Journal articles: 'Black-Scholes PDE'

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Özer, H. Ünsal, and Ahmet Duran. "The source of error behavior for the solution of Black–Scholes PDE by finite difference and finite element methods." International Journal of Financial Engineering 05, no. 03 (September 2018): 1850028. http://dx.doi.org/10.1142/s2424786318500287.

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Black–Scholes partial differential equation (PDE) is one of the most famous equations in mathematical finance and financial industry. In this study, numerical solution analysis is done for Black–Scholes PDE using finite element method with linear approach and finite difference methods. The numerical solutions are compared with Black–Scholes formula for option pricing. The numerical errors are determined for the finite element and finite difference applications to Black–Scholes PDE. We examine the error behavior and find the source of the corresponding errors under various market situations.

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RIGATOS, GERASIMOS G. "BOUNDARY CONTROL OF THE BLACK–SCHOLES PDE FOR OPTION DYNAMICS STABILIZATION." Annals of Financial Economics 11, no. 02 (June 2016): 1650009. http://dx.doi.org/10.1142/s2010495216500093.

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The objective of the paper is to develop a boundary control method for the Black–Scholes PDE which describes option dynamics. It is shown that the procedure for numerical solution of Black–Scholes PDE results into a set of nonlinear ordinary differential equations (ODEs) and an associated state equations model. For the local subsystems, into which a Black–Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the Black–Scholes PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the Black–Scholes PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the Black–Scholes PDE system so as to assure that all its state variables will converge to the desirable setpoints.

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Rigatos, G., and P. Siano. "Feedback control of the multi-asset Black–Scholes PDE using differential flatness theory." International Journal of Financial Engineering 03, no. 02 (June 2016): 1650008. http://dx.doi.org/10.1142/s2424786316500080.

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A method for feedback control of the multi-asset Black–Scholes PDE is developed. By applying semi-discretization and a finite differences scheme the multi-asset Black–Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations it is shown that differential flatness properties hold. This enables to solve the associated control problem and to succeed stabilization of the options’ dynamics. It is shown that the previous procedure results into a set of nonlinear ordinary differential equations (ODEs) and to an associated state equations model. For the local subsystems, into which a Black–Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the Black–Scholes PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-asset Black–Scholes PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the Black–Scholes PDE system so as to assure that all its state variables will converge to the desirable setpoints.

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4

Hu, Jinhao, and Siqing Gan. "High order method for Black–Scholes PDE." Computers & Mathematics with Applications 75, no. 7 (April 2018): 2259–70. http://dx.doi.org/10.1016/j.camwa.2017.12.002.

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El-Khatib, Youssef. "A Homotopy Analysis Method for the Option Pricing PDE in Post-Crash Markets." Mathematical Economics Letters 2, no. 3-4 (November 30, 2014): 45–50. http://dx.doi.org/10.1515/mel-2013-0014.

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AbstractWe investigate a solution for the option pricing partial differential equation (PDE) in a market suffering from a financial crisis. The post-crash model assumes that the volatility is stochastic. It is an extension of the famous Black and Scholes model. Therefore, the option pricing PDE for the crisis model is a generalization of the Black and Scholes PDE. However, to the best knowledge, it does not have a closed form solution for the general case. In this paper, we provide a solution for the pricing PDE of a European option during financial crisis using the homotopy analysis method.

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Ogunyebi, SN, SE Fadugba, TO Ogunlade, KJ Adebayo, BT Babalola, O. Faweya, and HO Emeka. "Direct Solution of the Black-Scholes PDE Models with Non-Integer Order." Journal of Physics: Conference Series 2199, no. 1 (February 1, 2022): 012003. http://dx.doi.org/10.1088/1742-6596/2199/1/012003.

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Abstract This paper presents a direct solution of Black-Scholes PDE models with non-integer order via the non-integer iterative method. The Black-Scholes PDE models of non-integer order became popular and accepted globally by option traders for the valuation of financial derivatives over the period of time and the study of its numerical approaches has a wide range of applications in the real world.. The performance of the method has been confirmed and measured.

