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Journal articles on the topic 'Martingale approach in option pricing'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Gerber, Hans U., and Elias S. W. Shiu. "Martingale Approach to Pricing Perpetual American Options." ASTIN Bulletin 24, no. 2 (November 1994): 195–220. http://dx.doi.org/10.2143/ast.24.2.2005065.

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AbstractThe method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the relationship between Samuelson's high contact condition and the first order condition for optimality.

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2

Wang, Ming-Chieh, Li-Jhang Huang, and Szu-Lang Liao. "Option Pricing Using the Martingale Approach with Polynomial Interpolation." Journal of Futures Markets 33, no. 5 (May 14, 2012): 469–91. http://dx.doi.org/10.1002/fut.21557.

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3

Yu, Xisheng, and Li Yang. "Pricing American Options Using a Nonparametric Entropy Approach." Discrete Dynamics in Nature and Society 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/369795.

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This paper studies the pricing problem of American options using a nonparametric entropy approach. First, we derive a general expression for recovering the risk-neutral moments of underlying asset return and then incorporate them into the maximum entropy framework as constraints. Second, by solving this constrained entropy problem, we obtain a discrete risk-neutral (martingale) distribution as the unique pricing measure. Third, the optimal exercise strategies are achieved via the least-squares Monte Carlo algorithm and consequently the pricing algorithm of American options is obtained. Finally, we conduct the comparative analysis based on simulations and IBM option contracts. The results demonstrate that this nonparametric entropy approach yields reasonably accurate prices for American options and produces smaller pricing errors compared to other competing methods.

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4

Nowak, Piotr, and Maciej Romaniuk. "A fuzzy approach to option pricing in a Levy process setting." International Journal of Applied Mathematics and Computer Science 23, no. 3 (September 1, 2013): 613–22. http://dx.doi.org/10.2478/amcs-2013-0046.

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Abstract In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets.We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.

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5

Liu, Guoxiang, Quanxin Zhu, Zhaowei Yan, and Yuanyao Ding. "The martingale approach for vulnerable binary option pricing under stochastic interest rate." Cogent Mathematics 4, no. 1 (January 1, 2017): 1340073. http://dx.doi.org/10.1080/23311835.2017.1340073.

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6

Lee, Jun Hui, and Kook Hyun Chang. "Volatility Smile Surface for Levy Option Pricing Model." Journal of Derivatives and Quantitative Studies 12, no. 1 (May 30, 2004): 73–86. http://dx.doi.org/10.1108/jdqs-01-2004-b0004.

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This paper discusses theoretical extensions of the implied volatility method of Dupire (1994) when the stock prices follow the Geometric Levy process. For the extensions of Kolmogorov forward equation for Levy process, this paper uses adjoint operator in L² spaces. This paper obtains similar results of Dupire (1994) and Andersen and Andreasan (2001). However, our results can be applied to more general semi-martingale processes such as well-known VG (Variance Gamma) model and NIG (Normal Inverse Gaussian) model with diffusion processes. This paper also applies the approach to the case of stochastic time changed Levy process, which generates the stochastic volatility models.

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7

Li, Qing, Songlin Liu, and Misi Zhou. "Nonparametric Estimation of Fractional Option Pricing Model." Mathematical Problems in Engineering 2020 (December 15, 2020): 1–8. http://dx.doi.org/10.1155/2020/8858821.

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The establishment of the fractional Black–Scholes option pricing model is under a major condition with the normal distribution for the state price density (SPD) function. However, the fractional Brownian motion is deemed to not be martingale with a long memory effect of the underlying asset, so that the estimation of the state price density (SPD) function is far from simple. This paper proposes a convenient approach to get the fractional option pricing model by changing variables. Further, the option price is transformed as the integral function of the cumulative density function (CDF), so it is not necessary to estimate the distribution function individually by complex approaches. Finally, it encourages to estimate the fractional option pricing model by the way of nonparametric regression and makes empirical analysis with the traded 50 ETF option data in Shanghai Stock Exchange (SSE).

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8

Siu, Tak Kuen. "Regime-Switching Risk: To Price or Not to Price?" International Journal of Stochastic Analysis 2011 (December 27, 2011): 1–14. http://dx.doi.org/10.1155/2011/843246.

