Pricing Of Financial Derivatives Research Articles

On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)] we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution for semilinear heat equations that the computational complexity is bounded by $O( d \, \epsilon^{-(4+\delta)})$ for any $\delta\in(0,\infty)$, where $d$ is the dimensionality of the problem and $\epsilon\in(0,\infty)$ is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of $100$-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for these 100-dimensional example PDEs are very satisfactory in terms of accuracy and speed. In addition, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the literature.

Pricing of stochastic volatility stock index option based on Feynman path integral

<sec>Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options. Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.</sec><sec>In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman's pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.</sec><sec>The remarkable advantages of Feynman path integral stock index option pricing model are as follows. Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.</sec>

Derivatives Pricing via Machine Learning

In this paper, we combine the theory of stochastic process and techniques of machine learning with the regression analysis, first proposed by [1] to solve for American option prices, and apply the new methodologies on financial derivatives pricing. Rigorous convergence proofs are provided for some of the methods we propose. Numerical examples show good applicability of the algorithms. More applications in finance are discussed in the Appendices.

Derivatives Pricing via Machine Learning

In this paper, we combine the theory of stochastic process and techniques of machine learning with the regression analysis, first proposed by Longstaff and Schwartz 2001 and apply the new methodologies on financial derivatives pricing. Rigorous convergence proofs are provided for some of the methods we propose. Numerical examples show good applicability of the algorithms.

Bilateral Defaultable Financial Derivatives Pricing and Credit Valuation Adjustment

The one-side defaultable financial derivatives valuation problems have been studied extensively, but the valuation of bilateral derivatives with asymmetric credit qualities is still lacking convincing mechanism. This paper presents an analytical model for valuing derivatives subject to default by both counterparties. The default-free interest rates are modeled by the Market Models, while the default time is modeled by the reduced-form model as the first jump of a time-inhomogeneous Poisson process. All quantities modeled are market-observable. The closed-form solution gives us a better understanding of the impact of the credit asymmetry on swap value, credit value adjustment, swap rate and swap spread.

An efficient exponential twisting importance sampling technique for pricing financial derivatives

This paper develops an efficient Exponential Twisting Importance Sampling technique for variance reduction when it is used to price financial derivatives. A general class of exponential change of measure is well explored. This is the generalization of the proportional exponential twisting function, , discussed as an Example 4.6.2 in Glasserman’s book (2004). Then a new framework to find the optimal importance sampling density function is proposed based on the Cauchy-Schwartz Inequality and the Least Square Approach. In the special Gaussian case, the theory of our general exponential change of measure means finding the new drift vector and covariance matrix simultaneously. The method proposed by the paper has little smoothness requirements for the payoff functions and doesn’t rely on the initial values. It is illustrated that this method is high efficient for pricing financial derivatives, such as Asian options, Straddle options, Volatility derivatives and the pricing under the stochastic volatility models. Furthermore, this method can be extended to more general importance sampling densities such as non-Gaussian or multi-modal distributions.

On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

Cox-Ingersoll-Ross (CIR) processes are extensively used in state-of-the-art models for the approximative pricing of financial derivatives. In particular, CIR processes are day after day employed to model instantaneous variances (squared volatilities) of foreign exchange rates and stock prices in Heston-type models and they are also intensively used to model short-rate interest rates. The prices of the financial derivatives in the above mentioned models are very often approximately computed by means of explicit or implicit Euler- or Milstein-type discretization methods based on equidistant evaluations of the driving noise processes. In this article we study the strong convergence speeds of all such discretization methods. More specifically, the main result of this article reveals that each such discretization method achieves at most a strong convergence order of $\delta/2$, where $0<\delta<2$ is the dimension of the squared Bessel process associated to the considered CIR process. In particular, we thereby reveal that discretization methods currently employed in the financial industry may converge with arbitrarily slow strong convergence rates to the solution of the considered CIR process. We thereby lay open the need of the development of other more sophisticated approximation methods which are capable to solve CIR processes in the strong sense in a reasonable computational time and which thus can not belong to the class of algorithms which use equidistant evaluations of the driving noise processes.

Option Pricing With Regime Switching Correlation: A Numerical PDE Approach

Modelling correlation between financial quantities is important in the accurate pricing of financial derivatives. In this paper, we introduce some stochasticity in correlation, by considering a regime-switching correlation model, in which the transition rates between regimes are given. We present a derivation of the associated Partial Differential Equation (PDE) problem. The problem involves a system of l PDEs, where l is the number of regimes. We formulate a finite difference method for the solution of the PDE system, and numerically demonstrate that it converges with second order. We study the effect of certain model parameters on the computed prices. We compare prices from this model, associated PDE and method with those from a stochastic correlation model, associated PDE and method in van Emmerich, 2006, Leung, 2017, Leung et al., 2016 and discuss advantages and disadvantages.

Quantum computational finance: Monte Carlo pricing of financial derivatives

Financial derivatives are contracts that can have a complex payoff dependent upon underlying benchmark assets. In this work, we present a quantum algorithm for the Monte Carlo pricing of financial derivatives. We show how the relevant probability distributions can be prepared in quantum superposition, the payoff functions can be implemented via quantum circuits, and the price of financial derivatives can be extracted via quantum measurements. We show how the amplitude estimation algorithm can be applied to achieve a quadratic quantum speedup in the number of steps required to obtain an estimate for the price with high confidence. This work provides a starting point for further research at the interface of quantum computing and finance.

