Some Explicitly Solvable SABR and Multiscale SABR Models : Option Pricing and Calibration

A multiscale SABR model that describes the dynamics of forward prices/rates is presented. New closed form formulae for the transition probability density functions of the normal and lognormal SABR and multiscale SABR models and for the prices of the corresponding European call and put options are deduced. The technique used to obtain these formulae is rather general and can be used to study other stochastic volatility models. A calibration problem for these models is formulated and solved. Numerical experiments with real data are presented.


Introduction
In this paper a multiscale SABR model that describes the time dynamics of forward prices/rates is presented.This model generalizes the well known SABR model introduced in 2002 by Hagan Kumar, Lesniewski, Woodward in [1].Under some hypotheses on the correlation structure of the model studied when we restrict our attention to the normal and lognormal multiscale SABR models it is possible to derive explicit (closed form) formulae to express the transition probability density functions of the stochastic processes implicitly defined by the models and of the prices of the corresponding European call and put options.Using the technique developed to derive the transition probability density function of the multiscale SABR model we deduce new explicit formulae for the transition probability density function of the normal and lognormal SABR models presented in [1].Specifically we show that the transition probability density functions of the normal and lognormal SABR models (with no assumptions on the correlation structure of the models) can be written as the inverse Fourier transform of explicitly known kernels.Moreover we show that for the multiscale 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 of two copies of these kernels.This property is interesting since it can be used to define easy to solve multiscale versions of other stochastic volatility models.
The multiscale SABR model introduced in this paper is motivated by the behaviour in the financial markets of equity prices, interest rates and currency exchange rates.In several circumstances empirical studies have shown that the dynamics of these quantities is described more satisfactorily by models that use at least two factors to describe the volatility dynamics than by models that use only one factor (see, for example, [2][3][4] and the reference therein).In [5][6][7] it has been shown that a generalized Heston model, that uses two stochastic volatilities varying on two different time scales, leads to satisfactory forecasts of the asset prices and of the corresponding European option prices.The prices considered in [5][6][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 volatility of the forward prices/rates variable.We limit our attention to "normal" and "lognormal" SABR and multiscale SABR models.In these models the instantaneous variation of the forward prices/rates depends only on the volatility or on the volatilities ("normal" models) or on the volatility or on the volatilities times the prices/rates itself ("lognormal" models).These models are special cases of more general SABR models where the variation of the forward prices/rates depends on the product between a sufficiently smooth function of the forward prices/rates and the volatility or the volatilities.Usually this function is chosen in the family of functions where x is a real variable and  is a parameter.That is the SABR models considered depend on the parameter  ,   0,1  0


, the normal models correspond to the choice   and the lognormal models correspond to the choice 1   .Let and be respectively the sets of real and of positive real numbers and let be a real variable that denotes time.Let us define the multiscale SABR model.To the forward prices/rates described by the stochastic process t    t x , , we associate two stochastic volatilities given by the stochastic processes 1, , , .The dynamics of the stochastic process t , is defined by the following system of stochastic differential equations: holds it is likely to observe abrupt changes in the forward rates/prices variable.The processes 0,1 0,2 , 1 , 2 , , are standard Wiener processes such that 1 2 , and 0,1 , 0,2 , 1 , , , are their stochastic differentials.The correlation structure of the model is defined by the following assumptions: where  denotes the expected value of  and the quantities are constants known as correlation coefficients.The autocorrelation coefficients of the previous stochastic differentials are equal to one.When the model is multiscale (i.e. when  0,1 0,2 , ) the meaning of the assumptions (4)- (9) is that the stochastic 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 initial conditions: where , are random variables that we assume b nc d in a point with probability one.For simplicity we identify the random variables 0 to e co entrate x  , ,0 , with the points where they are concentr d. ,0 Equations (1)co (3), the initial nditions ( 10)- (12), the assumptions on the correlation coefficients ( 4)- (9) and the conditions on the coefficients  , , define the multiscale SABR model.izes the SABR model introduced in 2002 by Hagan, Kumar, Lesniewski, Woodward [1] that is defined by the following stochastic differential equations: This model general d d , where   0,1   and 0   .The coefficients  and  of ( are known espectively as 13), ( where The Equa is a constant called correlation coefficien tions (13) and ( 14) are equipped with the initial conditions: where 0 x  and are random va 0 v  riables that we assume to be concentrated in a point with probability one and that, for simplicity, we identify with the points where they are concentrated.Moreover we assume that 0 0