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Should the regime-switching risk be priced? This is perhaps one of the important “normative” issues to be addressed in pricing contingent claims under a Markovian, regime-switching, Black-Scholes-Merton model. We address this issue using a minimal relative entropy approach. Firstly, we apply a martingale representation for a double martingale to characterize the canonical space of equivalent martingale measures which may be viewed as the largest space of equivalent martingale measures to incorporate both the diffusion risk and the regime-switching risk. Then we show that an optimal equivalent martingale measure over the canonical space selected by minimizing the relative entropy between an equivalent martingale measure and the real-world probability measure does not price the regime-switching risk. The optimal measure also justifies the use of the Esscher transform for option valuation in the regime-switching market.

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9

ELLIOTT, ROBERT J., TAK KUEN SIU, and LEUNGLUNG CHAN. "OPTION PRICING FOR GARCH MODELS WITH MARKOV SWITCHING." International Journal of Theoretical and Applied Finance 09, no. 06 (September 2006): 825–41. http://dx.doi.org/10.1142/s0219024906003846.

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In this paper we develop a method for pricing derivatives under a Markov switching version of the Heston-Nandi GARCH (1, 1) model by using a well known tool from actuarial science, namely the Esscher transform. We suppose that the dynamics of the GARCH process switch over time according to one of the regimes described by the states of an observable Markov chain process. By augmenting the conditional Esscher transform with the observable Markov switching process, a Markov switching conditional Esscher transform (MSCET) is developed to identify a martingale measure for option valuation in the incomplete market described by our model. We provide an alternative approach for the derivation of an analytical option valuation formula under the Markov switching Heston-Nandi GARCH (1, 1) model. The use of the MSCET can be justified by considering a utility maximization problem with respect to a power utility function associated with the Markov switching risk-averse parameters.

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10

Hainaut, Donatien. "Calendar Spread Exchange Options Pricing with Gaussian Random Fields." Risks 6, no. 3 (August 8, 2018): 77. http://dx.doi.org/10.3390/risks6030077.

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Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance introduces a bias in the valuation of calendar spread exchange options. As the payoff of these options depends on two asset values at different times, particular care must be taken for the modeling of covariance and autocorrelation. This article proposes a simple alternative model for asset prices with sub-exponential, exponential and hyper-exponential autocovariance structures. In the proposed approach, price processes are seen as conditional Gaussian fields indexed by the time. In general, this process is not a semi-martingale, and therefore, we cannot rely on stochastic differential calculus to evaluate options. However, option prices are still calculable by the technique of the change of numeraire. A numerical illustration confirms the important influence of the covariance structure in the valuation of calendar spread exchange options for Brent against WTI crude oil and for gold against silver.

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11

Ngo, M., T. Nguyen, and T. Duong. "Indifference pricing with counterparty risk." Bulletin of the Polish Academy of Sciences Technical Sciences 65, no. 5 (October 1, 2017): 695–702. http://dx.doi.org/10.1515/bpasts-2017-0074.

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Abstract We present counterparty risk by a jump in the underlying price and a structural change of the price process after the default of the counterparty. The default time is modeled by a default-density approach. Then we study an exponential utility-indifference price of an European option whose underlying asset is exposed to this counterparty risk. Utility-indifference pricing method normally consists in solving two optimization problems. However, by using the minimal entropy martingale measure, we reduce to solving just one optimal control problem. In addition, to overcome the incompleteness obstacle generated by the possible jump and the change in structure of the price process, we employ the BSDE-decomposition approach in order to decompose the problem into a global-before-default optimal control problem and an after-default one. Each problem works in its own complete framework. We demonstrate the result by numerical simulation of an European option price under the impact of jump’s size, intensity of the default, absolute risk aversion and change in the underlying volatility.

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12

Gerber, Hans U., and Hlias S. W. Shiu. "MARTINGALE APPROACH TO PRICING PERPETUAL AMERICAN OPTIONS ON TWO STOCKS." Mathematical Finance 6, no. 3 (July 1996): 303–22. http://dx.doi.org/10.1111/j.1467-9965.1996.tb00118.x.

