Ruin probability at a given time for a model with liabilities of the fractional Brownian motion type: A partial differential equation approach

In this paper we study the ruin probability at a given time for liabilities of diffusion type, driven by fractional Brownian motion with Hurst exponent in the range (0.5, 1). Using fractional Itô calculus we derive a partial differential equation the solution of which provides the ruin probability. An analytical solution is found for this equation and the results obtained by this approach are compared with the results obtained by Monte-Carlo simulation.     

Premium and reinsurance control of an ordinary insurance system with liabilities driven by a fractional Brownian motion

This paper investigates the problem of premium and reinsurance control of an ordinary insurance system when liabilities are driven by a fractional Brownian motion. The reserve equation is considered using two alternative routes: the first with no reinsurance option, and the second with some controllable proportional reinsurance coverage. Recent results from the theory of fractional linear-quadratic control (fractional calculus) are discussed, partially extended and utilized to derive compact analytical formulae for the optimal functionals of the safety loading (consequently for the respective premium rate), and the volume of the retained risk (or equivalently, for the proportion of the reinsurance coverage).     

A new closed-form formula for pricing European options under a skew Brownian motion

In this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.     

Option pricing under short-lived arbitrage: theory and tests

Models in financial economics derived from no-arbitrage assumptions have found great favour among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein–Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that idiosyncratic factors significantly effect arbitrage model parameters.     

Pricing under rough volatility

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyse in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.     

Non-ideal Brownian motion,generalized Langevin Equation and its application to the security market

Brownian motion has been extensively applied in the field of mathematical finance in modeling the stochastic processes of returns on securities. In this paper basic and generalized Langevin Equations with memory are used to augment Brownian motion to capture the well stylized facts of the financial market that frictions and imperfect information exist. The operator method of Fourier-Laplace transform with an appropriate kernel (influence function) is used to circumvent the difficulty associated with solving a time dependent nonlinear differential Equation, and a practical computational method is proposed.From the Langevin Equation, autocorrelation of the return process and the deviation of the return distribution from an ideal Brownian motion are extracted. It is also proven that the time-dependent differential Equation has a unique solution and that it is much more generalized than a martingale Brownian motion functional.     

Calibration of a nonlinear feedback option pricing model

We consider the option pricing model proposed by Mancino and Ogawa, where the implementation of dynamic hedging strategies has a feedback impact on the price process of the underlying asset. We present numerical results showing that the smile and skewness patterns of implied volatility can actually be reproduced as a consequence of dynamical hedging. The simulations are performed using a suitable semi-implicit finite difference method. Moreover, we perform a calibration of the nonlinear model to market data and we compare it with more popular models, such as the Black–Scholes formula, the Jump-Diffusion model and Heston's model. In judging the alternative models, we consider the following issues: (i) the consistency of the implied structural parameters with the times-series data; (ii) out-of-sample pricing; and (iii) parameter uniformity across different moneyness and maturity classes. Overall, nonlinear feedback due to hedging strategies can, at least in part, contribute to the explanation from a theoretical and quantitative point of view of the strong pricing biases of the Black–Scholes formula, although stochastic volatility effects are more important in this regard.     

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.     

Financial applications of semidefinite programming: a review and call for interdisciplinary research

Optimisation problems in finance commonly have non-linear constraints for which previous solutions have required unrealistic assumptions. However, many of these can be efficiently solved as semidefinite programming (SDP) problems, which have less restrictive assumptions. Through review of the literature that uses SDP in finance, two major research streams are identified: portfolio optimisation and option pricing. Nevertheless, many finance researchers are unaware of SDP. One possible reason is that this research is often published in non-finance journals. This paper aims to better integrate the SDP research to promote wider use of current findings and further interdisciplinary research, particularly in environmental finance.     

Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem

In this paper, we study a class of quadratic backward stochastic differential equations (BSDEs), which arises naturally in the utility maximization problem with portfolio constraints. We first establish the existence and uniqueness of solutions for such BSDEs and then give applications to the utility maximization problem. Three cases of utility functions, the exponential, power, and logarithmic ones, are discussed. This study is part of my PhD thesis supervised by Professor Ying Hu and defended at the University of Rennes 1 (in France) in October 2007.     

The role of stochastic volatility and return jumps: reproducing volatility and higher moments in the KOSPI 200 returns dynamics

This paper investigates the role of stochastic volatility and return jumps in reproducing the volatility dynamics and the shape characteristics of the Korean Composite Stock Price Index (KOSPI) 200 returns distribution. Using efficient method of moments and reprojection analysis, we find that stochastic volatility models, both with and without return jumps, capture return dynamics surprisingly well. The stochastic volatility model without return jumps, however, cannot fully reproduce the conditional kurtosis implied by the data. Return jumps successfully complement this gap. We also find that return jumps are essential in capturing the volatility smirk effects observed in short-term options.

A tale of two regimes: Theory and empirical evidence for a Markov-modulated jump diffusion model of equity returns and derivative pricing implications

We provide closed-form solutions for a continuous time, Markov-modulated jump diffusion model in a general equilibrium framework for options prices under a variety of jump diffusion specifications. We further demonstrate that the two-state model provides the leptokurtic return features, volatility smile, and volatility clustering observed empirically for the Dow Jones Industrial Average (DJIA) and its component stocks. Using 10 years of stock return data, we confirm the existence of jump intensity switching and clustering, illustrate transition probabilities, and verify superior empirical fit over competing Poisson-style models.