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Wilmott, Paul. "The two best ways to derive the Black–Scholes PDE." China Finance Review International 10, no. 2 (December 17, 2019): 168–74. http://dx.doi.org/10.1108/cfri-12-2018-0153.

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Purpose The purpose of this paper is to explain the Black–Scholes model with minimal technical requirements and to illustrate its impact from a business perspective. Design/methodology/approach The paper employs simple accounting concepts and an argument part based on business need. Findings The Black–Scholes partial differential equation can be derived in many ways, some easy to understand, some hard, some useful and others not. The two methods in this paper are extremely insightful. Originality/value The paper takes a big-picture view of derivatives valuation. As such, it is a simple accompaniment to more complex methods and aims to keep modelling grounded in reality.

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Han, Yuecai, and Chunyang Liu. "Asian Option Pricing under an Uncertain Volatility Model." Mathematical Problems in Engineering 2020 (April 21, 2020): 1–10. http://dx.doi.org/10.1155/2020/4758052.

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In this paper, we study the asymptotic behavior of Asian option prices in the worst-case scenario under an uncertain volatility model. We derive a procedure to approximate Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and splitting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximation method to solve a fully nonlinear PDE.

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Hossan, Md Shorif, Md Shafiqul Islam, and Md Kamrujjaman. "Efficient Numerical Schemes for Computations of European Options with Transaction Costs." European Journal of Mathematical Analysis 2 (February 17, 2022): 9. http://dx.doi.org/10.28924/ada/ma.2.9.

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This paper aims to find numerical solutions of the non-linear Black-Scholes partial differential equation (PDE), which often appears in financial markets, for European option pricing in the appearance of the transaction costs. Here we exploit the transformations for the computational purpose of a non-linear Black-Scholes PDE to modify as a non-linear parabolic type PDE with reliable initial and boundary conditions for call and put options. Several schemes are derived rigorously using the finite volume method (FVM) and finite difference method (FDM), which is the novelty of this paper. Stability and consistency analysis assure the convergence of these schemes. We apply these schemes to various volatility models, such as the Leland, Boyle and Vorst, Barles and Soner, and Risk-adjusted pricing methodology (RAPM). All the schemes are tested numerically. The convergence of the obtained results is observed, and we find that they are also reliable. Finally, we display all the approximate results together with the exact values through graphical and tabular representations.

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Prabakaran, Sellamuthu. "CONSTRUCTION OF THE BLACK-SCHOLES PDE WITH JUMP-DIFFUSION MODEL." Far East Journal of Mathematical Sciences (FJMS) 110, no. 1 (January 30, 2019): 131–63. http://dx.doi.org/10.17654/ms110010131.

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Tomas, Michael J., and Jun Yu. "An Asymptotic Solution for Call Options on Zero-Coupon Bonds." Mathematics 9, no. 16 (August 14, 2021): 1940. http://dx.doi.org/10.3390/math9161940.

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We present an asymptotic solution for call options on zero-coupon bonds, assuming a stochastic process for the price of the bond, rather than for interest rates in general. The stochastic process for the bond price incorporates dampening of the price return volatility based on the maturity of the bond. We derive the PDE in a similar way to Black and Scholes. Using a perturbation approach, we derive an asymptotic solution for the value of a call option. The result is interesting, as the leading order terms are equivalent to the Black–Scholes model and the additional next order terms provide an adjustment to Black–Scholes that results from the stochastic process for the price of the bond. In addition, based on the asymptotic solution, we derive delta, gamma, vega and theta solutions. We present some comparison values for the solution and the Greeks.

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Ďuriš, Karol, Shih-Hau Tan, Choi-Hong Lai, and Daniel Ševčovič. "Comparison of the Analytical Approximation Formula and Newton's Method for Solving a Class of Nonlinear Black–Scholes Parabolic Equations." Computational Methods in Applied Mathematics 16, no. 1 (January 1, 2016): 35–50. http://dx.doi.org/10.1515/cmam-2015-0035.

