Expansion method for pricing foreign exchange options under stochastic volatility and interest rates
EExpansion method for pricing foreign exchange optionsunder stochastic volatility and interest rates
Kenji Nagami † ‡
August 26, 2019
Abstract
Some expansion methods have been proposed for approximately pricing options whichhas no exact closed formula. Benhamou et al. (2010) presents the smart expansion methodthat directly expands the expectation value of payoff function with respect to the volatilityof volatility, then uses it to price options in the stochastic volatility model. In this paper,we apply their method to the stochastic volatility model with stochastic interest rates,and present the expansion formula for pricing options up to the second order. Then thenumerical studies are performed to compare our approximation formula with the Monte-Carlo simulation. It is found that our formula shows the numerically comparable resultswith the method proposed by Grzelak et al. (2012) which uses the approximation ofcharacteristic function. † Mitsubishi UFJ Trust Investment Technology Institute Co., Ltd. (MTEC), 4-2-6 Akasaka,Minato city, Tokyo 107-0052, Japan. ‡ The views expressed here are those of the author and do not represent the official viewsof the company. a r X i v : . [ q -f i n . C P ] A ug Introduction
The foreign exchange rate is expected to move depending on the interest rate of each currency.The higher interest rate currency is expected to be exchanged to the lower interest rate currencyat the lower exchange rate in a future than the spot exchange rate in the arbitrage free market.It is plausible to consider the FX model with stochastic interest rates. In fact, it would becertainly needed to price a kind of derivatives including exotic ones with payoff depending onboth FX rate and interest rates. One of those models is the so-called Heston-Hull-White modelthat represents the FX rate by the Heston model and incorporates the stochastic interest ratesdescribed by the Hull-White models. This FX model is thought to have no closed formula forpricing plain vanilla options, so it will be practically useful if a fast pricing formula is available.In this paper, we consider the option pricing problem in the Heston-Hull-White modeland present the approximation formula by using the smart expansion method. This method ispresented in [3] to obtain the approximation formula in the time dependent Heston model withthe deterministic interest rates. The expected value of payoff function is directly expanded withrespect to the volatility of volatility (vol-of-vol) and explicitly calculated by taking advantage ofthe Malliavin calculus. The smart expansion method appears to follow the apparently differentway from the asymptotic expansion method which expands the density function of underlyingasset, but both methods are likely to share a substantial part of concept as expanding byvol-of-vol and using the Malliavin calculus.The asymptotic expansion method is based on [15], [16], [7] and has been developed bymany authors including [17], [12], [13]. It is naturally applied to the option pricing problemin the Heston model and one can find the derivation of the approximation formula up to thesecond order in [14]. Other expansion methods applied for pricing options in the Heston modelcan be also found in some papers including [8], [1] and [9].The Heston model has the closed form expression of characteristic function, by which onecan exactly price plain vanilla options. For the Heston-Hull-White model, the approximationformula of characteristic function is obtained in [5] and they apply it to pricing options. Ourproposed method is also compared with their method in the numerical studies.This paper is organized as follows. Section 2 reviews the smart expansion method of [3]that is used for pricing options in the Heston model with the deterministic interest rates.In Section 3, we apply the expansion method to the Heston-Hull-White model and derive theapproximation formula for pricing options up to the second order of vol-of-vol. Section 4 showsthe results of numerical studies where our proposed method is compared with the Mote-Carlosimulation as benchmark. Section 5 gives the conclusion. The appendices give the results ofrepresentative calculations needed to derive and evaluate our formula.1