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13

SCHWEIZER, MARTIN. "RISKY OPTIONS SIMPLIFIED." International Journal of Theoretical and Applied Finance 02, no. 01 (January 1999): 59–82. http://dx.doi.org/10.1142/s0219024999000054.

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We study a general version of a quadratic approach to the pricing of options in an abstract financial market. The resulting price is the expectation of the option's discounted payoff under the variance-optimal signed martingale measure, and we give a very simple proof of this result. A conjecture of G. Wolczyńska essentially says that this measure coincides with the minimal signed martingale measure in a certain class of models. We show by a counterexample that this conjecture is false.

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14

Bienek, T., and M. Scherer. "VALUATION OF CONTINGENT GUARANTEES USING LEAST-SQUARES MONTE CARLO." ASTIN Bulletin 49, no. 1 (January 2019): 31–56. http://dx.doi.org/10.1017/asb.2018.43.

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AbstractWe consider the problem of pricing modern guarantee concepts in unit-linked life insurance, where the guaranteed amount grows contingent on the performance of an investment fund that acts simultaneously as the underlying security and the replicating portfolio. Using the Martingale Method, this nonstandard pricing problem can be transformed into a fixed-point problem, whose solution requires the evaluation of conditional expectations of highly path-dependent payoffs. By adapting the least-squares Monte Carlo method for American option pricing problems, we develop a new numerical approach to approximate the value of contingent guarantees and prove its convergence. Our valuation procedure can be applied to large-scale pricing problems, for which existing methods are infeasible, and leads to significant improvements in performance.

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15

BÄUERLE, NICOLE, and DANIEL SCHMITHALS. "CONSISTENT UPPER PRICE BOUNDS FOR EXOTIC OPTIONS." International Journal of Theoretical and Applied Finance 24, no. 02 (March 2021): 2150011. http://dx.doi.org/10.1142/s0219024921500114.

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We consider the problem of finding a consistent upper price bound for exotic options whose payoff depends on the stock price at two different predetermined time points (e.g. Asian option), given a finite number of observed call prices for these maturities. A model-free approach is used, only taking into account that the (discounted) stock price process is a martingale under the no-arbitrage condition. In case the payoff is directionally convex we obtain the worst case marginal pricing measures. The speed of convergence of the upper price bound is determined when the number of observed stock prices increases. We illustrate our findings with some numerical computations.

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16

Al-Hadad, Jonas, and Zbigniew Palmowski. "Pricing Perpetual American Put Options with Asset-Dependent Discounting." Journal of Risk and Financial Management 14, no. 3 (March 20, 2021): 130. http://dx.doi.org/10.3390/jrfm14030130.

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The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

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17

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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18

BELOMESTNY, DENIS, WOLFGANG KARL HÄRDLE, and EKATERINA KRYMOVA. "SIEVE ESTIMATION OF THE MINIMAL ENTROPY MARTINGALE MARGINAL DENSITY WITH APPLICATION TO PRICING KERNEL ESTIMATION." International Journal of Theoretical and Applied Finance 20, no. 06 (September 2017): 1750041. http://dx.doi.org/10.1142/s0219024917500418.

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We study the problem of nonparametric estimation of the risk-neutral densities from options data. The underlying statistical problem is known to be ill-posed and needs to be regularized. We propose a novel regularized empirical sieve approach for the estimation of the risk-neutral densities which relies on the notion of the minimal martingale entropy measure. The proposed approach can be used to estimate the so-called pricing kernels which play an important role in assessing the risk aversion over equity returns. The asymptotic properties of the resulting estimate are analyzed and its empirical performance is illustrated.

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19

McCauley, J. L., G. H. Gunaratne, and K. E. Bassler. "Martingale option pricing." Physica A: Statistical Mechanics and its Applications 380 (July 2007): 351–56. http://dx.doi.org/10.1016/j.physa.2007.02.038.

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20

Longstaff, Francis A. "Option Pricing and the Martingale Restriction." Review of Financial Studies 8, no. 4 (October 1995): 1091–124. http://dx.doi.org/10.1093/rfs/8.4.1091.

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21

Madan, Dilip B., and Frank Milne. "Option Pricing With V. G. Martingale Components." Mathematical Finance 1, no. 4 (October 1991): 39–55. http://dx.doi.org/10.1111/j.1467-9965.1991.tb00018.x.

