Martingale Measure Research Articles

On partially observed jump diffusions III: regularity of the filtering density

AbstractThe filtering equations associated to a partially observed jump diffusion model $$(Z_t)_{t\in [0,T]}=(X_t,Y_t)_{t\in [0,T]}$$ ( Z t ) t ∈ [ 0 , T ] = ( X t , Y t ) t ∈ [ 0 , T ] , driven by Wiener processes and Poisson martingale measures are considered. Building on results from two preceding articles on the filtering equations, the regularity of the conditional density of the signal $$X_t$$ X t , given observations $$(Y_s)_{s\in [0,t]}$$ ( Y s ) s ∈ [ 0 , t ] , is investigated, when the conditional density of $$X_0$$ X 0 given $$Y_0$$ Y 0 exists and belongs to a Sobolev space, and the coefficients satisfy appropriate smoothness and growth conditions.

Option Pricing Based on Neural Stochastic Differential Equations

Firstly, based on the Black-Scholes stock price model, the neural stochastic differential equation (NSDE) model was established by parameterizing the asset return rate and volatility as a drift network and a diffusion network, respectively. Secondly, in the empirical analysis, the underlying asset as a single stock option was used as the research object, and real stock data was used for the network training and testing. The experimental results show that the NSDE model can overcome the defects of the constant assumption of the Black-Scholes model. Finally, for the case where the price of the underlying asset of the option was unobservable, we proposed that the price of any target option and the price of a known option could be constrained within the Wasserstein distance of their risk-neutral equivalent martingale measure, and theoretically proved the method.

Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing.

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

Vulnerable options with regime switching and stochastic liquidity

Investigating default risk in pricing options holds significant practical importance, as nearly all market participants and institutions face credit risk. Additionally, economic cycles and asset liquidity are crucial factors that should be incorporated. This paper considers these factors and derives an analytical pricing formula. Specifically, we model the economic cycles through switching volatility driven by a continuous-time Markov chain, while we adopt a discounting factor based on market liquidity levels to model the asset liquidity. We establish a risk-neutral measure after embracing a regime-switching Esscher transform, and formulate a price representation to value vulnerable options analytically despite the complexity of the developed model. We conduct several numerical experiments to validate the model’s efficacy and flexibility.

A new pricing method for integrated energy systems based on geometric Brownian motions under the risk-neutral measure

An integrated energy system (IES) is a promising energy-saving technology. With the advancement of technology and deregulation of the energy market, numerous distributed energy systems (DESs) will be accessed in the IES with various trading modes in the future. This makes it even more challenging to evaluate the fair energy selling price of IES, which is of great concern to market regulators and participants. To address this issue, this study proposes a risk-neutral pricing method for IES based on geometric Brownian motions. First, geometric Brownian motions are introduced to characterize the uncertainty feature of the energy price of the DESs. Then, we propose a stochastic partial differential equation (SPDE) for the energy price of an IES by utilizing the Ito lemma and the properties of the martingale under the risk-neutral measure. Finally, sensitivity analyses of the proposed SPDE on the volatilities and two types of correlation coefficients are conducted by implementing the proposed model in the given scenario.

Pricing and Hedging Contingent Claims by Entropy Segmentation and Fenchel Duality

We present a new approach to the problem of characterizing and choosing equivalent martingale pricing measures for a contingent claim, in a finite-state incomplete market. This is the entropy segmentation method achieved by means of convex programming, thanks to which we divide the claim no-arbitrage prices interval into two halves, the buyer’s and the seller’s prices at successive entropy levels. Classical buyer’s and seller’s prices arise when the entropy level approaches 0. Next, we apply Fenchel duality to these primal programs to characterize the hedging positions, unifying in the same expression the cases of super (resp. sub) replication (arising when the entropy approaches 0) and partial replication (when entropy tends to its maximal value). We finally apply linear programming to our hedging problem to find in a price slice of the dual feasible set an optimal partial replicating portfolio with minimal CVaR. We apply our methodology to a cliquet style guarantee, using Heston’s dynamic with parameters calibrated on EUROSTOXX50 index quoted prices of European calls. This way prices and hedging positions take into account the volatility risk.