Case of deterministic interest rates
In this section, we briefly review Benhamou et al. (2010) [3] that uses the expansion methodto derive the approximation formula for pricing options in the Heston model under the deter-ministic interest rates.The Heston model can be described with the FX spot rate observed at time t denoted as S (cid:15)t and the stochastic variance v (cid:15)t that follows a CIR process. dS (cid:15)t = S (cid:15)t ( r d − r f ) dt + S (cid:15)t (cid:112) v (cid:15)t dW Qst , S (cid:15) = S , (1) dv (cid:15)t = k v ( θ v − v (cid:15)t ) dt + (cid:15)γ (cid:112) v (cid:15)t dW Qvt , v (cid:15) = v . (2) W Qst , W
Qvt are Brownian motions with correlation ρ sv on the domestic risk-neutral measure Q . r d , r f are the domestic and foreign interest rates respectively and assumed to be constantor deterministic. The FX rate is measured as amount of domestic currency exchanged with aunit of foreign currency.The stochastic differential equations are perturbed with a parameter (cid:15) ∈ [0 ,
1] whichaccompanies with the volatility of volatility (vol-of-vol) γ and is used to expand the optionpremium and specify the order of vol-of-vol.The FX forward rate with maturity T fixed is evaluated at t as F (cid:15)t = S (cid:15)t e (cid:82) Tt ( r d − r f ) ds , (3)which converges with the spot rate at the maturity T . The forward rate is martingale under Q . dF (cid:15)t /F (cid:15)t = (cid:112) v (cid:15)t dW Qst . (4)The plain vanilla put option with the maturity T and the strike K and a unit notional inforeign currency is considered and the price in the domestic currency is evaluated by using theexpectation on Q . P V ( (cid:15) ) = D d ( T ) E (cid:2) ( K − F (cid:15)T ) + (cid:3) , D d ( T ) = e − (cid:82) T r d dt . (5)If it is conditioned with the filtration F v generated by the volatility v (cid:15)t , the log forwardrate at the maturity is normally distributed. The relevant parameters are explicitly written as x (cid:15) ≡ E [log F (cid:15)T |F v ] + 12 V ar [log F (cid:15)T |F v ]= log F − (cid:90) T ρ sv v (cid:15)t dt + (cid:90) T ρ sv (cid:112) v (cid:15)t dW vt , (6) y (cid:15) ≡ V ar [log F (cid:15)T |F v ]= (cid:90) T (1 − ρ sv ) v (cid:15)t dt. (7)2he equation (5) can be written by using the conditional expectation with F v , which isexplicitly evaluated by the Black-Scholes formula. P V ( (cid:15) ) = D d ( T ) E (cid:2) E (cid:2) ( K − F (cid:15)T ) + |F v (cid:3)(cid:3) = E [ BS ( x (cid:15) , y (cid:15) )] . (8) BS ( x, y ) = D d ( T )( K Φ( − d ) − e x Φ( − d )) , d = x − log K + y/ √ y , d = d − √ y, (9)where Φ( x ) = (cid:82) x −∞ φ ( t ) dt, φ ( x ) = e − x √ π . This satisfies the following formula, which is useful toarrange the expansion formula later on. ∂BS ( x, y ) ∂y = 12 (cid:18) ∂ BS ( x, y ) ∂x − ∂BS ( x, y ) ∂x (cid:19) . (10)The option price can be expressed by expanding the BS term with respect to (cid:15) up to thesecond order as P V ( (cid:15) ) = E (cid:2) BS ( x (0) , y (0) ) (cid:3) + E (cid:20) ∂∂x BS ( x (0) , y (0) ) (cid:18) (cid:15)x (1) + (cid:15) x (2) (cid:19)(cid:21) + E (cid:20) ∂∂y BS ( x (0) , y (0) ) (cid:18) (cid:15)y (1) + (cid:15) y (2) (cid:19)(cid:21) + 12 E (cid:20) ∂ ∂x BS ( x (0) , y (0) ) (cid:15) x (cid:21) + 12 E (cid:20) ∂ ∂y BS ( x (0) , y (0) ) (cid:15) y (cid:21) + E (cid:20) ∂ ∂x∂y BS ( x (0) , y (0) ) (cid:15) x (1) y (1) (cid:21) + o ( (cid:15) ) , (11)where the arguments of BS formula x (cid:15) , y (cid:15) are expanded as x (cid:15) = x (0) + (cid:15)x (1) + (cid:15) x (2) + o ( (cid:15) ) , (12) y (cid:15) = y (0) + (cid:15)y (1) + (cid:15) y (2) + o ( (cid:15) ) . (13)The approximation formula of option price is obtained from (11) by collecting all of the termsup to the second order and replacing (cid:15) with 1.To specify the expansion coefficients, it is needed to expand the volatility v (cid:15)t with respectto (cid:15) . Assuming that the volatility is expanded as v (cid:15)t = v ,t + (cid:15)v ,t + (cid:15) v ,t + · · · , (14)then equating each side of the equation (2) with the same order of (cid:15) produces SDEs for theexpansion coefficients. dv ,t = k v ( θ v − v ,t ) dt, v , = v , (15) dv ,t = − k v v ,t dt + γ √ v ,t dW v,t , v , = 0 , (16)3 v ,t = − k v v ,t dt + γ v ,t √ v ,t dW v,t , v , = 0 . (17)These are solved to give the integral form representations. v ,t = θ v + ( v − θ v ) e − k v t , (18) v ,t = γe − k v t (cid:90) t e k v u √ v ,u dW v,u , (19) v ,t = γe − k v t (cid:90) t e k v u v ,u √ v ,u dW v,u . (20)The expansion coefficients x ( i ) , y ( i ) , i = 0 , , v = θ v hereafter.The zeroth order term is evaluated as the limit of vol-of-vol γ → E (cid:2) BS ( x (0) , y (0) ) (cid:3) = D d ( T ) E (cid:104)(cid:0) K − F T (cid:1) + (cid:105) = BS ( x , y ) , x = log F , y = v T. (21)Note that the BS formula is expanded around the parameters x (0) , y (0) inside the expectation,then the slightly different parameters x , y are used after evaluating directly the unconditionalexpectation. This equation can be also extended for derivatives of BS formula. E (cid:20) ∂ i + j ∂x i ∂y j BS ( x (0) , y (0) ) (cid:21) = ∂ i + j ∂x i ∂y j BS ( x , y ) , i, j = 0 , , . . . . (22)The higher order terms in the expansion formula (11) can be explicitly calculated by takingadvantage of the Malliavin calculus. Based on the lemma 1.2.1 in [10], the following lemmafor the Brownian motion W t is derived in [3]. E (cid:20) G (cid:18)(cid:90) T g ( t ) dW t (cid:19) (cid:90) T µ t dW t (cid:21) = E (cid:20) G (1) (cid:18)(cid:90) T g ( t ) dW t (cid:19) (cid:90) T g ( t ) µ t dt (cid:21) , (23)where G is a smooth function and g is a deterministic function and µ t is a square integrable andpredictable process. In the current case, G is identified with the BS formula or its derivativesand the argument of G corresponds to the stochastic integral appeared in x (0) with g = ρ sv √ v ,so the derivative of G is performed with respect to x .Finally the approximation formula up to the second order of vol-of-vol is obtained. P Happrox ( x , y ) = BS ( x , y ) + ρ sv v γ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u + ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s + v γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u (cid:90) Tu dse − k v s + 12 ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:18)(cid:90) T dte k v t (cid:90) Tt due − k v u (cid:19) . (24)4 Heston-Hull-White model
The previous section assumes the interest rates to be constant or deterministic. In this section,we consider the FX model under the stochastic interest rates, assuming that the domestic shortrate r dt and the foreign short rate r ft follow the Hull-White models. The FX spot rate S (cid:15)t follows the Heston model as before. So it is called the Heston-Hull-White model.The stochastic differential equations are written under the domestic risk-neutral measure Q as dS (cid:15)t = S (cid:15)t ( r dt − r ft ) dt + S (cid:15)t (cid:112) v (cid:15)t dW Qst , S (cid:15) = S , (25) dv (cid:15)t = k v ( θ v − v (cid:15)t ) dt + (cid:15)γ (cid:112) v (cid:15)t dW Qvt , v (cid:15) = v , (26) dr dt = k d ( θ d − r dt ) dt + η d dW Qdt , (27) dr ft = ( k f ( θ f − r ft ) − η f ρ Sf (cid:112) v (cid:15)t ) dt + η f dW Qft . (28)These equations are perturbed with a parameter (cid:15) ∈ [0 ,
