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Journal articles on the topic 'Equivalent martingale measure (EMM)'

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Published: 4 June 2021
Last updated: 16 February 2022

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1

SENGUPTA, INDRANIL. "GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING." International Journal of Theoretical and Applied Finance 19, no. 02 (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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2

HERDEGEN, MARTIN, and MARTIN SCHWEIZER. "STRONG BUBBLES AND STRICT LOCAL MARTINGALES." International Journal of Theoretical and Applied Finance 19, no. 04 (2016): 1650022. http://dx.doi.org/10.1142/s0219024916500229.

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In a numéraire-independent framework, we study a financial market with [Formula: see text] assets which are all treated in a symmetric way. We define the fundamental value ∗S of an asset [Formula: see text] as its super-replication price and say that the market has a strong bubble if ∗S and [Formula: see text] deviate from each other. None of these concepts needs any mention of martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numéraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local martingales under one ELMM, but true martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.
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3

HUBALEK, FRIEDRICH, and THOMAS HUDETZ. "CONVERGENCE OF MINIMUM ENTROPY OPTION PRICES FOR WEAKLY CONVERGING INCOMPLETE MARKET MODELS." International Journal of Theoretical and Applied Finance 03, no. 03 (2000): 559–60. http://dx.doi.org/10.1142/s0219024900000577.

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We study weak convergence of a sequence of (approximating) asset price models Sn to a limiting model S: both Sn and S are multi-dimensional asset price processes with some physical probability measures Pn resp. P, and a natural notion of process convergence is the weak convergence of the induced path probability measures, denoted by (Sn|Pn) resp. (S|P), on the abstract topological space of possible asset price trajectories. For the purpose of no-arbitrage pricing of options or more general derivatives on the model assets, there are two different aspects of this convergence: (i) convergence under the given physical probability measures, (Sn|Pn) → (S|P) and (ii) convergence under suitably chosen equivalent martingale measures (EMM) relevant for pricing derivatives, (Sn|Qn) → (S|Q). A simple example is the convergence of a sequence of discrete-time binomial models to the Black–Scholes model (geometric Brownian motion), where the model markets are complete and hence the choice of Qn resp. Q is unique. This example and the general case of complete limit markets (S|P) have been studied in [1]. In contrast we have several choices for Qn resp. Q when all the model markets are incomplete. A natural choice is the minimum entropy EMM [2, 3], defined as the (unique) EMM R minimizing the relative entropy H(R|P) to the physical measure P, among all EMMs. We prove the following: given that the approximating models converge under the physical measures, (Sn|Pn) → (S|P) — with some mild assumptions on P and on the minimum entropy EMMs Rn for Sn resp. R for S — entropy number convergence implies weak convergence of the minimum entropy option price processes: H(Rn|Pn) → H(R|P) implies (Sn|Rn) → (S|R). Several rigorous examples illustrate the result; cf. also [4].
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4

BENTH, FRED ESPEN, and FRANK PROSKE. "UTILITY INDIFFERENCE PRICING OF INTEREST-RATE GUARANTEES." International Journal of Theoretical and Applied Finance 12, no. 01 (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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5

Wong, Bernard, and C. C. Heyde. "On changes of measure in stochastic volatility models." Journal of Applied Mathematics and Stochastic Analysis 2006 (December 6, 2006): 1–13. http://dx.doi.org/10.1155/jamsa/2006/18130.

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Pricing in mathematical finance often involves taking expected values under different equivalent measures. Fundamentally, one needs to first ensure the existence of ELMM, which in turn requires that the stochastic exponential of the market price of risk process be a true martingale. In general, however, this condition can be hard to validate, especially in stochastic volatility models. This had led many researchers to “assume the condition away,” even though the condition is not innocuous, and nonsensical results can occur if it is in fact not satisfied. We provide an applicable theorem to check the conditions for a general class of Markovian stochastic volatility models. As an example we will also provide a detailed analysis of the Stein and Stein and Heston stochastic volatility models.
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6

Elliott, Robert J., and Dilip B. Madan. "A Discrete Time Equivalent Martingale Measure." Mathematical Finance 8, no. 2 (1998): 127–52. http://dx.doi.org/10.1111/1467-9965.00048.

