# Abstracts

**Option pricing and smile asymptotics in a multiscaling stochastic volatility model**

*Jacopo Corbetta (Università degli studi - Milano Bicocca, Italy)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

We consider a stochastic process for the log-price of a financial asset, recently introduced in [Andreoli et al., Adv. in Appl. Probab. 44 (2012), 1018-1051], which can be defined equivalently as a time-changed Brownian motion or as a stochastic volatility model. This process depends on 3 real parameters $(V, \lambda, D)$ with a precise economical meaning: $V$ is proportional to the average spot volatility, $\lambda$ is the intensity of a Poisson process which models shock times in the market, and $D$ describes the non-linear response of the market to shocks (after each shock time, the volatility increases as $t^{2D} >> t$).

Despite the few parameters, this model was shown to capture in a quantitative fashion some relevant stylized facts of financial series, such as the change in the log-return distribution from power-law tails (small time) to Gaussian (large time), the slow decay in the volatility autocorrelation, and the so-called multiscaling of moments. Here we continue the analysis of this model, with a focus on option pricing.

Our main theoretical result is a large deviations principle for the log-price in the small-time regime, with explicit (and non-trivial) rate and rate functions, based on a related large deviations principle for suitable functionals of a point process, which is of independent interest. This yields asymptotic formulas for the price of European options and the related implied volatility. In particular, despite the price having continuous paths, as the maturity $t$ vanishes the implied volatility of out-of-the-money options explodes as $C [ |x| / (t \sqrt{log(1/t)}) ]^{a}$, where $x$ is the log-moneyness and $C \in (0,\infty)$, $a \in (0,1)$ are constants that are functions of the model parameters.

We also enrich the model introducing jumps in the log-price, using the same Poisson process that drives the evolution of the volatility. This accounts for the so-called leverage effect and produces a skew in the log-return distribution. A Hull-White formula for the price of European options holds also in this correlated framework, averaging the usual Black-Scholes formula with a random volatility and a random initial price. This yields a fast Monte Carlo evaluation of option prices and allows for a numerical calibration of the model on real data, which shows a good agreement.

**Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives**

*Kristina R. Dahl (University of Oslo, Norway)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

In this paper we consider a stochastic optimal control problem where the dynamics of the state process, $X(t)$, is a controlled stochastic differential equation with jumps, delay and noisy memory. By this we mean that the dynamics of $X(t)$ depend on $\int_{t-\delta}^t X(s) dB(s)$ (where $B(t)$ is a Brownian motion). Hence, the dependence is noisy because of the Brownian motion, and it involves memory due to the influence from the previous values of the state process. We derive necessary and sufficient maximum principles for this stochastic control problem in two different ways, resulting in two sets of maximum principles. The first set of maximum principles is derived using Malliavin calculus techniques, while the second set comes from reduction to a discrete delay optimal control problem, and application of previously known results by Øksendal, Sulem and Zhang. Furthermore, we use these maximum principles to derive a method for solving noisy memory BSDEs. Finally, we present an example.

**A new hybrid Monte Carlo/Finite Difference method to obtain counterparty exposure profiles and sensitivities**

*Kees De Graaf (University of Amsterdam, The Netherlands)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

As a consequence of the recent financial crisis, quantities like Expected Exposure (EE), the expected amount of money that may be lost in the case of a counterparty default, and Potential Future Exposure (PFE) which measures the loss given a fixed confidence interval are now very important. Both EE and PFE can be deduced from the distribution of the option price at future time points. In this research, we show that this distribution can be obtained efficiently for a broad class of payoffs and stochastic models, by combining two popular methods in finance, namely: the Monte Carlo method for the generation of future scenarios for the risk factors and the finite difference method for solving the option pricing partial differential equation (PDE). Hence this novel method is named the Finite Difference Monte Carlo method (FDMC). An important feature of the method is that it can estimate sensitivities of EE to market risk factors highly efficient.