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AbstractMarket illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE-based option pricing models can be described by solutions to the generalized Black–Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. In this paper, different linearization techniques such as Newton's method and the analytic asymptotic approximation formula are adopted and compared for a wide class of nonlinear Black–Scholes equations including, in particular, the market illiquidity model and the risk-adjusted pricing model. Accuracy and time complexity of both numerical methods are compared. Furthermore, market quotes data was used to calibrate model parameters.

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Rigatos, Gerasimos, and Pierluigi Siano. "Stabilization of the multi-asset Black-Scholes PDE using differential flatness theory." IFAC-PapersOnLine 49, no. 8 (2016): 180–85. http://dx.doi.org/10.1016/j.ifacol.2016.07.434.

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Ramírez-Espinoza, Germán I., and Matthias Ehrhardt. "Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem." Advances in Applied Mathematics and Mechanics 5, no. 06 (December 2013): 759–90. http://dx.doi.org/10.4208/aamm.12-m1216.

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AbstractThis work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number-is high. For Asian options, additional similar problems arise when the “spatial” variable, the stock price, is close to zero.Here we focus on three methods: the exponentially fitted scheme, a modification of Wang’s finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

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Bajalan, Saeed, and Nastaran Bajalan. "Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation." Complexity 2021 (July 30, 2021): 1–15. http://dx.doi.org/10.1155/2021/5511396.

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The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

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Jayaraman, Amitesh S., Domenico Campolo, and Gregory S. Chirikjian. "Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group." Entropy 22, no. 4 (April 17, 2020): 455. http://dx.doi.org/10.3390/e22040455.

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The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

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Rigatos, Gerasimos G. "Stabilization of option price dynamics through feedback control of the Black-Scholes PDE." IFAC-PapersOnLine 48, no. 11 (2015): 574–80. http://dx.doi.org/10.1016/j.ifacol.2015.09.248.

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Roul, Pradip, and V. M. K. Prasad Goura. "A sixth order numerical method and its convergence for generalized Black–Scholes PDE." Journal of Computational and Applied Mathematics 377 (October 2020): 112881. http://dx.doi.org/10.1016/j.cam.2020.112881.

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LI, MINQIANG, and FABIO MERCURIO. "CLOSED-FORM APPROXIMATION OF PERPETUAL TIMER OPTION PRICES." International Journal of Theoretical and Applied Finance 17, no. 04 (June 2014): 1450026. http://dx.doi.org/10.1142/s0219024914500265.

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We develop an asymptotic expansion technique for pricing timer options in stochastic volatility models when the effect of volatility of variance is small. Based on the pricing PDE, closed-form approximation formulas have been obtained. The approximation has an easy-to-understand Black–Scholes-like form and many other attractive properties. Numerical analysis shows that the approximation formulas are very fast and accurate, especially when the volatility of variance is not large.

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Zhou, Liuwei, and Zhijie Wang. "Portfolio Strategy of Financial Market with Regime Switching Driven by Geometric Lévy Process." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/538041.

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The problem of a portfolio strategy for financial market with regime switching driven by geometric Lévy process is investigated in this paper. The considered financial market includes one bond and multiple stocks which has few researches up to now. A new and general Black-Scholes (B-S) model is set up, in which the interest rate of the bond, the rate of return, and the volatility of the stocks vary as the market states switching and the stock prices are driven by geometric Lévy process. For the general B-S model of the financial market, a portfolio strategy which is determined by a partial differential equation (PDE) of parabolic type is given by using Itô formula. The PDE is an extension of existing result. The solvability of the PDE is researched by making use of variables transformation. An application of the solvability of the PDE on the European options with the final data is given finally.

