### Some explicitly solvable SABR and multiscale SABR models: option pricing and calibration

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1. Abstract

A multiscale SABR model that describes the dynamics of forward prices/rates is introduced. The model is a system of three stochastic differential equations whose independent variable is time and whose dependent variables are the forward prices/rates and two stochastic volatilities varying on two different time scales. The two volatility equations are a two factor volatility model. This multiscale model generalizes the SABR model introduced in 2002 by Hagan, Kumar, Lesniewski, Woodward in [5]. The SABR model is a system of two stochastic differential equations whose dependent variables are the forward prices/rates and the stochastic volatility. The volatility equation of the SABR model is a one factor volatility model. Empirical studies have shown that in several circumstances two factors volatility models describe the behaviour of financial prices more accurately than one factor volatility models (see, for example, [1], [2], [3], [4], [8]). These studies motivate the introduction of the multiscale SABR models. The SABR model [5] is characterized by a correlation coefficient and by two parameters: the b volatility b Î [0,1] and the volatility of volatility e. The normal and lognormal SABR models correspond respectively to the choices b = 0 and b = 1. The normal and lognormal multiscale SABR models are the straightforward generalizations to the multiscale context of the corresponding SABR models. We study the normal and lognormal SABR and multiscale SABR models. Under some hypotheses on the correlation coefficients of the Wiener processes appearing in the models, ``easy to use" closed form formulae for the transition probability density functions of the state variables of the normal and lognormal multiscale SABR models and for the prices of the corresponding European call and put options are deduced. The formulae for the transition probability density functions are three dimensional integrals of explicit integrands. Due to the special form of the integrands these three dimensional integrals can be evaluated with a standard quadrature rule at the computational cost of a two dimensional integral. The method used to study the multiscale SABR models is applied to the study of the normal and lognormal SABR models. For these models some new closed form formulae for the transition probability density functions of their state variables and for the prices of the European call and put options are deduced. The transition probability density functions of the normal and lognormal SABR models are expressed as one dimensional integrals of explicit integrands. In the normal SABR model case the transition probability density function formula presented in Section 2 can be used instead of  formula (120) of [6]. This last formula is the one commonly used in mathematical finance and is based on the Mc Kean formula [5], [6] for the heat kernel of the Poincaré plane. In the lognormal SABR model case the formula presented in Section 3 for the transition probability density function is a special case of a new formula that gives the transition probability density function of the Hull and White stochastic volatility model [7] in presence of (possibly nonzero) correlation between the stochastic differentials appearing on the right hand side of the prices/rates and volatility equations. Moreover for the normal and lognormal SABR models formulae to price European call and put options are deduced. The formulae for the transition probability density functions presented here are based on some recent results of Yakubovic [10] on the heat kernel of the Lebedev Kontorovich Transform. A calibration problem for the normal and lognormal SABR and multiscale SABR models is studied. This calibration problem uses as data a set of option prices. The formulation of the calibration problem is based on the option pricing formulae mentioned above and on the least squares method. The procedure developed to solve the calibration problem is tested on real data. The real data studied are time series of exchange rates between currencies (euro/U.S. dollar, for short EUR/USD), of European call and put option prices on the Eurodollar futures price,  of   data of the U.S.A. five year interest rate swap futures prices and of the corresponding option prices. Forecast option prices obtained with the previous option pricing formulae are compared with prices actually observed. This comparison establishes the quality of the forecast prices, of the calibration procedure and of the models used. In particular it makes possible a comparison between SABR and multiscale SABR models that shows when the use of the multiscale SABR model is justified. This website contains some auxiliary material including some animations that helps the understanding of  the results presented in [4]. 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.

Keywords. Multiscale stochastic volatility models, option pricing, calibration problem, FX data, interest rate swap data.

AMS (MOS) Subject Classification. 62F10, 35R30, 91B70.

M.S.C. classification. 34M50, 60H10, 91B28.

