http://www.quant-press.com Library of articles about quantitative finance quant-press en http://www.quant-press.com/favicon.jpg http://www.quant-press.com Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the ?rst and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the ?rst order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusion's using probabilistic methods, i.e. the so called principle of not feeling the boundary. Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com By an appropriate change of variables, we prove here that the SVI implied volatility parameterisation proposed in [2] and the large-time asymptotic of the Heston implied volatility derived in [1] do agree algebraically, thus confirming a conjecture proposed by J. Gatheral in [2] as well as proposing a simpler expression for the asymptotic implied volatility under the Heston model. Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com Under the SABR stochastic volatility model, pricing and hedging contracts that are sensitive to forward smile risk (e.g., forward starting options, barrier options) require the joint transition density. In this paper, we address this problem by providing closed-form representations, asymptotically, of the joint transition density. Specifically, we construct an expansion of the joint density through a hierarchy of parabolic equations after applying total volatility-of-volatility scaling and a near-Gaussian coordinate transformation. We then established an existence result to characterize the truncation error and provide explicit joint density formulas for the first three orders. Our approach inherits the same spirit of a small total volatility-of-volatility assumption as in in the original SABR analysis. Our results for the joint transition density serve as a basis for managing forward smile risk. Through numerical experiments, we illustrate the accuracy of our expansion in terms of joint density, marginal density, probability mass and implied volatilities for European call options Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com We consider a stochastic local volatility model with domestic and foreign stochastic interest rates such that the volatilitydecomposes into a deterministic local volatility plus some bias terms. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, we are going to describe in detail the implementation of that model, focusing on the computation of a proper deterministic local volatility. To do so, we choose to generate an implied volatility surface without arbitrage in space and in time by parametrising a mixture of shifted lognormal densities under constraints and we use a Differential Evolution algorithm to calibrate the model's parameters to a finite set of option prices. We will therefore need to devise an evolutionary algorithm that handle constraints in a simple and efficient way. Using some of the improvements made to the DE algorithm combined with simple and robust constraints handling mechanisms we will propose a modified algorithm for solving our optimisation problem under constraints which greatly improves its performances. Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespecified negative power and a deterministic function of time. The empirical work uses a power of -1.5. It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for GM and FORD over the period October 2004 to September 2007. Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com The uncertain volatility model has long ago attracted the attention of practitioners as it provides worst-case pricing scenario for the sell-side. The valuation of a financial derivative based on this model requires solving a fully non-linear PDE. One can rely on finite difference schemes only when the number of variables (that is, underlyings and path-dependent variables) is small - in practice no more than three. In all other cases, numerical valuation seems out of reach. In this paper, we outline two accurate, easy-to-implement Monte-Carlo-like methods which hardly depend on dimensionality. The first method requires a parameterization of the optimal covariance matrix and consists in a series of backward low-dimensional optimizations. The second method relies heavily on a recently established connection between second-order backward stochastic differential equations and non-linear second-order parabolic PDEs. Both methods are illustrated by numerical experiments. Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com We follow a long path for Credit Derivatives and Collateralized Debt Obligations (CDOs) in particular, from the introduction of the Gaussian copula model and the related implied correlations to the introduction of arbitrage-free dynamic loss models capable of calibrating all the tranches for all the maturities at the same time. En passant, we also illustrate the implied copula, a method that can consistently account for CDOs with different attachment and detachment points but not for different maturities. The discussion is abundantly supported by market examples through history. The dangers and critics we present to the use of the Gaussian copula and of implied correlation had all been published by us, among others, in 2006, showing that the quantitative community was aware of the model limitations before the crisis. We also explain why the Gaussian copula model is still used in its base correlation formulation, although under some possible extensions such as random recovery. Overall we conclude that the modeling effort in this area of the derivatives market is unfinished, partly for the lack of an operationally attractive single-name consistent dynamic loss model, and partly because of the diminished investment in this research area. Tue, 02 Mar 2010 22:04:23 +0100 http://www.quant-press.com In this paper we address the relationship between the smile that stochastic volatility models produce and the dynamics they generate for implied volatilities. We introduce a new quantity, which we call the Skew Stickiness Ratio and show how, at order one in the volatility of volatility, it is linked to the rate at which the at-the-money-forward skew decays with maturity. We then focus on short maturity skews and (a) show that the difference between realized and implied SSR can be materialized as the PL of an option strategy, (b) introduce the notion of realized skew. Tue, 02 Mar 2010 22:04:23 +0100