Journal articles: 'Options (Finance) – Prices – Mathematical models'

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Abraham, Rebecca, and Hani El-Chaarani. "A Mathematical Formulation of the Valuation of Ether and Ether Derivatives as a Function of Investor Sentiment and Price Jumps." Journal of Risk and Financial Management 15, no. 12 (December 8, 2022): 591. http://dx.doi.org/10.3390/jrfm15120591.

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The purpose of this study was to create quantitative models to value ether, ether futures, and ether options based upon the ability of cryptocurrencies to transform existing intermediary-verified payments to non-intermediary-based currency transfers, the ability of ether as a late mover to displace bitcoin as the first mover, and the valuation of ether in the context of investor irrationality models. The risk-averse investor’s utility function is a combination of expectations of the performance of ether, expectations of cryptocurrencies’ transformative power, and expectations of ether superseding bitcoin. The moderate risk-taker’s utility function is an alt-Weibull distribution, along with a gamma distribution. Risk-takers have a utility function in the form of a Bessel function. Ether price functions consist of a Levy jump process. Ether futures are valued as the combination of current spot prices along with term prices. The value of spot prices is the product of a spot premium and a lognormal distribution of spot prices. The value of term prices is equal to the product of a term premium, and the Levy jump process of price fluctuations during the delivery period. For ether options, a less risky ether option portfolio offsets ether’s risk by a fixed-income trading strategy.

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CARMONA, RENÉ, and SERGEY NADTOCHIY. "TANGENT MODELS AS A MATHEMATICAL FRAMEWORK FOR DYNAMIC CALIBRATION." International Journal of Theoretical and Applied Finance 14, no. 01 (February 2011): 107–35. http://dx.doi.org/10.1142/s0219024911006280.

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Motivated by the desire to integrate repeated calibration procedures into a single dynamic market model, we introduce the notion of a "tangent model" in an abstract set up, and we show that this new mathematical paradigm accommodates all the recent attempts to study consistency and absence of arbitrage in market models. For the sake of illustration, we concentrate on the case when market quotes provide the prices of European call options for a specific set of strikes and maturities. While reviewing our recent results on dynamic local volatility and tangent Lévy models, we present a theory of tangent models unifying these two approaches and construct a new class of tangent Lévy models, which allows the underlying to have both continuous and pure jump components.

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Kumar Jaiswal, Jitendra, and Raja Das. "Artificial Neural Network Algorithms based Nonlinear Data Analysis for Forecasting in the Finance Sector." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 169. http://dx.doi.org/10.14419/ijet.v7i4.10.20829.

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The involvement of big populace in the quantitative trading has been increased remarkably since the wired and wireless systems have become quite ubiquitous in the fields of finance and economics. Statistical, mathematical and technical analysis in parallel with machine learning and artificial intelligence are frequently being applied to perceive prices moving pattern and forecasting. However stock price do not follow any deterministic regulatory function, factor or circumstances rather than many considerations such as economy and finance, political environments, demand and supply, buying and selling tendency, trading and investment, etc. Historical data assist remarkably for prices forecasting as an important option for mathematicians and researchers. In this paper, we have followed backpropagation and radial basis function neural network for predicting future prices by modifying these techniques as per requirements. We have also performed a comparative analysis of the two ANN techniques for existing and our modified models.

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Eissa, Mahmoud A., and M. Elsayed. "Improve Stock Price Model-Based Stochastic Pantograph Differential Equation." Symmetry 14, no. 7 (July 1, 2022): 1358. http://dx.doi.org/10.3390/sym14071358.

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Although the concept of symmetry is widely used in many fields, it is almost not discussed in finance. This concept appears to be relevant in relation, for example, to mathematical models that can predict stock prices to contribute to the decision-making process. This work considers the stock price of European options with a new class of the non-constant delay model. The stochastic pantograph differential equation (SPDE) with a variable delay is provided in order to overcome the weaknesses of using stochastic models with constant delay. The proposed model is constructed to improve the evaluation process and prediction accuracy for stock prices. The feasibility of the proposed model is introduced under relatively weak conditions imposed on its volatility function. Furthermore, the sensitivity of time lag is discussed. The robust stochastic theta Milstein (STM) method is combined with the Monte Carlo simulation to compute asset prices within the proposed model. In addition, we prove that the numerical solution can preserve the non-negativity of the solution of the model. Numerical experiments using real financial data indicate that there is an increasing possibility of prediction accuracy for the proposed model with a variable delay compared to non-linear models with constant delay and the classical Black and Scholes model.

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Fernández, Lexuri, Peter Hieber, and Matthias Scherer. "Double-barrier first-passage times of jump-diffusion processes." mcma 19, no. 2 (July 1, 2013): 107–41. http://dx.doi.org/10.1515/mcma-2013-0005.

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Abstract. Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution; extending single-barrier results by [Journal of Derivatives 10 (2002), 43–54]. In mathematical finance, the double-barrier first-passage time is required to price exotic derivatives, for example corridor bonus certificates, (step) double barrier options, or digital first-touch options, that depend on whether or not the underlying asset price exceeds certain threshold levels. Furthermore, it is relevant in structural credit risk models if one considers two exit events, e.g., default and early repayment.

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Aghabeygi, Mona, Kamel Louhichi, and Sergio Gomez y Paloma. "Impacts of fertilizer subsidy reform options in Iran: an assessment using a Regional Crop Programming model." Bio-based and Applied Economics 11, no. 1 (July 20, 2022): 55–73. http://dx.doi.org/10.36253/bae-10981.