1] next to the volatility of volatility(vol-of-vol) γ , which is again used to expand the option premium. The foreign interest ratehas an additional drift term originating from the measure change from the foreign risk-neutralmeasure to the domestic one. The parameters θ d , θ f are deterministic functions of time andto be determined with the initial values r d , r f by using the observed curves.The FX forward rate with maturity T fixed is evaluated at t as F (cid:15)t = S (cid:15)t P f ( t, T ) P d ( t, T ) , (29)where the price of the domestic discount bond with maturity T observed at t is denoted by P d ( t, T ), and the foreign one is denoted as well.As the previous section, the put option with the maturity T and the strike K and a unitnotional in foreign currency is considered and priced in the domestic currency. It is convenientto express the put option premium in terms with the forward rate and the domestic forwardmeasure Q T which uses P d ( t, T ) as the num´eraire. P V
HHW ( (cid:15) ) = E (cid:104) e − (cid:82) T dtr dt ( K − S (cid:15)T ) + (cid:105) = D d ( T ) E T (cid:2) ( K − F (cid:15)T ) + (cid:3) . (30)The expectation symbol with the superscript T means to be evaluated under Q T . D d ( T ) = P d (0 , T ) is the domestic discount bond price with maturity T observed at t = 0.The forward rate is martingale under Q T as explicitly derived in [5]. dF (cid:15)t /F (cid:15)t = (cid:112) v (cid:15)t dW Tst − η d B d ( t, T ) dW Tdt + η f B f ( t, T ) dW Tft ≡ σ F ( t, v (cid:15)t ) dW TF t , (31)where the deterministic functions specific to the Hull-White model are defined by B d ( t, T ) = 1 k d ( e − k d ( T − t ) − , B f ( t, T ) = 1 k f ( e − k f ( T − t ) − . (32)5he variance of forward rate is explicitly written as σ F ( t, v (cid:15)t ) = v (cid:15)t + η d B d ( t, T ) + η f B f ( t, T ) − ρ Sd η d B d ( t, T ) (cid:112) v (cid:15)t +2 ρ Sf η f B f ( t, T ) (cid:112) v (cid:15)t − ρ df η d η f B d ( t, T ) B f ( t, T ) . (33)If the correlation between forward rate and volatility is expressed as dW TF t dW Tvt = ρ F v ( t, v (cid:15)t ) dt ,it satisfies σ F ( t, v (cid:15)t ) ρ F v ( t, v (cid:15)t ) = ρ Sv (cid:112) v (cid:15)t − η d ρ vd B d ( t, T ) + η f ρ vf B f ( t, T ) . (34)The equation (31) includes no term that is explicitly dependent on the short rates. If theinterest rates are described by other models, the short rates may appear in the equation.In the current case, the equation shows that the log forward rate at the maturity is normallydistributed with the filtration F v conditioned. Assuming that the expectations are replacedwith those under Q T , the option price is evaluated as in the equation (8) and expanded as inthe formula (11). The arguments of BS formula are expressed as x (cid:15) ≡ E T [log F (cid:15)T |F v ] + 12 V ar T [log F (cid:15)T |F v ]= log F − (cid:90) T σ F ( t, v (cid:15)t ) ρ F v ( t, v (cid:15)t ) dt + (cid:90) T σ F ( t, v (cid:15)t ) ρ F v ( t, v (cid:15)t ) dW Tvt , (35) y (cid:15) ≡ V ar T [log F (cid:15)T |F v ]= (cid:90) T σ F ( t, v (cid:15)t )(1 − ρ F v ( t, v (cid:15)t )) dt. (36)The SDE of volatility is expressed under Q T as dv (cid:15)t = ( k v ( θ v − v (cid:15)t ) + (cid:15)γρ vd η d B d ( t, T ) (cid:112) v (cid:15)t ) dt + (cid:15)γ (cid:112) v (cid:15)t dW Tvt , v (cid:15) = v , (37)where the additional drift term is appeared through the measure change from Q to Q T . Weassume no correlation between the FX volatility and the interest rates ρ vd = 0 , ρ vf = 0hereafter, since they appear to be redundant in the usual practical