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7

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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8

Hussein, Boushra Y. "Equivalent Locally Martingale Measure for the Deflator Process on Ordered Banach Algebra." Journal of Mathematics 2020 (June 9, 2020): 1–7. http://dx.doi.org/10.1155/2020/5785098.

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This paper aims at determining the measure of Q under necessary and sufficient conditions. The measure is an equivalent measure for identifying the given P such that the process with respect to P is the deflator locally martingale. The martingale and locally martingale measures will coincide for the deflator process discrete time. We define s-viable, s-price system, and no locally free lunch in ordered Banach algebra and identify that the s-price system C,π is s-viable if and only a character functional ψC≤π exists. We further demonstrate that no locally free lunch is a necessary and sufficient condition for the equivalent martingale measure Q to exist for the deflator process and the subcharacter ϕ∈Γ such that φC=π. This paper proves the existence of more than one condition and that all conditions are equivalent.
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9

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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10

Kabanov, Yuri. "In discrete time a local martingale is a martingale under an equivalent probability measure." Finance and Stochastics 12, no. 3 (2008): 293–97. http://dx.doi.org/10.1007/s00780-008-0063-y.

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11

Magdon-Ismail, M. "The equivalent martingale measure: an introduction to pricing using expectations." IEEE Transactions on Neural Networks 12, no. 4 (2001): 684–93. http://dx.doi.org/10.1109/72.935082.

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12

JARROW, ROBERT. "THE THIRD FUNDAMENTAL THEOREM OF ASSET PRICING." Annals of Financial Economics 07, no. 02 (2012): 1250007. http://dx.doi.org/10.1142/s2010495212500078.

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The importance of market efficiency to derivative pricing is not well understood. The purpose of this paper is to explain this connection using the third fundamental theorem of asset pricing. The third fundamental theorem of asset pricing characterizes the conditions under which an equivalent martingale probability measure exists in an economy. Noting that the existence of an equivalent martingale probability measure is both necessary and sufficient for the market being informationally efficient, we prove that in a complete market, the market being efficient is both necessary and sufficient for the validity of the risk neutral valuation methodology.
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13

Jianqi, Yang, Yan Haifeng, and Liu Limin. "Martingale measures in the market with restricted information." Journal of Applied Mathematics and Decision Sciences 2006 (July 3, 2006): 1–7. http://dx.doi.org/10.1155/jamds/2006/74864.

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This paper considers the problem of the market with restricted information. By constructing a restricted information market model, the explicit relation of arbitrage and the minimal martingale measure between two different information markets are discussed. Also a link among all equivalent martingale measures under restricted information market is given.
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14

Mania, M., and R. Tevzadze. "A Semimartingale Bellman Equation and the Variance-Optimal Martingale Measure." Georgian Mathematical Journal 7, no. 4 (2000): 765–92. http://dx.doi.org/10.1515/gmj.2000.765.

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Abstract We consider a financial market model, where the dynamics of asset prices is given by an Rm -valued continuous semimartingale. Using the dynamic programming approach we obtain an explicit description of the variance optimal martingale measure in terms of the value process of a suitable problem of an optimal equivalent change of measure and show that this value process uniquely solves the corresponding semimartingale backward equation. This result is applied to prove the existence of a unique generalized solution of Bellman's equation for stochastic volatility models, which is used to determine the variance-optimal martingale measure.
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15

Hamza, Kais, Saul Jacka, and Fima Klebaner. "The equivalent martingale measure conditions in a general model for interest rates." Advances in Applied Probability 37, no. 02 (2005): 415–34. http://dx.doi.org/10.1017/s0001867800000240.