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The aim of this paper is to assess the potential impacts of different fertilizer subsidy reform options on the performance of the Iranian crops production sector. This is achieved using a Regional Crop Programming (RCP) model, based on Positive Mathematical Programming, which includes in total 14 crop activities and encompasses 31 administrative regions. The RCP model is a collection of micro-economic models, working with exogenous prices, each representing the optimal crop allocation at the regional level. The model is calibrated against observed data on crop acreage, yield responses to nitrogen application, and exogenous supply elasticities. Simulation results show that a total removal of nitrogen fertilizer subsidies would affect the competitiveness of crops with the highest nitrogen application rates and lead to a slight reduction of national agricultural income, at approximately 1%. This effect, which is more pronounced at the regional level, is driven by area reallocation rather than land productivity. The reallocation of nitrogen fertilizer subsidy to only strategic crops boost their production and income but increase disparity among regions and affects negatively welfare compared to the current universal fertilizer program. The transfer efficiency analysis shows that both target and universal simulated options are inefficient with an efficiency score below one.

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Giribone, Pier Giuseppe, and Roberto Revetria. "Certificate pricing using Discrete Event Simulations and System Dynamics theory." Risk Management Magazine 16, no. 2 (August 18, 2021): 75–93. http://dx.doi.org/10.47473/2020rmm0092.

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The study proposes an innovative application of Discrete Event Simulations (DES) and System Dynamics (SD) theory to the pricing of a certain kind of certificates very popular among private investors and, more generally, in the context of wealth management. The paper shows how numerical simulation software mainly used in traditional engineering, such as industrial and mechanical engineering, can be successfully adapted to the risk analysis of structured financial products. The article can be divided into three macro-sections: in the first part a synthetic overview of the most widespread option pricing models in the quantitative finance branch is given to the readers together with the fundamental technical-instrumental background of the implemented DES and SD simulator. After dealing with some of the most popular models adopted for Equity and Equity index options, which are the most common underlying assets for the certificates structuring, we move, in the second part, to describe how the mathematical models can be integrated into a general simulation environment able to provide both DES and SD extensively used in the engineering field. The core stochastic differential equation (SDE) will therefore be translated, together with all its input parameters, into a visual block model which allows an immediate quantitative analysis of how market parameters and the other model variables can change over time. The possibility for the structurer to observe how the variables evolve day-by-day gives a strong sensitivity to evaluate how the price and the associated risk measures can be directly affected. The third part of the study compares the results obtained from the simulator designed by the authors with the more traditional pricing approaches, which consist in programming Matlab® codes for the numerical integration of the core stochastic dynamics through a Euler-Maruyama scheme. The comparison includes a price check using the Bloomberg® DLIB pricing module and a check directly against the valuation provided by the counterparty. In this section, real market cases will therefore be examined with a complete quantitative analysis of two of the most widespread categories of certificates in wealth management: Multi-asset Barrier Reverse Convertible with Issuer Callability and Multi-asset Express Certificate with conditional memory fixed coupon.

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Nguyen, Ngoc Quynh Anh, and Thi Ngoc Trang Nguyen. "Risk measures computation by Fourier inversion." Journal of Risk Finance 18, no. 1 (January 16, 2017): 76–87. http://dx.doi.org/10.1108/jrf-03-2016-0034.

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Purpose The purpose of this paper is to present the method for efficient computation of risk measures using Fourier transform technique. Another objective is to demonstrate that this technique enables an efficient computation of risk measures beyond value-at-risk and expected shortfall. Finally, this paper highlights the importance of validating assumptions behind the risk model and describes its application in the affine model framework. Design/methodology/approach The method proposed is based on Fourier transform methods for computing risk measures. The authors obtain the loss distribution by fitting a cubic spline through the points where Fourier inversion of the characteristic function is applied. From the loss distribution, the authors calculate value-at-risk and expected shortfall. As for the calculation of the entropic value-at-risk, it involves the moment generating function which is closely related to the characteristic function. The expectile risk measure is calculated based on call and put option prices which are available in a semi-closed form by Fourier inversion of the characteristic function. We also consider mean loss, standard deviation and semivariance which are calculated in a similar manner. Findings The study offers practical insights into the efficient computation of risk measures as well as validation of the risk models. It also provides a detailed description of algorithms to compute each of the risk measures considered. While the main focus of the paper is on portfolio-level risk metrics, all algorithms are also applicable to single instruments. Practical implications The algorithms presented in this paper require little computational effort which makes them very suitable for real-world applications. In addition, the mathematical setup adopted in this paper provides a natural framework for risk model validation which makes the approach presented in this paper particularly appealing in practice. Originality/value This is the first study to consider the computation of entropic value-at-risk, semivariance as well as expectile risk measure using Fourier transform method.

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Madan, Dilip B., and King Wang. "Risk Neutral Jump Arrival Rates Implied in Option Prices and Their Models." Applied Mathematical Finance 28, no. 3 (May 4, 2021): 201–35. http://dx.doi.org/10.1080/1350486x.2021.2007145.

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SKIADOPOULOS, GEORGE. "VOLATILITY SMILE CONSISTENT OPTION MODELS: A SURVEY." International Journal of Theoretical and Applied Finance 04, no. 03 (June 2001): 403–37. http://dx.doi.org/10.1142/s021902490100105x.