To compute the EE of an option, we first need scenarios of the underlying that can be generated by a Monte Carlo simulation. At any time point, during the life of the option, an option value for every path is needed. From these prices a distribution can be computed and the EE results as the mean, whereas the PFE can be extracted as a quantile. This calls for an efficient pricing technique in particular when exotic options and nonstandard dynamics such as the Heston stochastic volatility model are in scope. In the FDMC method, the option prices are obtained by solving the PDE by the finite difference method. It fully benefits from the advantages of the individual methods, from the Monte Carlo method, the scenario generation is used, while the finite difference method gives us a grid of option prices. The generated scenarios are interpolated on the solution grid to obtain a distribution of the price.

The validity of the FDMC method is done by a comparative study. This study is presented in a forthcoming paper together with Q. Feng and C.W. Oosterlee where also SGBM is tested. The FDMC method is applied to Bermudan put options under the Heston dynamics and the results are validated by comparison with the semi-analytic COS method.

In modern risk management, CVA needs to be hedged, therefore the sensitivities play an important role. The bump and revalue approach is computationally unattractive. By combining the FDMC method with the pathwise Monte Carlo method, these sensitivities can be obtained fast and accurate.

**Markovian random walks to approximate Janssen's ALM risk model**

*Sarah Dendievel (Université libre de Bruxelles, Belgium)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

In Janssen's ALM risk model, the asset $A$ and the liability $B$ of a company are modeled by the stochastic differential equations $dA(t)= A(t)\mu_A dt + A(t)\sigma_A dZ_A(t)$ and $dB(t) = B(t)\mu_B dt + B(t) \sigma_B dZ_B(t)$, where $Z_A(t)$ and $Z_B(t)$ are standard Brownian motions. Recently, it has been shown by Latouche and Nguyen that Brownian motion can be approximated by a Markovian random walk $\{(X(t),\varphi(t)):t\in\mathbb{R}^{+}\}$, i.e. a two dimensional stochastic process where the phase $\varphi(t)$ is the state at time $t$ of a Markov process which controls the level $X(t)$. The level takes continuous values and varies linearly with a rate depending on the phase. Markovian random walks are widely used in applied probability (telecommunication modeling, risk process,...) because of their tractability and computational efficiency.

In the literature, time-dependent distributions of the level have been studied using Laplace transforms: Ahn and Ramaswami derive time-dependent distributions of a Markovian fluid queue (i.e. random walks with a reflecting boundary at $0$) for the levels $x>0$ in terms of the transform matrix of the busy period duration, i.e. the matrix for which the $(i,j)$-th component is $\mathbb{E}[\exp(-s\tau)1_{\{\varphi(t)=j\}}|X(0)=0,\varphi(0)=i]$, with $\tau=\inf\{t>0:X(t)=0\}$, for $i\in \mathcal{S}_{+}$ and $j\in \mathcal{S}_{-}$.

Here, we use arguments based on the Erlangization method, suggested by Asmussen et al. in a fluid queue context related to ruin theory problems, and so avoid Laplace transform calculations. Furthermore, this method is practical for the computation of joint probabilities thanks to the memoryless property of the exponential distribution.

In this talk, we present the Markovian random walk approximation for the process $\alpha(t)=\ln(A(t)/B(t))$, we perform numerical approximations for $R(u,\theta)=\mathbb{P}[\tau_0>\theta|\alpha(0)=u]$, the non-ruin probability before a given time $\theta$, with $\tau_0=\inf\{t>0:\alpha(t)<0\}$; we provide numerical illustrations for ruin indicators such as $I(u,x,[0,\theta])=\mathbb{P}[\max_{0<t<\theta} \alpha(t)>x|\alpha(0)=u]$; and we show how phase-type models can lead to more elaborate analysis of the process $\alpha(t)$.

**Continuously-generated Jump Processes: A framework for efficient pricing with jumps and stochastic volatility**

*David Derrett (University of Cambridge, UK)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

We construct a flexible and numerically tractable class of asset models by firstly choosing a bivariate diffusion process $(X,Y)$, and then defining the price of the asset at time $t$ to be the value of $Y$ when $X$ first exceeds $t$. Such price processes will typically have jumps; conventional pricing methodologies would try to solve a PIDE, which can be numerically problematic, but using the fact that the pricing problem is embedded in a two-dimensional diffusion, we are able to exploit efficient methods for two-dimensional diffusion equations to find prices. Models with time dependence (that is, where the bivariate diffusion is $X$-dependent) are no more difficult in this approach.