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Assuming that the forward rates f t u are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes f t u =∫0 uf t v dv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.
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16

Hamza, Kais, Saul Jacka, and Fima Klebaner. "The equivalent martingale measure conditions in a general model for interest rates." Advances in Applied Probability 37, no. 2 (2005): 415–34. http://dx.doi.org/10.1239/aap/1118858632.

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Assuming that the forward rates ftu are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes ftu=∫0uftvdv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.
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17

HE, XIN-JIANG, and SONG-PING ZHU. "Pricing European options with stochastic volatility under the minimal entropy martingale measure." European Journal of Applied Mathematics 27, no. 2 (2015): 233–47. http://dx.doi.org/10.1017/s0956792515000510.

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In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.
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18

Taqqu, Murad S., and Walter Willinger. "The analysis of finite security markets using martingales." Advances in Applied Probability 19, no. 01 (1987): 1–25. http://dx.doi.org/10.1017/s0001867800016360.

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The theory of finite security markets developed by Harrison and Pliska [1] used the separating hyperplane theorem to establish the relationship between the lack of arbitrage opportunities and the existence of a certain martingale measure. In this paper we treat this theory by examining certain geometric properties of the sample paths of the price process, that is, we focus on the price increments of the stocks between one time period to the next and convert them to martingale differences through an equivalent change of measure. Thus, in contrast to Harrison and Pliska&s functional analytic derivation, our approach is based on probabilistic methods and allows a geometric interpretation which not only provides a connection to linear programming but also yields an algorithm for analyzing finite security markets. Moreover, we can make precise the connection between diverse expressions of economic equilibrium such as ‘absence of arbitrage’, ‘martingale property, and ‘complementary slackness property’.
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19

Taqqu, Murad S., and Walter Willinger. "The analysis of finite security markets using martingales." Advances in Applied Probability 19, no. 1 (1987): 1–25. http://dx.doi.org/10.2307/1427371.

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The theory of finite security markets developed by Harrison and Pliska [1] used the separating hyperplane theorem to establish the relationship between the lack of arbitrage opportunities and the existence of a certain martingale measure. In this paper we treat this theory by examining certain geometric properties of the sample paths of the price process, that is, we focus on the price increments of the stocks between one time period to the next and convert them to martingale differences through an equivalent change of measure. Thus, in contrast to Harrison and Pliska&s functional analytic derivation, our approach is based on probabilistic methods and allows a geometric interpretation which not only provides a connection to linear programming but also yields an algorithm for analyzing finite security markets. Moreover, we can make precise the connection between diverse expressions of economic equilibrium such as ‘absence of arbitrage’, ‘martingale property, and ‘complementary slackness property’.
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20

Azevedo, N., D. Pinheiro, S. Z. Xanthopoulos, and A. N. Yannacopoulos. "Who would invest only in the risk-free asset?" International Journal of Financial Engineering 05, no. 03 (2018): 1850024. http://dx.doi.org/10.1142/s242478631850024x.

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Within the setup of continuous-time semimartingale financial markets, we show that a multiprior Gilboa–Schmeidler minimax expected utility maximizer forms a portfolio consisting only of the riskless asset if and only if among the investor’s priors there exists a probability measure under which all admissible wealth processes are supermartingales. Furthermore, we show that under a certain attainability condition (which is always valid in finite or complete markets) this is also equivalent to the existence of an equivalent (local) martingale measure among the investor’s priors. As an example, we generalize a no betting result due to Dow and Werlang.
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21

Thierbach, F. "Mean-Variance Hedging Under Additional Market Information." International Journal of Theoretical and Applied Finance 06, no. 06 (2003): 613–36. http://dx.doi.org/10.1142/s0219024903002092.