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ESIPOV, SERGEI, and IGOR VAYSBURD. "ON THE PROFIT AND LOSS DISTRIBUTION OF DYNAMIC HEDGING STRATEGIES." International Journal of Theoretical and Applied Finance 02, no. 02 (April 1999): 131–52. http://dx.doi.org/10.1142/s0219024999000108.

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Hedging a derivative security with non-risk-neutral number of shares leads to portfolio profit or loss. Unlike in the Black–Scholes world, the net present value of all future cash flows till maturity is no longer deterministic, and basis risk may be present at any time. The key object of our analysis is probability distribution of future P & L conditioned on the present value of the underlying. We consider time dynamics of this probability distribution for an arbitrary hedging strategy. We assume log-normal process for the value of the underlying asset and use convolution formula to relate conditional probability distribution of P & L at any two successive time moments. It leads to a simple PDE on the probability measure parameterized by a hedging strategy. For risk-neutral replication the P & L probability distribution collapses to a delta-function at the Black–Scholes price of the contingent claim. Therefore, our approach is consistent with the Black–Scholes one and can be viewed as its generalization. We further analyze the PDE and derive formulae for hedging strategies targeting various objectives, such as minimizing variance or optimizing distribution quantiles. The developed method of computing the profit and loss distribution for a given hedging scheme is applied to the classical example of hedging a European call option using the "stop-loss" strategy. This strategy refers to holding 1 or 0 shares of the underlying security depending on the market value of such security. It is shown that the "stop-loss" strategy can lead to a loss even for an infinite frequency of re-balancing. The analytical method allows one to compute profit and loss distributions without relying on simulations. To demonstrate the strength of the method we reproduce the Monte Carlo results on "stop-loss" strategy given in Hull's book, and improve the precision beyond the limits of regular Monte-Carlo simulations.

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LYUKOV, ALEXANDER. "OPTION PRICING WITH FEEDBACK EFFECTS." International Journal of Theoretical and Applied Finance 07, no. 06 (September 2004): 757–68. http://dx.doi.org/10.1142/s0219024904002633.

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The paper provides a continuous time model for order-driven stock market. The model allows to derive a nonlinear PDE as a modification of Black–Scholes equation for option pricing with a local volatility as a function of the stock price. The solution can be expanded in series in the parameter, which relates to the size of option market. The first-order correction for the option price increases the price of a European call. The second-order correction for volatility allows to describe the "volatility smile".

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TAYLOR, STEPHEN, and SCOTT GLASGOW. "A NOVEL REDUCTION OF THE SIMPLE ASIAN OPTION AND LIE-GROUP INVARIANT SOLUTIONS." International Journal of Theoretical and Applied Finance 12, no. 08 (December 2009): 1197–212. http://dx.doi.org/10.1142/s0219024909005634.

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We develop the complete 6-dimensional classical symmetry group of the partial differential equation (PDE) that governs the fair price of a simple Asian option within a simple market model. The symmetries we expose include the 5-dimensional symmetry group partially noted by Rogers and Shi, and communicated implicitly by the change of numéraire arguments of Večeř (in which symmetries reduce the original 2 + 1 dimensional simple Asian option PDE to a 1 + 1 dimensional PDE). Going beyond this previous work, we expose a new 1-dimensional space of symmetries of the Asian PDE that cannot reasonably be found by inspection. We demonstrate that the new symmetry could be used to formulate a new, "nonlinear" derivative security that has a 1 + 1 dimensional PDE formulation. We indicate that this nonlinear security has a closed-form pricing formula similar to that of the Black–Scholes equation for a particular market dependent payoff, and show that hedging the short position in this particular exotic option is stable for all market parameters. We also demonstrate the patently Lie-algebraic method for obtaining the already well-known "Rogers–Shi–Večeř" reduction.

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Lin, Sha, and Song-Ping Zhu. "Pricing resettable convertible bonds using an integral equation approach." IMA Journal of Management Mathematics 31, no. 4 (December 24, 2019): 417–43. http://dx.doi.org/10.1093/imaman/dpz015.

