifically, the interpretation leads us to demonstrate that any contingent claim — possibly high dimensional and path-dependent — under Markovian and no- arbitrage assumptions, can be semi-statically hedged using a portfolio of short maturity options. We show how the method can be used to obtain an upper and lower bound to the true price, where the lower bound is obtained by following a sub-optimal policy, while the upper bound by exploiting the dual formulation. Unlike other duality based upper bounds where one typically has to resort to nested simulation for constructing super-martingales, the martingales in the current approach come at no extra cost, without the need for any sub-simulations. We demonstrate through numerical examples the simplicity and efficiency of the method for both pricing and semi-static hedging of path-dependent options.

Game-theoretic derivation of upper hedging prices of multivariate contingent claims and submodularity

We investigate upper and lower hedging prices of multivariate contingent claims from the viewpoint of game-theoretic probability and submodularity. By considering a game between "Market" and "Investor" in discrete time, the pricing problem is reduced to a backward induction of an optimization over simplexes. For European options with payoff functions satisfying a combinatorial property called submodularity or supermodularity, this optimization is solved in closed form by using the Lov\'asz extension and the upper and lower hedging prices can be calculated efficiently. This class includes the options on the maximum or the minimum of several assets. We also study the asymptotic behavior as the number of game rounds goes to infinity. The upper and lower hedging prices of European options converge to the solutions of the Black-Scholes-Barenblatt equations. For European options with submodular or supermodular payoff functions, the Black-Scholes-Barenblatt equation is reduced to the linear Black-Scholes equation and it is solved in closed form. Numerical results show the validity of the theoretical results.

Modern Perspectives on Reinforcement Learning in Finance

We give an overview and outlook of the field of reinforcement learning as it applies to solving financial applications of intertemporal choice. In finance, common problems of this kind include pricing and hedging of contingent claims, investment and portfolio allocation, buying and selling a portfolio of securities subject to transaction costs, market making, asset liability management and optimization of tax consequences, to name a few. Reinforcement learning allows us to solve these dynamic optimization problems in an almost model-free way, relaxing the assumptions often needed for classical approaches. A main contribution of this article is the elucidation of the link between these dynamic optimization problem and reinforcement learning, concretely addressing how to formulate expected intertemporal utility maximization problems using modern machine learning techniques.

Optimal Investment and Contingent Claim Valuation With Exponential Disutility Under Proportional Transaction Costs

We consider indifference pricing of contingent claims consisting of payment flows in a discrete time model with proportional transaction costs and under exponential disutility. This setting covers utility maximisation as a special case. A dual representation is obtained for the associated disutility minimisation problem, together with a dynamic procedure for solving it. This leads to an efficient and convergent numerical procedure for indifference pricing which applies to a wide range of payoffs, a large range of time steps and all magnitudes of transaction costs.

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.