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The developing literature on "smile consistent" no-arbitrage models has emerged from the need to price and hedge exotic options consistently with the prices of standard European options. This survey paper describes the steps through which this literature has evolved by providing a taxonomy of the various models. It highlights the main ideas behind the different models, and it outlines their advantages and limitations. Practical issues in implementing the models are also discussed.

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Ekström, Erik, and Johan Tysk. "PROPERTIES OF OPTION PRICES IN MODELS WITH JUMPS." Mathematical Finance 17, no. 3 (July 2007): 381–97. http://dx.doi.org/10.1111/j.1467-9965.2007.00308.x.

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LORENZO, MERCURI. "PRICING ASIAN OPTIONS IN AFFINE GARCH MODELS." International Journal of Theoretical and Applied Finance 14, no. 02 (March 2011): 313–33. http://dx.doi.org/10.1142/s0219024911006371.

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We derive recursive relationships for the m.g.f. of the geometric average of the underlying within some affine Garch models [Heston and Nandi (2000), Christoffersen et al. (2006), Bellini and Mercuri (2007), Mercuri (2008)] and use them for the semi-analytical valuation of geometric Asian options. Similar relationships are obtained for low order moments of the arithmetic average, that are used for an approximated valuation of arithmetic Asian options based on truncated Edgeworth expansions, following the approach of Turnbull and Wakeman (1991). In both cases the accuracy of the semi-analytical procedure is assessed by means of a comparison with Monte Carlo prices. The results are quite good in the geometric case, while in the arithmetic case the proposed methodology seems to work well only in the Heston and Nandi case.

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MERINO, R., J. POSPÍŠIL, T. SOBOTKA, and J. VIVES. "DECOMPOSITION FORMULA FOR JUMP DIFFUSION MODELS." International Journal of Theoretical and Applied Finance 21, no. 08 (December 2018): 1850052. http://dx.doi.org/10.1142/s0219024918500528.

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In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

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ERIKSSON, JONATAN. "MONOTONICITY IN THE VOLATILITY OF SINGLE-BARRIER OPTION PRICES." International Journal of Theoretical and Applied Finance 09, no. 06 (September 2006): 987–96. http://dx.doi.org/10.1142/s0219024906003822.

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We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases: • If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general. • If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general. We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.

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Henderson, Vicky. "ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS." Mathematical Finance 15, no. 1 (January 2005): 49–59. http://dx.doi.org/10.1111/j.0960-1627.2005.00210.x.

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SCHOUTENS, WIM, and STIJN SYMENS. "THE PRICING OF EXOTIC OPTIONS BY MONTE–CARLO SIMULATIONS IN A LÉVY MARKET WITH STOCHASTIC VOLATILITY." International Journal of Theoretical and Applied Finance 06, no. 08 (December 2003): 839–64. http://dx.doi.org/10.1142/s0219024903002249.

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Recently, stock price models based on Lévy processes with stochastic volatility were introduced. The resulting vanilla option prices can be calibrated almost perfectly to empirical prices. Under this model, we will price exotic options, like barrier, lookback and cliquet options, by Monte–Carlo simulation. The sampling of paths is based on a compound Poisson approximation of the Lévy process involved. The precise choice of the terms in the approximation is crucial and investigated in detail. In order to reduce the standard error of the Monte–Carlo simulation, we make use of the technique of control variates. It turns out that there are significant differences with the classical Black–Scholes prices.

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TAKAHASHI, AKIHIKO, and KOHTA TAKEHARA. "FOURIER TRANSFORM METHOD WITH AN ASYMPTOTIC EXPANSION APPROACH: AN APPLICATION TO CURRENCY OPTIONS." International Journal of Theoretical and Applied Finance 11, no. 04 (June 2008): 381–401. http://dx.doi.org/10.1142/s0219024908004853.

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This paper develops a Fourier transform method with an asymptotic expansion approach for option pricing. The method is applied to European currency options with a libor market model of interest rates and jump-diffusion stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas of the characteristic functions of log-prices of the underlying assets and the prices of currency options based on a third order asymptotic expansion scheme; we use a jump-diffusion model with a mean-reverting stochastic variance process such as in Heston [7]/Bates [1] and log-normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

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BENTH, FRED ESPEN, and RODWELL KUFAKUNESU. "PRICING OF EXOTIC ENERGY DERIVATIVES BASED ON ARITHMETIC SPOT MODELS." International Journal of Theoretical and Applied Finance 12, no. 04 (June 2009): 491–506. http://dx.doi.org/10.1142/s0219024909005324.

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Based on a non-Gaussian Ornstein–Uhlenbeck model for energy spot, we derive prices for Asian and spread options using Fourier techniques. The option prices are expressed in terms of the Fourier transform of the payoff function and the characteristic functions of the driving noises, being independent increment processes. In many relevant situations, these functions are explicitly available, and fast Fourier transform can be used for efficient numerical valuation. The arithmetic nature of our model implies that only a one-dimensional Fourier transform is required in the computation of the price, contrary to geometric models where transformation along both underlying variables is necessary.

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EKSTRÖM, ERIK, and JOHAN TYSK. "OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS." International Journal of Theoretical and Applied Finance 07, no. 07 (November 2004): 901–7. http://dx.doi.org/10.1142/s0219024904002694.

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There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

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BIELECKI, TOMASZ R., IGOR CIALENCO, ISMAIL IYIGUNLER, and RODRIGO RODRIGUEZ. "DYNAMIC CONIC FINANCE: PRICING AND HEDGING IN MARKET MODELS WITH TRANSACTION COSTS VIA DYNAMIC COHERENT ACCEPTABILITY INDICES." International Journal of Theoretical and Applied Finance 16, no. 01 (February 2013): 1350002. http://dx.doi.org/10.1142/s0219024913500027.

