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arXiv2016

A BSDE arising in an exponential utility maximization problem in a pure jump market model

We consider the problem of utility maximization with exponential preferences in a market where the traded stock/risky asset price is modelled as a Lévy-driven pure jump process (i.e. the driving Lévy process has no Brownian component). In this setting, we study the terminal utility optimization problem in the presence of a European contingent claim. We consider in detail the BSDE (backward stochastic differential equation) characterising the value function when using an exponential utility function. First we analyse the well-definedness of the generator. This leads to some conditions on the market model related to conditions for the market to admit no free lunches. Then we give bounds on the candidate optimal strategy. Thereafter, we discuss the example of a cross-hedging problem and, under severe assumptions on the structure of the claim, we give explicit solutions. Finally, we establish an explicit solution for a related BSDE with a suitable terminal condition but a simpler generator.

math.PR↗

arXiv2017

Maximizing expected utility in the Arbitrage Pricing Model

We consider an infinite dimensional optimization problem motivated by mathematical economics. Within the celebrated "Arbitrage Pricing Model", we use probabilistic and functional analytic techniques to show the existence of optimal strategies for investors who maximize their expected utility.

q-fin.MF↗

arXiv2017

Liquidity Effects of Trading Frequency

In this article, we present a discrete time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the agents choose to provide liquidity in equilibrium. On the other hand, it also makes markets more fragile, in the sense that the agents choose to provide liquidity in equilibrium only if they are market-neutral (i.e., their beliefs satisfy certain martingale property). Even a very small deviation from market-neutrality may cause the agents to stop providing liquidity, if the trading frequency is sufficiently high, which represents an endogenous liquidity crisis (aka flash crash) in the market. This framework enables us to provide more insight into how such a liquidity crisis unfolds, connecting it to the so-called adverse selection effect.

q-fin.TR↗

arXiv2015

Optimal liquidation of an asset under drift uncertainty

We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift parameter by allowing an arbitrary probability distribution to characterise the uncertainty about the drift parameter. Filtering theory is used to describe the evolution of the posterior beliefs about the drift once the price process is being observed. An optimal stopping time is determined as the first passage time of the posterior mean below a monotone boundary, which can be characterised as the unique solution to a non-linear integral equation. We also study monotonicity properties with respect to the prior distribution and the asset volatility.

q-fin.MF↗

arXiv2017

Correction to Black-Scholes formula due to fractional stochastic volatility

Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a function of maturity to a fractional power.

q-fin.MF↗

arXiv2017

Utility Maximisation for Exponential Levy Models with option and information processes

We consider expected utility maximisation problem for exponential Levy models and HARA utilities in presence of illiquid asset in portfolio. This illiquid asset is modelled by an option of European type on another risky asset which is correlated with the first one. Under some hypothesis on Levy processes, we give the expressions of information processes figured in maximum utility formula. As applications, we consider Black-Scholes models with correlated Brownian Motions, and also Black-Scholes models with jump part represented by Poisson process.

q-fin.MF↗

arXiv2021

Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the 'instantaneous arbitrage capability' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow bundle or of the Dirac Laplacian of the twisted cash flow bundle with the exterior algebra bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenböck formula, the non vanishing of the homology group of the cash flow bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.

q-fin.MF↗

arXiv2015

Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions

We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as asset pricing models in mathematical finance. The generator of this Markov process with killing is a second-order, degenerate, elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the $2α$-power of the distance to the boundary of the half-plane, with $α\in(0,1]$. Our stochastic representation formulas provide the unique solutions to the elliptic boundary value and obstacle problems, when we seek solutions which are suitably smooth up to the boundary portion $Γ_{0}$ contained in the boundary of the upper half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formulas provide the unique solutions which are not guaranteed to be any more than continuous up to the boundary portion $Γ_{0}$.

math.PR↗

arXiv2016

The pricing of contingent claims and optimal positions in asymptotically complete markets

We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semi-martingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to Large Deviations theory, and in particular, the celebrated Gärtner-Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed markets with vanishing risk aversion, the basis risk model with high correlation, models of large markets with vanishing trading restrictions and the Black-Scholes-Merton model with either vanishing default probabilities or vanishing transaction costs. Lastly, we show that the large claim regime could naturally arise in partial equilibrium models.

