Quantitative Finance

We consider explicit approximations for European put option prices within the Stochastic Verhulst model with time-dependent parameters, where the volatility process follows the dynamics d V t = κ t ( θ t − V t ) V t dt+ λ t V t d B t . Our methodology involves writing the put option price as an expectation of a Black-Scholes formula, reparameterising the volatility process and then performing a number of expansions. The difficulties faced are computing a number of expectations induced by the expansion procedure explicitly. We do this by appealing to techniques from Malliavin calculus. Moreover, we deduce that our methodology extends to models with more general drift and diffusion coefficients for the volatility process. We obtain the explicit representation of the form of the error generated by the expansion procedure, and we provide sufficient ingredients in order to obtain a meaningful bound. Under the assumption of piecewise-constant parameters, our approximation formulas become closed-form, and moreover we are able to establish a fast calibration scheme. Furthermore, we perform a numerical sensitivity analysis to investigate the quality of our approximation formula in the Stochastic Verhulst model, and show that the errors are well within the acceptable range for application purposes.

This paper considers an initial market model, specified by its underlying assets S and its flow of information F , and an arbitrary random time τ which might not be an F -stopping time. As the death time and the default time (that τ might represent) can be seen when they occur only, the progressive enlargement of F with τ sounds tailor-fit for modelling the new flow of information G that incorporates both F and τ . In this setting of informational market, the first principal goal resides in describing as explicitly as possible the set of all deflators for ( S τ ,G) , while the second principal goal lies in addressing the No-Free-Lunch-with-Vanishing-Risk concept (NFLVR hereafter) for ( S τ ,G) . Besides this direct application to NFLVR, the set of all deflators constitutes the dual set of all "admissible" wealth processes for the stopped model ( S τ ,G) , and hence it is vital in many hedging and pricing related optimization problems. Thanks to the results of Choulli et al. [7], on martingales classification and representation for progressive enlarged filtration, our two main goals are fully achieved in different versions, when the survival probability never vanishes. The results are illustrated on the two particular cases when (S,F) follows the jump-diffusion model and the discrete-time model.

We show that in a financial market given by semimartingales an arbitrage opportunity, provided it exists, can only be exploited through short selling. This finding provides a theoretical basis for differences in regulation for financial services providers that are allowed to go short and those without short sales. The privilege to be allowed to short sell gives access to potential arbitrage opportunities, which creates by design a bankruptcy risk.

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous Lévy processes. With a view of option pricing, we provide a systematic analysis of the existence of equivalent martingale measures, under which the model remains analytically tractable. This includes the existence of Esscher martingale measures and martingale measures having minimal distance to the physical probability measure. Moreover, we provide pricing formulae for European call options and perform a case study.

In this paper we propose a general framework for modeling an insurance liability cash flow in continuous time, by generalizing the reduced-form framework for credit risk and life insurance. In particular, we assume a nontrivial dependence structure between the reference filtration and the insurance internal filtration. We apply these results for pricing non-life insurance liabilities in hybrid financial and insurance markets, while taking into account the role of inflation under the benchmark approach. This framework offers at the same time a general and flexible structure, and explicit and treatable pricing formula.

In this paper we study utility maximization with proportional transaction costs. Assuming extended weak convergence of the underlying processes we prove the convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended weak convergence theory developed in [1] and the Meyer--Zheng topology introduced in [24].

We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models.

This paper considers a time-inconsistent stopping problem in which the inconsistency arises from non-constant time preference rates. We show that the smooth pasting principle, the main approach that has been used to construct explicit solutions for conventional time-consistent optimal stopping problems, may fail under time-inconsistency. Specifically, we prove that the smooth pasting principle solves a time-inconsistent problem within the intra-personal game theoretic framework if and only if a certain inequality on the model primitives is satisfied. We show that the violation of this inequality can happen even for very simple non-exponential discount functions. Moreover, we demonstrate that the stopping problem does not admit any intra-personal equilibrium whenever the smooth pasting principle fails. The "negative" results in this paper caution blindly extending the classical approaches for time-consistent stopping problems to their time-inconsistent counterparts.

We use the theory of cooperative games for the design of fair insurance contracts. An insurance contract needs to specify the premium to be paid and a possible participation in the benefit (or surplus) of the company. It results from the analysis that when a contract is exposed to the default risk of the insurance company, ex-ante equilibrium considerations require a certain participation in the benefit of the company to be specified in the contracts. The fair benefit participation of agents appears as an outcome of a game involving the residual risks induced by the default possibility and using fuzzy coalitions.