Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

We propose a simple model of inter-bank borrowing and lending where the evolution of the logmonetary reserves of N banks is described by a system of diffusion processes coupled through their drifts in such a way that stability of the system depends on the rate of inter-bank borrowing and lending. Systemic risk is characterized by a large number of banks reaching a default threshold by a given time horizon. Our model incorporates a game feature where each bank controls its rate of borrowing/lending to a central bank. The optimization reflects the desire of each bank to borrow from the central bank when its monetary reserve falls below a critical level or lend if it rises above this critical level which is chosen here as the average monetary reserve. Borrowing from or lending to the central bank is also subject to a quadratic cost at a rate which can be fixed by the regulator. We solve explicitly for Nash equilibria with finitely many players, and we show that in this model the central bank acts as a clearing house, adding liquidity to the system without affecting its systemic risk. We also study the corresponding Mean Field Game in the limit of large number of banks in the presence of a common noise.

We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of high-frequency S&P 500 index data con rms that volatility reverts slowly to its mean in comparison to the tick-by-tick uctuations of the index value, but it is fast mean-reverting when looked at over the time scale of a derivative contract many months. This motivates an asymptotic analysis of the partial di erential equation satis ed by derivative prices, utilizing the distinction between these time scales.The analysis yields pricing and implied volatility formulas, and the latter is used to " t the smile" from European index option prices. The theory identi es the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log-moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure, and the data produces estimates of the three important parameters which are found to be stable within periods where the underlying volatility is close to being stationary. These segments of stationarity are identi ed using a wavelet-based tool.The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk can be roughly estimated, but are not needed for the asymptotic pricing formulas for European derivatives. The extension to American and path-dependent contingent claims is the subject of future work.

Abstract. After the celebrated Black-Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [2,3] that, in the presence of a separation of time scales, between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of partial differential equations this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff function inherent to option pricing. We present the case of call options for which the payoff function forms an angle at the strike price. This case is important since these are the typical instruments used in the calibration of pricing models. We establish the pointwise accuracy of the corrected Black-Scholes price by using an appropriate payoff regularization which is removed simultaneously as the asymptotics is performed.1. Introduction. Stochastic volatility models in financial mathematics can be thought of as a Brownian-type particle (the stock price) moving in an environment where the diffusion coefficient is randomly fluctuating in time according to some ergodic (mean-reverting) diffusion process. We then have two Brownian motions, one driving the motion of the particle and the other driving the fluctuations of the medium. In the context of Physics there is no natural correlation between these two Brownian motions since they do not "live" in the same space. In the context of Finance they jointly define the dynamics of the stock price under its physical probability measure or an equivalent risk-neutral martingale measure. Correlation between them is perfectly natural. There are economic arguments for a negative correlation or leverage effect between stock price and volatility shocks, and from common experience and empirical studies, asset prices tend to go down when volatility goes up.The diffusion equation appears as a contingent claim pricing equation, its terminal condition being the payoff of the claim. We refer to [5] or [6] for surveys on stochastic volatility. When volatility is fast mean-reverting, on a time-scale smaller than typical maturities, one can perform a singular perturbation analysis of the pricing PDE. As we have shown in [2], this expansion reveals a first correction made of two terms: one is directly associated with the market price of volatility risk and the other is proportional to the correlation coefficient between the two Brownian motions involved. We refer to [2] for a detailed account of evidence of a fast scale in volatility and the use of this asymptotics to parametrize the evolution ...

We specify a general methodological framework for systemic risk measures via multidimensional acceptance sets and aggregation functions. Existing systemic risk measures can usually be interpreted as the minimal amount of cash needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions before aggregating the individual risks. An important feature of our approach is the possibility of allocating cash according to the future state of the system (scenario-dependent allocation). We also provide conditions that ensure monotonicity, convexity, or quasiconvexity of our systemic risk measures. K E Y W O R D Sacceptance set, aggregation, systemic risk, risk measures INTRODUCTIONThe financial crisis has dramatically demonstrated that traditional risk management strategies of financial systems, which predominantly focus on the solvency of individual institutions as if they were in isolation, insufficiently capture the perilous systemic risk that is generated by the interconnectedness of the system entities and the corresponding contagion effects. This has brought awareness of the urgent need for novel approaches that capture systemic riskiness. A large part of the current literature on systemic financial risk is concerned with the modeling structure of financial networks, the analysis of the contagion, and the spread of a potential exogenous (or even endogenous) shock into the system. For a given financial (possibly random) network and a given random shock, one then Mathematical Finance. 2019;29:329-367.wileyonlinelibrary.com/journal/mafi