Martingale Measure Research Articles

Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales

The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles for a sequence of Markov processes.

On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Levy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential Levy models: Merton models and variance gamma models.

An Approach to Measure the Expectation of Generic Functions of the Market Return

This paper proposes an approach that associates the risk-neutral probability measure with option prices and then computes the expectation of quantities under the real world probability measure, exploiting the form of the stochastic discount factor. This approach deviates from foundational approaches that embrace assumptions about preferences and economic primitives to propose formulas for prices and risk premiums. Our method is analytically tractable, absolved of distributional assumptions, and we use the approach to elaborate on empirical questions regarding disaster probabilities, conditional return moments, and conditional return asymmetries.

Distributional divergence, statistical experiments and consequences in option pricing

ABSTRACTDistributional Divergence and Statistical Experiments are used herein for a positive stochastic process. This framework provides, under mild assumptions, Risk Neutral Probability (-ies) for a stock price process which does not have necessarily either to satisfy a Stochastic Differential Equation or to follow a model, both non-realistic assumptions. The results contribute in understanding the relation between statistical contiguity and market's informational efficiency. -price of European option is obtained, confirming the universal quote of the Black–Scholes–Merton price for the class of calm stock prices that includes log-normal price. Other consequences are presented.

SUPER-HEDGING AMERICAN OPTIONS WITH SEMI-STATIC TRADING STRATEGIES UNDER MODEL UNCERTAINTY

We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price [Formula: see text] is given by the supremum over the prices of the American option under randomized models. That is, [Formula: see text], where [Formula: see text] and the martingale measure [Formula: see text] are chosen such that [Formula: see text] and [Formula: see text] prices the European options correctly, and [Formula: see text] is the price of the American option under the model [Formula: see text]. Our result generalizes the example given in Hobson & Neuberger (2016) that the highest model-based price can be considered as a randomization over models.

SIEVE ESTIMATION OF THE MINIMAL ENTROPY MARTINGALE MARGINAL DENSITY WITH APPLICATION TO PRICING KERNEL ESTIMATION

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.

On the Theoretical Valuation of V-KOSPI 200 Futures

Although the V-KOSPI 200 Futures markets opened in November 2014, trading has not been active until recently. One of the reasons for the illiquidity is due to the lack of a market consensus on the stochastic process model for the underlying volatility index (V-KOSPI 200). Given this fact, there is no theoretical pricing model that can be used for the determination of the benchmark price for the V-KOSPI 200 Futures. In this paper, we use the generalized method of moments method to search for a model that fits well with the time series of V-KOSPI 200 under the historical measure. In addition, we compare the performance of each model for the pricing of the V-KOSPI 200 Futures under the risk neutral measure. In the empirical analysis, we find that the CEV (constant elasticity of variance) parameter with the value about 1.5 is needed to price both the underlying V-KOSPI 200 process (under the physical measure) and the V-KOSPI 200 Futures (under the risk neutral measure). We also find that the mean reversion property is necessary to explain the dynamics of V-KOSPI 200.

Unit-linked life insurance policies: Optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.

New results on the existence of interpolating and weakly interpolating martingale measures

New results on the existence of interpolating and weakly interpolating martingale measures

Practical Applications of What Goes into Risk-Neutral Volatility? Empirical Estimates of Risk and Subjective Risk Preferences

The Black–Scholes option-pricing model doesn’t work in the real world because option prices embed both the market’s estimate of the empirical returns distribution as well as investors’ risk attitudes. These influences are reflected in the risk-neutral probability distribution, which can be extracted from option prices without restrictive assumptions from a pricing model. In What Goes into Risk-Neutral Volatility? Empirical Estimates of Risk and Subjective Risk Preferences, published in the Fall 2016 issue of The Journal of Portfolio Management, Stephen Figlewski, professor of finance at NYU’s Stern School of Business, finds that risk-neutral volatility is strongly influenced by investors’ projections of future realized volatility and by the risk-neutralization process. Several significant variables, including the daily trading range and tail risk, are connected in different ways to realized volatility, while other variables reflect risk attitudes, such as the levels of consumer and investor confidence. This report and the original article will be of particular interest to investors and academics who seek to bridge the gap between theory and practice. TOPICS:Options, quantitative methods, tail risks

Martingale Measures and Collateral

This paper discusses the notion of martingale measures in the context of the pricing and hedging of perfectly collateralized derivatives in discrete time. We define a precise mathematical framework to deal with this paradigm. We then prove the main result of equivalence between absence of arbitrage and existence of a measure such that assets discounted at their own collateral rate behave like martingales. This result implies that martingale measures are no longer associated to a specific numeraire. The result is proven in a general framework and doesn't assume specific model dynamics.