Pricing a European option for a model in this class consists of solving a second order elliptic PDE. This problem is amenable to highly optimized and robust numerical PDE solving techniques such as adaptive meshing, solution error estimates and the finite element approach.

Models in this class range from the most parsimonious, with few parameters, to those which can match the observed term structure of implied volatility. This allows flexibility. We construct an example model which accounts for so-called volatility events, caused by the scheduled release of pertinent information, such as unemployment figures, inflation rates and economic growth rates.

Finally, we discuss the computation of parameter sensitivities and calibration of models in this class.

**Revisiting Value-at-Risk backtesting using Bayesian statistics**

*Alexandra Dias (University of Leicester, UK)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

Value-at-Risk (VaR) models are an essential tool for risk management in the banking industry. We use Bayesian statistics methods to evaluate the performance of VaR models. We compare the results produced by Bayesian methods with the results given by classical methods. We find that the posterior probability density function for the probability of exceeding VaR is skewed to the right. The probability of observing a loss larger than VaR is larger than 50\%. Regulators advocate conservatism in estimating VaR but VaR models are producing VaR estimates which are biased towards optimism. Classical backtesting methods presently used don’t reveal this bias but Bayesian methods can do it.

**Diffusion Approximation of Bivariate Dynamic Contagion Processes and Applications in Filtering**

*Xin Dong (Imperial College London, UK)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

In this project, we derive the diffusion approximation of the Bivariate Dynamic Contagion Processes (BDCP). The BDCP is a broad class of bivariate point processes including shot-noise Cox processes and Hawkes processes and it can be used in modelling high frequency events under the impact from both an external factor and an internal factor with contagion effects. We show that the BDCP converges weakly to a bivariate Ornstein-Uhlenbeck diffusion process based on the martingale central limit theorem. With the limiting diffusion system, we provide an approximation solution to filtering problems with point process observations. We apply the result on some problems in insurance.

**Double reflected BSDEs with jumps and generalized Dynkin games**

*Roxana Dumitrescu (Université Paris Dauphine, France)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

We study DBBSDEs with jumps and RCLL barriers, and their links with generalized Dynkin games. We provide existence and uniqueness results and prove that for any Lipschitz driver, the solution of the DBBSDE coincides with the value function of a game problem, which can be seen as a generalization of the classical Dynkin problem to the case of $g$-conditional expectations. Using this characterization, we prove some new results on DBBSDEs with jumps, such as comparison theorems and a priori estimates. We then study DBBSDEs with jumps and RCLL obstacles in the Markovian case and their links with parabolic partial integro-differential variational inequalities (PIDVI) with two obstacles.

**Modeling Sovereign Credit Risk under Partial-information**

*Zehra Eksi-Altay (WU Vienna, Austria)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

In this study we model the dynamics of sovereign credit spreads. Specifically, we interpret credit spreads as the noisy observations of some state process which represents the financial strength of the sovereigns under consideration. This naturally gives rise to a partial-information setting in which, the default history of the sovereigns constitutes an additional piece of information. In order to incorporate that information in the analysis, we extend the estimation methodology given in Elliot (1993) and obtain an EM algorithm for the setting where the state variable follows a Markov chain observed via diffusive and point processes information. We test the speed, efficiency and accuracy of the algorithm by running a simulation analysis.

**Ergodic BSDEs with jumps and their applications**

*Victor Fedyashov (Oxford, UK)*

Wednesday June 4, 11:00-11:30 | session P3 | Poster session | room lobby

In this paper we look at ergodic BSDEs in the case where the forward dynamics are given by a solution to a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck SDE with Lévy noise, taking values in a separable Hilbert space. We establish the existence of a unique bounded solution to an infinite horizon discounted BSDE. We then use the vanishing discount approach together with coupling techniques to obtain a Markovian solution to the EBSDE. We also prove uniqueness under certain growth conditions. Applications are then given, in particular to risk-averse ergodic optimal control and power plant evaluation under uncertainty.