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Some Explicitly Solvable SABR And Multiscale SABR Models .

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Journal of Mathematical Finance, 2013, 3, 10-32http://dx.doi.org/10.4236/jmf.2013.31002 Published Online February 2013 (http://www.scirp.org/journal/jmf)Some Explicitly Solvable SABR and Multiscale SABRModels: Option Pricing and CalibrationLorella Fatone1, Francesca Mariani2, Maria Cristina Recchioni3, Francesco Zirilli41Department of Mathematics and Information Technology, University of Camerino, Camerino, Italy2Department of Economics, University of Verona, Verona, Italy3Department of Management, University Politecnica Marche, Ancona, Italy4Department of Mathematics, “G. Castelnuovo”, University of Rome “La Sapienza”, Rome, ItalyEmail: lorella.fatone@unicam.it, francesca.mariani@univr.it, m.c.recchioni@univpm.it, f.zirilli@caspur.itReceived September 20, 2012; revised November 16, 2012; accepted November 29, 2012ABSTRACTA multiscale SABR model that describes the dynamics of forward prices/rates is presented. New closed form formulaefor the transition probability density functions of the normal and lognormal SABR and multiscale SABR models and forthe prices of the corresponding European call and put options are deduced. The technique used to obtain these formulaeis rather general and can be used to study other stochastic volatility models. A calibration problem for these models isformulated and solved. Numerical experiments with real data are presented.Keywords: Multiscale Stochastic Volatility Models; Option Pricing; Calibration Problem; FX Data1. IntroductionIn this paper a multiscale SABR model that describes thetime dynamics of forward prices/rates is presented. Thismodel generalizes the well known SABR model introduced in 2002 by Hagan Kumar, Lesniewski, Woodwardin [1]. Under some hypotheses on the correlation structure of the model studied when we restrict our attentionto the normal and lognormal multiscale SABR models itis possible to derive explicit (closed form) formulae toexpress the transition probability density functions of thestochastic processes implicitly defined by the models andof the prices of the corresponding European call and putoptions. Using the technique developed to derive thetransition probability density function of the multiscaleSABR model we deduce new explicit formulae for thetransition probability density function of the normal andlognormal SABR models presented in [1]. Specificallywe show that the transition probability density functionsof the normal and lognormal SABR models (with noassumptions on the correlation structure of the models)can be written as the inverse Fourier transform of explicitly known kernels. Moreover we show that for themultiscale models (under some assumptions on the correlation structure of the models) the corresponding multiscale transition probability density functions can be expressed as the inverse Fourier transform of the product oftwo copies of these kernels. This property is interestingsince it can be used to define easy to solve multiscaleversions of other stochastic volatility models.Copyright 2013 SciRes.The multiscale SABR model introduced in this paperis motivated by the behaviour in the financial markets ofequity prices, interest rates and currency exchange rates.In several circumstances empirical studies have shownthat the dynamics of these quantities is described moresatisfactorily by models that use at least two factors todescribe the volatility dynamics than by models that useonly one factor (see, for example, [2-4] and the referencetherein). In [5-7] it has been shown that a generalizedHeston model, that uses two stochastic volatilities varying on two different time scales, leads to satisfactoryforecasts of the asset prices and of the correspondingEuropean option prices. The prices considered in [5-7]are the S&P 500 index, the associated European call andput option prices and some spot electric power prices.These findings motivate the use in the multiscale SABRmodel of two factors (i.e. two stochastic volatilities)varying on two different time scales to describe the volatility of the forward prices/rates variable. We limit ourattention to “normal” and “lognormal” SABR and multiscale SABR models. In these models the instantaneousvariation of the forward prices/rates depends only on thevolatility or on the volatilities (“normal” models) or onthe volatility or on the volatilities times the prices/ratesitself (“lognormal” models). These models are specialcases of more general SABR models where the variationof the forward prices/rates depends on the product between a sufficiently smooth function of the forwardprices/rates and the volatility or the volatilities. UsuallyJMF