Variance-Optimal Martingale Measures for Diffusion Processes with Stochastic Coefficients

In this paper we present the solution of the optimal variance optimal martingale measure for stochastic volatility models, when the noises are correlated. It is proved that the value function of the dual problem is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. The method to develop our results is based on a Bernstein’s type of argument. The dual problem of the quadratic hedging problem is studied analyzing the expression obtained after a change of measure, which corresponds to some class of risk-sensitive control problems.

A new closed-form formula for pricing European options under a skew Brownian motion

ABSTRACTIn this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.

The Volatility-of-Volatility Term Structure

We apply the model-free implied risk-neutral measure of variance to VIX options to investigate the volatility-of-volatility (VVIX) term structure. We find that the information content of the VVIX term structure is fairly distinct from the VIX term structure. Both carry different pieces of information about market uncertainty. The second PCA component (SlopeVVIX) of the VVIX term structure is a significant risk factor. In joint regressions SlopeVVIX predicts straddle returns of S&P500 and VIX options incremental to the VIX term structure and the variance risk premium. We propose an affine approximation method of the VVIX and use parameter estimates of a VIX option pricing model to identify three factors that describe most of the term structure. Continuous volatility-of-volatility adds most to the term structure, followed by variance jumps and a lower bound component. These systematic changes of the risk-factors can be put in economic context by the variance-of-variance to variance ratio.

The Two Envelope Problem: Insurance for the Not So Greener Envelope

In the two envelope problem the expected pay-off appears to be higher than the amount of money in the envelope at hand. Here the holder of the envelope is allowed to buy insurance and so realize the expected pay-off. Pricing the insurance premium fairly shows that the expected pay-off is the amount in the envelope. It is shown that pricing the insurance option one must use linked probabilities or alternatively risk-neutral probabilities.

Pricing European options and risk measurement under exponential Lévy models — a practical guide

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.

VIX Forecast Under Different Volatility Specifications

Stochastic volatility (SV) models are theoretically more attractive than the GARCH type of models as it allows additional randomness. The classical SV models deduce a continuous probability distribution for volatility so that it does not admit a computable likelihood function. The estimation requires the use of Bayesian approach. A recent approach considers discrete stochastic autoregressive volatility models for a bounded and tractable likelihood function. Hence, a maximum likelihood estimation can be achieved. This paper proposes a general approach to link SV models under the physical probability measure, both continuous and discrete types, to their processes under a martingale measure. Doing so enables us to deduce the close-form expression for the VIX forecast for the both SV models and GARCH type models. We then carry out an empirical study to compare the performances of the continuous and discrete SV models using GARCH models as benchmark models.

Options Prices in Incomplete Markets

In this paper we consider the valuation of an option with time to expiration T and pay-off function g which is a convex function (as is a European call option), and constant interest rate r = 0, for a variety of underlying price process models constructed from two independent Poisson processes, and an independent Brownian motion. This gives rise to incomplete market models with an infinite number of risk neutral measures. The collection of risk neutral measures gives rise to different prices, which comprise intervals that we calculate. The intervals can vary dramatically depending on the model parameters.

Systematic Credit Risk: CDX Index Correlation and Extreme Dependence

Dependence is an important issue in credit risk portfolio modeling and pricing. We discuss a straightforward common factor model of credit risk dependence, which is motivated by intensity models such as Duffie and Singleton (1998), among others. In the empirical analysis, we study dependence under the risk-neutral measure using credit default swap (CDS) spread data of liquid large-cap U.S. obligors. The proxy for the commonfactor is the DJ CDX.NA.IG index. We document that (i) the CDX factor is significant but has low explanatory power, (ii) factor sensitivities show distinct time-varying nature and that (iii) systematic credit risk shows asymmetric extreme factor dependence, where extreme dependence is present for upward CDX movements only. This finding from an EVT-copula approach is what is predicted by various intensity models of joint defaults.

Real option analysis case with dynamic risk neutral probability

Real option analysis case with dynamic risk neutral probability