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In this paper we present a theoretical framework for determining dynamic ask and bid prices of derivatives using the theory of dynamic coherent acceptability indices in discrete time. We prove a version of the First Fundamental Theorem of Asset Pricing using the dynamic coherent risk measures. We introduce the dynamic ask and bid prices of a derivative contract in markets with transaction costs. Based on these results, we derive a representation theorem for the dynamic bid and ask prices in terms of dynamically consistent sequence of sets of probability measures and risk-neutral measures. To illustrate our results, we compute the ask and bid prices of some path-dependent options using the dynamic Gain-Loss Ratio.

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GAPEEV, PAVEL V., and MONIQUE JEANBLANC. "FIRST-TO-DEFAULT AND SECOND-TO-DEFAULT OPTIONS IN MODELS WITH VARIOUS INFORMATION FLOWS." International Journal of Theoretical and Applied Finance 24, no. 04 (June 2021): 2150022. http://dx.doi.org/10.1142/s0219024921500229.

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We continue to study a credit risk model of a financial market introduced recently by the authors, in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions that are not independent of the ones describing the default intensity rates. We obtain closed form expressions for the no-arbitrage prices of some first-to-default and second-to-default European style contingent claims given the reference filtration initially and progressively enlarged by the two successive default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

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TENG, LONG, MATTHIAS EHRHARDT, and MICHAEL GÜNTHER. "QUANTO PRICING IN STOCHASTIC CORRELATION MODELS." International Journal of Theoretical and Applied Finance 21, no. 05 (August 2018): 1850038. http://dx.doi.org/10.1142/s0219024918500383.

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Correlation plays an important role in pricing multi-asset options. In this work we incorporate stochastic correlation into pricing quanto options which is one special and important type of multi-asset option. Motivated by the market observations that the correlations between financial quantities behave like a stochastic process, instead of using a constant correlation, we allow the asset price process and the exchange rate process to be stochastically correlated with a parameter which is driven either by an Ornstein–Uhlenbeck process or a bounded Jacobi process. We derive an exact quanto option pricing formula in the stochastic correlation model of the Ornstein–Uhlenbeck process and a highly accurate approximated pricing formula in the stochastic correlation model of the bounded Jacobi process, where correlation risk has been hedged. The comparison between prices using our pricing formula and the Monte-Carlo method are provided.

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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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MARABEL, JACINTO. "PRICING DIGITAL OUTPERFORMANCE OPTIONS WITH UNCERTAIN CORRELATION." International Journal of Theoretical and Applied Finance 14, no. 05 (August 2011): 709–22. http://dx.doi.org/10.1142/s0219024911006425.

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Multi-asset options exhibit sensitivity to the correlations between the underlying assets and these correlations are notoriously unstable. Moreover, some of these options such as the digital outperformance options, have a cross-gamma that changes sign depending on the relative evolution of the underlying assets. In this paper, I present a model to price digital outperformance options when there is uncertainty about correlation, but it is assumed to lie within a certain range. Under the assumption that assets prices follow a Geometric Brownian motion with constant instantaneous volatilities I present an analytic expression for the price of the digital outperformance option under the constant correlation assumption, as well as the partial differential equation corresponding to the uncertain correlation model. The comparison of the prices obtained using both models shows that there is no constant correlation which allows attaining the price obtained under the uncertain correlation model. This fact shows that it can be dangerous to assume a constant instantaneous correlation for products with a cross-gamma that changes sign.

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ALÒS, E., F. ANTONELLI, A. RAMPONI, and S. SCARLATTI. "CVA AND VULNERABLE OPTIONS IN STOCHASTIC VOLATILITY MODELS." International Journal of Theoretical and Applied Finance 24, no. 02 (March 2021): 2150010. http://dx.doi.org/10.1142/s0219024921500102.

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This work aims to provide an efficient method to evaluate the Credit Value Adjustment (CVA) for a vulnerable European option, which is an option subject to some default event concerning the issuer solvability. Financial options traded in OTC markets are of this type. In particular, we compute the CVA in some popular stochastic volatility models such as SABR, Hull et al., which have proven to fit quite well market derivatives prices, admitting correlation with the default event. This choice covers the relevant case of Wrong Way Risk (WWR) when a credit deterioration determines an increase in the claim value. Contrary to the structural modeling adopted in [G. Wang, X. Wang & K. Zhu (2017) Pricing vulnerable options with stochastic volatility, Physica A 485, 91–103; C. Ma, S. Yue & Y. Ma (2020) Pricing vulnerable options with Stochastic volatility and Stochastic interest rate, Computational Economics 56, 391–429], we use the reduced-form intensity-based approach to provide an explicit representation formula for the vulnerable option price and related CVA. Later, we specialize the evaluation formula and construct its approximation for the three models mentioned above. Assuming a CIR model for the default intensity process, we run a numerical study to test our approximation, comparing it with Monte Carlo simulations. The results show that for moderate values of the correlation and maturities not exceeding one year, the approximation is very satisfactory as of accuracy and computational time.

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BOSSENS, FRÉDÉRIC, GRÉGORY RAYÉE, NIKOS S. SKANTZOS, and GRISELDA DEELSTRA. "VANNA-VOLGA METHODS APPLIED TO FX DERIVATIVES: FROM THEORY TO MARKET PRACTICE." International Journal of Theoretical and Applied Finance 13, no. 08 (December 2010): 1293–324. http://dx.doi.org/10.1142/s0219024910006212.

