Black-Scholes Framework Research Articles

Numerical solutions of PDEs for corporate bond pricing: A computational finance approach

Abstract. In this paper, we will see how one can use numerical algorithms to solve PDEs, containing partial derivative terms, for corporate bond pricing. We utilise the Black-Scholes framework which involves adding stochastic interest rates and credit risk into a PDE model. Then, we will take a quick look at some different numerical methods to solve the PDE model, like finite difference methods (FDM), finite element methods (FEM) as well as standard Monte Carlo (MC) simulations. Above all, we provide some simulations to show how well they work for PDEs, and how much time it takes. We would like to highlight that numerical methods lead to more accurate and efficient bond pricing with these methods, and also that we can analyse together how well each method works under different market scenarios. The paper will end with future directions for numerical methods, which should allow including more sophisticated algorithms and machine-learning techniques that improve scalability and can be used in a real-time manner in financial markets.

Super-replication of life-contingent options under the Black–Scholes framework

Abstract We consider the super-replication problem for a class of exotic options known as life-contingent options within the framework of the Black–Scholes market model. The option is allowed to be exercised if the death of the option holder occurs before the expiry date, otherwise there is a compensation payoff at the expiry date. We show that there exists a minimal super-replication portfolio and determine the associated initial investment. We then give a characterisation of when replication of the option is possible. Finally, we give an example of an explicit super-replicating hedge for a simple life-contingent option.

Multi-regime foreign exchange rate model: Calibration and pricing

To price exotic foreign exchange (FX) options, a model needs to be selected for FX spot rate dynamics. The classic approach of modelling spot rates with Black–Scholes framework makes inappropriate assumptions of constant drift and volatility, resulting in mispricing. In this article, we investigate multi-regime Black–Scholes (MRBS) model for FX rate, with regime-switching behaviour of drift and volatility governed by a Markov chain. We derive an analytic formula for European FX call options via Fourier transform and present a Monte Carlo simulation algorithm for FX barrier option pricing. Further, we propose a calibration strategy with penalty. Finally, the empirical study shows that volatility smile can be recovered by MRBS model; calibrating MRBS with penalty gives stable calibrated parameters and small out-of-sample mean squared errors; combined use of our calibration methods and pricing algorithms provides reasonable prices for FX barrier options. Thus, MRBS model, together with proposed calibration-with-penalty strategy and pricing algorithm, provides a promising approach for FX option pricing.

Pricing of Al-Urbun and a Class of Al-Istijrar Islamic Contracts under the Black–Scholes Framework

Islamic financial contracts necessarily need to abide by Shariah principles. As such, some contracts have been introduced for risk-hedging real transactions that differ from those seen in conventional financial markets. In this paper, we examine two such products, Al-Urbun and Al-Istijrar, and determine fair prices for both the Al-Urbun and a class of Al-Istijrar under the Black–Scholes framework.

Pricing a resettable convertible bond based on decomposition method and PDE models

In this paper, a partial differential equation approach based on the underlying stock price path decomposition is developed to price an American-style resettable convertible bond. The American-style resettable convertible bond is viewed as a mixture of three simple securities, which can be used to replicate the feature of payoffs of the resettable convertible bond completely. The partial differential equations under the Black–Scholes framework are established to price these simple securities. An implicit Euler method is used to discretize the first-order time derivative while a central finite difference method on a piecewise uniform mesh is used to discretize the spatial derivatives. The error estimates are developed by using the maximum principle in two mesh sets both for the time semi-discretization scheme and the spatial discretization scheme, respectively. It is proved that the scheme is first-order convergent for the time variable and second-order convergent for the spatial variable. Numerical experiments support these theoretical results.

Valuing rebate options and equity-linked products

In this article, we propose rebate options with multi-step barriers, which are an extension of rebate options with a constant barrier. Despite the applicability and marketability of rebate options, there have been only a few research on obtaining analytical formulas. Accordingly, in this paper, we derive closed-form pricing formulas for these options under the Black–Scholes framework. The rebate options with multi-step barriers allow a flexible barrier structure, and thus we propose complex equity-linked products embedded with rebate options with barriers and derive pricing formulas for them. We conduct numerical studies on the pricing of rebate options with multi-step barriers, equity-linked securities, and equity-indexed annuities. The numerical studies validate the prices obtained from the pricing formulas.

A simple European option pricing formula with a skew Brownian motion

Zhu and He [(2018). A new closed-form formula for pricing European options under a skew Brownian motion. The European Journal of Finance 24(12): 1063–1074] provided an innovative closed-form solution by replacing the standard Brownian motion in the Black–Scholes framework using a particular skew Brownian motion. Their formula involves numerically integrating the product of the Guassian density and corresponding distribution function. Being different from their pricing formula, we derive a much simpler formula that only involves the Gaussian distribution function and Owen's $T$ function.