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22

Ruan, Xinfeng, Wenli Zhu, Jiexiang Huang, and Shuang Li. "Continuous-Time Portfolio Selection and Option Pricing under Risk-Minimization Criterion in an Incomplete Market." Journal of Applied Mathematics 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/175269.

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We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset are governed by a jump diffusion equation. We obtain the Radon-Nikodym derivative in the minimal martingale measure and a partial integrodifferential equation (PIDE) of European call option. In a special case, we get the exact solution for European call option by Fourier transformation methods. Finally, we employ the pricing kernel to calculate the optimal portfolio selection by martingale methods.

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23

Zhu, Yonggang. "Equivalent Martingale Measure in Asian Geometric Average Option Pricing." Journal of Mathematical Finance 04, no. 04 (2014): 304–8. http://dx.doi.org/10.4236/jmf.2014.44027.

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24

Artzner, Philippe, and Freddy Delbaen. "Credit Risk and Prepayment Option." ASTIN Bulletin 22, no. 1 (May 1992): 81–96. http://dx.doi.org/10.2143/ast.22.1.2005128.

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AbstractThe paper examines a type of insurance contract for which secondary markets do exist: default risk insurance is implicit in corporate bonds and other risky debts. It applies risk neutral martingale measure pricing to evaluate the option for a borrower with default risk, to prepay a fixed rate loan. A simple “matchbox” example is presented with a spreadsheet treatment.

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25

Carr, Peter. "First-order calculus and option pricing." Journal of Financial Engineering 01, no. 01 (March 2014): 1450009. http://dx.doi.org/10.1142/s2345768614500093.

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The modern theory of option pricing rests on Itô calculus, which is a second-order calculus based on the quadratic variation of a stochastic process. One can instead develop a first-order stochastic calculus, which is based on the running minimum of a stochastic process, rather than its quadratic variation. We focus here on the analog of geometric Brownian motion (GBM) in this alternative stochastic calculus. The resulting stochastic process is a positive continuous martingale whose laws are easy to calculate. We show that this analog behaves locally like a GBM whenever its running minimum decreases, but behaves locally like an arithmetic Brownian motion otherwise. We provide closed form valuation formulas for vanilla and barrier options written on this process. We also develop a reflection principle for the process and use it to show how a barrier option on this process can be hedged by a static postion in vanilla options.

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26

Liang, Yijuan, and Xiuchuan Xu. "Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities." Sustainability 11, no. 3 (February 4, 2019): 815. http://dx.doi.org/10.3390/su11030815.

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Pricing multi-asset options has always been one of the key problems in financial engineering because of their high dimensionality and the low convergence rates of pricing algorithms. This paper studies a method to accelerate Monte Carlo (MC) simulations for pricing multi-asset options with stochastic volatilities. First, a conditional Monte Carlo (CMC) pricing formula is constructed to reduce the dimension and variance of the MC simulation. Then, an efficient martingale control variate (CV), based on the martingale representation theorem, is designed by selecting volatility parameters in the approximated option price for further variance reduction. Numerical tests illustrated the sensitivity of the CMC method to correlation coefficients and the effectiveness and robustness of our martingale CV method. The idea in this paper is also applicable for the valuation of other derivatives with stochastic volatility.

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27

Liu, Nan, Mei Ling Wang, and Xue Bin Lü. "Multi-Asset Option Pricing Based on Exponential Lévy Process." Applied Mechanics and Materials 380-384 (August 2013): 4537–40. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.4537.

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The multi-dimensional Esscher transform was used to find a locally equivalent martingale measure to price the options based on multi-asset. An integro-differential equation was driven for the prices of multi-asset options. The numerical method based on the Fourier transform was used to calculate some special multi-asset options in exponential Lévy models. As an example we give the calculation of extreme options.

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28

HUBALEK, FRIEDRICH, and CARLO SGARRA. "QUADRATIC HEDGING FOR THE BATES MODEL." International Journal of Theoretical and Applied Finance 10, no. 05 (August 2007): 873–85. http://dx.doi.org/10.1142/s0219024907004433.

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In the present paper we give some preliminary results for option pricing and hedging in the framework of the Bates model based on quadratic risk minimization. We provide an explicit expression of the mean-variance hedging strategy in the martingale case and study the Minimal Martingale measure in the general case.