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We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black–Scholes price through the so-called "probability of survival" and the "expected first exit time". Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.

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DIA, BAYE M. "A REGULARIZED FOURIER TRANSFORM APPROACH FOR VALUING OPTIONS UNDER STOCHASTIC DIVIDEND YIELDS." International Journal of Theoretical and Applied Finance 13, no. 02 (March 2010): 211–40. http://dx.doi.org/10.1142/s0219024910005747.

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This paper studies the option pricing problem in a class of models in which dividend yields follow a time-homogeneous diffusion. Within this framework, we develop a new approach for valuing options based on the use of a regularized Fourier transform. We derive a pricing formula for European options which gives the option price in the form of an inverse Fourier transform and propose two methods for numerically implementing this formula. As an application of this pricing approach, we introduce the Ornstein-Uhlenbeck and the square-root dividend yield models in which we explicitly solve the pricing problem for European options. Finally we highlight the main effects of a stochastic dividend yield on option prices.

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Maller, Ross A., David H. Solomon, and Alex Szimayer. "A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS." Mathematical Finance 16, no. 4 (September 1, 2006): 613–33. http://dx.doi.org/10.1111/j.1467-9965.2006.00286.x.

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Yu, Cindy L., Haitao Li, and Martin T. Wells. "MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES." Mathematical Finance 21, no. 3 (October 19, 2010): 383–422. http://dx.doi.org/10.1111/j.1467-9965.2010.00439.x.

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VON HAMMERSTEIN, ERNST AUGUST, EVA LÜTKEBOHMERT, LUDGER RÜSCHENDORF, and VIKTOR WOLF. "OPTIMALITY OF PAYOFFS IN LÉVY MODELS." International Journal of Theoretical and Applied Finance 17, no. 06 (September 2014): 1450041. http://dx.doi.org/10.1142/s0219024914500411.

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In this paper, we determine the lowest cost strategy for a given payoff in Lévy markets where the pricing is based on the Esscher martingale measure. In particular, we consider Lévy models where prices are driven by a normal inverse Gaussian (NIG)- or a variance Gamma (VG)-process. Explicit solutions for cost-efficient strategies are derived for a variety of vanilla options, spreads, and forwards. Applications to real financial market data show that the cost savings associated with these strategies can be quite substantial. The empirical findings are supplemented by a result that relates the magnitude of these savings to the strength of the market trend. Moreover, we consider the problem of hedging efficient claims, derive explicit formulas for the deltas of efficient calls and puts and apply the results to German stock market data. Using the time-varying payoff profile of efficient options, we further develop alternative delta hedging strategies for vanilla calls and puts. We find that the latter can provide a more accurate way of replicating the final payoff compared to their classical counterparts.

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ERIKSSON, BJORN, and MARTIJN PISTORIUS. "METHOD OF MOMENTS APPROACH TO PRICING DOUBLE BARRIER CONTRACTS IN POLYNOMIAL JUMP-DIFFUSION MODELS." International Journal of Theoretical and Applied Finance 14, no. 07 (November 2011): 1139–58. http://dx.doi.org/10.1142/s0219024911006644.

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We present a method of moments approach to pricing double barrier contracts when the underlying is modelled by a polynomial jump-diffusion. By general principles the price is linked to certain infinite dimensional linear programming problems. Subsequently approximating these by finite dimensional linear programming problems, upper and lower bounds for the prices of such options are found. We derive theoretical convergence results for this algorithm, and provide numerical illustrations by applying the method to the valuation of several double barrier-type contracts (double barrier knock-out call, American corridor and double-no-touch options) under a number of different models, also allowing for a deterministic short rate.

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BELOMESTNY, DENIS, ANASTASIA KOLODKO, and JOHN SCHOENMAKERS. "PRICING CMS SPREAD OPTIONS IN A LIBOR MARKET MODEL." International Journal of Theoretical and Applied Finance 13, no. 01 (February 2010): 45–62. http://dx.doi.org/10.1142/s021902491000567x.

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We present two approximation methods for the pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.

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Lindström, Erik. "Implications of Parameter Uncertainty on Option Prices." Advances in Decision Sciences 2010 (May 5, 2010): 1–15. http://dx.doi.org/10.1155/2010/598103.

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Financial markets are complex processes where investors interact to set prices. We present a framework for option valuation under imperfect information, taking risk neutral parameter uncertainty into account. The framework is a direct generalization of the existing valuation methodology. Many investors base their decisions on mathematical models that have been calibrated to market prices. We argue that the calibration process introduces a source of uncertainty that needs to be taken into account. The models and parameters used may differ to such extent that one investor may find an option underpriced; whereas another investor may find the very same option overpriced. This problem is not taken into account by any of the standard models. The paper is concluded by presenting simulations and an empirical study on FX options, where we demonstrate improved predictive performance (in sample and out of sample) using this framework.

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PELLEGRINO, TOMMASO. "SECOND-ORDER STOCHASTIC VOLATILITY ASYMPTOTICS AND THE PRICING OF FOREIGN EXCHANGE DERIVATIVES." International Journal of Theoretical and Applied Finance 23, no. 03 (May 2020): 2050021. http://dx.doi.org/10.1142/s0219024920500211.