with probability one im
We consider the normal and the lognormal SABR and m plies that , 0 i t v  with ability one for 0 t  .A similar statement holds for 0 v  and t v , 0 t  .prob ultiscale SABR models.These models are obtained from the previous ones choosing respectively in Equations (13) and (1) 0 we consider the logn models we assume that 0 0 x   .In the lognormal models the assumption 0 0 x   obability one implies that   can be studied using the following es erical methods, series expansions in the parameter approach : num  or hybrid methods.The last approach combines series expansions and numerical methods.The SABR models with   can be studied using integral formulae inv in ergeometric functions for their transition probability density functions [8].The models with 0.5 olv g hyp   deserve special attention.These models will b elsewhere.We begin our analysis e studied with of ormal m the study the n ultiscale SABR model (see Section 2) for three reasons.The first reason is that under the previous hypotheses on its correlation structure the normal multiscale SABR model can be solved explicitly.The second reason is that the normal multiscale SABR model can be considered as an improvement not only of the normal SABR model but also of SABR models with  different from zero, sufficiently small.In fact the use of two volatilities makes the normal multiscale model more "flexible" than the SABR models.For example the normal multiscale SABR model reproduces both balanced and skewed probability distributions of prices/rates and can forecast satisfactorily the option prices even when the options considered have strike price near to zero and are at the money.In these circumstances the normal SABR model fails to explain the observed prices.The third reason is that in the class of SABR models parametrized by  , , the normal models are the simplest ones a y is useful to understand the other models.For example in Section 3 we use the results obtained in the study of the normal models to study the lognormal models.
The e nd their stud xplicit formulae of the transition probability de ale SA for the normal and lognormal SA d in the calibration problem and in th nsity functions associated with the normal and lognormal models are one (SABR models) or three dimensional (multiscale SABR models) integrals of explicitly known integrands.The formulae are closed form and "easy to use" in the sense that their numerical evaluation can be done with elementary methods.These formulae are used to derive explicit (closed form) formulae for the corresponding prices of European call and put options.The option pricing formulae are integrals of explicitly known integrands.Due to the special form of the integrands the numerical evaluation of the multi-dimensional integrals involved in the formulae of the transition probability density functions and of the option prices can be done very efficiently with ad hoc quadrature rules.
Moreover from the formula for the normal multisc BR transition probability density function we derive a formula for the transition probability density function of the normal SABR model.This formula is expressed as a one dimensional integral of a (regular) explicitly known integrand, is an elementary formula that can used instead of the formula deduced in [9] (Formula (120) in [9]).This last formula (Formula (120) in [9]) is based on the McKean formula for the heat kernel of the Poincaré plane.In a similar way a new formula for the transition probability density function of the lognormal SABR model is deduced.This last formula is a special case of an explicit, "easy to use" formula for the transition probability density function of the stochastic process implicitly defined by the Hull and White model [10] when there is a possibly nonzero correlation between the stochastic differentials appearing on the right hand side of the forward prices/rates and volatility equations.These are two interesting formulae since up to now in the case of nonzero correlation for the transition probability density functions of the lognormal SABR model and of the Hull and White model only asymptotic expansions in the correlation coefficient were known (see for example [10,11] and the references therein).The results relative to the Hull and White model will be presented elsewhere.The formulae presented in this paper are obtained using the Fourier transform, the method of separation variables and the results of Yakubovic [12] about the Lebedev Kontorovich Transform.
A calibration problem BR and multiscale SABR models is considered.These models are calibrated using option price data, the option pricing formulae mentioned above and the least squares method.The calibration problem is formulated as a constrained optimization problem for the least squares error function.Given the forward prices/rates the calibrated models are used to forecast option prices.We discuss some numerical experiments with real data where observed and forecast option prices are compared.These experiments confirm the validity of the procedure used to forecast option prices, of the calibration procedure and of the models presented.In particular they make possible a comparison between SABR and multiscale SABR models that shows when the use of the multiscale SABR models is justified.
The real data use e forecasting experiments are discrete time observations of the euro/US dollar (EUR/USD) exchange rate (futures prices), of the futures prices of the USA five year interest rate swap and of the prices of the corresponding European put and call options (i.e.European foreign exchange options on EUR/USD futures prices and options on USA five year interest rate swap futures).That is we consider Foreign eXchange (FX) data and interest rates data.Note that forward/futures prices are quantities stated in the contracts stipulated to buy or to sell currencies in a future date and that they remain unchanged during the life time of the contracts.For the convenience of the reader let us recall some facts about the derivatives mentioned above.A foreign exchange option is a derivative that gives to the owner the right but not the obligation to exchange a given quantity of money denominated in one currency into money denominated in another currency in a specified date at a pre-agreed exchange rate.Exchange rate derivatives are widely traded and serve different needs, for example, they serve the needs of firms active in the international trade arena that want 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% per floating interest rate payments based on 3-month LIBOR interest rate.These swaps are widely traded.In fact they are excellent tools for duration management and asset/ liability gap management for bank treasuries, insurers and financial services companies.Note that the use of futures prices/rates instead of forward prices/rates in our numerical experiments is due to the fact that only the latter ones are over the counter prices.Moreover recall that when during the time period considered the risk free interest rates are deterministic forward and futures prices/ rates coincide (see [13], Proposition 3.1 and [14]).
The numerical experiment on the EUR/USD exchange rate shows that, once calibrated using call and put option prices relative to a given date, the normal multiscale SABR model, given the asset price at the time of the forecast, is able to produce forecasts of call and put option prices that outperform those obtained with the normal 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 factor two.This means that the calibrated multiscale model has really a multiscale behaviour and as a consequence the interpretation of the data benefits from the presence of the second time scale.
In the next experiment the lognormal models are used to interpret interest rate swaps data.In this case the futures price has abrupt changes so that the improvement in the data interpretation obtained introducing the multiscale model is significant.In particular when the lognormal multiscale SABR model is considered the values of the parameters