L. FATONE ET AL.this function is chosen in the family of functions x 0,1 where x is a real variable and is aparameter. That is the SABR models considered dependon the parameter , 0,1 , the normal modelscorrespond to the choice 0 and the lognormalmodels correspond to the choice 1 .Let and be respectively the sets of real andof positive real numbers and let t be a real variable thatdenotes time. Let us define the multiscale SABR model.To the forward prices/rates described by the stochasticprocess xt , t 0 , we associate two stochastic volatilities given by the stochastic processes v1,t , v2,t , t 0 .The dynamics of the stochastic process xt , v1,t , v2,t ,t 0 , is defined by the following system of stochasticdifferential equations: dxt xt v1,t dWt 0,1 v2,t dWt 0,2 , t 0,dv1,t 1v1,t dWt1 ,t 0,dv2,t 2 v2,t dWt 2 ,t 0,(1)(2)(3)where the quantities i , i 1, 2 , and are realpositive constants such that 1 2 and 0,1 . Thefact that v1,t , v2,t , t 0 , vary on different time scalesis expressed by the condition 0 1 2 . When thecondition 0 1 2 holds it is likely to observeabrupt changes in the forward rates/prices variable. Theprocesses Wt 0,1 , Wt 0,2 , Wt1 , Wt 2 , t 0 , are standardWiener processes such that W00,1 W00,2 W01 W02 0 ,and dWt 0,1 , dWt 0,2 , dWt1 , dWt 2 , t 0 , are their stochastic differentials. The correlation structure of themodel is defined by the following assumptions:dWt1dWt 2 0, t 0,(4)dWt 0,1dWt1 0,1dt , t 0,(5)dWt 0,1dWt 2 0, t 0,(6) 0, t 0,(7)0,21tdWt dWdWt 0,2dWt 2 0,2dt , t 0,(8)dWt 0,1dWt 0,2 0, t 0,(9)where denotes the expected value of and thequantities 0,1 , 0,2 1,1 are constants known ascorrelation coefficients. The autocorrelation coefficientsof the previous stochastic differentials are equal to one.When the model is multiscale (i.e. when 0 1 2 )the meaning of the assumptions (4)-(9) is that thestochastic differentials on the right hand side of (1)-(3) associated to the two (long and short) time scales are independent.The Equations (1)-(3) are equipped with the initialCopyright 2013 SciRes.11conditions:x0 x 0 ,(10)v1,0 v 1,0 ,(11)v2,0 v 2,0 ,(12)where x 0 , v i ,0 , i 1, 2 , are random variables that weassume to be concentrated in a point with probability one.For simplicity we identify the random variables x 0 , v i ,0 ,i 1, 2 , with the points where they are concentrated. Weassume v i ,0 0 , i 1, 2 .The stochastic differential Equations (1)-(3), the initialconditions (10)-(12), the assumptions on the correlationcoefficients (4)-(9) and the conditions on the coefficients , i 0 , i 1, 2 , define the multiscale SABR model.This model generalizes the SABR model introduced in2002 by Hagan, Kumar, Lesniewski, Woodward [1] thatis defined by the following stochastic differential equations: dxt xt vt dWt , t 0,(13)dvt vt dQt ,(14)t 0,where 0,1 and 0 . The coefficients and of (13), (14) are known respectively as -volatilityand as volatility of volatility. Moreover Wt , Qt , t 0 ,are standard Wiener processes such that W0 Q0 0,dWt , dQt , t 0 , are their stochastic differentials andwe have:dWt dQt dt , t 0,(15)where 1,1 is a constant called correlation coefficient. The Equations (13) and (14) are equipped withthe initial conditions:x0 x 0 ,(16)v0 v 0 ,(17)where x 0 and v 0 are random variables that we assumeto be concentrated in a point with probability one andthat, for simplicity, we identify with the points wherethey are concentrated. Moreover we assume that v 0 0 .Note that for i 1, 2 the assumption v i ,0 0 withprobability one implies that vi ,t 0 with probabilityone for t 0 . A similar statement holds for v 0 and vt ,t 0.We consider the normal and the lognormal SABR andmultiscale SABR models. These models are obtainedfrom the previous ones choosing respectively in Equations (13) and (1) 0 (normal models) or 1(lognormal models). When we consider the lognormalmodels we assume that x 0 0 . In the lognormal modelsthe assumption x 0 0 with probability one implies thatxt 0 with probability one for t 0 . Under theassumptions (4)-(9) for the normal and lognormal SABRJMF

12L. FATONE ET AL.and multiscale SABR models the transition probabilityden- sity functions associated to the state variables ofthese models are expressed via integral formulae with explicitly known integrands. In this sense the normal andlognormal models are explicitly solvable. The SABR andmultiscale SABR models with 0,1 , 1 2 canbe studied using the following approaches: numericalmethods, series expansions in the parameter or hybrid methods. The last approach combines series expansions and numerical methods. The SABR modelswith 0,1 , 0 can be studied using integralformulae involving hypergeometric functions for theirtransition probability density functions [8]. The modelswith 0.5 deserve special attention. These modelswill be studied elsewhere.We begin our analysis with the study of the normalmultiscale SABR model (see Section 2) for three reasons.The first reason is that under the previous hypotheses onits correlation structure the normal multiscale SABRmodel can be solved explicitly. The second reason is thatthe normal multiscale SABR model can be considered asan improvement not only of the normal SABR model butalso of SABR models with different from zero,sufficiently