Arbitrage-Free XVA (Invited Session talk –Stéphane Crépey)
Agostino Capponi*, Columbia University; Stephan Sturm, WPI; Maxim Bichuch, Johns Hopkins University
We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no-arbitrage arguments, we derive backward stochastic differential equations (BSDEs) associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer’s and seller’s XVA, which in turn identify a no-arbitrage interval.In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the uniquely determined XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo and collateral rates, we study the semilinear partial differential equation (PDE) characterizing buyer’s and seller’s XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.
Option pricing under time-varying risk-aversion with applications to risk forecasting
Ruediger Kiesel*, University Duisburg-Essen; Florentin Rahe, University Ulm
We present a new option-pricing model, which explicitly captures the difference in the persistence of volatility under historical and risk-neutral probabilities. The model also allows to capture the empirical properties of pricing kernels, such as time-variation and the typical S-shape. We apply our model for two purposes. First, we analyze the risk preferences of market participants invested in S&P 500 index options during 2001 – 2009. We find that risk-aversion strongly increases during stressed market conditions and relaxes during normal market conditions. Second, we extract forward-looking information from S&P 500 index options and perform out-of-sample Value-at-Risk (VaR) forecasts during the period of the subprime mortgage crises. We compare the VaR forecasting performance of our model with four alternative VaR models and find that 2-Factor Stochastic Volatility models have the best forecasting performance.
Log-Skew-Normal Mixture Model For Option Valuation
VISWANATHAN ARUNACHALAM*, UNIV. NACIONAL DE COLOMBIA; Jose Jimenéz, Univ Nacional de Colombia
There is good empirical evidence to show that the financial series, whether stocks or indices, currencies or interest rates do not follow the log-normal random walk underlying the Black-Scholes model, which is the basis for most of the theory of options valuation. In this article, we present an alternative approach for calculating the price of the call and put options when stock return distribution follows a Log-Skew-Normal mixture distribution. We obtain explicit expression for the price of the European options and formula for Greeks. We also analyze the effect the implied volatility at-the-money and a derivation of an expression for the implied volatilities at and around-the-money and discuss its asymptotic behavior. We present some numerical results for the calibration to real market option data.
On American swaptions under the linear-rational framework
Yerkin Kitapbayev*, Boston University; Damir Filipovic, EPFL
In this paper we study the American version of the swaptions under the linear-rational term structure model (Filipovic, Larsson and Trolle (2014)). This framework enables us to simplify
the pricing problem significantly and formulate
corresponding optimal stopping problem for a diffusion process. The latter problem reduces to a free-boundary problem which we tackle by local time-space
calculus (Peskir (2005)). We characterize the optimal stopping boundary as the unique solution to nonlinear integral equation and using this we obtain the arbitrage-free price of the American swaption and the optimal exercise strategies in terms of swap rates for both fixed-rate payer and receiver.