The Normal SABR and Multiscale SABR
Le normal multiscale SABR model.This ance/w14 contains some auxiliary material including animations that help the understanding of this paper.A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
The remainder of the paper is organized as f ction 2 we study the normal SABR and multiscale SABR models.In Section 3 we study the lognormal SABR and multiscale SABR models.In Section 4 we formulate a calibration problem for these models and we present a numerical method to solve it.In Section 5 a procedure that, given the asset prices at the time of the forecast, forecasts option prices using the calibrated models is presented.The calibration and forecasting procedures are applied to study real data.Currency exchange rates and interest rates derivatives data and the corresponding option price data are studied.The forecast option 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.

Models
t us consider the model is obtained choosing 0   in (1), (2), (3) and is given by the following stocha ifferential equations: with the initial conditions: where e mu s 0 , and t t when , , , , , , , , , , , , , , , of the at they > 0 t t forward pr an be ices/rates negative.We vary on the real axis,  and th , , , , , , , , considered as a function of the variables x v v t .The function MN p  is the solution of the Fokker Planck equation: , , , denotes the Dirac's delta and in (25) the operator   L  is given by: .  25) are completed with appropriate boundary conditions.Th dey of tic part of the Fokker Planck equation implies that boundary conditions must be specified with care.For simplicity we omit these boundary conditions.Note that the transition probability density function , , , , , , , , , , , , , , , where τ is the imaginary unit and the function , h y y  ebedev trans-is the "heat kernel for the Kontorovich-L form" (see [12] p. 748).That is: where and      sinh K  denote respectively the h lic sine a he sec type modified Bessel function of order yperbond t ond  (see [ 5).With a simple change of 15] p. variables Formula (27) can be rewritten as follows: , , , , ˆMN p be the Fourier transform of MN p Let with respect to the x x   variable, we have: , , , , 29), (31) and the properties of the Fourier transform we have: , , , , , , , where N g  is the Fourier transform of N g with respect to the k variable, that is: 1 e e π π 2 Note that Formula (27) and similarly Formulae (29), (32) give the transition probability density function n in-as a one dimensional Fourier integral of a know tegrand.The integrand has a special form, in fact it con tains the product of two copies of a function evaluated in two different points.This function, defined in (30) or in (33), is a one dimensional integral of an explicitly known integrand.That is Formulae (27), (29), (32) are three dimensional integrals.However the special form of their integrands mentioned above implies that the evaluation of these three dimensional integrals with an elemen quadrature rule can be done at the computational co a tary st of two dimensional integral.Note that the function N g  defined in (33) when is the transition probability density function of the normal SABR model.This can be seen proceeding as done at the end of this Section to deduce (27) or simply verifying that N g  satisfies the Fokker Planck equation associated to (13), (14) when 0   with the appropriate initial and boundary conditions.Formula (33) for the transition probability density function of the normal SABR model is a new and useful formula that can substitute the formula commonly used in the mathematical finance literature, that is Form (1 the heat ke ré ula ned 20) of [9], that is based on McKean formula for the plane.The integral that appears in (33) is a one dimensional integral of a smooth function whose numerical evaluation is easier than the evaluation of the integral of a singular function contai in Formula (120) of [9].Moreover Formulae ( 27) and (33) are deduced using elementary tools, that is: the Fourier transform, the me rnel of the Poinca thod of separation of variables and the results of [12] on the Kontorovich Lebedev transform.The McKean formula is derived using the differential geometry of the Poincaré plane.That is Formula (33) and its elementary derivation simplify the study of the normal SABR model.Formula ( 27) can be used to deduce some useful consequences.For example from Formula (27) it is possible to deduce an explicit formula for the marginal probability distribution of the forward prices/rates stochastic process defined by ( 18)-(23) under the assumptions (4)- (9), that is: N m is given by: N m can be rewritten as follows: where 2 N a  is given by: An alternative expression of the marginal probability distribution (34) can be obtained using Formula (32), that is: , , , , , From (27) using the no arbitrage pricing theory formulae to price in the normal multiscale SABR model European call and put options can be derived.The assumption that the risk free interest rate is deterministic during the life time of the priced option implies that the forward prices/rates coincide with the futures prices/rates.In fact the forward price is a martingale under the (for-ward) measure associated to ( 18)-( 23) and the futures price is a martingale under the risk-neutral measure.However if the risk free interest rate is deterministic the forward measure and the risk neutral measure coincide (see [13], Proposition 3.1).Hence we can assume that we are working with futures prices/rates instead of with forward prices/rates and we can exploit the fact that these prices/rates are martingales under the risk-neutral measure.That is the risk neutral measure used to compute the option prices and the "physical" measure used to describe the underlying dynamics defined by ( 18)-( 23) are the same (see [13], Proposition 3.1).
Under the previous assumption on the risk free int ra erest te manipulating Formulae ( 27), (34) and using the results contained in [12] we obtain formulae to price European call and put options in the normal multiscale SABR model.That is the formulae for the prices MN C and MN P at time 0 t  of respectively European call and put options with strike price K   and maturity time 0 T  (i.e.time to maturity maturity 0 0 T     since 0 t  is assumed to be "today") when the forward price of the underlying and the values of the stochastic volatilities at time 0 t  (that is today) are respectively , , , , , , ,0 , , where     max , 0     .Note that since in the normal multiscale SABR model the forward prices/rates can be negative we have chosen instead of as it is done when models with positive asset prices are considered.Moreover in (40), (41) we have chosen the discount factor equal to one, that is we have chosen the risk free interest rate equal to zero.This choice is due to the desire of keeping the expression of Formulae (40), (41) simple and can be easily removed.
reover in (46) the integral with re the Mo spect to  variable can be computed explicitly, we have: , , , where A formula analogous to (48) can be obtained for the put option price , (49) N P integrating (47) with respect to the  variable.
Note that also the integrals in the  variable ing in Formulae ( 40 x  vari es.In 6)-(4 th unt factor has been chosen equal to one.This assumption can be easily removed. Let us derive Formula (27).The reader not interested in this derivation can move to Section 3. We begin deducing Formula (27) when , , ,

MN
Equation (50) must be equipped with an initial condition in 0 s t t    , that is: , , , , , , with the initial and the appropriate boundary conditions.To solve Problem (52) and (53), we proceed by separation of variables, that is we assume: (54) , , s to be determined.Substitu g (52) we have: where G are function ng initial value problems: That is the assumption (54) reduces the solution of ( 52) and (53) to the solution of the followi with the appropriate boundary conditions.The constant appearing in (56) comes from the separation of ables.
To solve problems ( 56) and (57) we assume that the functions j G , 1, 2 j  , have the following form: where , are functions to be determined.Note that for with the boundary condition: (61) The boundary condition (61) is derived from the boun-lim 0, 1, 2.
dary conditions imposed to the solution of (50) (and as a consequence to the solution of (56)) and follows from the fact that we are looking for solutions of (56) that are probability density functions.Imposing the boundary condition (61) to the general solution of (60) we have: where for (62) the function   , , , is an "arbitrar tant" of the solutio is independe y cons nt of c j v , that can be d ), etermined through (58 (59) imposing the initial conditions (57) to the parabolic Equation (56).In fact we have: j j j j j j j j j j j j j j j 3 using the inversion formula of the Kontorovich Le Transform (see Formula (3) of [12] and the references therein) we have: bedev Using ( 63) and ( 64) we obtain the following formula for the function , , , , , , , , , , , When the boundary conditions are chosen appropriately the solution (65) of the backward Kolmogorov Equation (50) with x , x , k respectively with 0,1 0,2