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We consider models for the pricing of foreign exchange derivatives, where the underlying asset volatility as well as the one for the foreign exchange rate are stochastic. Under this framework, singular perturbation methods have been used to derive first-order approximations for European option prices. In this paper, based on a previous result for the calibration and pricing of single underlying options, we derive the second-order approximation pricing formula in the two-dimensional case and we apply it to the pricing of foreign exchange options.

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Li, Yu. "A mean bound financial model and options pricing." International Journal of Financial Engineering 04, no. 04 (December 2017): 1750047. http://dx.doi.org/10.1142/s2424786317500475.

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Most of financial models, including the famous Black–Scholes–Merton options pricing model, rely upon the assumption that asset returns follow a normal distribution. However, this assumption is not justified by empirical data. To be more concrete, the empirical observations exhibit fat tails or heavy tails and implied volatilities against the strike prices demonstrate U-shaped curve resembling a smile, which is the famous volatility smile. In this paper we present a mean bound financial model and show that asset returns per time unit are Pareto distributed and assets are log Gamma distributed under this model. Based on this we study the sensitivity of the options prices to a change in underlying parameters, which are commonly called the Greeks, and derive options pricing formulas. Finally, we reveal the relation between correct volatility and implied volatility in Black–Scholes model and provide a mathematical explanation of volatility smile.

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Derman, Emanuel, and Iraj Kani. "Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility." International Journal of Theoretical and Applied Finance 01, no. 01 (January 1998): 61–110. http://dx.doi.org/10.1142/s0219024998000059.

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In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options prices and their associated local volatility surface, we show how to construct a family of continuous time stochastic processes which define the arbitrage-free evolution of this local volatility surface through time. The no-arbitrage conditions are similar to, but more involved than, the HJM conditions for arbitrage-free stochastic movements of the interest rate curve. They guarantee that even under a general stochastic volatility evolution the initial options prices, or their equivalent Black–Scholes implied volatilities, remain fair. We introduce stochastic implied trees as discrete implementations of our family of continuous time models. The nodes of a stochastic implied tree remain fixed as time passes. During each discrete time step the index moves randomly from its initial node to some node at the next time level, while the local transition probabilities between the nodes also vary. The change in transition probabilities corresponds to a general (multifactor) stochastic variation of the local volatility surface. Starting from any node, the future movements of the index and the local volatilities must be restricted so that the transition probabilities to all future nodes are simultaneously martingales. This guarantees that initial options prices remain fair. On the tree, these martingale conditions are effected through appropriate choices of the drift parameters for the transition probabilities at every future node, in such a way that the subsequent evolution of the index and of the local volatility surface do not lead to riskless arbitrage opportunities among different option and forward contracts or their underlying index. You can use stochastic implied trees to value complex index options, or other derivative securities with payoffs that depend on index volatility, even when the volatility surface is both skewed and stochastic. The resulting security prices are consistent with the current market prices of all standard index options and forwards, and with the absence of future arbitrage opportunities in the framework. The calculated options values are independent of investor preferences and the market price of index or volatility risk. Stochastic implied trees can also be used to calculate hedge ratios for any contingent index security in terms of its underlying index and all standard options defined on that index.

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FUNAHASHI, HIDEHARU. "REPLICATION SCHEME FOR THE PRICING OF EUROPEAN OPTIONS." International Journal of Theoretical and Applied Finance 24, no. 03 (May 2021): 2150014. http://dx.doi.org/10.1142/s021902492150014x.

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This paper proposes an efficient method for calculating European option prices under local, stochastic, and fractional volatility models. Instead of directly calculating the density function of a target underlying asset, we replicate it from a simpler diffusion process with a known analytical solution for the European option. For this purpose, we derive six functions that characterize the density function of a diffusion process, for both the original and simpler processes and match these functions so that the latter mimics the former. Using the analytical formula, we then approximate the option price of the target asset. By comparison with previous works and numerical experiments, we show that the accuracy of our approximation is high, and the calculation is fast enough for practical purposes; hence, it is suitable for calibration purposes.

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Liu, David, and An Wei. "Regulated LSTM Artificial Neural Networks for Option Risks." FinTech 1, no. 2 (June 2, 2022): 180–90. http://dx.doi.org/10.3390/fintech1020014.

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This research aims to study the pricing risks of options by using improved LSTM artificial neural network models and make direct comparisons with the Black–Scholes option pricing model based upon the option prices of 50 ETFs of the Shanghai Securities Exchange from 1 January 2018 to 31 December 2019. We study an LSTM model, a mathematical option pricing model (BS model), and an improved artificial neural network model—the regulated LSTM model. The method we adopted is first to price the options using the mathematical model—i.e., the BS model—and then to construct the LSTM neural network for training and predicting the option prices. We further form the regulated LSTM network with optimally selected key technical indicators using Python programming aiming at improving the network’s predicting ability. Risks of option pricing are measured by MSE, RMSE, MAE and MAPE, respectively, for all the models used. The results of this paper show that both the ordinary LSTM and the traditional BS option pricing model have lower predictive ability than the regulated LSTM model. The prediction ability of the regulated LSTM model with the optimal technical indicators is superior, and the approach adopted is effective.

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CHANG, CHIA-LIN, SHING-YANG HU, and SHIH-TI YU. "RECENT DEVELOPMENTS IN QUANTITATIVE FINANCE: AN OVERVIEW." Annals of Financial Economics 09, no. 02 (September 2014): 1402002. http://dx.doi.org/10.1142/s2010495214020023.