A consumption-investment model with state-dependent lower bound constraint on consumption

This paper studies a life-time consumption-investment problem under the Black-Scholes framework, where the consumption rate is subject to a lower bound constraint that linearly depends on the investor's wealth. It is a stochastic control problem with state-dependent control constraint to which the standard stochastic control theory cannot be directly applied. We overcome this by transforming it into an equivalent stochastic control problem in which the control constraint is state-independent so that the standard theory can be applied. We give an explicit optimal consumption-investment strategy when the constraint is homogeneous. When the constraint is non-homogeneous, it is shown that the value function is third-order continuously differentiable by a differential equation approach, and a feedback form optimal consumption-investment strategy is provided. According to our findings, if the investor is concerned with long-term more than short-term consumption, then she should always consume as few as possible; otherwise, she should consume optimally when her wealth is above a threshold, and consume as few as possible when her wealth is below the threshold.

Calibrating with a smile: A Mellin transform approach to volatility surface calibration

The implied volatility in the Black-Scholes framework is not a constant but a function of both the strike price (“smile/skew”) and the time to expiry. A popular approach to recovering the volatility surface is through the use of deterministic volatility function models via Dupire’s equation. A new method for volatility surface calibration based on the Mellin transform is proposed. An explicit formula for the volatility surface is obtained in terms of the Mellin transform of the call option price with respect to the strike price, and a numerical algorithm is provided. Results of numerical simulations are presented and the stability of the method is numerically verified. The proposed Mellin transform approach provides a simpler and more direct fitting of generalised forms of the volatility surface given previously in the literature.

Skew-Brownian motion and pricing European exchange options

This article derives a closed-form pricing formula for European exchange options under a non-Gaussian framework for the underlying assets, intending to resolve mispricing associated with a geometric Brownian motion. The dynamics of each of the two correlated underlying assets are assumed to be governed by the exponential of a skew-Brownian motion, which is specified as a sum of a standard Brownian motion and an independent reflected Brownian motion. The proposed pricing formula does not incur additional computational costs than the standard Black–Scholes framework, which one can quickly recover as a particular case of the proposed framework. Finally, we present some numerical experiments followed by a valuable discussion on the results.

Multi-step barrier products and static hedging

This paper examines multi-step barrier options with an arbitrary payoff function using extended static hedging methods. Although there have been studies using extended reflection principles to obtain joint distribution functions for barrier options with complex barrier conditions, and static hedging methods to evaluate limited barrier options with well-known payoff functions, we obtain an explicit expression of barrier option price which has a general payoff function under the Black–Scholes framework assumption. The explicit multi-step barrier options prices we discuss in this paper are not only useful in that they can handle different levels and time steps barrier and all types of payoff functions, but can also extend to pricing of barrier options under finite discrete jump–diffusion models with a simple barrier. In the last part, we supplement the theory with numerical examples of various multi-step barrier options under the Black–Scholes or discrete jump–diffusion model for comparison purposes.

A simple and efficient numerical method for pricing discretely monitored early-exercise options

We present a simple, fast, and accurate method for pricing a variety of discretely monitored options in the Black-Scholes framework, including autocallable structured products, single and double barrier options, and Bermudan options. The method is based on a quadrature technique, and it employs only elementary calculations and a fixed one-dimensional uniform grid. The convergence rate is O(1/N4) and the complexity is O(MNlogN), where N is the number of grid points and M is the number of observation dates. Besides Black-Scholes, our method is also applicable to more general frameworks such as Merton’s jump diffusion model.

The valuation of knock-out power calls under Black–Scholes framework

Knock-out power calls are options that incorporate barriers to the valuation of power calls. Introducing barriers to power calls reduces the costs to hold power calls which are known to have higher leverage than the standard vanillas. In this paper, we model the valuation of knock-out power calls using Crank–Nicolson and Monte Carlo simulation under Black–Scholes environment. Results show that Crank–Nicolson is more accurate and more efficient than Monte Carlo simulation for pricing knock-out power calls.

Deep learning for efficient frontier calculation in finance

We propose deep neural network algorithms to calculate the efficient frontier in mean–variance and mean–conditional value-at-risk (mean–CVaR) portfolio optimization problems. Starting with the mean–variance portfolio optimization problem in the Black–Scholes framework, we first compare the analytical solution with the computed one, and we show the efficiency of the methodology in high dimensions. Adding additional constraints, we compare different formulations and, obtaining the same frontier, show that the “no short selling” and “no borrowing” problem can be solved with all formulations. Then, a new projected feedforward network is shown to be able to deal with some local and global constraints on the weights of the portfolio while outperforming classical penalization methods. We extend our numerical results to assets following the Heston model, and we show that the results obtained in the Black–Scholes method still hold. Finally, we present numerical results for the more difficult mean–CVaR optimization problem, which we show are realistic only for algorithms with a global resolution of the problem.

