Quantitative Finance

In this paper we study the problem of nonlinear pricing of an American option with a right-continuous left-limited (RCLL) payoff process in an incomplete market with default, from the buyer's point of view. We show that the buyer's price process can be represented as the value of a stochastic control/optimal stopping game problem with nonlinear expectations, which corresponds to the maximal subsolution of a constrained reflected Backward Stochastic Differential Equation (BSDE). We then deduce a nonlinear optional decomposition of the buyer's price process. To the best of our knowledge, no dynamic dual representation (resp. no optional decomposition) of the buyer's price process can be found in the literature, even in the case of a linear incomplete market and brownian filtration. Finally, we prove the "infimum" and the "supremum" in the definition of the stochastic game problem can be interchanged. Our method relies on new tools, as simultaneous nonlinear Doob-Meyer decompositions of processes which have a Y ν -submartingale property for each admissible control ν .

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow & Protter for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas & Shreve (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay \theta>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as the delay \theta> 0. This implies that there is no arbitrage in the market in that case. However, when \theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al and the references therein.

We study the origins of the dt − − √ effect in finance and SDE. In particular, we show, in the game-theoretic framework, that market volatility is a consequence of the absence of riskless opportunities for making money and that too high volatility is also incompatible with such opportunities. More precisely, riskless opportunities for making money arise whenever a traded security has fractal dimension below or above that of the Brownian motion and its price is not almost constant and does not become extremely large. This is a simple observation known in the measure-theoretic mathematical finance. At the end of the article we also consider the case of non-zero interest rate. This version of the article was essentially written in March 2005 but remains a working paper.

We analyze the VIX futures market with a focus on the exchange-traded notes written on such contracts, in particular we investigate the VXX notes tracking the short-end part of the futures term structure. Inspired by recent developments in commodity smile modelling, we present a multi-factor stochastic-local volatility model that is able to jointly calibrate plain vanilla options both on VIX futures and VXX notes, thus going beyond the failure of purely stochastic or simply local volatility models. We discuss numerical results on real market data by highlighting the impact of model parameters on implied volatilities.

This paper investigates a hybrid stochastic differential reinsurance and investment game between one reinsurer and two insurers, including a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset. The two insurers, as the followers of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. The competitive relationship between two insurers is modeled by the non-zero-sum game, and their decision making will consider the relative performance measured by the difference in their terminal wealth. We consider wealth processes with delay to characterize the bounded memory feature. This paper aims to find the equilibrium strategy for the reinsurer and insurers by maximizing the expected utility of the reinsurer's terminal wealth with delay and maximizing the expected utility of the combination of insurers' terminal wealth and the relative performance with delay. By using the idea of backward induction and the dynamic programming approach, we derive the equilibrium strategy and value functions explicitly. Then, we provide the corresponding verification theorem. Finally, some numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium strategy. We find the delay factor discourages or stimulates investment depending on the length of delay. Moreover, competitive factors between two insurers make their optimal reinsurance-investment strategy interact, and reduce reinsurance demand and reinsurance premium price.

We derive a closed form solution for an optimal control problem related to an interbank lending schemes subject to terminal probability constraints on the failure of banks which are interconnected through a financial network. The derived solution applies to a real banks network by obtaining a general solution when the aforementioned probability constraints are assumed for all the banks. We also present a direct method to compute the systemic relevance parameter for each bank within the network.

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions.

We propose a multi-factor polynomial framework to model and hedge long-term electricity contracts with delivery period. This framework has several advantages: the computation of forwards, risk premium and correlation between different forwards are fully explicit, and the model can be calibrated to observed electricity forward curves easily and well. Electricity markets suffer from non-storability and poor medium- to long-term liquidity. Therefore, we suggest a rolling hedge which only uses liquid forward contracts and is risk-minimizing in the sense of Föllmer and Schweizer. We calibrate the model to over eight years of German power calendar year forward curves and investigate the quality of the risk-minimizing hedge over various time horizons.

The Black-Scholes-Merton (BSM) theory for price variation has been well established in mathematical financial engineering. However, it has been recognized that long-term outcomes in practice may divert from the Black-Scholes formula, which is the expected value of the stochastic process of price changes. While the expected value is expected for the long-run average of infinite realizations of the same stochastic process, it may give an erroneous picture of nearly every realization when the probability distribution is skewed, as is the case for prices. Here we propose a new formula of the BSM theory, which is based on the median of the stochastic process. This formula makes a more realistic prediction for the long-term outcomes than the current Black-Scholes formula.

Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant elasticity of variance (CEV) option prices in the same manner without resorting to the small-strike volatility smile asymptotics of De Marco et al. [SIAM J. Financ. Math., 2017, 8(1), 709-737]. For the uncorrelated SABR model, the new option pricing method is accurate and arbitrage-free across all strike prices.