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29

LUDKOVSKI, MICHAEL, and QUNYING SHEN. "EUROPEAN OPTION PRICING WITH LIQUIDITY SHOCKS." International Journal of Theoretical and Applied Finance 16, no. 07 (November 2013): 1350043. http://dx.doi.org/10.1142/s021902491350043x.

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We study the valuation and hedging problem of European options in a market subject to liquidity shocks. Working within a Markovian regime-switching setting, we model illiquidity as the inability to trade. To isolate the impact of such liquidity constraints, we focus on the case where the market is completely static in the illiquid regime. We then consider derivative pricing using either equivalent martingale measures or exponential indifference mechanisms. Our main results concern the analysis of the semi-linear coupled Hamilton–Jacobi–Bellman (HJB) equation satisfied by the indifference price, as well as its asymptotics when the probability of a liquidity shock is small. A comparative analysis between the model price and the classical Black–Scholes benchmark is given using the concepts of implied and adjusted time to maturity. We then present several numerical studies of the liquidity risk premia obtained in our models leading to practical guidelines on how to adjust for liquidity risk in option valuation and hedging.

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30

Ouyang, Yanmin, Jingyuan Yang, and Shengwu Zhou. "Valuation of the Vulnerable Option Price Based on Mixed Fractional Brownian Motion." Discrete Dynamics in Nature and Society 2018 (December 3, 2018): 1–16. http://dx.doi.org/10.1155/2018/4047350.

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The pricing problem of a kind of European vulnerable option was studied. The mixed fractional Brownian motion and the jump process were used to characterize the evolution of stock prices. The closed-form solution to European option pricing was obtained by applying martingale measure transformation method. At the end of this paper, some numerical experiments were adopted to compare the new pricing formula introduced in this paper with the classical Black-Scholes pricing formula. The result showed that the new pricing formula conformed to the actual financial market. In fact, the option value is positively correlated with the underlying asset price and the company’s asset price and the jump process has significant influence on the value of option.

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31

SENGUPTA, INDRANIL. "GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING." International Journal of Theoretical and Applied Finance 19, no. 02 (March 2016): 1650014. http://dx.doi.org/10.1142/s021902491650014x.

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In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.

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32

Peng, Bo, and Zhi Hui Wu. "Pricing Option on Jump Diffusion and Stochastic Interest Rates Model." Applied Mechanics and Materials 50-51 (February 2011): 723–27. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.723.

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This paper assumed that the stock price jump process for a special kind of renewal jump process, that is incident time interval for independent and subordinate to Gamma distribution random variable sequence. We obtain the European bi-direction option pricing formulas on jump diffusion model under the stochastic interest rates by simply mathematical induce by means of martingale method.

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33

Fischer, Klaus P. "Pricing Pension Fund Guarantees: A Discrete Martingale Approach." Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration 16, no. 3 (April 8, 2009): 256–66. http://dx.doi.org/10.1111/j.1936-4490.1999.tb00200.x.

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34

Zhang, Hui, and Wen Yu Meng. "Dynamic Robust Pricing Model of European Call Option and Empirical Research in Fractional Market." Advanced Materials Research 368-373 (October 2011): 3226–29. http://dx.doi.org/10.4028/www.scientific.net/amr.368-373.3226.

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The fractional financial market with Knightian uncertainty is studied. We get the dynamic robust pricing model of European call option. Using the important theories of the quasi conditional expectation and the quasi martingale, we get the explicit solution of the model. By making empirical research on the financial product of Chinese bank ahead 09004, we depict the important impacts of the Knightian uncertainty on the robust pricing of European call option.

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35

Fouque, Jean-Pierre, and Chuan-Hsiang Han. "A martingale control variate method for option pricing with stochastic volatility." ESAIM: Probability and Statistics 11 (February 2007): 40–54. http://dx.doi.org/10.1051/ps:2007005.

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36

López, Oscar, and Nikita Ratanov. "Option Pricing Driven by a Telegraph Process with Random Jumps." Journal of Applied Probability 49, no. 03 (September 2012): 838–49. http://dx.doi.org/10.1017/s0021900200009578.