JEL classification: C53, G12, C61

2.  Normal SABR model: new closed form formulae for the transition probability density function and for the European option prices

In this Section we give closed form formulae for the transition probability density function and for the European option prices of the normal SABR model.  The option prices are expressed as expected values of the discounted payoff when the dynamics of the underlying asset price is given by the normal SABR model. The formulae presented are elementary and can be used instead of the formulae deduced by Hagan, Kumar, Lesniewski, Woodward in [5].

Let R and R+ be the sets of real and of positive real numbers respectively and t be a real variable that denotes time.

Let the forward prices/rates be described by the stochastic process  xt, t ≥ 0, and the stochastic volatility associated to the forward prices/rates be given by the stochastic process vt, t>0. The SABR model introduced in 2002 by Hagan, Kumar, Lesniewski, Woodward [5]   is defined by the following equations:

where the quantities  and  are real constants satisfying the following conditions: bÎ[0,1],  e > 0. The stochastic processes  Wt, Qt , t>0, are standard Wiener processes such that W0= Q0= 0,  dWt, dQt , t>0, are their stochastic differentials and we assume that   <dWt dQt >= rdt, t>0, where  <· >  denotes the expected value of · and  rÎ(-1,1) is a constant  known as correlation coefficient. Equations (1) and (2) are equipped with the initial conditions:

where ,   are random variables assumed to be concentrated in a point with probability one. We assume that >0. This assumption implies that the process , t>0, remains positive with probability one for t>0. Note that  is not observable in the financial market and must be interpreted as a parameter of the model. The normal SABR model is the model obtained choosing  b=0 in equations (1).  That is the normal SABR model is defined by the following  equations:

equipped with the initial conditions (3), (4). Let   be the transition probability density function  of the normal SABR model (5), (6), (3), (4), that is let  be the probability density of having ,  when  , , t, t’³0,  t-t’>0. Note that when t’=0 we must choose x’=, v’=. In  starting from the backward Kolmogorov equation associated to the process defined by (5), (6), using elementary analysis,  the following formula is deduced:

where i is the imaginary unit,  the function  gN is given by:

sinh is the hyperbolic sine and is the second type modified Bessel function of  purely imaginary order.

The function  is known as the ``heat kernel for the Kontorovich-Lebedev transform" (see [10] p. 748). The function  given in (7) can be rewritten in terms of elementary functions as follows:

where cosh is the hyperbolic cosine. To deduce the option pricing formulae that follow from (9) we give the expression of the  marginal distribution MN of the random variable xt,  t>0, of having xt=x,  t>0, given that xt’ =x’,  vt’=v’, t, t’³0, t-t’>0, that is:

where mN is the following function:

and the function  is given by:

Let us consider in the normal SABR model a European call option expiring at time t=T>0 and having strike price K. The value CN of this option  at time t=0, expressed as expected value of the discounted payoff,  when at time t=0  the underlying prices/rates value is   and  the stochastic volatility is , is given by:

where (x)+ is the maximum between x and zero.  Using (11) we can  rewrite (13) as follows:

where

Note that since we have <xt>= , t>0,  the price at time t=0 of a European  put option expiring at time t=T and having strike price K when   and  can be obtained via the put-call parity relation, that is via the relation:

In formulae (13), (14), (16) we have assumed that the discount factor is equal to one and this corresponds to choose the risk free interest rate equal to zero. This assumption can be easily removed. Note that the expected values contained in formulae (13), (14), (16) are computed with respect to the measure whose density is given by (9), that is with respect to the physical measure. This is correct when during the life time of the options the risk free interest rate is deterministic.  In fact in this case the physical and the risk neutral measures coincide (see [9]).

3.  Lognormal SABR model: new closed form formulae for the transition probability density function and for the European option prices

In this Section we give closed form formulae for the transition probability density function and for the European option prices of the lognormal SABR model. The lognormal SABR  model is obtained from (1), (2) when b=1, that is it is given by:

where  >0 is a real constant. As mentioned in Section 2 Wt, Qt , t>0, are standard Wiener processes such that W0= Q0= 0,  dWt, dQt , t>0, are their stochastic differentials and we assume that  <dWt dQt >= rdt, t>0, where  rÎ(-1,1) is a constant  known as correlation coefficient. Equations (17) and (18) are equipped with the initial conditions:

where ,   are random variables  concentrated in a point with probability one. We assume that ,  are positive, this implies that the stochastic processes ,  , t>0, remain positive with probability one for t>0.

