Local Volatility Under Stochastic Interest Rates Using Mixture Models PDF Download

Author: Mark S. Joshi
Publisher:
ISBN:
Category :
Languages : en
Pages : 22

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Book Description
A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. To this end, we present two methods for calibrating a local volatility model under correlated stochastic interest rates. This is achieved by first fitting a mixture model to market prices, and then determining the local volatility function that is consistent with this mixture model.

Author: Mark S. Joshi
Publisher:
ISBN:
Category :
Languages : en
Pages : 20

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Stochastic volatility, local volatility and stochastic interest rates are three of the most important extensions to the standard Black-Scholes framework. Although much work has been done on models incorporating one or two of these extensions, very little has been done on the combination of all three. We show how to efficiently calibrate and simulate such a model by utilizing a mixture diffusion based approach, which takes advantage of the multidimensional fractional FFT.

Author: Eric Benhamou
Publisher:
ISBN:
Category :
Languages : en
Pages : 10

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This paper studies the impact of stochastic interest rates for local volatility hybrids. Our research shows that it is possible to explicitly determine the bias between the local volatility of a model with stochastic interest rates and the local volatility of the same model, but with deterministic interest rates as a function between the correlation of the stochastic interest rates and the digital at the local strike. The paper will show that this bias can be expressed in a simpler form under the assumption of a diffusion of the stochastic interest rates, enabling us to compute a fast calibration for a hybrid model with stochastic interest rates. This bias leads to a decrease in the value of the local volatility as a result of the induced volatility caused by the stochastic drift. Numerical results illustrate the importance of the bias and confirm that some stochastic noise arises from the stochastic drift.

Author: Bing Hu
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Author: Mingyang Xu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.

Author: Eric Benhamou
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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This paper presents new approximation formulae of European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1275872 for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot and to obtain accurate approximations with tight estimates of the error terms. This approach can also be used in the case of stochastic dividends or stochastic convenience yields. We finally provide numerical results to illustrate the accuracy with real market data.

Author: Damiano Brigo
Publisher: Springer Science & Business Media
ISBN: 354034604X
Category : Mathematics
Languages : en
Pages : 1016

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Book Description
The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. The fast-growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.

Author: Daniel Alexandre Bloch
Publisher:
ISBN:
Category :
Languages : en
Pages : 37

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Book Description
We establish the need for local volatility coupled with domestic and foreign stochastic interest rates to properly manage some exotic hybrid options. We then compute such a local volatility and identify a bias with respect to the local volatility with deterministic rates. Performing variance-covariance analysis on the logarithm of the underlying price together with the domestic and foreign spot rates we estimate that bias by calculating the variances of the logarithm of the underlying price with and without stochastic rates at fixed points in time and in space. Equating the resulting variances we express the local volatility with stochastic rates in terms of the one with deterministic rates plus a bias obtaining an exact, fast and robust way of calibrating any local volatility with stochastic rates to market prices. We calculate it by using a bootstrapping method requiring solving a quadratic equation at each maturity and strike and present results on the Japanese market.

Author: Oleg Kovrizhkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 36

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Book Description
We consider the following models:1. Generalization of a local volatility model rolled with a moving average of the spot: dS = mu Sdt + sigma(S/A)SdW$ where A(t) is a moving average of spot S.2. Generalization of Heston pure stochastic volatility model rolled with a moving average of the stochastic volatility: dS = mu Sdt + sigma SdW, dsigma^2 = k(theta - sigma^2)dt + gamma sigma dZ where theta(t) is a moving average of variance sigma^2.3. Generalization of a full stochastic volatility with the process for volatility depending on both sigma and S and rolled with a moving average of S: dS = mu Sdt + sigma SdW, dsigma = a(sigma, S/A)dt + b(sigma, S/A)dZ,corr(dW, dZ) = rho(sigma, S/A)$, where A(t) is a moving average of the spot S. We will generalize these and other ideas further and show that they lead to a 2-factor pure stochastic volatility model: dS = mu Sdt + sigma SdW$, sigma = sigma(v_1, v_2), dv_1 = a_1(v_1, v_2)dt + b_1(v_1, v_2)dZ_1,dv_2 = a_2(v_1, v_2)dt + b_2(v_1, v_2)dZ_2, corr(dW, dZ_1) = rho_1(v_1, v_2), corr(dW, dZ_2) = rho_2(v_1, v_2), corr(dZ_1, dZ_2) = rho_3(v_1, v_2) and give examples of analytically solvable models, applicable for multicurrency models consistent with cross currency pairs dynamics in FX. We also consider jumps and stochastic interest rates.

Author: Yu Tian
Publisher:
ISBN:
Category :
Languages : en
Pages : 146

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Book Description
This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.