The Lognormal SABR and Multiscale SABR Models
t us consider the lognormal multiscale SABR model, ls (1)-(3) when 1, we have: with the initial conditions: , , , , , , , , , , , where 81) and the properties of the Fourier transform we have: , , , , , ,  , , , ,  d e  , , , , , where we assume  for example, [10], [11] and the references therei Note that the lognormal SABR model is a special case of the Hull and White stochastic volatility model.It is easy to see that the analysis presented here for the lognormal SABR model can be extended to the study of e Hull and White model in presence of a nonzero co p ensity function of the Hull and White model and for the corresponding European call and put option prices.These formulae will be presented elsewhere.
The previous formulae for the transition p density functions  , , ,  , , , ,  ,   , , , where the function L m is given by: . Formula (89) can be rewritten as fo , is by llows: The numerical experience presented in Section 5 has shown that in the evaluation of Formula (90) the complex square root that defines must be computed very accurately.For this purpose i numerical experim we have found useful to exploit the results of [20].
Note that the integrands of the integrals appearing in Formulae (87) and (88) have the same special form of the integrands of Formulae (40) and (41).This implies that evaluating Formulae (87) and (88) has the same computational cost than evaluating Formulae (40) and (41).This last cost has been discussed in Section 2.
In the case of the lognormal SABR model Formulae (87) and (88) reduce respectively to the following formulae: , 1) the risk free interest rate has been c equal to zero and as a consequence the discount factor has been chosen equal to one.This choice is made to simplify the formulae and can be removed easily.Under the assumptions ( 4)-( 9) the normal and lognormal multiscale SABR models ( 18)-( 20) and ( 69)-( 71) together with the associated option pricing Formu (41) and ( 87), (88) are parameterized by six real quantities, that is: the parameters lae (40  41), (87), (88).That is all together when we consider the normal and lognormal multiscale SABR models there are seven real parameters that must be determined in the calibration problem.We introduce the vector 7    and the set 7     defined as follows:

A Calibration Problem for the Normal d Lo norm
In the calibration problem for the (normal and lognormal) multiscale SABR models the vector 7    is   ibrati the unknown that must be determined and defines the set of the "feasible" vectors of the cal on problem.That is   is the set of vectors th is "physical" constraints that follow from th ing of the parameters in the model equations.
Similarly when we consider the (normal and lognormal) SABR models the unknown of the calibration problem is the vector at sat e mean fy the  and t of the "feasible" vectors of the calibration problem is defined as follows: To keep the notation simple in the formulation of the calibration problems for the SABR and multiscale SABR models we denote with the same symbol  a vector belonging to , , , , T and strike price  , are not necessarily distinct.For example options having the same maturity time and several strike prices can be considered, in this case in the previous sets some of the maturity times are repeated.Moreover let , , where the objective function