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Abstract In this paper, the fair price of an American-style resettable convertible bond (CB) under the Black–Scholes model with a particular reset clause is calculated. This is a challenging problem because an unknown optimal conversion price needs to be determined together with the bond price. There is also an additional complexity that the value of the conversion ratio will change when the underlying price touches the reset price. Because of the additional reset clause, the bond price is not always a monotonically increasing function with the underlying price, which is impossible for other types of the CBs. Of course, the problem can be dealt with using the Monte-Carlo simulation. But, a partial differential equation (PDE)/integral equation approach is far superior in terms of computational efficiency. Fortunately, after establishing the PDE system governing the bond price, we are able to present an integral equation representation by applying the incomplete Fourier transform on the PDE system.

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Song, Qingshuo, and Pengfei Yang. "Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution." Quantitative Finance 15, no. 5 (November 25, 2013): 901–8. http://dx.doi.org/10.1080/14697688.2013.838634.

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Rigatos, G., and N. Zervos. "Detection of Mispricing in the Black–Scholes PDE Using the Derivative-Free Nonlinear Kalman Filter." Computational Economics 50, no. 1 (April 16, 2016): 1–20. http://dx.doi.org/10.1007/s10614-016-9575-2.

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Mahatma, Yudi, and Ibnu Hadi. "Stochastic Volatility Estimation of Stock Prices using the Ensemble Kalman Filter." InPrime: Indonesian Journal of Pure and Applied Mathematics 3, no. 2 (November 10, 2021): 136–43. http://dx.doi.org/10.15408/inprime.v3i2.20256.

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AbstractVolatility plays important role in options trading. In their seminal paper published in 1973, Black and Scholes assume that the stock price volatility, which is the underlying security volatility of a call option, is constant. But thereafter, researchers found that the return volatility was not constant but conditional to the information set available at the computation time. In this research, we improve a methodology to estimate volatility and interest rate using Ensemble Kalman Filter (EnKF). The price of call and put option used in the observation and the forecasting step of the EnKF algorithm computed using the solution of Black-Scholes PDE. The state-space used in this method is the augmented state space, which consists of static variables: volatility and interest rate, and dynamic variables: call and put option price. The numerical experiment shows that the EnKF algorithm is able to estimate accurately the estimated volatility and interest rates with an RMSE value of 0.0506.Keywords: stochastic volatility; call option; put option; Ensemble Kalman Filter. AbstrakVolatilitas adalah faktor penting dalam perdagangan suatu opsi. Dalam makalahnya yang dipublikasikan tahun 1973, Black dan Scholes mengasumsikan bahwa volatilitas harga saham, yang merupakan volatilitas sekuritas yang mendasari opsi beli, adalah konstan. Akan tetapi, para peneliti menemukan bahwa volatilitas pengembalian tidaklah konstan melainkan tergantung pada kumpulan informasi yang dapat digunakan pada saat perhitungan. Pada penelitian ini dikembangkan metodologi untuk mengestimasi volatilitas dan suku bunga menggunakan metode Ensembel Kalman Filter (EnKF). Harga opsi beli dan opsi jual yang digunakan pada observasi dan pada tahap prakiraan pada algoritma EnKF dihitung menggunakan solusi persamaan Black-Scholes. Ruang keadaan yang digunakan adalah ruang keadaan yang diperluas yang terdiri dari variabel statis yaitu volatilitas dan suku bunga, dan variabel dinamis yaitu harga opsi beli dan harga opsi jual. Eksperimen numerik menunjukkan bahwa algoritma ENKF dapat secara akurat mengestimasi volatiltas dan suku bunga dengan RMSE 0.0506.Kata kunci: volatilitas stokastik; opsi beli; opsi jual; Ensembel Kalman Filter.

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Debnath, Tanmoy Kumar, and ABM Shahadat Hossain. "A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing." GANIT: Journal of Bangladesh Mathematical Society 40, no. 1 (July 14, 2020): 13–27. http://dx.doi.org/10.3329/ganit.v40i1.48192.