case. The issue about theirroles would remain as a future subject. Then the volatility has the same expansion coefficientsas in Q . It is also assumed that v = θ v as before.The expansion coefficients as defined in the equations (12) (13) are explicitly written here. x (0) = log F − ρ sv v T + ρ sv √ v (cid:90) T dW Tv,t , (38) x (1) = − ρ sv (cid:90) T v ,t dt + ρ sv √ v (cid:90) T v ,t dW Tv,t , (39) x (2) = − ρ sv (cid:90) T v ,t dt + ρ sv √ v (cid:90) T v ,t dW Tv,t − ρ sv v / (cid:90) T v ,t dW Tv,t , (40) y (0) = (cid:90) T σ F ( t, v )(1 − ρ F v ( t, v )) dt, (41) y (1) = (cid:90) T (cid:0) − ρ sv + α ( t ) (cid:1) v ,t dt (42)6 (2) = (cid:90) T (cid:0) − ρ sv + α ( t ) (cid:1) v ,t dt − v (cid:90) T α ( t ) v ,t dt, (43)where we note that the additional terms specific to the stochastic interest rates are appeared, α ( t ) ≡ ρ sd η d √ v − e − k d ( T − t ) k d − ρ sf η f √ v − e − k f ( T − t ) k f . (44)The zeroth order term is evaluated as the limit of vol-of-vol γ → E T (cid:2) BS ( x (0) , y (0) ) (cid:3) = D d ( T ) E T (cid:104)(cid:0) K − F T (cid:1) + (cid:105) = BS ( x , y ) , x = log F , y = (cid:90) T σ F ( t, v ) dt. (45)This equation corresponds to the extended formula of (21) to allow for the stochastic interestrates and is evaluated by the expectation under Q T . It can be also extended to the cases forderivatives of BS formula as in the equation (22) with y defined above. The arguments of theBS formula may be abbreviated.The variance integral of the forward rate is explicitly written as (cid:90) T σ F ( t, v ) dt = v T + η d (cid:18) Tk d − k d + 2 e − k d T k d − e − k d T k d (cid:19) + η f (cid:32) Tk f − k f + 2 e − k f T k f − e − k f T k f (cid:33) − √ v ρ sd η d (cid:18) − Tk d + 1 − e − k d T k d (cid:19) + 2 √ v ρ sf η f (cid:32) − Tk f + 1 − e − k f T k f (cid:33) − ρ df η d k d η f k f (cid:32) T + 1 − e − ( k d + k f ) T k d + k f − − e − k d T k d − − e − k f T k f (cid:33) . (46)In what follows, we show the expressions of higher order terms which are evaluated byinvoking the brute force calculation and arranged by using the equation (10). The results forthe representative terms are summarized in the appendix A. The partial derivatives may beabbreviated as ∂ nx = ∂ n ∂x n , ∂ ny = ∂ n ∂y n , n = 1 , , . . . . The integrals may be represented in themanner that the derivative symbol for variable of integration is set next to the integral as (cid:82) dxf ( x ) = (cid:82) f ( x ) dx , so that it would be easy to read the integration interval of each variableespecially for multiple integrals. The abbreviation β ( t ) ≡ − ρ sv + α ( t ) is also used. E T (cid:2) ∂ x BSx (1) (cid:3) = ρ sv v γ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u . (47) E T (cid:2) ∂ y BSy (1) (cid:3) = ρ sv v γ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u (1 − ρ sv + α ( u )) . (48) E T (cid:2) ∂ x BSx (2) (cid:3) = ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s ρ sv γ ∂ x BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u − ρ sv v γ ∂ x BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due k v u (cid:90) Tu dse − k v s . (49) E T (cid:2) ∂ y BSy (2) (cid:3) = ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s (1 − ρ sv + α ( s )) − γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u α ( u ) − ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due k v u (cid:90) Tu dse − k v s α ( s ) . (50) E T (cid:104) ∂ x BSx (cid:105) = ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:18)(cid:90) T dte k v t (cid:90) Tt due − k v u (cid:19) + 2 ρ sv v γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u (cid:90) Tu dse − k v s + ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s + 12 ρ sv v γ ∂ x BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due k v u (cid:90) Tu dse − k v s + 14 ρ sv γ ∂ x BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u . (51) E T (cid:104) ∂ y BSy (cid:105) = ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:18)(cid:90) T dte k v t (cid:90) Tt due − k v u β ( u ) (cid:19) + 2 v γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u β ( u ) (cid:90) Tu dse − k v s β ( s ) . (52) E T (cid:2) ∂ x ∂ y BSx (1) y (1) (cid:3) = ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:18)(cid:90) T dte k v t (cid:90) Tt due − k v u (cid:19) (cid:18)(cid:90) T dse k v s (cid:90) Ts dre − k v r β ( r ) (cid:19) + ρ sv v γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u (cid:90) Tu dse − k v s β ( s )+ ρ sv v γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u β ( u ) (cid:90) Tu dse − k v s ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s β ( s ) . (53)These expressions are to be substituted into the right hand side of equation (11) withexpectations under Q T used and (cid:15) = 1. Then we can finally obtain the approximation formulato evaluate the price up to the second order of vol-of-vol. This can be calculated by using theexplicit expressions summarized in the appendix B. P HHWapprox ( x , y ; α ) = BS ( x , y ) + ρ sv v γ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u (1 + α ( u ))+ ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s (1 + α ( s ))+ v γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u (1 + α ( u )) (cid:90) Tu dse − k v s (1 + α ( s ))+ 12 ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:18)(cid:90) T dte k v t (cid:90) Tt due − k v u (1 + α ( u )) (cid:19) − ρ sv v γ ∂ x ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due k v u (cid:90) Tu dse − k v s α ( s ) − γ ∂ y BS ( x , y ) (cid:90) T dte k v t (cid:90) Tt due − k v u α ( u ) . (54)The effect of stochastic interest rates is incorporated in the α dependent terms and y specified in the equation (46). If the interest rates are deterministic, we obtain the formulawith α terms disappeared and y replaced with v T , which corresponds with the previoussection’s result, P HHWapprox ( x , v T ; 0) = P Happrox ( x , v T ).The pure effect of stochastic interest rates is determined by∆ P approx = P HHWapprox ( x , y ; α ) − P Happrox ( x , v T ) . (55)If the interest rates are deterministic, the option price is accurately calculated by usingthe characteristic function of the Heston model and it is denoted as P ChF . Therefore we canpresent the other approximation formula that is defined by the hybrid of the expansion methodand the characteristic function method.˜ P HHWapprox ( x , y ; α ) = P ChF + P HHWapprox ( x , y ; α ) − P Happrox ( x , v T ) (56)It also appears that the price of pure Heston model takes a role as control variate. In this section, we study the accuracy of the approximation formula (54) (56) against theMonte-Carlo simulation using the QE scheme in [2], which we use as benchmark. We alsocompare the accuracy with another method presented in [4], [5], which uses the approximationof characteristic function. 9he model parameters are set to be the hypothetical values in [3], [5]. The interest ratesmodel parameters are η d = 0 . , η f = 1 . , k d = 1% , k f = 5% , (57)and zero rates are 0% for each currency. We could also use any zero rates, which simply changethe forward rate and the deterministic part of FX drift term in the MC simulation.The FX model parameters are v = 0 . , γ = 0 . , k v = 3 . (58)The correlation matrix is ρ sv ρ sd ρ sf ρ vd ρ vf ρ df = − . − . − .
151 0 01 0 . , (59)where we assume no correlation between the FX volatility and the interest rates ρ vd = ρ vf = 0as explained before, though [5] can approximately allow for finite correlations.The Monte-Carlo simulation is performed with the number of scenarios 10 and the timeinterval 0 .
