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 Sat, 08 Jan 2011 23:24:22 +0100 http://www.quant-press.com Convexity correction arises when one computes the expected value of an interest rate index under a probability measure other than its own nat- ural martingale measure. As a typical example, the natural martingale measure of the swap rate is the swap measure with annuity as the nu- meraire. However, the evaluation of the discounted expectation of the payoŽ in a constant maturity swap (CMS) derivative is performed under the forward measure corresponding to the payment date. In this paper, we propose an extension of Carr-Madan's static replication approach by exploring the linkage between replication, convexity correction and nu- meraire change. We illustrate how the static replication of a CMS caplet by a portfolio of payer swaptions is related to convexity correction associ- ated with the bond-annuity numeraire ratio. We also demonstrate the use of the extended static replication approach for hedging in-arrears clean index principal swaps and annuity options. Sat, 08 Jan 2011 23:24:22 +0100 http://www.quant-press.com For general time-dependent local volatility models, we propose new ap- proximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well. Sat, 08 Jan 2011 23:24:22 +0100 http://www.quant-press.com We address the problem of defining and calculating forward volatility implied by option prices when the underlying asset is driven by a stochastic volatility process. We examine alternative notions of forward implied volatility and the information required to extract these measures from the prices of European options at fixed maturities. We then specialize to the SABR model and show how the asymptotic expansion of the bivariate transition density in Wu {Wu10} allows calibration of the SABR model with piecewise constant parameters and calculation of forward volatility. We then investigate empirically whether current option prices at multiple maturities contain useful information in predicting future option prices and future implied volatility. We undertake this investigation using data on options on the euro-dollar, sterling-dollar, and dollar-yen exchange rates. We find that prices across maturities do indeed have predictive value. Moreover, we find that model-based forward volatility extracts this predicative information better than a standard "model-free'' measure of forward volatility and better than spot implied volatility. The enhancement to out-of-sample forecasting accuracy gained from model-based forward volatility is greatest at longer forecasting horizons. Sat, 08 Jan 2011 23:24:22 +0100