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In this paper, we have applied the finite difference methods (FDMs) for the valuation of European put option (EPO). We have mainly focused the application of Implicit finite difference method (IFDM) and Crank-Nicolson finite difference method (CNFDM) for option pricing. Both these techniques are used to discretized Black-Scholes (BS) partial differential equation (PDE). We have also compared the convergence of the IFDM and CNFDM to the analytic BS price of the option. This turns out a conclusion that both these techniques are fairly fruitful and excellent for option pricing. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 13-27

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in 't Hout, K. J., and K. Volders. "Stability and convergence analysis of discretizations of the Black-Scholes PDE with the linear boundary condition." IMA Journal of Numerical Analysis 34, no. 1 (May 16, 2013): 296–325. http://dx.doi.org/10.1093/imanum/drs050.

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El kharrazi, Zaineb, Nouh Izem, Mustapha Malek, and Sahar Saoud. "A Partition of unity finite element method for valuation American option under Black-Scholes model." Moroccan Journal of Pure and Applied Analysis 7, no. 2 (January 29, 2021): 324–36. http://dx.doi.org/10.2478/mjpaa-2021-0021.

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Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.

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Salvador, Beatriz, Cornelis W. Oosterlee, and Remco van der Meer. "European and American Options Valuation by Unsupervised Learning with Artificial Neural Networks." Proceedings 54, no. 1 (August 19, 2020): 14. http://dx.doi.org/10.3390/proceedings2020054014.

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Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). In this work, the classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option, we solve the linear complementarity problem formulation.

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Fan, Congyin, Kaili Xiang, and Peimin Chen. "Efficient Option Pricing in Crisis Based on Dynamic Elasticity of Variance Model." Discrete Dynamics in Nature and Society 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/7496539.

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Market crashes often appear in daily trading activities and such instantaneous occurring events would affect the stock prices greatly. In an unstable market, the volatility of financial assets changes sharply, which leads to the fact that classical option pricing models with constant volatility coefficient, even stochastic volatility term, are not accurate. To overcome this problem, in this paper we put forward a dynamic elasticity of variance (DEV) model by extending the classical constant elasticity of variance (CEV) model. Further, the partial differential equation (PDE) for the prices of European call option is derived by using risk neutral pricing principle and the numerical solution of the PDE is calculated by the Crank-Nicolson scheme. In addition, Kalman filtering method is employed to estimate the volatility term of our model. Our main finding is that the prices of European call option under our model are more accurate than those calculated by Black-Scholes model and CEV model in financial crashes.

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Bonotto, E. M., M. Federson, and P. Muldowney. "The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral." Proceedings of the Singapore National Academy of Science 15, no. 01 (March 2021): 45–59. http://dx.doi.org/10.1142/s2591722621400068.

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The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

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HOOGLAND, J. K., and C. D. D. NEUMANN. "LOCAL SCALE INVARIANCE AND CONTINGENT CLAIM PRICING." International Journal of Theoretical and Applied Finance 04, no. 01 (February 2001): 1–21. http://dx.doi.org/10.1142/s0219024901000857.

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Prices of tradables can only be expressed relative to one another at any instant of time. This fundamental fact should therefore also hold for contingent claims, i.e. tradable instruments, whose prices depend on the prices of other tradables. We show that this property induces a local scale invariance in the problem of pricing contingent claims. Due to this symmetry we do not require any martingale techniques to arrive at the price of a claim. If the tradables are driven by Brownian motion, we find, in a natural way, that this price satisfies a PDE. Both possess a manifest gauge invariance. A unique solution can only be given when we impose restrictions on the drifts and volatilities of the tradables, i.e. the underlying market structure. We give some examples of the application of this PDE to the pricing of claims. In the Black–Scholes world we show the equivalence of our formulation with the standard approach. It is stressed that the formulation in terms of tradables leads to a significant conceptual simplification of the pricing-problem.