05 year. The initial FX forward rate is F = 100. We evaluate the put options withthe maturities T = 1 , , , ,
10 years and the unit notional in the foreign currency. The strikesare based on [11] and given by K i ( T ) = F exp(0 . δ i √ T ) , { δ i } i =1 ,..., = {− . , − , − . , , . , , . } . (60)The implied volatilities and the prices of put options are shown in Table 1 and Table 2respectively. It is found that the differences of implied volatilities from the Monte-Carlosimulation are a few basis points for each method (1bp = 0.01%). The expansion basedmethods have a comparable accuracy with the approximate characteristic function method.The difference of implied volatility is appeared to grow for long maturity, but it is also foundthat the difference of price remains to be comparable with the standard error of the Monte-Carlo simulation. It is shown in [3] that the price error is estimated as o ( γ T ) for theexpansion method with the deterministic interest rates, so we can expect that the expansionmethod with the stochastic interest rates may have the equal or larger price error. Theapproximate characteristic function method is obtained by replacing the non-affine √ v termswith their expectation values in the Kolmogorov backward equation, so it is likely to have abetter accuracy for shorter maturity. 10 Method K1 K2 K3 K4 K5 K6 K71Y MC 22.94% 22.60% 22.27% 21.96% 21.66% 21.39% 21.15%Exp -0.01% -0.02% -0.03% -0.03% -0.03% -0.03% -0.04%ExpChF -0.03% -0.02% -0.01% -0.01% -0.00% +0.01% +0.01%AppChF -0.03% -0.02% -0.01% -0.01% -0.00% +0.01% +0.02%3Y MC 22.90% 22.66% 22.43% 22.20% 21.98% 21.77% 21.57%Exp -0.01% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02%ExpChF -0.02% -0.02% -0.02% -0.01% -0.00% -0.00% +0.01%AppChF -0.02% -0.02% -0.02% -0.01% -0.00% +0.00% +0.01%5Y MC 23.02% 22.83% 22.64% 22.45% 22.27% 22.11% 21.94%Exp -0.03% -0.04% -0.03% -0.02% -0.02% -0.02% -0.02%ExpChF -0.04% -0.04% -0.03% -0.02% -0.01% -0.01% +0.00%AppChF -0.04% -0.04% -0.03% -0.02% -0.01% -0.00% +0.01%7Y MC 23.19% 23.02% 22.87% 22.71% 22.56% 22.42% 22.28%Exp -0.02% -0.02% -0.01% -0.01% -0.00% -0.00% -0.01%ExpChF -0.02% -0.02% -0.01% -0.00% +0.00% +0.00% +0.01%AppChF -0.02% -0.02% -0.01% -0.00% +0.01% +0.01% +0.01%10Y MC 23.53% 23.40% 23.27% 23.14% 23.02% 22.90% 22.78%Exp -0.00% +0.00% +0.01% +0.01% +0.02% +0.03% +0.03%ExpChF -0.01% +0.00% +0.01% +0.02% +0.03% +0.03% +0.04%AppChF -0.01% +0.00% +0.01% +0.02% +0.03% +0.04% +0.05%
Table 1: Implied volatilities of put options calculated by the Monte-Carlo simulation (MC)and the differences from MC for the expansion method (Exp), the expansion & characteris-tic function hybrid method (ExpChF) and the approximate characteristic function method(AppChF). Strikes K i , i = 1 , . . . , T Method K1 K2 K3 K4 K5 K6 K71Y MC 3.27 4.63 6.43 8.74 11.64 15.18 19.38s.e. (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02)Exp -0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01ExpChF -0.01 -0.01 -0.01 -0.00 -0.00 +0.00 +0.00AppChF -0.01 -0.01 -0.01 -0.00 -0.00 +0.00 +0.013Y MC 5.31 7.72 10.97 15.24 20.70 27.50 35.74s.e. (0.01) (0.01) (0.02) (0.02) (0.02) (0.03) (0.03)Exp -0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01ExpChF -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 +0.00AppChF -0.01 -0.01 -0.01 -0.01 -0.00 +0.00 +0.015Y MC 6.62 9.78 14.09 19.82 27.26 36.65 48.23s.e. (0.01) (0.02) (0.02) (0.02) (0.03) (0.03) (0.04)Exp -0.02 -0.03 -0.03 -0.02 -0.02 -0.02 -0.01ExpChF -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 +0.00AppChF -0.02 -0.03 -0.02 -0.01 -0.01 -0.00 +0.007Y MC 7.65 11.42 16.62 23.61 32.79 44.53 59.21s.e. (0.01) (0.02) (0.02) (0.03) (0.03) (0.04) (0.04)Exp -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.01ExpChF -0.02 -0.01 -0.01 -0.00 +0.00 +0.00 +0.01AppChF -0.02 -0.01 -0.01 -0.00 +0.01 +0.01 +0.0110Y MC 8.97 13.51 19.88 28.56 40.10 55.10 74.19s.e. (0.01) (0.02) (0.02) (0.03) (0.04) (0.04) (0.05)Exp -0.00 +0.00 +0.01 +0.02 +0.03 +0.04 +0.04ExpChF -0.00 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05AppChF -0.00 +0.00 +0.01 +0.02 +0.04 +0.05 +0.06