The formulae presented in this Section hold for rÎ(-1,1) and are closed form formulae  (see [4] for further details). Note that up to now closed form formulae for the lognormal SABR model known in the scientific literature are restricted to the case r=0, when r ¹0 the known formulae are power series expansions.

We define the variable , t>0. Using the new variable ,  t>0,  the stochastic differential equations (17), (18) are rewritten as follows:

and the initial conditions (19), (20) are rewritten as follows:

Let   be the transition probability density function  of the lognormal SABR model (21), (22), (23), (24), that is the probability density of having ,  when  , , t, t³0,  t-t>0. Note that when t=0 we must choose, . Starting from the backward Kolmogorov equation associated to the process defined by (21), (22), using elementary analysis, in [4]  the following formula is deduced:

where the function  gL is given by:

and

The function gL is defined through the ``heat kernel for the Kontorovich-Lebedev transform" (see [10] p. 748).

The function pL can be rewritten in terms of elementary functions as follows:

To deduce the option price formulae that follow from (28) we give the expression of  the  marginal distribution ML of the random variable xt, t>0, of having xt=x  given that xt=x, vt=v,  t, t³0, t-t>0, that is:

where mL is the following function:

and the function  is given by:

Let us consider in the lognormal SABR model a European call option expiring at time t=T>0 and having strike price K>0.  The value CL of this option at time t=0 when at time t=0 the underlying prices/rates value is   and  the stochastic volatility is  is given by:

Similarly in the lognormal SABR model let PL denote the value at time t=0 of the European put option expiring at time t=T>0 and having strike price K>0 when at time t=0 the underlying prices/rates value is   and the stochastic volatility is  , its value is given by:

Note that as done in Section 2 in formulae (32), (33) we have assumed that the discount factor is equal to one and, as a consequence, that the risk free interest  rate is equal to zero. Moreover we have assumed that the risk free interest rate is deterministic during the lifetime of the options considered.

4. Normal and lognormal multiscale SABR models: closed form formulae for the transition probability density function and for the European option prices

Let us introduce the multiscale SABR model.  We associate to the forward prices/rates variable xt, t ≥ 0, two stochastic volatilities v1,t, v2,t, t ≥ 0, one fluctuating on a fast time scale and the other one fluctuating on a long time scale. The dynamics of the stochastic process xt, v1,t, v2,t, t > 0, is defined by the following system of stochastic differential equations:

where the quantities  and , i = 1,2, are real constants such that: bÎ[0,1],  ei > 0, i=1,2, e1 £ e2 .

Moreover Wt0,1, Wt0,2 ,Wt1, Wt2, t>0, are standard Wiener processes such that W00,1= W00,2=W01=W02=0,  dWt0,1, dWt0,2 ,dWt1, dWt2, t>0, are their stochastic differentials and we assume that  <dWt0,1dWt0,2 >= 0,  <dWt0,1dWt1 >= r0,1dt, <dWt0,2dWt2 >= r0,2dt,   <dWt1dWt2>=0 , t>0, r0,1, r0,2Î(-1,1) are  constants  known as correlation coefficients.   The fact that the model is a two scale stochastic volatility model is translated in the assumption that . The equations (34), (35), (36)  must be equipped with the initial conditions:

The random variables , ,   are assumed to be concentrated in a point with probability one.  We  assume that , >0. The assumption ,  >0 with probability one implies that ,  v2,t are positive with probability one for t > 0. Note that the quantities ,  are not observable in the financial market and that they must be regarded as parameters of the model.

The multiscale SABR model (34),  (35),  (36)  generalizes the SABR model (1), (2).

We consider the normal and the lognormal multiscale SABR models. These models are obtained from (34), (35), (36) choosing in (34) respectively b = 0 and b = 1. When we consider the lognormal model we assume  > 0. In the lognormal model the assumption   > 0 with probability one (together with the assumptions ,  >0) implies that xt > 0 with probability one for t > 0. Under the previous assumptions on the correlation structure for the normal and lognormal multiscale SABR models the transition probability density functions of the state variables of these models can be expressed via closed form integral formulae with explicit integrands. In this sense the normal and lognormal models are explicitly solvable.