small. In fact the use of two volatilitiesmakes the normal multiscale model more “flexible” thanthe SABR models. For example the normal multiscaleSABR model reproduces both balanced and skewedprobability distributions of prices/rates and can forecastsatisfactorily the option prices even when the optionsconsidered have strike price near to zero and are at themoney. In these circumstances the normal SABR modelfails to explain the observed prices. The third reason isthat in the class of SABR models parametrized by , 0,1 , the normal models are the simplest ones andtheir study is useful to understand the other models. Forexample in Section 3 we use the results obtained in thestudy of the normal models to study the lognormalmodels.The explicit formulae of the transition probabilitydensity functions associated with the normal and lognormal models are one (SABR models) or three dimensional (multiscale SABR models) integrals of explicitlyknown integrands. The formulae are closed form and“easy to use” in the sense that their numerical evaluationcan be done with elementary methods. These formulaeare used to derive explicit (closed form) formulae for thecorresponding prices of European call and put options.The option pricing formulae are integrals of explicitlyknown integrands. Due to the special form of the integrands the numerical evaluation of the multi-dimensionalintegrals involved in the formulae of the transitionprobability density functions and of the option prices canbe done very efficiently with ad hoc quadrature rules.Moreover from the formula for the normal multiscaleCopyright 2013 SciRes.SABR transition probability density function we derive aformula for the transition probability density function ofthe normal SABR model. This formula is expressed as aone dimensional integral of a (regular) explicitly knownintegrand, is an elementary formula that can used insteadof the formula deduced in [9] (Formula (120) in [9]).This last formula (Formula (120) in [9]) is based on theMcKean formula for the heat kernel of the Poincaréplane. In a similar way a new formula for the transitionprobability density function of the lognormal SABRmodel is deduced. This last formula is a special case ofan explicit, “easy to use” formula for the transition probability density function of the stochastic process implicitly defined by the Hull and White model [10] whenthere is a possibly nonzero correlation between the stochastic differentials appearing on the right hand side ofthe forward prices/rates and volatility equations. Theseare two interesting formulae since up to now in the caseof nonzero correlation for the transition probability density functions of the lognormal SABR model and of theHull and White model only asymptotic expansions in thecorrelation coefficient were known (see for example[10,11] and the references therein). The results relative tothe Hull and White model will be presented elsewhere.The formulae presented in this paper are obtained usingthe Fourier transform, the method of separation variablesand the results of Yakubovic [12] about the LebedevKontorovich Transform.A calibration problem for the normal and lognormalSABR and multiscale SABR models is considered. Thesemodels are calibrated using option price data, the optionpricing formulae mentioned above and the least squaresmethod. The calibration problem is formulated as aconstrained optimization problem for the least squareserror function. Given the forward prices/rates the calibrated models are used to forecast option prices. We discuss some numerical experiments with real data whereobserved and forecast option prices are compared. Theseexperiments confirm the validity of the procedure used toforecast option prices, of the calibration procedure and ofthe models presented. In particular they make possible acomparison between SABR and multiscale SABR models that shows when the use of the multiscale SABRmodels is justified.The real data used in the calibration problem and inthe forecasting experiments are discrete time observations of the euro/US dollar (EUR/USD) exchange rate(futures prices), of the futures prices of the USA fiveyear interest rate swap and of the prices of the corresponding European put and call options (i.e. Europeanforeign exchange options on EUR/USD futures pricesand options on USA five year interest rate swap futures). That is we consider Foreign eXchange (FX) dataand interest rates data. Note that forward/futures pricesJMF