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In this paper we analyze the mean-variance hedging approach in an incomplete market under the assumption of additional market information, which is represented by a given, finite set of observed prices of non-attainable contingent claims. Due to no-arbitrage arguments, our set of investment opportunities increases and the set of possible equivalent martingale measures shrinks. Therefore, we obtain a modified mean-variance hedging problem, which takes into account the observed additional market information. Solving this we obtain an explicit description of the optimal hedging strategy and an admissible, constrained variance-optimal signed martingale measure, that generates both the approximation price and the observed option prices.
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22

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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23

Bender, Christian, Mikko S. Pakkanen, and Hasanjan Sayit. "Sticky Continuous Processes have Consistent Price Systems." Journal of Applied Probability 52, no. 02 (2015): 586–94. http://dx.doi.org/10.1017/s0021900200012651.

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Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni (2006).
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24

Bender, Christian, Mikko S. Pakkanen, and Hasanjan Sayit. "Sticky Continuous Processes have Consistent Price Systems." Journal of Applied Probability 52, no. 2 (2015): 586–94. http://dx.doi.org/10.1239/jap/1437658617.

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Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni (2006).
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25

Kegnenlezom, M., P. Takam Soh, M. L. D. Mbele Bidima, and Y. Emvudu Wono. "A jump-diffusion model for pricing electricity under price-cap regulation." Mathematical Sciences 13, no. 4 (2019): 395–405. http://dx.doi.org/10.1007/s40096-019-00308-6.

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Abstract In this paper, we derive a new jump-diffusion model for electricity spot price from the “Price-Cap” principle. Next, we show that the model has a non-classical mean-reverting linear drift. Moreover, using this model, we compute a new exact formula for the price of forward contract under an equivalent martingale measure and we compare it to Cartea et al. (Appl Math Finance 12(4):313–335, 2005) formula.
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26

Nowak, Piotr, and Michal Pawlowski. "Option Pricing With Application of Levy Processes and the Minimal Variance Equivalent Martingale Measure Under Uncertainty." IEEE Transactions on Fuzzy Systems 25, no. 2 (2017): 402–16. http://dx.doi.org/10.1109/tfuzz.2016.2637372.

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27

Tan, Xiaoyu, Shenghong Li, and Shuyi Wang. "Pricing European-Style Options in General Lévy Process with Stochastic Interest Rate." Mathematics 8, no. 5 (2020): 731. http://dx.doi.org/10.3390/math8050731.

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This paper extends the traditional jump-diffusion model to a comprehensive general Lévy process model with the stochastic interest rate for European-style options pricing. By using the Girsanov theorem and Itô formula, we derive the uniform formalized pricing formulas under the equivalent martingale measure. This model contains not only the traditional jump-diffusion model, such as the compound Poisson model, the renewal model, the pure-birth jump-diffusion model, but also the infinite activities Lévy model.
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28

Jensen, Jens Ledet, and Jan Pedersen. "Ornstein–Uhlenbeck type processes with non-normal distribution." Journal of Applied Probability 36, no. 02 (1999): 389–402. http://dx.doi.org/10.1017/s0021900200017204.

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We analyse a class of diffusion models that (i) allows an explicit expression for the likelihood function of discrete time observation, (ii) allows the possibility of heavy-tailed observations, and (iii) allows an analysis of the tails of the increments. The class simply consists of transformed Ornstein–Uhlenbeck processes and is of relevance for heavy-tailed time series. We also treat the question of the existence of an equivalent martingale measure for the class of models considered.
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Jensen, Jens Ledet, and Jan Pedersen. "Ornstein–Uhlenbeck type processes with non-normal distribution." Journal of Applied Probability 36, no. 2 (1999): 389–402. http://dx.doi.org/10.1239/jap/1032374460.

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We analyse a class of diffusion models that (i) allows an explicit expression for the likelihood function of discrete time observation, (ii) allows the possibility of heavy-tailed observations, and (iii) allows an analysis of the tails of the increments. The class simply consists of transformed Ornstein–Uhlenbeck processes and is of relevance for heavy-tailed time series. We also treat the question of the existence of an equivalent martingale measure for the class of models considered.
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30

YANG, BEN-ZHANG, JIA YUE, and NAN-JING HUANG. "EQUILIBRIUM PRICE OF VARIANCE SWAPS UNDER STOCHASTIC VOLATILITY WITH LÉVY JUMPS AND STOCHASTIC INTEREST RATE." International Journal of Theoretical and Applied Finance 22, no. 04 (2019): 1950016. http://dx.doi.org/10.1142/s021902491950016x.