Monte Carlo Option Pricing in Tensorflow

We look at pricing financial derivatives on gpus. After some general comparison of frameworks, we focus on pricing via Monte Carlo simulations and compare numpy, tensorflow cpu and tensorflow gpu using the Black Scholes model, the Heston model and the Heston model with local volatility.

A New Model for Pricing Collateralized Financial Derivatives

This paper presents a new model for pricing financial derivatives subject to collateralization. It allows for collateral arrangements adhering to bankruptcy laws. As such, the model can back out the market price of a collateralized contract. This framework is very useful for valuing outstanding derivatives. Using a unique dataset, we find empirical evidence that credit risk alone is not overly important in determining credit-related spreads. Only accounting for both collateral posting and credit risk can sufficiently explain unsecured credit costs. This finding suggests that failure to properly account for collateralization may result in significant mispricing of derivatives. We also empirically gauge the impact of collateral agreements on risk measurements. Our findings indicate that there are important interactions between market and credit risk.

Arbitrage-free pricing of derivatives in nonlinear market models

The objective of this paper is to provide a comprehensive study of the no-arbitrage pricing of financial derivatives in the presence of funding costs, the counterparty credit risk and market frictions affecting the trading mechanism, such as collateralization and capital requirements. To achieve our goals, we extend in several respects the nonlinear pricing approach developed in (El Karoui and Quenez 1997) and (El Karoui et al. 1997), which was subsequently continued in (Bielecki and Rutkowski 2015).

Pricing Financial Derivatives Subject to Counterparty Risk and Credit Value Adjustment

This article presents a generic model for pricing financial derivatives subject to counterparty credit risk. Both unilateral and bilateral types of credit risks are considered. Our study shows that credit risk should be modeled as American style options in most cases, which require a backward induction valuation. To correct a common mistake in the literature, we emphasize that the market value of a defaultable derivative is actually a risky value rather than a risk-free value. Credit value adjustment (CVA) is also elaborated. A practical framework is developed for pricing defaultable derivatives and calculating their CVAs at a portfolio level.

Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube

This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $[0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only $o(n^{-1/2})$ for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of $O(n^{-1/2-1/(4d-2)+\epsilon})$ for arbitrarily small $\epsilon>0$. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains $O(n^{-1+\epsilon})$ if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.

Litigation Funding: An Economic Analysis

Basic economic analysis of litigation funding shows that risk neutral plaintiffs without budget constraints will not accept funding unless they are pessimistic relative to the funder. Risk aversion makes a plaintiff who shares probabilistic beliefs with the funder act observationally equivalent to a pessimistic, risk neutral plaintiff, so she will accept funding as well. An important benefit of litigation funding - evident from the application of a change of measure to risk neutral probabilities, an analytical approach widely used in the pricing of financial derivatives - is that it moves litigation outcomes closer to risk neutral outcomes and therefore closer to actions consistent with the plaintiff's perceived merits, something that is of underemphasized importance in law and procedure. The best funding outcomes (for investors) are likely when plaintiffs are risk averse or budget constrained. Poor outcomes are more likely when funded plaintiffs are risk neutral and unconstrained.

Identifying the implied volatility using the total variation regularization

This paper studies an inverse problem of determining the implied volatility in the financial products linked with gold price, which has important application in financial derivatives pricing. Based on the total variation regularization strategy, the existence and necessary condition of the minimum for the control function are addressed, and the local uniqueness of the solution is also given by a modified case. Furthermore, the stability and convergence for the regularized approach are discussed. The results obtained in this paper may be useful for those who engage in hedging or risk management.

Polynomial upper and lower bounds for financial derivative price functions under regime-switching

We present a new approach to bounding financial derivative prices in regime-switching market models from both above and below. We derive sufficient conditions under which a particular class of functions act as bounds for the prices of financial derivatives in regime-switching market models. Using these sufficient conditions, we then formulate, in a general setting, optimization problems whose solutions can be identified with tight upper and lower bounds. The problems are made numerically tractable by imposing polynomial structures and employing results from the theory of sum-of-squares polynomials to arrive at a semidefinite programming problem that is implementable by existing software. The bounds obtained take the form of smooth polynomial functions and are valid for a continuous range of initial times and states. Moreover, they are obtained without recourse to sample path simulation or discretization of the temporal or spatial variables. We demonstrate the effectiveness of the proposed method on European-, barrier- and American-style options in several regime-switching settings with and without jumps.

An Implicit-Explicit Computational Method Based on Time Semi-Discretization for Pricing Financial Derivatives with Jumps

This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.

Pricing 50ETF Option Based on Genetic Algorithm BP Model

Pricing financial derivatives is focus in finance theory and practice. Comparing to the traditional parameter model pricing method, the neural network method has obvious advantages in solving this problem. In this paper,we will price the option of Shanghai 50ETF based on the improved BP neural network model (GABP). The results show that the effect of neural network is better than that of B-S model, and the accuracy of GABP model is higher than that of BP neural network model and B-S model.

(Avoiding Systemic Financial Risk)

Chinese Abstract: 避免系統性的金融風險，跟一個金融機構怎麼避免它所面對的風險，是兩個不同的概念，最重要的考慮是系統性金融風險裡包含著的“溢出效應”（spillover effect）。本文討論中國現時面對的幾種金融風險：資產價格泡沫的風險、不良貸款的風險、金融衍生產品的風險，以及匯率過度波動性的風險。 English Abstract: Avoiding systemic financial risk and how a financial institution avoids the risk it is exposed to are two distinct concepts; the most critical consideration is the spillover effect involved in systemic financial risk. This paper discusses a few financial risks China currently confronts: the risks of asset price bubbles, non-performing loans, financial derivatives, and excess exchange rate volatility.