A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets

We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$-pure jump additive processes and its application to portfolio optimization

AbstractWe derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for $$ L^1 $$ L 1 -canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in terms of their Malliavin derivatives. In mathematical finance, the COH formula under a change of measure is crucial for representing financial derivatives under a risk-neutral probability measure. To prove our main results, we use the Malliavin-Skorohod calculus in $$ L^0 $$ L 0 and $$ L^1 $$ L 1 for additive processes, as developed by Di Nunno and Vives (2017). An application of our results is solving the local risk minimization (LRM) problem in financial markets driven by pure jump additive processes. LRM, a prominent hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. By applying our COH formula, we obtain explicit expressions for locally risk-minimizing hedging strategies in terms of Malliavin derivatives under the market model underlying the additive process. These formulas provide practical tools for managing risks in financial market price fluctuations with $$L^1$$ L 1 -additive processes.

Time-Varying Deterministic Volatility Model for Options on Wheat Futures

This study introduces a robust model that captures wheat futures’ volatility dynamics, influenced by seasonality, time to maturity, and storage dynamics, with minimal calibratable parameters. Our approach reduces error-proneness and enhances plausibility checks, offering a reliable alternative to models that are difficult to calibrate. Transferring estimated parameters from liquid to illiquid markets is feasible, which is challenging for models with numerous parameters. This is of practical importance as it improves the modeling of volatility in illiquid markets, where price discovery is less efficient. In liquid markets, on the other hand, where speculative activity is high, we find that implied volatility is usually the best measure. Additionally, the introduced volatility model is suitable for pricing options on wheat futures as a risk-neutral measure.

Option Pricing in an Incomplete Market

The purpose of this paper is to illustrate the pricing of options in an incomplete market using the new consistent uplifted martingale measure methodology introduced by Grigorian and Jarrow [2024, Filtration Reduction and Incomplete Markets, Frontiers of Mathematical Finance, 3(1), 78–105; 2023, Filtration Reduction and Completeness in Brownian Motion Models. Working Paper, Cornell University; 2024, Filtration Reduction and Completeness in Jump-Diffusion Models. Working Paper, Cornell University]. We apply it to an incomplete market where a stock has stochastic volatility. Two valuation formulas are generated, depending upon whether the trader is more concerned about volatility or price risk in the construction of a partial replicating portfolio for the option’s payoff.

Quantum computational finance for martingale asset pricing in incomplete markets

A derivative is a financial asset whose future payoff is a function of underlying assets. Pricing a financial derivative involves setting up a market model, finding a martingale (“fair game”) probability measure for the model from the existing asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. In this work, we first formulate the pricing problem in a linear algebra setting, including the realistic setting of incomplete markets where derivatives do not have a unique price. We show that the problem can be solved with a variety of quantum techniques such as quantum linear programming and the quantum linear systems algorithm. While in previous works the martingale measure is assumed to be given, here it is extracted from market variables akin to bootstrapping, a common practice among financial institutions. We discuss the quantum zero-sum game algorithm and the quantum simplex algorithm as viable subroutines. For quantum linear systems solvers, we formalize a new market assumption milder than market completeness, which allows the potential for large speedups. Towards prototype use cases, we conduct numerical experiments motivated by the Black–Scholes–Merton model.

Change of measure in a Heston–Hawkes stochastic volatility model

Abstract We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed. Finally, a practical application to efficient computation of exposures is discussed.

Portfolio optimization beyond utility maximization: the case of driftless markets

This paper presents a novel perspective on portfolio optimization by recognizing that prices can be expressed as a scaled likelihood ratio of state price densities. This insight leads to the immediate conclusion that the optimal portfolio has a simple representation in terms of the likelihood ratio between the agent-defined physical measure and the risk-neutral measure, eliminating the need for utility maximization. The agent only needs to specify her choice of the physical measure, and we demonstrate both frequentist and Bayesian approaches for this selection. Utility maximization can be seen as a specific method for choosing the physical measure. The resulting likelihood ratio is log-utility optimal with respect to all benchmarks, aligning our approach with finding the growth optimal portfolio described in the literature. Notably, the expected log return corresponds to the relative entropy between the physical and risk-neutral measures, establishing a fundamental link to information theory. As a proof of concept, we explore previously unexplored territory in portfolio optimization, specifically addressing perceived mean reversion in specific driftless markets, such as foreign exchange (FX) markets.