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This paper focuses on the pricing of variance swaps in incomplete markets where the short rate of interest is determined by a Cox–Ingersoll–Ross model and the stock price is determined by a Heston model with simultaneous Lévy jumps. We obtain the pricing kernel and the equivalent martingale measure in an equilibrium framework. We also give new closed-form solutions for the delivery prices of discretely sampled variance swaps under the forward measure, as opposed to the risk neural measure, by employing the joint moment generating function of underlying processes. Theoretical results and numerical examples are provided to illustrate how the values of variance swaps depend on the jump risks and stochastic interest rate.
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31

Siu, Tak Kuen, John W. Lau, and Hailiang Yang. "Pricing Participating Products under a Generalized Jump-Diffusion Model." Journal of Applied Mathematics and Stochastic Analysis 2008 (July 13, 2008): 1–30. http://dx.doi.org/10.1155/2008/474623.

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We propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diffusion models and Markovian regime-switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.
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32

Intarasit, Arthit. "Markov Regime Switching of Stochastic Volatility Lévy Model on Approximation Mode." Journal of Applied Mathematics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/549304.

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This paper deals with financial modeling to describe the behavior of asset returns, through consideration of economic cycles together with the stylized empirical features of asset returns such as fat tails. We propose that asset returns are modeled by a stochastic volatility Lévy process incorporating a regime switching model. Based on the risk-neutral approach, there exists a large set of candidates of martingale measures due to the driving of a stochastic volatility Lévy process in the proposed model which renders the market incomplete in general. We first establish an equivalent martingale measure for the proposed model introduced in risk-neutral version. Regime switching of stochastic volatility Lévy process is employed in an approximation mode for model calibration and the calibration of parameters model done based on EM algorithm. Finally, some empirical results are illustrated via applications to the Bangkok Stock Exchange of Thailand index.
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33

Eddahbi, M., J. L. Solé, and J. Vives. "A Stroock formula for a certain class of Lévy processes and applications to finance." Journal of Applied Mathematics and Stochastic Analysis 2005, no. 3 (2005): 211–35. http://dx.doi.org/10.1155/jamsa.2005.211.

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We find a Stroock formula in the setting of generalized chaos expansion introduced by Nualart and Schoutens for a certain class of Lévy processes, using a Malliavin-type derivative based on the chaotic approach. As applications, we get the chaotic decomposition of the local time of a simple Lévy process as well as the chaotic expansion of the price of a financial asset and of the price of a European call option. We also study the behavior of the tracking error in the discrete delta neutral hedging under both the equivalent martingale measure and the historical probability.
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34

Vazifedan, Mehdi, and Qiji Jim Zhu. "No-Arbitrage Principle in Conic Finance." Risks 8, no. 2 (2020): 66. http://dx.doi.org/10.3390/risks8020066.

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In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid–ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “no-arbitrage” principle in financial models with the bid–ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.
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35

Maris, Florian, and Hasanjan Sayit. "Consistent Price Systems in Multiasset Markets." International Journal of Stochastic Analysis 2012 (August 27, 2012): 1–14. http://dx.doi.org/10.1155/2012/687376.

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Let be any d-dimensional continuous process that takes values in an open connected domain in . In this paper, we give equivalent formulations of the conditional full support (CFS) property of in . We use them to show that the CFS property of X in implies the existence of a martingale M under an equivalent probability measure such that M lies in the neighborhood of for any given under the supremum norm. The existence of such martingales, which are called consistent price systems (CPSs), has relevance with absence of arbitrage and hedging problems in markets with proportional transaction costs as discussed in the recent paper by Guasoni et al. (2008), where the CFS property is introduced and shown sufficient for CPSs for processes with certain state space. The current paper extends the results in the work of Guasoni et al. (2008), to processes with more general state space.
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36

Han, Miao, Xuefeng Song, Huawei Niu, and Shengwu Zhou. "Pricing Vulnerable Options with Market Prices of Common Jump Risks under Regime-Switching Models." Discrete Dynamics in Nature and Society 2018 (2018): 1–15. http://dx.doi.org/10.1155/2018/8545841.

