Coupling and option price comparisons in a jump-diffusion model

The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability v_f associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of v_f—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of v_f consists of additive probabilities below v_f whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below v_f, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.

We now return to general continuous-time financial market models in the setting of §6.1, i.e. there are d+1 primary traded assets whose price processes are given by stochastic processes S 0,... , S d , which are assumed to be adapted, right-continuous with left-limits (RCLL) and strictly positive semimartin-gales on a filtered probability space (Ω, F ℙ, F) (as usual F = (F t ) t≤T ). We assume that the market is free of arbitrage, in the sense that there exist equivalent martingale measures, but it contains non-attainable contingent claims, i.e. there are cash flows which cannot be replicated by self-financing trading strategies. In view of Theorem 6.1.5 this means that we do not have a unique equivalent martingale measures. We try to answer the obvious questions in this setting: how should we price the non-attainable contingent claims, i.e. which of the possible equivalent martingale measures should we pick for our valuation formula based on expectation, and, how can we construct hedging strategies for the nonattainable contingent claims to ‘minimise the risk? We try to answer these two questions in the general setting and then consider a prominent example of an incomplete market, a market with stochastic volatility, in more detail.KeywordsTrading StrategyStochastic VolatilityContingent ClaimMartingale MeasureIncomplete MarketThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Forward Measures in a Ho and Lee Jump Diffusion Model

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.

We provide a framework for learning risk-neutral measures (Martingale measures) for pricing options from high frequency financial data. In a simple geometric Brownian motion model, a price volatility, a fixed interest rate and a no-arbitrage condition suffice to determine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these assumptions to obtain a class of allowable risk-neutral measures. We then propose a framework for learning the appropriate risk-neural measure. Since the riskneutral measure prices all options simultaneously, we can use all the option contracts on a particular underlying stock for learning. We demonstrate the performance of these models on historical data. In particular, we show that both learning without a no-arbitrage condition and a no-arbitrage condition without learning are worse than our framework; however the combination of learning with a no-arbitrage condition has the best result. These results indicate the potential to learn Martingale measures with a no-arbitrage condition providing just the right constraint. We also compare our approach to standard Binomial models with volatility estimates (historical volatility and GARCH volatility predictors). Finally, we illustrate the power of such a framework by developing a real time trading system based upon these pricing methods.

In a viable single-period model with one stock and k ≥ 2 scenarios the completeness of the market is equivalent to the uniqueness of the risk neutral probability; this equivalence allows to price every derivative security with a unique fair price. When the market is incomplete, the set of all possible risk neutral probabilities is not a singleton and for every non attainable derivative security we have a bid-ask interval of possible prices. In literature, different methods have been proposed in order to select a unique risk neutral probability starting with the real world probability p. Contrary to the complete case, in all these models \(\textbf{p}\) is really used for the option pricing and its elicitation is a crucial point for every criterion used to select a risk neutral probability. We propose a method for the valuation problem in incomplete markets which can be used when p is a partial conditional probability assessment as well as when we have different expert opinions expressed through conditional probability assessments. In fact, it is not always possible to elicit a probability distribution p over all the possible states of the world: the information that we have could be partial, conditional or even not coherent. Therefore we will select a risk neutral probability by minimizing a discrepancy measure introduced in [2] and analized in [3] between p and the set of all possible risk neutral probability, where p can be a partial conditional probability assessments or it can be given by the fusion of different expert opinions.KeywordsExpert OpinionCall OptionConditional EventMartingale MeasureStrike PriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper gives an ordering on option prices under various well‐known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q‐optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer (Mathematical Finance, 11(4), 2001) in the special case of the Heston model.

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Levy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed.

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

ABSTRACTWe present a finite difference method to solve a system of two Partial-Integro Differential Equations which arise from pricing an option under a Jump-Telegraph Diffusion Model for the underlying asset, considering the risk-neutral valuation formula under an equivalent martingale measure. This system is fully discretized using an Implicit–Explicit two-time level scheme and quadrature formulas. The resulting two tridiagonal algebraic linear systems are solved recursively using the Thomas Algorithm. Some numerical results are presented and the numerical order of convergence for the method is estimated. Finally, the robustness of the method is validated against an exact solution obtained for a perturbed problem.

Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (Ft)t∈T, P) with T⊆R+. Under the assumption that the set Me of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let $$V(S_t, t)$$ be the price of an option at time t when the share price of the underlying stock is $$S_t$$ . See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond $$B_t$$ and a risky asset stock $$S_t$$ . They evolve as where r is the interest rate, $$\mu $$ is the drift, $$\sigma $$ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price $$B_t^{-1}V(S_t,t)$$ . By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure . Also, under the risk-neutral measure $$\mu =r$$ . We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.

Arbitrage-Free Prediction of the Implied Volatility Smile

The minimal relative entropy criterion for the selection of an equivalent martingale measure in an incomplete market seems to still hold some mystique in its financial interpretation. In this paper we work toward this interpretation by suggesting and exploring the idea of relating a martingale measure selection criterion to a CAPM-like pricing scheme. We examine this idea in the case of the minimal relative entropy criterion and we present some preliminary results. We work within a one-period financial market and show that the minimal relative entropy pricing criterion is equivalent to some CAPM-like pricing scheme where the classical beta coefficient formula has been replaced by some “entropic beta” and the market portfolio by some “appropriate” reference portfolio. Furthermore, we show that if the assets involved have returns that are jointly normal, then this “entropic beta” formula coincides with the classical beta coefficient. Additionally and for comparison reasons, we briefly illustrate that if our criterion for the choice of the martingale measure was the minimization of the variance of the Radon–Nikodym derivative, then the resulting martingale pricing and the pricing implied by the classical CAPM scheme would be the same.

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.