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Let Xij be the forward FX rate for currency I in terms of currency j. Suppose that the volatility smiles for Xik and Xjk modeled with the log normal (β= 1) SABR model. We show that the cross FX rate Xij = Xik/Xjk is then governed by the log normal SABR model, and find explicit formulas for its SABR parameters β, ρ, ν in terms of the SABR parameters for Xik and Xjk. These results are not exact, but...
Here we analyze the mean‐reverting SABR model:
We use asympotic methods to derive an effective forward equation for the marginal density
This effective forward equation is not exact, but is accurate through O(ϵ2), the same accuracy as the SABR implied volatility formulas. Since the effective forward equation has one spatial dimension (F) instead of two (F, A), it can be solved efficiently numerically...
We analyze the valuation of European digital call and put options in the market standard SABR stochastic volatility model. Asymptotic methods developed for the arbitrage‐free SABR model are used to obtain explicit, closed‐form formulae for the valuation of European digital call and put options under the SABR model. Results derived in this paper have the same order of accuracy as the closed‐form SABR...
We revise the standard analysis of constant maturity swaps, caps, and floors to account for dual forecast and discount curves. This reduces the pricing of these deals to evaluation of quadratic swaplets, caplets, and floorlets. We use the explicit, closed‐form expressions for the value of these quadratic options under the SABR model. This enables us to use swaption smile information to improve CMS...
Many deal types, such as constant maturity swaps, caps, and floors, contain convexity corrections. Valuing these convexity corrections eventually boils down to evaluating quadratic swaps or options.The values of quadratic swaps and options are known exactly under normal and lognormal volatility models. However, flat normal and lognormal models are inappropriate, since derivatives with quadratic payoffs...
The presence of stochastic volatility in an option model impacts the values of the hedge ratios (the “greeks”), and in particular the option delta. In the context of the SABR model, the greeks were calculated in [1] based on the asymptotic expression for the implied volatility derived there. In [2], the option delta of [1] was modified to take into account the effects of the correlation between the...
We combine singular perturbation techniques with an effective media argument to analyze the general Heston model:
We first show that the marginal probability density Q(T, F) satisfies an effective 1‐d forward equation through O(ε2). We analyze this 1‐d forward equation using an effective media approach. For any given expiry date Tex, this analysis yields effective SABR parameters αeff, ρeff, νeff...
We compare two interpolation methods which are widely used to construct discount curves, forecast (projection) curves, basis curves, and other financial curves. We find that the area‐preserving, quadratic‐spline method is superior to the “smart quadratic” method, yielding smoother, more natural looking forward curves with few of the artifacts exhibited by the smart quadratic curves. We also show how...
Analysis of the standard SABR model leads to an effective forward equation which has time‐independent coefficients, and analysis of this reduced‐dimensionality equation leads to explicit asymptotic formulas for the implied normal volatilities of European options. These formulas are accurate to within O(ε2), and are used extensively in practice for pricing and managing the risks of European options...
In “The SABR Chronicles,” Patrick S. Hagan charts the development of the model from its inception as an introduction to “Managing Vol Surfaces” by Patrick S. Hagan, Andrew S. Lesniewski, and Diana E. Woodward.
Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by
where dZL is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot...
The SABR model has two variables, the forward asset price and the local volatility . A singular perturbation analysis has shown that the marginal density Q(T, F) defined by
can be found through by solving a one‐dimensional (1D) effective forward equation of the form
where and
This reduces the valuation of European options to one spatial dimension from two. Recently, similar asymptotic analyses...
A book's delta risks are usually calculated by bumping the rate of each stripping instrument, re‐stripping the discount curve, and re‐valuing the book using the new discount curve. The difference between the new and old value of the book is the book's bucket delta risk with respect to that stripping instrument. After finding the bucket delta risks with respect to all the stripping instruments, the...
Smile risk is often managed using the explicit implied volatility formulas developed for the SABR model [1]. These asymptotic formulas are not exact, and this can lead to arbitrage for low strike options. Here we provide an alternate method for pricing options under the SABR model: We use asymptotic techniques to reduce the SABR model from two dimensions to one dimension. This leads to an effective...
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