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This paper investigates the valuation of vulnerable European options considering the market prices of common systematic jump risks under regime-switching jump-diffusion models. The way of regime-switching Esscher transform is adopted to identify an equivalent martingale measure for pricing vulnerable European options. Explicit analytical pricing formulae for vulnerable European options are derived by risk-neutral pricing theory. For comparison, the other two cases are also considered separately. The first case considers all jump risks as unsystematic risks while the second one assumes all jumps risks to be systematic risks. Numerical examples for the valuation of vulnerable European options are provided to illustrate our results and indicate the influence of the market prices of jump risks on the valuation of vulnerable European options.
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37

Lototsky, Sergey V., Henry Schellhorn, and Ran Zhao. "An infinite-dimensional model of liquidity in financial markets." Probability, Uncertainty and Quantitative Risk 6, no. 2 (2021): 117. http://dx.doi.org/10.3934/puqr.2021006.

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<p style='text-indent:20px;'>We consider a dynamic market model of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand surface constitutes the sole input to the model. We model demand using a two-parameter Brownian motion because (i) different points on the demand curve correspond to orders motivated by different information, and (ii) in general, the market price of risk equation of no-arbitrage theory has no solutions when the demand curve is driven by a finite number of factors, thus allowing for arbitrage. We prove that if the driving noise is infinite-dimensional, then there is no arbitrage in the model. Under the equivalent martingale measure, the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. We consider several parameterizations of the model and show advantages of specifying the demand curve as a quantity that is a function of price, as opposed to price as a function of quantity. An online appendix presents a basic empirical analysis of the model: calibration using information from actual order books, computation of option prices using Monte Carlo simulations, and comparison with observed data.</p>
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38

BRANGER, NICOLE. "PRICING DERIVATIVE SECURITIES USING CROSS-ENTROPY: AN ECONOMIC ANALYSIS." International Journal of Theoretical and Applied Finance 07, no. 01 (2004): 63–81. http://dx.doi.org/10.1142/s0219024904002335.

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This paper analyses two implied methods to determine the pricing function for derivatives when the market is incomplete. First, we consider the choice of an equivalent martingale measure with minimal cross-entropy relative to a given benchmark measure. We show that the choice of the numeraire has an impact on the resulting pricing function, but that there is no sound economic answer to the question which numeraire to choose. The ad-hoc choice of the numeraire introduces an element of arbitrariness into the pricing function, thus contradicting the motivation of this method as the least prejudiced way to choose the pricing operator. Second, we propose two new methods to select a pricing function: the choice of the stochastic discount factor (SDF) with minimal extended cross-entropy relative to a given benchmark SDF, and the choice of the Arrow–Debreu (AD) prices with minimal extended cross-entropy relative to some set of benchmark AD prices. We show that these two methods are equivalent in that they generate identical pricing functions. They avoid the dependence on the numeraire and replace it by the dependence on the benchmark pricing function. This benchmark pricing function, however, can be chosen based on economic considerations, in contrast to the arbitrary choice of the numeraire.
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39

Beissner, Patrick. "Coherent-Price Systems and Uncertainty-Neutral Valuation." Risks 7, no. 3 (2019): 98. http://dx.doi.org/10.3390/risks7030098.