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Quantitative finance combines mathematical finance, financial statistics, financial econometrics and empirical finance to provide a solid quantitative foundation for the analysis of financial issues. The purpose of this special issue on "Recent developments in quantitative finance" is to highlight some areas of research in which novel methods in quantitative finance have contributed significantly to the analysis of financial issues, specifically fast methods for large-scale non-elliptical portfolio optimization, the impact of acquisitions on new technology stocks: the Google–Motorola case, the effects of firm characteristics and recognition policy on employee stock options prices after controlling for self-selection, searching for landmines in equity markets, whether CEO incentive pay improves bank performance, using a quantile regression analysis of U.S. commercial banks, testing price pressure, information, feedback trading, and smoothing effects for energy exchange traded funds, actuarial implications of structural changes in El Niño-Southern Oscillation Index dynamics, credit spreads and bankruptcy information from options data, QMLE of a standard exponential ACD model: asymptotic distribution and residual correlation, and using two-part quantile regression to analyze how earnings shocks affect stock repurchases.

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PAGLIARANI, STEFANO, and ANDREA PASCUCCI. "LOCAL STOCHASTIC VOLATILITY WITH JUMPS: ANALYTICAL APPROXIMATIONS." International Journal of Theoretical and Applied Finance 16, no. 08 (December 2013): 1350050. http://dx.doi.org/10.1142/s0219024913500507.

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We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function, which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.

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MERINO, RAÚL, JAN POSPÍŠIL, TOMÁŠ SOBOTKA, TOMMI SOTTINEN, and JOSEP VIVES. "DECOMPOSITION FORMULA FOR ROUGH VOLTERRA STOCHASTIC VOLATILITY MODELS." International Journal of Theoretical and Applied Finance 24, no. 02 (March 2021): 2150008. http://dx.doi.org/10.1142/s0219024921500084.

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The research presented in this paper provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, even simple nonlinear financial derivatives as vanilla European options are typically priced by means of Monte–Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility — a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models with rough fractional volatility and we derive an explicit semi-closed approximation formula. Numerical properties of the approximation for a popular model — the rBergomi model — are studied and we propose a hybrid calibration scheme which combines the approximation formula alongside MC simulations. This scheme can significantly speed up the calibration to financial markets as illustrated on a set of AAPL options.

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Wang, Xingchun. "Valuation of options on the maximum of two prices with default risk under GARCH models." North American Journal of Economics and Finance 57 (July 2021): 101422. http://dx.doi.org/10.1016/j.najef.2021.101422.

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Carr, Peter, Andrey Itkin, and Dmitry Muravey. "Semi-Closed Form Prices of Barrier Options in the Time-Dependent CEV and CIR Models." Journal of Derivatives 28, no. 1 (July 10, 2020): 26–50. http://dx.doi.org/10.3905/jod.2020.1.113.

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Singh, Vipul Kumar. "Pricing competitiveness of jump-diffusion option pricing models: evidence from recent financial upheavals." Studies in Economics and Finance 32, no. 3 (August 3, 2015): 357–78. http://dx.doi.org/10.1108/sef-08-2012-0099.

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Purpose – The purpose of this paper is to investigate empirically the forecasting performance of jump-diffusion option pricing models of (Merton and Bates) with the benchmark Black–Scholes (BS) model relative to market, for pricing Nifty index options of India. The specific period chosen for this study canvasses the extreme up and down limits (jumps) of the Indian capital market. In addition, equity markets keep on facing high and low tides of financial flux amid new economic and financial considerations. With this backdrop, the paper focuses on finding an impeccable option-pricing model which can meet the requirements of option traders and practitioners during tumultuous periods in the future. Design/methodology/approach – Envisioning the fact, the all option-pricing models normally does wrong valuation relative to market. For estimating the structural parameters that governs the underlying asset distribution purely from the underlying asset return data, we have used the nonlinear least-square method. As an approach, we analyzed model prices by dividing the option data into 15 moneyness-maturity groups – depending on the time to maturity and strike price. The prices are compared analytically by continuously updating the parameters of two models using cross-sectional option data on daily basis. Estimated parameters then used to figure out the forecasting performance of models with corresponding BS and market – for pricing day-ahead option prices and implied volatility. Findings – The outcomes of the paper reveal that the jump-diffusion models are a better substitute of classical BS, thus improving the pricing bias significantly. But compared to jump-diffusion model of Merton’s, the model of Bates’ can be applied more uniquely to find out the pricing of three popularly traded categories: deep-out-of-the-money, out-of-the-money and at-the-money of Nifty index options. Practical implications – The outcome of this research work reveals that the jumps are important components of pricing dynamics of Nifty index options. Incorporation of jump-diffusion process into option pricing of Nifty index options leads to a higher pricing effectiveness, reduces the pricing bias and gives values closer to the market. As the models have been tested in extreme conditions to determine the dominant effectuality, the outcome of this paper helps traders in keeping the investment protected under normal conditions. Originality/value – The specific period chosen for this study is very unique; it canvasses the extreme up and down limits (jumps) of the Indian capital market and provides the most apt situation for testifying the pricing competitiveness of the models in question. To testify the robustness of models, they have been put into a practical implication of complete cycle of financial frame.

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SIDENIUS, JAKOB, VLADIMIR PITERBARG, and LEIF ANDERSEN. "A NEW FRAMEWORK FOR DYNAMIC CREDIT PORTFOLIO LOSS MODELLING." International Journal of Theoretical and Applied Finance 11, no. 02 (March 2008): 163–97. http://dx.doi.org/10.1142/s0219024908004762.