Let us consider  the normal multiscale SABR model, we have:

with the initial conditions (37), (38), (39).

Let   be the transition probability density function  of the normal multiscale SABR model  (40), (41), (42), (37), (38), (39), that is the probability density of having , , when  , , , t, t³0, t-t>0. Note that when t=0 we must choose , , ,  Starting from  the backward Kolmogorov equation associated to the process defined by (40), (41), (42) , using elementary analysis, in [4]  the following formula is deduced:

where the function  gN is given in  (7).

Note that the transition probability density function of the normal multiscale SABR model given in (43) is expressed as the Fourier transform of the product of two copies of the function gN evaluated in two different points. The two copies of the function gN are coupled through the integration in the variable k.  The variable k is the conjugate variable of the variable x-x.

To deduce the option price formulae that follow from (43) we give the expression of the marginal distribution MMN in the normal multiscale SABR model of the variable xt, t>0, of having xt =x  given that , , , t, t³0, t-t>0, that is:

where mN is the function given in  (11).

Let us consider in the normal multiscale SABR model a European call option expiring at time t=T>0 and having strike price K.  The value CMN of this option  at time t=0 when at time t=0 the underlying prices/rates value is    and  the stochastic volatilities are  , is given by:

Let PMN denote the price in the normal multiscale SABR model at time t=0 of the European put option expiring at time t=T>0 and having strike price K when at time t=0 the underlying prices/rates value is   and the stochastic volatilities are   , , we have:

Note that, as already done in Section 2 and 3, in formulae (45), (46) we have assumed the discount factor equal to one and, as a consequence, the risk free interest rate equal to zero. Moreover we have assumed that the risk free interest rate is deterministic during the lifetime of the options considered.

We conclude this Section studying the lognormal multiscale SABR model, that is model  (34), (35), (36) with b=1. We have:

with the initial conditions (37), (38), (39). When we consider the lognormal multiscale SABR model we assume >0. The assumption >0 with probability one (together with the assumption , >0) implies that  is positive with probability one for t>0. Using the variable , t>0,  we can rewrite the stochastic differential equation (47) as done in (21).

Let   be the transition probability density function  of the lognormal multiscale SABR model (47), (48), (49), (37), (38), (39) expressed in the variable, t>0, that is the probability density of having , , when  , , ,  t, t³0,  t-t>0. Note that when t=0 we must choose , , . Using elementary analysis applied to the backward Kolmogorov equations associated to the process (47), (48), (49) expressed in the variable , t>0, we deduce the following formula (see [4] for further details):

where the function  gL is defined in (26).

To deduce the option price formulae that follow from (50) we give the marginal distribution MML of the variables xt=x,  given that , , ,  t, t³0,  t-t>0, that is:

where mL is the function given in (30).

Let us consider in the lognormal multiscale SABR model a European call option expiring at time t=T>0 and having strike price K>0.  The value CML of this option  at time t=0 when at time t=0 the underlying prices/rates value is   (recall x0=ln(x0 / x0)=0)  and  the stochastic volatilities are  , is given by:

where mL is the function given in (30).

Let PML denote the price in the lognormal multiscale SABR model at time t=0 of the European put option expiring at time t=T>0 and having strike price K>0  when at time t=0 the underlying prices/rates value is   and the stochastic volatilities are    , ,  we have:

Note that as done in Section 2 and 3 in formulae (52), (53) we have assumed the discount factor equal to one and, as a consequence, the risk free interest rate equal to zero. Moreover we have assumed that the risk free interest rate is deterministic during the lifetime of the options considered.