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This paper considers fundamental questions of arbitrage pricing that arises when the uncertainty model incorporates ambiguity about risk. This additional ambiguity motivates a new principle of risk- and ambiguity-neutral valuation as an extension of the paper by Ross (1976) (Ross, Stephen A. 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13: 341–60). In the spirit of Harrison and Kreps (1979) (Harrison, J. Michael, and David M. Kreps. 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20: 381–408), the paper establishes a micro-economic foundation of viability in which ambiguity-neutrality imposes a fair-pricing principle via symmetric multiple prior martingales. The resulting equivalent symmetric martingale measure set exists if the uncertain volatility in asset prices is driven by an ambiguous Brownian motion.
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40

Ching, Wai-Ki, Tak-Kuen Siu, and Li-Min Li. "Pricing Exotic Options under a High-Order Markovian Regime Switching Model." Journal of Applied Mathematics and Decision Sciences 2007 (October 8, 2007): 1–15. http://dx.doi.org/10.1155/2007/18014.

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We consider the pricing of exotic options when the price dynamics of the underlying risky asset are governed by a discrete-time Markovian regime-switching process driven by an observable, high-order Markov model (HOMM). We assume that the market interest rate, the drift, and the volatility of the underlying risky asset's return switch over time according to the states of the HOMM, which are interpreted as the states of an economy. We will then employ the well-known tool in actuarial science, namely, the Esscher transform to determine an equivalent martingale measure for option valuation. Moreover, we will also investigate the impact of the high-order effect of the states of the economy on the prices of some path-dependent exotic options, such as Asian options, lookback options, and barrier options.
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41

Li, Peng, Wei Wang, Lin Xie, and Zhixin Yang. "Premium Valuation of the Pension Benefit Guaranty Corporation with Regime Switching." Mathematical Problems in Engineering 2021 (May 24, 2021): 1–15. http://dx.doi.org/10.1155/2021/9966515.

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The Pension Benefit Guaranty Corporation (PBGC) provides insurance coverage for single-employer and multiemployer pension plans in private sector. It has played an important role in protecting the retirement security for over 1.5 million people since it was established about half a decade ago. PBGC collects insurance premiums from employers that sponsor insured pension plans for its coverage and receives funds from pension plans that it takes over. To address the issue of underfunded plans that the PBGC has, this work studies how to evaluate risk-based premiums for the PBGC. Inspired by a couple of existing work in which the premature termination of pension fund and distress termination of sponsor assets are analyzed separately, our work examines the two types of terminations under one framework and considers the occurrence of each termination dynamically. Given that market regime might have a big impact on the dynamics of both pension fund and sponsor’s assets, we thus formulate our model using a continuous-time two-state Markov chain in which bull market and bear market are delineated. We thus formulate our model using a continuous-time two-state Markov Chain in which bull market and bear market are delineated. In other words, the pension fund and sponsor assets are market dependent in our work. Given that this additional uncertainty described by regime switching makes the market incomplete, we therefore utilize the Esscher transform to determine an equivalent martingale measure and apply the risk neutral pricing method to obtain the closed-form expressions for premium of PBGC. In addition, we carry out numerical analysis to demonstrate our results and observe that premium increases according to the retirement benefit irrespective of the type of terminations. In comparison to the case of early distress termination of sponsor assets, the premium goes up more quickly when premature termination of pension funds occurs first due to the fact that pension fund is the first venue of retirement security. Furthermore, we look at how the premium changes with respect to other key parameters as well and make some detailed observations in the section of numerical analysis.
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42

Jarrow, Robert, Philip Protter, and Jaime San Martin. "Asset price bubbles: Invariance theorems." Frontiers of Mathematical Finance, 2021, 0. http://dx.doi.org/10.3934/fmf.2021006.

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<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>
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43

Choulli, Tahir, and Martin Schweizer. "Stability of Sigma-Martingale Densities in L Log L Under an Equivalent Change of Measure." SSRN Electronic Journal, 2011. http://dx.doi.org/10.2139/ssrn.1986855.

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44

Feinstein, Zachary, and Birgit Rudloff. "Scalar Multivariate Risk Measures with a Single Eligible Asset." Mathematics of Operations Research, September 16, 2021. http://dx.doi.org/10.1287/moor.2021.1153.

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In this paper, we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model. Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs that the usual scalar concept of time consistency is too strong and not satisfied. We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.
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