 
L  is given by: , ,  and evaluating the objective function on this set of points.The initial guess of the minimization method is chosen among these points using a heuristic rule.The m iz on method used is a variable metric steepest descent method (see [2 .This method is an iterative procedure that, given an initial vector , , where are given positive constants.Details o the im f the variable me steepest des cent method used to solve the cali c found in [6]. he erical expe ents presented th ), ( 87), ( 88), ( 92), (93) and completing the results obtained with the appropriate discount compared with the option prices actually observed.The ormed u osite midpoint quadrature rule with 1000 nodes in each coordinate direction.These choices guarantee approximately six significant digits correct in the option prices.
We   In the firs In the fir periment we use the normal SABR and m ultiscale SABR models to interpret these data.In par-  ticular for these models we solve the calibration problem posed in Section 4. We solve Problem (96) when and for 18 , wher .Recall that the option ntained R models co ulae that hold whe in [1], Tables 1 and 2 show respectively the parameter values obtained as solution of the calibration problems (96) for the normal SABR and multiscale SABR models when we consider the data relative to . those with the The calibrated models, that is parameter values given in Tables 1 and 2, are used to forecast option prices one day ahead of the current date, that is ahead of the observation day of the prices used to calibrate the model.The forecasts are made evaluating formulae (40) and (41) when Q MN  en Q N  ors.These form price the fut tilities problem and evaluating formulae (46) and (47) wh multiplied by the appropriate discount fact ulae are evaluated using as futures res price observed the day of the forecast.The vola and obtained from the calibration of the volatilities the day of the forecast.
Let us define the moneyness of an option a given day as the ratio between the strike price of the option and the futures price on the EUR/USD exchange rate of that day.Figures 4 and 5 show the forecast option prices one day in the future (i.e. at time with equal one day) the observed option prices and the relative errors of show that in this experiment the normal multiscale SABR model outperforms the normal SABR model.This is probably due to the fact that the use of two volatilities in the multiscale SABR model captures efficiently the "smile" effect contained in the option prices.In fact the values of the constants 1  and 2  resulting from the solution of the calibration problem shown in Table 2 differ approximately of a factor two showing that the presence of the second volatility is  shown in Table 2 do not differ of one or mo in Hesto re or of magnitude as found for similar constants udies [6,7].In [6] and [7] a multiscale n odel has been used to study electric power prices.Electric power prices show severe spikes and abrupt cha ges that justify the huge difference in the In the second experiment we consider the daily observed values of the USA five-year interest rate swap (see Figure 6   100 = 6 that is 600 base points, this corresponds to an interest rate of per year (see Figures 7 and 8).We consider two dates and Tables 3 and 4 show the parameter values obtained calibrating the lognormal SABR and multiscale SABR models on the data discussed above relative to the USA five-year interest rate swap futures price and its options observed at .In particular      This numerical experiment shows that the use of two volatilities is justified when the forward/futures prices present significant changes in their behaviour.Note that the calibration done using the he beginning of the futures price fall, already provides two volatilities of volatilities significantly different (i.e ) and     , 2  forward prices/rates variable.Finally Figure 14 shows the relative errors on the forecast option prices one day

Conclusion
The closed form for ulae fo the transition probability tiscale SABR model improves normal SABR model especially in forecasting option es and in particular prices of at the money options.density function of the normal and lognormal SABR and and using these formulae "easy to use" formulae of European options on futures prices/rates have been deduced and have been used to study the prices of European call and put options on the Eurodollar futures price and on the USA five year interest rate swap futures price.Using these option pricing formulae a calibration problem based on the least squares method is formulated and solved numerically.The models are used to study real data time series.The numerical experiments compare the performance of the SABR and multiscale SABR models in forecasting option prices.The comparison suggests that in the circumstances studied in Section 5 the lognormal SABR model outperforms the normal SABR model and the SABR multiscale models outperform the corresponding SABR models.In general we could say that the multiscale SABR models outperform the corresponding SABR models when the change in time of the data interpreted by the models is sufficiently big.Finally let us point out that the potential of the technique used to derive these formulae can be exploited in other circumstances.In fact the idea of expressing the transition probability density function of a two factor volatility model as a kind of convolution of two copies of the kernel of the corresponding one factor volatility model can be exploited to study the multiscale generalization of other stochastic volatility models.Moreover the closed form formulae for the transition probability density functions of the normal and lognormal SABR and multiscale SABR models presented in this paper deserve further investigation and can be exploited, for example, to price exotic derivatives or to solve new calibration problems.


) and (41) can b done explicitly.(41 fu appeare However in the case of Formulae (40) and ) this integration leads to formulae computationally useless.In fact in Formulae (40) and (41) the evaluation of the nctions N M in a point of the  grid implies the computation of a two dimension ntegralvariable couples K ,  and 0 let us consider the backward Kolmosociated to the stochastic differential Equations (18)-(20) satisfied by the functi n o j

5
"past" variables x , 1 v , 2 v , t with the final condition (51) is the solution of the Fokker Planck Equation (24as forward Kolmogorov equation) in the "future" variables x , 1 v is a new formula, in fact up to now when 0   only series expansions in powers of  with base point 0   have been known for L p (see, n).