The evaluation of the option pricing formulae of Section 2, 3, 4, that is of the formulae (13), (32), (33), (45), (46), (52), (53), can take advantage from several special features of these formulae. In particular formulae (13), (32), (33), (45), (46), (52), (53) are Fourier integrals that can be computed using the FFT algorithm. Moreover the computational cost of evaluating these formulae for several choices of the strike price K and of the spot price  can be reduced noting that the functions  mN  and mL  given respectively in (11) and (30) that appear in formulae  (13),  (32), (33),  (45), (46), (52), (53)  are independent of the strike price K and of the spot price  so that  they can be computed once and forever on a grid of the k variable and then used to evaluate the option prices for several values of the strike price K and of the spot prices . Recall that formula (14) to price European call options in the normal SABR model can be used instead of formula (13) and that formulae analogous to (14) can be deduced for the remaining option price formulae derived in Section 2, 3, 4.

5.1 Calibration problem

The calibration problems considered in this Section use as data a set of option prices observed at a given time. The option price data are fitted in the least squares sense with the option pricing formulae presented in Section 2,  3  and 4  imposing the constraints on the model parameters that are dictated from their “physical” meaning. In this Section the risk free interest rate r is not assumed to be zero. In the case of the normal and lognormal SABR models the constraints define the set M* of the feasible points as follows:

and in the case of the normal and lognormal  multiscale SABR model the constraints define the set M** of the feasible points as follows:

where R4 and R7 are respectively the four and  seven dimensional real Euclidean space. The calibration problem is formulated as a nonlinear constrained least squares problem whose feasible set is M* or M**. The independent variable of this least squares problem is   (SABR models) or (multiscale SABR models).

Note that the vector  contains the model parameters and the risk free interest rate and that, for simplicity, we use the same symbol  to denote a vector belonging to M* or to M**. The risk free interest rate appears in the discount factor that multiplies the option pricing formulae given in the previous Sections.

When we calibrate the model parameters of the normal SABR model the objective function is defined as the square of the differences between the observed European call and put option prices and the corresponding call and put option prices computed with formulae (14), (16) completed with the appropriate discount factor. The same procedure is used to calibrate all the remaining models presented in Section 3 and 4. That is formulae (32), (33) are used to calibrate the log-normal SABR model and formulae (45), (46) and (52), (53) are used to calibrate respectively the normal and lognormal multiscale SABR models. Formulae (32), (33), (45), (46), (52), (53) are completed with the appropriate discount factor depending on the risk free interest rate that has been omitted in the previous Sections.

5.2 Numerical Results on future prices on the EUR/USD exchange rate

In this  experiment we consider the daily values of the futures price on the EUR/USD currency's exchange rate having maturity September 16th, 2011, (the third Friday of September 2011) (ticker YTU1 Curncy of Figure 1) and the daily prices of the corresponding European call and put options with expiry date September 9th, 2011 and strike prices KC,i = KP,i = Ki = 1.375+0.005*(i-1), i = 1,2,¼,18. The strike prices Ki, i = 1,2,¼, 18, are expressed in USD. These prices are observed in the time period that goes from September 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 20-22 trading days. Figure1 shows the futures price (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 Ki, i = 1,2,¼,18, as a function of time.

Figure 1: YTU1 (blue line) and EUR/USD currency's exchange rate (pink line) versus time.

Figure 2: Call option prices on YTU1 with strike price Ki = 1.375+0.005*(i-1), i = 1,2,¼,18, and expiry date T= September 9th, 2011 versus time.

Figure 3: Put option prices on YTU1 with strike price Ki = 1.375+0.005*(i-1), i = 1,2,¼,18, and expiry date T= September 9th, 2011 versus time.

We use the normal SABR and  multiscale SABR models to study these data.

In particular for these models we solve the calibration problem posed in Section 5.1 for t= tj, j = 1,2,3, where t1 = September 27th, 2010, t2 = October 25th, 2010, t3 = November 4th, 2010 using as data the EUR/USD futures prices (YTU1 ticker of Figure 1) and the prices of the previously mentioned eighteen call and eighteen put options (Figure 2 and Figure 3) available at t = tj, j = 1,2,3. The choice of the dates t= tj, j = 1,2,3, exploits the fact that the option pricing formulae deduced in Section 2,  3 and 4 are closed form formulae. That is these formulae do not involve asymptotic expansions and can be used always. In particular they can be used when the products e2 (T-t), ei2 (T-t), i = 1,2,  where t is the current time and T is the expiry date of the options are not small and this is the case when we consider the choices made previously.

show respectively the parameter values obtained as solution of the calibration problems considered above for the normal SABR and multiscale SABR models.