q-fin.MF↗

arXiv2015

On the no-arbitrage market and continuity in the Hurst parameter

We consider a market with fractional Brownian motion with stochastic integrals generated by the Riemann sums. We found that this market is arbitrage free if admissible strategies that are using observations with an arbitrarily small delay. Moreover, we found that this approach eliminates the discontinuity of the stochastic integrals with respect to the Hurst parameter H at H=1/2.

q-fin.MF↗

arXiv2015

Representation and approximation of ambit fields in Hilbert space

We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fields can be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation. This stochastic partial differential equation is formulated on a suitable Hilbert space of functions on the positive real line with values in the state space of the Hambit field. We provide an explicit construction of such a space. Finally, we apply this interpretation of Hambit fields to develop a finite difference scheme, for which we prove convergence under some Lipschitz conditions.

math.PR↗

arXiv2017

Sticky processes, local and true martingales

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance.

q-fin.MF↗

arXiv2016

Dynamics of multivariate default system in random environment

We consider a multivariate default system where random environmental information is available. We study the dynamics of the system in a general setting and adopt the point of view of change of probability measures. We also make a link with the density approach in the credit risk modelling. In the particular case where no environmental information is concerned, we pay a special attention to the phenomenon of system weakened by failures as in the classical reliability system.

q-fin.RM↗

arXiv2016

Retarded action principle and self-financing portfolio dynamics

We derive a consistent differential representation for the dynamics of a self-financing portfolio for different hedging strategies. In the basis of the derivation there is the so called "retarded action principle", which represents the causality in the evolution of dependent stochastic variables. We demonstrate this principle on example of a vanilla and a storage option.

q-fin.MF↗

arXiv2016

Semi-static completeness and robust pricing by informed investors

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework where a set of non-dominated models is given. The concept of semi-static completeness is introduced: it corresponds to having exact replication by means of semi-static strategies. We show that semi-static completeness is equivalent to an extremality property, and give a characterization of the induced filtration structure. Furthermore, we consider investors with additional information and, for specific types of extra information, we characterize the models that are semi-statically complete for the informed investors. Finally, we provide some examples where robust pricing for informed and uninformed agents can be done over semi-statically complete models.

math.PR↗

arXiv2015

Endogenous Current Coupons

We consider the problem of identifying current coupons for Agency backed To-be-Announced (TBA) Mortgage Backed Securities. In a doubly stochastic factor based model which allows for prepayment intensities to depend upon current and origination mortgage rates, as well as underlying investment factors, we identify the current coupon with solutions to a degenerate elliptic, non-linear fixed point problem. Using Schaefer's theorem we prove existence of current coupons. We also provide an explicit approximation to the fixed point, valid for compact perturbations off a baseline factor-based intensity model. Numerical examples are provided which show the approximation performs remarkably well in estimating the current coupon.

q-fin.PR↗

arXiv2015

An example of short-term relative arbitrage

Long-term relative arbitrage exists in markets where the excess growth rate of the market portfolio is bounded away from zero. Here it is shown that under a time-homogeneity hypothesis this condition will also imply the existence of relative arbitrage over arbitrarily short intervals.

q-fin.MF↗

arXiv2015

On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry

In this paper, we study the multi-asset Black-Scholes model in terms of the importance that the correlation parameter space (equivalent to an $N$ dimensional hypercube) has in the solution of the pricing problem. We show that inside of this hypercube there is a surface, called the Kummer surface $Σ_K$, where the determinant of the correlation matrix $ρ$ is zero, so the usual formula for the propagator of the $N$ asset Black-Scholes equation is no longer valid. Worse than that, in some regions outside this surface, the determinant of $ρ$ becomes negative, so the usual propagator becomes complex and divergent. Thus the option pricing model is not well defined for these regions outside $Σ_K$. On the Kummer surface instead, the rank of the $ρ$ matrix is a variable number. By using the Wei-Norman theorem, we compute the propagator over the variable rank surface $Σ_K$ for the general $N$ asset case. We also study in detail the three assets case and its implied geometry along the Kummer surface.

q-fin.MF↗