,
the stochastic differentials of the Wiener rocesses of the model.In this way it is possible to obtain new (closed form) formulae for the transition probability d robability ML p and L p written using the logreturn variable can be easily rewritten in the ori ariable  gin , al forward prices/rates v 0 space.We formulate a calibration problem for the models studied in Section 2 and 3.
goe ber 27th, 2010, to July 19th, 2011.The observations are made daily and the prices considered are the closing prices of the day.Recall that a year is made of about 250 -260 trading days and a month is made of about 21 trading days.

Figure 1
shows the futures price EUR/USD (ticker YTU1 Curncy) (blue line) and the EUR/USD currency's exchange rate (pink line) as a function of time.Figures 2 and 3 show respectively the prices (in USD) of the corresponding call and put options with maturity time September 9th, 2011 and strike price i K , 1, 2, ,18 i   , as a function of time.e option prices computed evaluating with numerical quadratures the integrals contained in Formulae (40), (41), (46), (47 factors are numerical quadratures are perf sing the compThe computation of thirty-six option prices using the midpoint quadrature rule as specified previously requires three and half seconds on the Intel CORE Duo CPU T6400

Figure 3 .
Figure 3. Put option prices on YTU1 with strike price

Figure 4 .Figure 5 .
Figure 4. Relative errors obtained using the normal SABR September 27th, 2010 versus moneyness (FX experi (a) and m ltiscale SABR (b) models calibrated ment).u at  t 1 = = t interpret the data.Note that the values of 1  and 2 (a)), the corresponding futures prices having maturity September 30th, 2011 (the ticker DSU1 in Figure 6(b)) and the prices of the corresponding European call and put options with expiry date September 19th, 2011 and strike prices iffe the forecasting ability of the normal SABR and mulcale SABR models as shown in Figures 4 and 5. ti   106 0.5 1

Figure 6 .
Figure 6.Observed USA five-year interest rate swap (a) and the corresponding futures price DSU1 having maturity September, 2011 (b) versus time.

1 t 6
sider two dates the first one re the oscillations of the e second one 2 t selected at the be e futures price (see Figure2010, where the values of e selected in a period futures price are small and ginning of the fall of 6(b)).Note that from Novem-the corresponding futures prices (ticker DSU1 Figure2010to December 15th, 2010 the futures price goes from the value of 110 to the value of 104.Recall that these futures prices are expressed in hundreds of base points.

Figures 9 (
a), 10(a), 11(a) and 12(a) show the observed option prices and the forecast option prices as a using the normal SABR model (see Figures 9(a) and 11(a)) and the multiscale SABR models (see Figures 10(a) and 12(a)).Figures 9(b), 10(b), 11(b) and 12(b) show the the ative e ors committed on the forecast option prices one day ahead of the current day as a function of the moneyness.In particular we use the values of the model parameters obtained calibrating the model using the data at 1 t t  to forecast the option prices at ), where one day t   .Figures 9(b), 10(b), 11(b) and 12(b) show tha the use f t o

s
with the results obtained with the lognormal and 12(b)) are smaller than the corresponding relative errors of the SABR model (see Figures 9(b) and 11(b)).In particular the lognormal multiscale SABR mode oves substantially the lognormal SABR model in the forecasting of the prices of at the money options (see Figures 9(b), 10(b) and 11(b), 12(b)).

Figure 7 .
Figure 7.Call option prices on DSU1 with strike price

Figure 8 .
Figure 8.Put option prices on DSU1 with strike price

Figure 9 .and 2 Figure 11 .
Figure 9. Observed and forecast prices one day in the future o the lognormal SABR model calibrated at ˆ1 = = October 12t  t t

Figure 13 .
Figure 13.We can see that the values of the para-meters remain substantially unchanged in the two months period except for the values of the parameters

Figure 12 .
Figure 12.Observed and forecast prices one day in the future of call and put options (a) and relative errors (b) obtained using the lognormal multiscale SABR model calibrated at versus moneyness (interest rate swap experiment).ˆ2 = = November 15th, 2010  t t

Figure 13 .Figure 14 .
Figure 13.Parameter values obtained calibrating the logno months in the period going from September 14th, 2010 to Nov rm em 0 of the  present two numerical experiments based calibration problem of Section 4. The stopping meters of the minimization algorithm introduced in (98) have been chosen as follows:

Table 4 . Solution of the calibration problem: Lognormal multiscale SABR model (interest rate swap experiment).
In fact the relative errors on the forecast option prices of the multiscale SABR model (see Figures10(b)

Table 4 )
, that is a date before t .