 T* - date of calibration e r R September 27th, 2010 0.514 0.161 -0.415 0.0 October 25th, 2010 0.844 0.186 -0.355 0.033 November 4th, 2010 0.844 0.181 -0.267 0.019

Table 1: Solution of the calibration problem: normal SABR model

 t*- date of calibration e1 r0,1 R e2 r0,2 September 27th, 2010 0.514 0.143 -0.0164 0.0069 0.8842 0.083 -0.558 October 25th, 2010 0.513 0.164 -0.046 0.0077 0.892 0.087 -0.550 November 4th, 2010 0.507 0.158 -0.084 0.0112 0.894 0.087 -0.561

Table 2: Solution of the calibration problem: normal multiscale SABR model

The calibrated models, that is those with the parameter values given in Table 1 and Table 2, are used to forecast option prices one day, two days,..., up to twenty one days ahead of the observation day of the prices that are used to calibrate the model. Recall that we are considering trading days and that twenty one trading days are approximately one month. The forecasts are made evaluating formulae (45) and (46) when we consider the normal multiscale SABR model and evaluating formulae (14) and (16) when we consider the normal SABR model multiplied by the appropriate discount factor. Note that in the forecasting experiment these formulae are evaluated using as underlying asset value the futures price observed the day of the forecast. The volatilities  and  obtained from the calibration problem are taken as proxies of the volatilities the day of the forecast.

In detail we forecast the option prices day by day up to twenty one (trading) days ahead of t*=tj, j=1,2,3,  (i.e. we consider t*+ Dt, t*+2Dt,...,t*+21Dt , where Dt = one day) using in the option pricing  formulae evaluated at the futures price observed the day of the forecast.

MOVIE 1 is made of twenty one frames and shows the futures prices Figure 1 (window on the upper-left corner), the observed and forecast prices of the eighteen call options considered in Figure 2 (window on the upper-right corner) one day, two days,…, twenty one days ahead of the day of calibration and the corresponding relative errors obtained using the normal SABR (window on the lower-left corner) and normal multiscale SABR  (window on the lower-right corner) models.

MOVIE 2 is made of twenty one frames and shows the futures prices Figure 1 (window on the upper-left corner), the observed and  forecast prices  of the eighteen put options considered in Figure 3 (window on the upper-right corner)  one day, two days,…, twenty one days ahead of the day of calibration and the corresponding relative errors obtained using the normal SABR (window on the lower-left corner) and normal multiscale SABR  (window on the lower-right corner) models..

Note that the mean relative errors in the twenty one day forecasts obtained using the normal multiscale SABR model show that this model guarantees about two significant digits correct in the option prices. This is true also when the options are at the money, that is when the moneyness is close to one. The fact that the normal multiscale model outperforms the normal SABR model is evident when t* = t2, but it is true also when t* = t1  or t* = t3. Figure 4 shows the mean relative errors obtained with the calibration done at t*=t2. Note that when t*=t2  the futures price of the EUR/USD exchange rate is 1.3901 and the options with strike price Ki = 1.375+0.005*(i-1), i = 1,2,¼,6, can be considered at the money.

(a)(b)

Figure 4: Mean relative error on the forecast option prices over a period of one month (21 trading days). The forecasts are obtained  using the model parameters resulting from the calibration at t* = t2 = October 25th 2010 with normal SABR model (a) and normal multiscale SABR model (b) versus strike price (FX experiment).