Utility-indifference pricing of European options with proportional transaction costs

Pricing options in a market with transaction costs is an important research topic in quantitative finance. There are basically two categories of approaches to deal with transaction costs. One is discrete hedging strategies under the Black–Scholes framework, another is utility indifference approaches. Hedging approaches are easy to implement, but preference-independent, whereas utility indifference approaches incorporate risk preferences, in general involving lengthy calculations. In a market with transaction costs, trading stocks involves risk so investors’ risk preferences must be taken into consideration. In this paper, we price European options with proportional transaction costs using a utility indifference approach which updates the fraction of one’s total wealth in the risky asset at regular intervals. Our HJB equation for the portfolio without option is two-dimensional instead of three-dimensional as in standard utility indifference approaches. Since the HJB equations could only be solved by a numerical method, our approach provides saving of computing time. Results from our numerical experiments via the finite difference method are presented to illustrate the effects of key option parameters on option prices.

Valuation of piecewise linear barrier options

This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closed-form prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.

The price of liquidity in the reinsurance of fund returns

We consider a new type of contract for insuring the returns of hedge funds and aim to extend downside protection to an investment portfolio beyond the first tranche of losses insured by first-loss fee structures, which have become increasingly popular in the market. By considering a second tranche, we suggest an up-front premium to a reinsurance party, in exchange for which the investor gains full protection against all losses, not just those occurring in the first tranche. We identify a fund’s underlying liquidity as a key parameter in deriving the price for the additional reinsurance and provide a method for computing the premium using two approaches: an analytic closed-form solution based on the Black–Scholes framework, and a numerical simulation using a Markov-switching model. In addition, a simplified backtesting method is implemented to evaluate the practical application and performance of the concept for both mathematical models.

Pricing multiple barrier derivatives under stochastic volatility

This work generalizes existing one- and two-dimensional pricing formulas with an equal number of barriers to a setting of n dimensions and up to two barriers in the presence of stochastic volatility. This allows for the consideration of multidimensional single-barrier derivatives with, for example, a collateral triggered by a barrier default of the issuing company. We introduce stochastic volatility to a multidimensional Black–Scholes framework via the common Cox–Ingersoll–Ross process and present semianalytical solutions for collateralized structured products with two barriers, one representing default and the other a market-related option. Our model accommodates implied volatility skew along the lines of displaced diffusions. We show that our semianalytical formulas are more efficient in terms of computational speed than Monte Carlo simulations, particularly for tail scenarios. Moreover, our proposed analytical simplifications permit a twenty-fivefold gain in time savings compared with the results given by the main theorem. These multidimensional structured products gained increasing popularity after the subprime and financial crises. Therefore, we perform comprehensive sensitivity analyses with respect to stochastic volatility parameters and contribute to a better understanding of multidimensional barrier derivatives in a stochastic volatility framework.

Learning agents in Black-Scholes financial markets.

Black–Scholes (BS) is a remarkable quotation model for European option pricing in financial markets. Option prices are calculated using an analytical formula whose main inputs are strike (at which price to exercise) and volatility. The BS framework assumes that volatility remains constant across all strikes; however, in practice, it varies. How do traders come to learn these parameters? We introduce natural agent-based models, in which traders update their beliefs about the true implied volatility based on the opinions of other agents. We prove exponentially fast convergence of these opinion dynamics, using techniques from control theory and leader-follower models, thus providing a resolution between theory and market practices. We allow for two different models, one with feedback and one with an unknown leader.

QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds

This article presents a discrete-time option pricing model that is rooted in reinforcement learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model can go model-free and learn to price and hedge an option directly from data, without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for the optimal pricing and hedging of options. Once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based reinforcement learning. Further, due to the simplicity and tractability of our model, which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relationship to the original BSM model, we suggest that our model could be used in the benchmarking of different RL algorithms for financial trading applications. TOPICS:Derivatives, options Key Findings • Reinforcement learning (RL) is the most natural way for pricing and hedging of options that relies directly on data and not on a specific model of asset pricing. • The discrete-time RL approach to option pricing generalizes classical continuous-time methods, enables tracking mis-hedging risk, which disappears in the formal continuous-time limit, and provides a consistent framework for using options for both hedging and speculation. • A simple quadratic reward function, which presents a minimal extension of the classical Black-Scholes framework when combined with the Q-learning method of RL, gives rise to a particularly simple computational scheme where option pricing and hedging are semi-analytical, as they amount to multiple uses of a conventional least-squares regression.