Valuation and hedging of cryptocurrency inverse options

Currently, the most liquidly traded options on the crypto underlying are the so-called inverse options. An inverse option contract is quoted and traded in the units of the underlying cryptocurrency. The main economic reason for the popularity of inverse contracts in the crypto exchanges (such as Deribit) is that inverse contracts enable traders to operate without maintaining fiat cash accounts. For the theoretical part, we show that inverse options are just regular vanilla options considered under the martingale measure using the forward of the underlying as the numéraire. This measure requires an adjustment to option delta. For the empirical part, we use Deribit options data of past five years to backtest delta-hedged option strategies. We introduce USD and Coin accounting of trading Profit&Loss (P&L) which is important for designing strategies in crypto options. We show empirically that USD and Coin accounting rules are equivalent when performance is measured is Coin and USD units, respectively. We establish that the risk-premia observed in options on Deribit is negative and significant so that strategies selling volatility are expected to generate positive risk-adjusted performance in the long-term.

Deep learning for quadratic hedging in incomplete jump market

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

Pricing vulnerable options under cross-asset markov-modulated jump-diffusion dynamics

In this study, the dynamics of the underlying asset price and the counterparty's asset price are governed by a cross-asset Markov-modulated jump-diffusion model that captures the time-inhomogeneity and systematic cojumps. Additionally, the forward interest rate processes are driven by a Markov-modulated Heath–Jarrow–Morton model to depict stochastic volatilities. Under an incomplete-market setting, we apply the Markov-modulated Esscher transform technique to determine a risk-neutral martingale measure. After determining the Markov-modulated Esscher parameters, we obtain an integral expression on the prices of vulnerable European-style Black–Scholes options. The numerical illustrations indicate that the findings are consistent with Klein (1996) and contribute to the extant literature on cojumping impacts on vulnerable option prices.

Risk management under weighted limited expected loss

We present and solve an optimal asset allocation problem under a weighted limited expected loss (WLEL) constraint. This formulation encompasses the risk management problem with a limited expected loss (LEL) constraint as a specialized instance and offers a pertinent internal risk management instrument for firms. We observe that a WLEL constraint makes the optimizing investor pursue less volatile payoffs than the unconstrained Merton solution. Compared to the LEL-constrained problem with the same weighted default threshold, the WLEL optimal terminal wealth displays a less dispersed distribution with a smaller variance, suggesting a more secure risk management framework. Conducting a comprehensive equilibrium analysis in the presence of a WLEL risk manager, we validate the relatively conservative investment approach undertaken by the WLEL manager. Subsequently, we expand our findings to encompass broader incomplete market settings, wherein the uniqueness of the equivalent local martingale measure is not assured.

Asset pricing and hedging in financial markets with fixed and proportional transaction costs

We establish the asset pricing and hedging principle in a financial market model, which is a specific case of the von Neumann-Gale dynamical system, with both fixed and proportional transaction costs and trading constraints. The main results are hedging criteria stated in terms of consistent valuation systems, generalizing the notion of an equivalent martingale measure.

A Discrete Risk-Theory Approach to Manage Equity-Linked Policies in an Incomplete Market

We construct a model where, at each time instance, risky securities can only take a limited number of values and the equity-linked policy sold by the insurer to policyholders pays benefits linked to these securities. Since the number of states in the model exceeds the number of securities in the (incomplete) market, the martingale measure is not unique, posing a problem in pricing insurance instruments. In this framework, we consider how a super-replicating strategy violates the assumption of absence of arbitrage, yet simultaneously allows the insurance company to fully hedge against financial risk. Since the super-replicating strategy, when considered alone, would be too costly for any rational insured person, through the definition of the safety loading, we demonstrate how the insurer can still hedge against financial risk, albeit at the expense of increasing its exposure to demographic risk. This approach does not aim to show how the pricing of the index-linked policy can actually be performed but rather highlights how risk theory-based approaches (via the definition of the profit and loss random variable) enable the management of the trade-off between financial risk and demographic risk.