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In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

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37

López, Oscar, and Nikita Ratanov. "Option Pricing Driven by a Telegraph Process with Random Jumps." Journal of Applied Probability 49, no. 3 (September 2012): 838–49. http://dx.doi.org/10.1239/jap/1346955337.

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In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

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38

Ruan, Xinfeng, Wenli Zhu, Shuang Li, and Jiexiang Huang. "Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/165727.

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We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.

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Yao, Luogen, and Gang Yang. "Option Pricing by Probability Distortion Operator Based on the Quantile Function." Mathematical Problems in Engineering 2019 (June 16, 2019): 1–9. http://dx.doi.org/10.1155/2019/5831569.

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A new class of distortion operators based on quantile function is proposed for pricing options. It is shown that option prices obtained with our distortion operators are just the prices under mean correcting martingale measure in exponential Lévy models. In particular, Black-Scholes formula can be recuperated by our distortion operator. Simulation analysis shows that our distortion operator is superior to normal distortion operator and NIG distortion operator.

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Salhi, Khaled. "Pricing European options and risk measurement under exponential Lévy models — a practical guide." International Journal of Financial Engineering 04, no. 02n03 (June 2017): 1750016. http://dx.doi.org/10.1142/s2424786317500165.

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This paper provides a thorough survey of the European option pricing, with new trends in the risk measurement, under exponential Lévy models. We develop all steps of pricing from equivalent martingale measures construction to numerical valuation of the option price under these measures. We then construct an algorithm, based on Rockafellar and Uryasev representation and fast Fourier transform, to compute Risk indicators, like the VaR and the CVaR of derivatives. The results are illustrated with an example of each exponential Lévy class. The main contribution of this paper is to build a comprehensive study from the theoretical point of view to practical numerical illustration and to give a complete characterization of the studied equivalent martingale measures by discussing their similarity and their applicability in practice. Furthermore, this work proposes applications to the Fourier inversion technique in risk measurement.

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Kiesel, Rüdiger. "Nonparametric statistical methods and the pricing of derivative securities." Journal of Applied Mathematics and Decision Sciences 6, no. 1 (January 1, 2002): 1–22. http://dx.doi.org/10.1155/s1173912602000019.

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In this review paper we summarise several nonparametric methods recently applied to the pricing of financial options. After a short introduction to martingale-based option pricing theory, we focus on two possible fields of application for nonparametric methods: the estimation of risk-neutral probabilities and the estimation of the dynamics of the underlying instruments in order to construct an internally consistent model.

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Zhang, Hui, and Wen Yu Meng. "Dynamic Robust Pricing Model of European Call Option under the Fractional Market with Knightian Uncertainty." Advanced Materials Research 271-273 (July 2011): 675–78. http://dx.doi.org/10.4028/www.scientific.net/amr.271-273.675.

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The fractional financial market with Knightian uncertainty is studied. Using the important theories of the quasi conditional expectation and the quasi martingale, we establish the dynamic robust pricing model of European call option and get the explicit solution of the model.

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Zhao, Jun, Ru Zhou, and Peibiao Zhao. "Existence and Uniqueness of Martingale Solutions to Option Pricing Equations with Noise." Lithuanian Mathematical Journal 60, no. 4 (October 2020): 562–76. http://dx.doi.org/10.1007/s10986-020-09499-1.

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Wang, Chao, Jianmin He, and Shouwei Li. "The European Vulnerable Option Pricing with Jumps Based on a Mixed Model." Discrete Dynamics in Nature and Society 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/8035746.

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In this paper, we combine the reduced-form model with the structural model to discuss the European vulnerable option pricing. We define that the default occurs when the default process jumps or the corporate goes bankrupt. Assuming that the underlying asset follows the jump-diffusion process and the default follows the Vasicek model, we can have the expression of European vulnerable option. Then we use the measure transformation and martingale method to derive the explicit solution of it.

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Kim, Moo Sung, and Tae Hun Kang. "The Pricing and Hedging using the Implied Information Conditioned on Martingale Restriction and Market Efficiency." Journal of Derivatives and Quantitative Studies 17, no. 4 (November 30, 2009): 1–42. http://dx.doi.org/10.1108/jdqs-04-2009-b0001.