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We present the SPA framework, a novel approach to the modeling of the dynamics of portfolio default losses. In this framework, models are specified by a two-layer process. The first layer models the dynamics of portfolio loss distributions in the absence of information about default times. This background process can be explicitly calibrated to the full grid of marginal loss distributions as implied by initial CDO tranche values indexed on maturity, as well as to the prices of suitable options. We give sufficient conditions for consistent dynamics. The second layer models the loss process itself as a Markov process conditioned on the path taken by the background process. The choice of loss process is non-unique. We present a number of choices, and discuss their advantages and disadvantages. Several concrete model examples are given, and valuation in the new framework is described in detail. Among the specific securities for which algorithms are presented are CDO tranche options and leveraged super-senior tranches.

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CUTHBERTSON, CHARLES, GRIGORIOS PAVLIOTIS, AVRAAM RAFAILIDIS, and PETTER WIBERG. "ASYMPTOTIC ANALYSIS FOR FOREIGN EXCHANGE DERIVATIVES WITH STOCHASTIC VOLATILITY." International Journal of Theoretical and Applied Finance 13, no. 07 (November 2010): 1131–47. http://dx.doi.org/10.1142/s0219024910006145.

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We consider models for the valuation of derivative securities that depend on foreign exchange rates. We derive partial differential equations for option prices in an arbitrage-free market with stochastic volatility. By use of standard techniques, and under the assumption of fast mean reversion for the volatility, these equations can be solved asymptotically. The analysis goes further to consider specific examples for a number of options, and to a considerable degree of complexity.

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Vedran Uran. "THE PRINCIPLE OF EXERCISING OPTIONS ON THE ELECTRICITY MARKET." Journal of Energy - Energija 56, no. 1 (November 14, 2022): 114–33. http://dx.doi.org/10.37798/2007561349.

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Continual price fluctuations are possible to hedge by using various financial instruments, including options. An option buyer buys the right to the settlement price of an underlying asset such as electricity. Due to the volatility of the asset price, the buyer is not obliged to exercise the option. In such a case, the buyer’s only loss is the purchased right or the option premium, which is equal to the option price. Mathematical models for option pricing have been developed in the last hundred years. These models were very popular during the 1970s, owing to the application of the Black-Scholes formula for the calculation of theoretical option prices. In this article, options are distinguished according to various criteria, specific exercising methods are described and the historical development of option pricing models is reviewed. An example of option exercising with closing out of the margins on the largest Middle European electricity exchange, EEX, is presented.

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LO, HARRY, and ALEKSANDAR MIJATOVIĆ. "VOLATILITY DERIVATIVES IN MARKET MODELS WITH JUMPS." International Journal of Theoretical and Applied Finance 14, no. 07 (November 2011): 1159–93. http://dx.doi.org/10.1142/s0219024911006656.

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It is well documented that a model for the underlying asset price process that seeks to capture the behaviour of the market prices of vanilla options needs to exhibit both diffusion and jump features. In this paper we assume that the asset price process S is Markov with càdlàg paths and propose a scheme for computing the law of the realized variance of the log returns accrued while the asset was trading in a prespecified corridor. We thus obtain an algorithm for pricing and hedging volatility derivatives and derivatives on the corridor-realized variance in such a market. The class of models under consideration is large, as it encompasses jump-diffusion and Lévy processes. We prove the weak convergence of the scheme and describe in detail the implementation of the algorithm in the characteristic cases where S is a CEV process (continuous trajectories), a variance gamma process (jumps with independent increments) or an infinite activity jump-diffusion (discontinuous trajectories with dependent increments).

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MADAN, DILIP B., and KING WANG. "OPTION IMPLIED VIX, SKEW AND KURTOSIS TERM STRUCTURES." International Journal of Theoretical and Applied Finance 24, no. 05 (August 2021): 2150030. http://dx.doi.org/10.1142/s0219024921500308.

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Comparisons are made of the Chicago Board of Options Exchange (CBOE) skew index with those derived from parametric skews of bilateral gamma models and from the differentiation of option implied characteristic exponents. Discrepancies can be due to strike discretization in evaluating prices of powered returns. The remedy suggested employs a finer and wider set of strikes obtaining additional option prices by interpolation and extrapolation of implied volatilities. Procedures of replicating powered return claims are applied to the fourth power and the derivation of kurtosis term structures. Regressions of log skewness and log excess kurtosis on log maturity confirm the positivity of decay in these higher moments. The decay rates are below those required by processes of independent and identically distributed increments.

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Lee, C. F., Ta-Peng Wu, and Ren-Raw Chen. "The Constant Elasticity of Variance Models: New Evidence from S&P 500 Index Options." Review of Pacific Basin Financial Markets and Policies 07, no. 02 (June 2004): 173–90. http://dx.doi.org/10.1142/s021909150400010x.

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The seminal work by Cox (1975, 1996), MacBeth and Merville (1979, 1980) and Emanuel and Macbeth (1982) show that, both theoretically and empirically, the constant elasticity of variance option model (CEV) is superior to the Black–Scholes model in explaining market prices. In this paper, we extend the MacBeth and Merville (1979, 1980) research by using a European contract (S&P 500 index options). We find supportive evidence to the MacBeth and Merville results although our sample is not subject to American premium biases. Furthermore, we reduce the approximation errors by using the non-central chi-square probability functions proposed by Shroder (1989).

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Journal articles on the topic 'Options (Finance) – Prices – Mathematical models'

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