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arXiv2020

Polynomial term structure models

In this article, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipovic's maximal degree theorem for exponential polynomial models, as well as an explicit characterisation of the set of feasible parameters in the case when the factor process is bounded. Extensions to time-inhomogeneous and multi-factor polynomial models are also considered.

q-fin.MF↗
arXiv2016

Pathwise super-replication via Vovk's outer measure

Since Hobson's seminal paper [D. Hobson: Robust hedging of the lookback option. In: Finance Stoch. (1998)] the connection between model-independent pricing and the Skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing-hedging duality for financial derivatives which are susceptible to the Skorokhod approach. Using Vovk's approach to mathematical finance we derive a model-independent super-replication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.

q-fin.MF↗
arXiv2015

Dirac Processes and Default Risk

We introduce Dirac processes, using Dirac delta functions, for short-rate-type pricing of financial derivatives. Dirac processes add spikes to the existing building blocks of diffusions and jumps. Dirac processes are Generalized Processes, which have not been used directly before because the dollar value of non-Real numbers is meaningless. However, short-rate pricing is based on integrals so Dirac processes are natural. This integration directly implies that jumps are redundant whilst Dirac processes expand expressivity of short-rate approaches. Practically, we demonstrate that Dirac processes enable high implied volatility for CDS swaptions that has been otherwise problematic in hazard rate setups.

q-fin.PR↗
arXiv2015

A Posteriori Error Estimator for a Front-Fixing Finite Difference Scheme for American Options

For the numerical solution of the American option valuation problem, we provide a script written in MATLAB implementing an explicit finite difference scheme. Our main contribute is the definition of a posteriori error estimator for the American options pricing which is based on Richardson's extrapolation theory. This error estimator allows us to find a suitable grid where the computed solution, both the option price field variable and the free boundary position, verify a prefixed error tolerance.

q-fin.MF↗
arXiv2015

Optimal Multiple Trading Times Under the Exponential OU Model with Transaction Costs

This paper studies the timing of trades under mean-reverting price dynamics subject to fixed transaction costs. We solve an optimal double stopping problem to determine the optimal times to enter and subsequently exit the market, when prices are driven by an exponential Ornstein-Uhlenbeck process. In addition, we analyze a related optimal switching problem that involves an infinite sequence of trades, and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Among our results, we find that the investor generally enters when the price is low, but may find it optimal to wait if the current price is sufficiently close to zero. In other words, the continuation (waiting) region for entry is disconnected. Numerical results are provided to illustrate the dependence of timing strategies on model parameters and transaction costs.

q-fin.TR↗
arXiv2015

Population viewpoint on Hawkes processes

This paper focuses on a class of linear Hawkes processes with general immigrants. These are counting processes with shot noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of age pyramid which evolves according to immigration and births. The virtue if this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of linear Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

math.PR↗
arXiv2015

Network Structure and Counterparty Credit Risk

In this paper we offer a novel type of network model which can capture the precise structure of a financial market based, for example, on empirical findings. With the attached stochastic framework it is further possible to study how an arbitrary network structure and its expected counterparty credit risk are analytically related to each other. This allows us, for the first time, to model the precise structure of an arbitrary financial market and to derive the corresponding expected exposure in a closed-form expression. It further enables us to draw implications for the study of systemic risk. We apply the powerful theory of characteristic functions and Hilbert transforms. The latter concept is used to express the characteristic function (c.f.) of the random variable (r.v.) $\max(Y, 0)$ in terms of the c.f. of the r.v. $Y$. The present paper applies this concept for the first time in mathematical finance. We then characterise Eulerian digraphs as distinguished exposure structures and show that considering the precise network structures is crucial for the study of systemic risk. The introduced network model is then applied to study the features of an over-the-counter and a centrally cleared market. We also give a more general answer to the question of whether it is more advantageous for the overall counterparty credit risk to clear via a central counterparty or classically bilateral between the two involved counterparties. We then show that the exact market structure is a crucial factor in answering the raised question.