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Alobaidi, Ghada, and Roland Mallier. "Asymptotic analysis of American call options." International Journal of Mathematics and Mathematical Sciences 27, no. 3 (2001): 177–88. http://dx.doi.org/10.1155/s0161171201005701.

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American call options are financial derivatives that give the holder the right but not the obligation to buy an underlying security at a pre-determined price. They differ from European options in that they may be exercised at any time prior to their expiration, rather than only at expiration. Their value is described by the Black-Scholes PDE together with a constraint that arises from the possibility of early exercise. This leads to a free boundary problem for the optimal exercise boundary, which determines whether or not it is beneficial for the holder to exercise the option prior to expiration. However, an exact solution cannot be found, and therefore by using asymptotic techniques employed in the study of boundary layers in fluid mechanics, we find an asymptotic expression for the location of the optimal exercise boundary and the value of the option near to expiration.

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Mei, Shu-Li. "Faber-Schauder Wavelet Sparse Grid Approach for Option Pricing with Transactions Cost." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/168630.

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Transforming the nonlinear Black-Scholes equation into the diffusion PDE by introducing the log transform ofSand(T−t)→τcan provide the most stable platform within which option prices can be evaluated. The space jump that appeared in the transformation model is suitable to be solved by the sparse grid approach. An adaptive sparse approximation solution of the nonlinear second-order PDEs was constructed using Faber-Schauder wavelet function and the corresponding multiscale analysis theory. First, we construct the multiscale wavelet interpolation operator based on the definition of interpolation wavelet theory. The operator can be used to discretize the weak solution function of the nonlinear second-order PDEs. Second, using the couple technique of the variational iteration method (VIM) and the precision integration method, the sparse approximation solution of the nonlinear partial differential equations can be obtained. The method is tested on three classical nonlinear option pricing models such as Leland model, Barles-Soner model, and risk adjusted pricing methodology. The solutions are compared with the finite difference method. The present results indicate that the method is competitive.

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Salvador, Beatriz, Cornelis W. Oosterlee, and Remco van der Meer. "Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks." Mathematics 9, no. 1 (December 28, 2020): 46. http://dx.doi.org/10.3390/math9010046.

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Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

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Jandačka, Martin, and Daniel Ševčovič. "On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile." Journal of Applied Mathematics 2005, no. 3 (2005): 235–58. http://dx.doi.org/10.1155/jam.2005.235.

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We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the variance of the synthesized portfolio. Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval. In this model, prices of vanilla options can be computed from a solution to a fully nonlinear parabolic equation in which a diffusion coefficient representing volatility nonlinearly depends on the solution itself giving rise to explaining the volatility smile analytically. We derive a robust numerical scheme for solving the governing equation and perform extensive numerical testing of the model and compare the results to real option market data. Implied risk and volatility are introduced and computed for large option datasets. We discuss how they can be used in qualitative and quantitative analysis of option market data.

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Kalife, Aymeric, and Saad Mouti. "On Optimal Options Book Execution Strategies with Market Impact." Market Microstructure and Liquidity 02, no. 03n04 (December 2016): 1750002. http://dx.doi.org/10.1142/s2382626617500022.

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We consider the optimal execution of a book of options when market impact is a driver of the option price. We aim at minimizing the mean-variance risk criterion for a given market impact function. First, we develop a framework to justify the choice of our market impact function. Our model is inspired from Leland’s option replication with transaction costs where the market impact is directly part of the implied volatility function. The option price is then expressed through a Black– Scholes-like PDE with a modified implied volatility directly dependent on the market impact. We set up a stochastic control framework and solve an Hamilton–Jacobi–Bellman equation using finite differences methods. The expected cost problem suggests that the optimal execution strategy is characterized by a convex increasing trading speed, in contrast to the equity case where the optimal execution strategy results in a rather constant trading speed. However, in such mean valuation framework, the underlying spot price does not seem to affect the agent’s decision. By taking the agent risk aversion into account through a mean-variance approach, the strategy becomes more sensitive to the underlying price evolution, urging the agent to trade faster at the beginning of the strategy.