When t =, i=1,2,3,  the values e1 and e2 of the two volatilities of volatilities resulting from the calibration of the normal multiscale SABR model differ substantially  (i.e. the ratio e2/ e1  is greater than 1.6-1.7,  see Table 1 and Table 2) and  the quality of the forecast option prices obtained using the normal multiscale SABR model is higher than the quality of the forecast option prices obtained using the normal SABR model. We have observed that when the multiscale SABR model improves significantly the quality of the forecast option prices obtained with the SABR model the value of the ratio e2/ e1   of the parameter values resulting from the calibration is greater 1.6-1.7 (see [4]  for further details). In fact in [4] we have repeated this experiment using the same data and the lognormal SABR and multiscale SABR models instead of the normal SABR and multiscale SABR models. We observe that the lognormal SABR model provides forecast option prices of high quality and that the use of the multiscale SABR model does not improve substantially the quality of the forecasts. In the experiment with the lognormal models the ratio e2/ e1 resulting from the calibration is approximately one.

5.3 Numerical Results on the U.S.A. five-Year Interest Rate Swap

In this experiment we consider the daily observed values of the U.S.A. five-Year Interest Rate Swap expressed in per cent (the ticker USSWAP5 curncy in Figure 5(a)), the corresponding futures prices having maturity September 30th, 2011 (the ticker DSU1 in Figure 5(b)) and the prices of the corresponding European call and put options with expiry date September 19th, 2011 and strike prices Ki = 106+0.5*(i-1), i = 1,2,¼,18. These prices are observed in the period going from September 14th, 2010, to July 20th, 2011.  The strike prices Ki, i = 1,2,¼, 18, are expressed in hundreds of base points. For example, Ki = 106 corresponds to an interest rate of 106-100 = 6 hundreds of base points, that is corresponds to an interest rate of 6% per year (see Figure 5).

Note that from November 12th, 2010 to December 15th, 2010 the futures price goes from the value of 110 to the value of 104 hundreds of base points.

(a) (b)

Figure 5: Observed U.S.A. five-Year Interest Rate Swap (a) and the corresponding future price DSU1 having maturity September, 2011 (b) versus time.

Figure 6: Call option prices on DSU1 with strike price Ki = 106+0.5*(i-1), i = 1,2,¼,18, and expiry date T= September 19th, 2011 versus time.

Figure 7: Put option prices on DSU1 with strike price Ki = 106+0.5*(i-1), i = 1,2,¼,18, and expiry date T= September 19th, 2011 versus time.

We use the lognormal SABR and multiscale SABR models to study the data shown in Figure 5, 6, 7 [4].

This study shows that the lognormal  multiscale  SABR model improves substantially the quality of the forecast option prices obtained using the lognormal SABR model and we observe that in this case the ratio e2/ e1  resulting from the calibration is greater than 1.6. That is the experiment on the U.S.A. five-year interest rate swap confirms the finding of the experiment presented in Subsection 5.2.

We present now an experiment that tackles the question of how the results of the calibration reflect the model dynamics.

The experiment consists in calibrating the lognormal models every day for approximately two months,  that is we calibrate  the lognormal models in the period going from September 14th, 2010 to November 15th, 2010 using the daily data of the interest rate swap futures prices considered in this Subsection.

Note that in this period we can observe significant oscillations in the value of the futures prices. In fact, for example, on October 12th, 2010  the futures price (ticker DSU1 Figure 5(b)) is 112.093 and on November  15th, 2010 the futures price is down to 107.875.

Note that in the first part of the period (September 14th, 2010- October 14th, 2010)  the oscillations of the futures price are small

and in the second part  (October 14th, 2010- November 15th, 2010) in particular around the end of October  2010 there is a fall of the futures price (see Figure 5).

The parameter values obtained from the calibration of the lognormal models are shown in Figure 8. We can see that the values of the

parameters remain substantially unchanged in the two months period except for the values of the parameters e and e2 that show a significant

change at the end of October 2010 and during the first fifteen days of November 2010. Moreover we use the parameter values obtained with the calibration of the lognormal SABR and  multiscale SABR models to compute the forecast option prices one day in the future during the two months period mentioned above. Movie 3 and Movie 4 show the forecast option prices, the observed option prices and the relative errors. The two movies show that the use of the lognormal multiscale SABR model instead of the lognormal SABR model improves substantially the forecast prices.

Figure 8: Parameter values obtained calibrating the lognormal SABR and multiscale SABR models every day for two months in the period going from September 14th, 2010 to November 15th, 2010 versus time

## References

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