q-fin.MF↗
arXiv2015

Pricing and hedging game options in currency models with proportional transaction costs

The pricing, hedging, optimal exercise and optimal cancellation of game or Israeli options are considered in a multi-currency model with proportional transaction costs. Efficient constructions for optimal hedging, cancellation and exercise strategies are presented, together with numerical examples, as well as probabilistic dual representations for the bid and ask price of a game option.

q-fin.MF↗
arXiv2015

A general Doob-Meyer-Mertens decomposition for $g$-supermartingale systems

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) version for right-continuous $g$-supermartingales. As examples of application, we prove an optional decomposition theorem for $g$-supermartingale systems, and also obtain a general version of the well-known dual formation for BSDEs with constraint on the gains-process, using very simple arguments.

math.PR↗
arXiv2015

On statistical indistinguishability of complete and incomplete discrete time market models

We investigate the possibility of statistical evaluation of the market completeness for discrete time stock market models. It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into a incomplete one. The paper shows that market incompleteness is also non-robust. We show that, for any incomplete market from a wide class of discrete time models, there exists a complete market model with arbitrarily close stock prices. This means that incomplete markets are indistinguishable from the complete markets in the terms of the market statistics.

q-fin.MF↗
arXiv2015

Non-Arbitrage Under Additional Information for Thin Semimartingale Models

This paper completes the two studies undertaken in \cite{aksamit/choulli/deng/jeanblanc2} and \cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) when the stock price processes are quasi-left-continuous (do not jump on predictable stopping times). Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and the progressive enlargement with a random time. For this flow of information, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. This also generalizes \cite{choulli/deng} to the case when the jump times are not ordered in anyway. Furthermore, for the current context, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration.

q-fin.MF↗
arXiv2015

Structural default model with mutual obligations

This paper considers mutual obligations in the interconnected bank system and analyzes their influence on joint and marginal survival probabilities as well as CDS and FTD prices for the individual banks. To make the role of mutual obligations more transparent, a simple structural default model with banks' assets driven by correlated multidimensional Brownian motion with drift is considered. This model enables a closed form representation for many quantities of interest, at least in a 2D case, to be obtained, and moreover, model calibration is provided. Finally, we demonstrate that mutual obligations have to be taken into account in order to get correct values for model parameters.

q-fin.PR↗
arXiv2016

Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion

While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Ito process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Ito process in a one-sided manner without reflection.

q-fin.MF↗
arXiv2015

An explicit solution for optimal investment in Heston model

In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model.

q-fin.MF↗
arXiv2015

Small-time expansions for state-dependent local jump-diffusion models with infinite jump activity

In this article, we consider a Markov process X, starting from x and solving a stochastic differential equation, which is driven by a Brownian motion and an independent pure jump component exhibiting state-dependent jump intensity and infinite jump activity. A second order expansion is derived for the tail probability P[X(t)>x+y] in small time t, for y>0. As an application of this expansion and a suitable change of the underlying probability measure, a second order expansion, near expiration, for out-of-the-money European call option prices is obtained when the underlying stock price is modeled as the exponential of the jump-diffusion process X under the risk-neutral probability measure.

math.PR↗
arXiv2015

Dynamics of Order Positions and Related Queues in a Limit Order Book

Order positions are key variables in algorithmic trading. This paper studies the limiting behavior of order positions and related queues in a limit order book. In addition to the fluid and diffusion limits for the processes, fluctuations of order positions and related queues around their fluid limits are analyzed. As a corollary, explicit analytical expressions for various quantities of interests in a limit order book are derived.

q-fin.TR↗
arXiv2016

Small-time asymptotics for Gaussian self-similar stochastic volatility models

We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter $H$, its first Karhunen-Loeve eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for $H$ result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1/2 and 3/2, and the estimator for $H$ sees an affine adjustment, while remaining model-free.

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