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GRISHCHENKO, OLESYA, XIAO HAN, and VICTOR NISTOR. "A VOLATILITY-OF-VOLATILITY EXPANSION OF THE OPTION PRICES IN THE SABR STOCHASTIC VOLATILITY MODEL." International Journal of Theoretical and Applied Finance 23, no. 03 (May 2020): 2050018. http://dx.doi.org/10.1142/s0219024920500181.

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We propose a new type of asymptotic expansion for the transition probability density function (or heat kernel) of certain parabolic partial differential equations (PDEs) that appear in option pricing. As other, related methods developed by Costanzino, Hagan, Gatheral, Lesniewski, Pascucci, and their collaborators, among others, our method is based on the computation of the truncated asymptotic expansion of the heat kernel with respect to a “small” parameter. What sets our method apart is that our small parameter is possibly different from the time to expiry and that the resulting commutator calculations go beyond the nilpotent Lie algebra case. In favorable situations, the terms of this asymptotic expansion can quickly be computed explicitly leading to a “closed-form” approximation of the solution, and hence of the option price. Our approximations tend to have much fewer terms than the ones obtained from short time asymptotics, and are thus easier to generalize. Another advantage is that the first term of our expansion corresponds to the classical Black-Scholes model. Our method also provides equally fast approximations of the derivatives of the solution, which is usually a challenge. A full theoretical justification of our method seems very difficult at this time, but we do provide some justification based on the results of (Siyan, Mazzucato, and Nistor, NWEJ 2018). We therefore mostly content ourselves to demonstrate numerically the efficiency of our method by applying it to the solution of the mean-reverting SABR stochastic volatility model PDE, commonly referred to as the [Formula: see text]SABR PDE, by taking the volatility of the volatility parameter [Formula: see text] (vol-of-vol) as a small parameter. For this PDE, we provide extensive numerical tests to gauge the performance of our method. In particular, we compare our approximation to the one obtained using Hagan’s formula and to the one obtained using a new, adaptive finite difference method. We provide an explicit asymptotic expansion for the implied volatility (generalizing Hagan’s formula), which is what is typically needed in concrete applications. We also calibrate our model to observed market option price data. The resulting values for the parameters [Formula: see text], [Formula: see text], and [Formula: see text] are realistic, which provides more evidence for the conjecture that the volatility is mean-reverting.

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41

Kolman, Marek. "Galerkin FEM for Black-Scholes PDE." SSRN Electronic Journal, 2017. http://dx.doi.org/10.2139/ssrn.3081892.

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Le Floc'h, Fabien. "Pitfalls of Exponential Fitting on the Black-Scholes PDE." SSRN Electronic Journal, 2016. http://dx.doi.org/10.2139/ssrn.2711720.

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Nuugulu, S. M., F. Gideon, and K. C. Patidar. "A robust numerical solution to a time-fractional Black–Scholes equation." Advances in Difference Equations 2021, no. 1 (February 24, 2021). http://dx.doi.org/10.1186/s13662-021-03259-2.

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AbstractDividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options.

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Mehra, Mani, Kuldip Singh Patel, and Ankita Shukla. "Wavelet-optimized compact finite difference method for convection–diffusion equations." International Journal of Nonlinear Sciences and Numerical Simulation, December 3, 2020. http://dx.doi.org/10.1515/ijnsns-2018-0295.

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AbstractIn this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

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Davey, Ashley, and Harry Zheng. "Deep Learning for Constrained Utility Maximisation." Methodology and Computing in Applied Probability, November 26, 2021. http://dx.doi.org/10.1007/s11009-021-09912-3.

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AbstractThis paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.

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Journal articles on the topic 'Black-Scholes PDE'

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