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This article empirically tests the time-correlation of implied information reflecting the return dynamics of KOSPI 200 markets in the view of the decision making and market efficiency. Because option prices are not perfectly correlated with each other and with the underlying asset, the information contents of the option are different from those of the underlying market price. And, under the non-complete of the market and the limited arbitrage, the information implied in option (underlying) market price may be more useful in the option (underlying) market than in the underlying (option) market. The estimation results show that the time-correlation of incremental information are existed in performance of out-of-sample pricing and delta hedging conditioned on MR, a result which is not suggestive of the informational efficiency of the KOSPI 200 market. But, the decision marking using the systematic pattern may not be useful due to the option pricing models that allows moments of higher order than two reflecting the source of which the risk-neutrality assumption is strongly rejected by the data.

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BENTH, FRED ESPEN, and FRANK PROSKE. "UTILITY INDIFFERENCE PRICING OF INTEREST-RATE GUARANTEES." International Journal of Theoretical and Applied Finance 12, no. 01 (February 2009): 63–82. http://dx.doi.org/10.1142/s0219024909005117.

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We consider the problem of utility indifference pricing of a put option written on a non-tradeable asset, where we can hedge in a correlated asset. The dynamics are assumed to be a two-dimensional geometric Brownian motion, and we suppose that the issuer of the option have exponential risk preferences. We prove that the indifference price dynamics is a martingale with respect to an equivalent martingale measure (EMM) Q after discounting, implying that it is arbitrage-free. Moreover, we provide a representation of the residual risk remaining after using the optimal utility-based trading strategy as the hedge. Our motivation for this study comes from pricing interest-rate guarantees, which are products usually offered by companies managing pension funds. In certain market situations the life company cannot hedge perfectly the guarantee, and needs to resort to sub-optimal replication strategies. We argue that utility indifference pricing is a suitable method for analysing such cases. We provide some numerical examples giving insight into how the prices depend on the correlation between the tradeable and non-tradeable asset, and we demonstrate that negative correlation is advantageous, in the sense that the hedging costs become less than with positive correlation, and that the residual risk has lower volatility. Thus, if the insurance company can hedge in assets negatively correlated with the pension fund, they may offer cheaper prices with lower Value-at-Risk measures on the residual risk.

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Kawaguchi, Yuichiro, and Kazuhiro Tsubokawa. "The pricing of real options in discrete time models." Journal of Property Investment & Finance 19, no. 1 (February 1, 2001): 9–34. http://dx.doi.org/10.1108/14635780110365334.

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This paper proposes a discrete time real options model with time‐dependent and serial correlated return process for a real estate development problem with waiting options. Based on a Martingale condition, the paper claims to be able to relax many unrealistic assumptions made in the typical real option pricing methodology. Our real option model is a new one without assuming the return process as “Ito Process”, specifically, without assuming a geometric Brownian motion. We apply the model to the condominium market in Tokyo metropolitan area in the period 1971‐1997 and estimate the value of waiting to invest in 1998‐2007. The results partly provide realistic estimates of the parameters and show the applicability of our model.

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Zimmer, Christian Johannes. "The Use of Martingale Theory for the Superreplication of Exotic Options in Incomplete Markets." Brazilian Review of Econometrics 23, no. 2 (November 2, 2003): 323. http://dx.doi.org/10.12660/bre.v23n22003.2728.

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In this article we show the importance of modern martingale theory for the pricing and hedging of exotic options, especially in incomplete markets. When emitting an exotic option, the seller firstly has to ask himself whether there exists a hedging strategy for this title or not. Especially, when he wants to use a more realistic model than the simple Black-Scholes framework, the answer is not always obvious. We show in this article how to analyze this problem in the case of an exotic option, the Generalized Bermudian Option, which will turn out to be a generalization of the American option.

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Lee, Jaewoo. "Option Pricing Approach to International Reserves." Review of International Economics 17, no. 4 (September 2009): 844–60. http://dx.doi.org/10.1111/j.1467-9396.2009.00849.x.

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Cutland, Nigel, Ekkehard Kopp, and Walter Willinger. "A Nonstandard Approach to Option Pricing." Mathematical Finance 1, no. 4 (October 1991): 1–38. http://dx.doi.org/10.1111/j.1467-9965.1991.tb00017.x.

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