On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates PDF Download
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Author: Lech A. Grzelak
Publisher:
ISBN:
Category :
Languages : en
Pages : 26
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Book Description
We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to efficient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity-FX-interest rate hybrid payoffs.
Author: Lech A. Grzelak
Publisher:
ISBN:
Category :
Languages : en
Pages : 26
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Book Description
We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to efficient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity-FX-interest rate hybrid payoffs.
Author: Lech Aleksander Grzelak
Publisher:
ISBN:
Category :
Languages : en
Pages : 27
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Book Description
Author: Alessandro Gnoatto
Publisher:
ISBN:
Category :
Languages : en
Pages : 43
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Book Description
We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed efficiently through the FFT methodology thanks to the affinity of the model. A joint calibration exercise of the implied volatility surfaces of a triangle of FX rates shows the flexibility of our framework in dealing with the typical symmetries that characterize the FX market. Our framework is also able to describe many non trivial links between FX rates and interest rates: a second calibration exercise highlights the ability of the model to fi t simultaneously FX implied volatilities while being coherent with interest rate products.
Author:
Publisher:
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Category :
Languages : da
Pages : 74
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Book Description
Author: Shijun Liu
Publisher:
ISBN:
Category :
Languages : en
Pages : 37
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Book Description
We first derive closed form solutions for currency options, currency futures, future options and the term structures of interest rates in a diffusion-jump model of stochastic interest rate, stochastic volatility and time varying jump intensity in currency price. We demonstrate that the introduction of constant jump intensity in the nominal stochastic discount factor shifts the whole term structure of interest rates vertically but has no influence on its shape. However, when the jump intensity is endogenous (time varying) the shape of the term structure is influenced through the factor sensitivity of interest rates. We also document considerable improvement in currency option pricing precision over alternative models if the true model is diffusion-jump with endogenous intensity in a simulation experiment. We conclude that allowing for multidimensional interaction is of significant qualitative and quantitative importance for the pricing of currency options and for understanding the shape of the term structure.
Author: Thadavillil Jithendranathan
Publisher:
ISBN:
Category : Financial futures
Languages : en
Pages : 210
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Book Description
Author: Anders B. Trolle
Publisher:
ISBN:
Category : Interest rates
Languages : en
Pages : 62
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Book Description
We develop a tractable and flexible stochastic volatility multi-factor model of the term structure of interest rates. It features correlations between innovations to forward rates and volatilities, quasi-analytical prices of zero-coupon bond options and dynamics of the forward rate curve, under both the actual and risk-neutral measure, in terms of a finite-dimensional affine state vector. The model has a very good fit to an extensive panel data set of interest rates, swaptions and caps. In particular, the model matches the implied cap skews and the dynamics of implied volatilities. The model also performs well in forecasting interest rates and derivatives.
Author: Riccardo Rebonato
Publisher: Princeton University Press
ISBN: 1400829321
Category : Business & Economics
Languages : en
Pages : 486
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Book Description
In recent years, interest-rate modeling has developed rapidly in terms of both practice and theory. The academic and practitioners' communities, however, have not always communicated as productively as would have been desirable. As a result, their research programs have often developed with little constructive interference. In this book, Riccardo Rebonato draws on his academic and professional experience, straddling both sides of the divide to bring together and build on what theory and trading have to offer. Rebonato begins by presenting the conceptual foundations for the application of the LIBOR market model to the pricing of interest-rate derivatives. Next he treats in great detail the calibration of this model to market prices, asking how possible and advisable it is to enforce a simultaneous fitting to several market observables. He does so with an eye not only to mathematical feasibility but also to financial justification, while devoting special scrutiny to the implications of market incompleteness. Much of the book concerns an original extension of the LIBOR market model, devised to account for implied volatility smiles. This is done by introducing a stochastic-volatility, displaced-diffusion version of the model. The emphasis again is on the financial justification and on the computational feasibility of the proposed solution to the smile problem. This book is must reading for quantitative researchers in financial houses, sophisticated practitioners in the derivatives area, and students of finance.
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: Lech A. Grzelak
Publisher:
ISBN:
Category :
Languages : en
Pages : 26
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Book Description
We present an extension of stochastic volatility equity models by a stochastic Hull-White interest rate component while assuming non-zero correlations between the underlying processes. We place these systems of stochastic differential equations in the class of affine jump diffusion - linear quadratic jump-diffusion processes (Duffie, Pan and Singleton, Cheng and Scaillet) so that the pricing of European products can be efficiently done within the Fourier cosine expansion pricing framework. We compare the new stochastic volatility Schobel-Zhu-Hull-White hybrid model with a Heston-Hull-White model, and also apply the models to price some hybrid structured derivatives that combine the equity and interest rate asset classes.