L. FATONE ET AL.are quantities stated in the contracts stipulated to buy orto sell currencies in a future date and that they remainunchanged during the life time of the contracts. For theconvenience of the reader let us recall some facts aboutthe derivatives mentioned above. A foreign exchangeoption is a derivative that gives to the owner the right butnot the obligation to exchange a given quantity of moneydenominated in one currency into money denominated inanother currency in a specified date at a pre-agreed exchange rate. Exchange rate derivatives are widely tradedand serve different needs, for example, they serve theneeds of firms active in the international trade arena thatwant to reduce their exposure to exchange rate variations.The USA five year interest rate swap exchanges semiannual interest rate payments at the fixed rate of 4% perfloating interest rate payments based on 3-month LIBORinterest rate. These swaps are widely traded. In fact theyare excellent tools for duration management and asset/liability gap management for bank treasuries, insurersand financial services companies. Note that the use offutures prices/rates instead of forward prices/rates in ournumerical experiments is due to the fact that only thelatter ones are over the counter prices. Moreover recallthat when during the time period considered the risk freeinterest rates are deterministic forward and futures prices/rates coincide (see [13], Proposition 3.1 and [14]).The numerical experiment on the EUR/USD exchangerate shows that, once calibrated using call and put optionprices relative to a given date, the normal multiscaleSABR model, given the asset price at the time of theforecast, is able to produce forecasts of call and putoption prices that outperform those obtained with thenormal SABR model. Note that the values of the parameters 1 and 2 of the normal multiscale model obtained in the calibration differ of approximately a factortwo. This means that the calibrated multiscale model hasreally a multiscale behaviour and as a consequence theinterpretation of the data benefits from the presence ofthe second time scale.In the next experiment the lognormal models are usedto interpret interest rate swaps data. In this case thefutures price has abrupt changes so that the improvementin the data interpretation obtained introducing the multiscale model is significant. In particular when the lognormal multiscale SABR model is considered the valuesof the parameters 1 and 2 obtained in the calibration differ for about a factor two. The results obtainedon the interest rate swap data with the lognormal modelsconfirm the findings of the experiments on the EUR/USD exchange rate data with the normal models. Finallythe stability of the parameter values obtained in the calibration is investigated. It is shown that calibrating themodels daily with the option price data collected in oneday during a period of about two months (that is caliCopyright 2013 SciRes.13brating the models approximately forty times) the modelsparameters obtained remain substantially unchanged during the two months period. Recall that there are approximately twenty trading days in a month.The website: http://www.econ.univpm.it/recchioni/finance/w14 contains some auxiliary material includinganimations that help the understanding of this paper. Amore general reference to the work of the authors and oftheir coauthors in mathematical finance is the e.The remainder of the paper is organized as follows. InSection 2 we study the normal SABR and multiscaleSABR models. In Section 3 we study the lognormalSABR and multiscale SABR models. In Section 4 weformulate a calibration problem for these models and wepresent a numerical method to solve it. In Section 5 aprocedure that, given the asset prices at the time of theforecast, forecasts option prices using the calibratedmodels is presented. The calibration and forecasting procedures are applied to study real data. Currency exchangerates and interest rates derivatives data and the corresponding option price data are studied. The forecastoption prices are compared with the prices actually observed. The comparison shows the relevance of the multiscale SABR models. Finally in Section 6 some conclusions are drawn.2. The Normal SABR and Multiscale SABRModelsLet us consider the normal multiscale SABR model. Thismodel is obtained choosing 0 in (1), (2), (3) and isgiven by the following stochastic differential equations:dxt v1,t dWt 0,1 v2,t dWt 0,2 , t 0,(18)dv1,t 1v1,t dWt1 ,(19)dv2,t 2 v2,t dWt 2 ,t 0,t 0,(20)with the initial conditions:x0 x 0 ,(21)v1,0 v 1,0 ,(22)v2,0 v 2,0 ,(23)where x 0 , v i ,0 , i 1, 2 , are random variables that weassume to be concentrated in a point with probability one.The quantities i , i 1, 2 , are positive constants suchthat 1 2 . Moreover we assume that conditions (4)-(9)hold. In a similar way starting from (13), (14), (16), (17)and choosing 0 in (13) we can write the normalSABR model.Let us consider the transition probability density function of the stochastic process xt , v1,t , v2,t , t 0 , implicitly defined by (18)-(23), that is the probabilityJMF

L. FATONE ET AL.14density function of having xt x , v1,t v1 , v2,t v2given the fact that xt x

European option prices. The prices considered in [5-7] are the S&P 500 index, the associated European call and put option prices and some spot electric power prices. These findings motivate the use in the multiscale SABR model of two factors (i.e. two stochastic volatilities) varying on two different time scales to describe the vola-