Table 2: Prices of put options calculated by the Monte-Carlo simulation (MC) with the stan-dard errors(s.e.), the differences from MC for the expansion method (Exp), the expansion &characteristic function hybrid method (ExpChF) and the approximate characteristic functionmethod (AppChF). 11he expansion methods assume that the vol-of-vol is small enough to expand the optionprice with respect to it. It is observed that the difference tends to grow with the vol-of-volraised up in Table 3 where ATM options are used. We note that the expansion & characteristicfunction hybrid method has smaller difference than the pure expansion method. The expansion& ChF hybrid method partially takes advantage of the characteristic function method for thecase of deterministic interest rates, so it is plausible that the hybrid method has the higheraccuracy than the pure expansion one for high vol-of-vol cases.Note that the approximate characteristic function method is assumed to use for the lowvol-of-vol cases. The method replaces the non-affine √ v terms in PDE with the expectationvalue, which is evaluated by the approximation formula subject to 8 k v ¯ v ≥ γ in [4]. Hence itis not simply applicable to the high vol-of-vol cases. We would expect that the expansion &ChF hybrid method can be practically used even for the somewhat high vol-of-vol cases.In Table 4, we report the cases with the interest rate volatility raised up. The differenceof implied volatility of options can be grown up as the interest rate volatility increases. It isobserved that the difference remains to be a few basis points if the interest rate volatility is afew percent.As for the computational time, it takes 36 ms for the pure expansion method to evaluate35 prices in Table 2, while it takes 845 ms for the approximate characteristic function method.The Core i3 CPU 2.0GHz is used. The pure expansion method is faster than the approxi-mate characteristic function method by a factor 20. The expansion & ChF hybrid methodtakes 855 ms, which is comparable to the approximate characteristic function method. Themethods using the characteristic function need the numerical integration involving exponentialfunctions and trigonometric functions, which dominates the computational cost. The order ofcomputational time for each method would be schematically written as(Heston/Exp) < (HHW/Exp) (cid:28) (Heston/ChF) < (HHW/AppChF) ∼ (HHW/ExpChF) , where the left/right hand sides stand for the used model/method respectively.12 Method 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11Y MC 22.32% 22.17% 21.96% 21.69% 21.39% 21.05% 20.70% 20.33% 19.96% 19.58%Exp +0.00% -0.01% -0.03% -0.06% -0.13% -0.24% -0.40% -0.63% -0.91% -1.27%ExpChF +0.00% -0.00% -0.01% -0.02% -0.02% -0.02% -0.03% -0.03% -0.03% -0.03%AppChF +0.00% -0.00% -0.01% -0.01% -0.02% -0.02% -0.02% -0.03% -0.03% -0.03%3Y MC 22.48% 22.35% 22.20% 22.01% 21.80% 21.56% 21.31% 21.05% 20.78% 20.51%Exp -0.01% -0.01% -0.02% -0.04% -0.07% -0.13% -0.20% -0.31% -0.45% -0.62%ExpChF -0.01% -0.01% -0.01% -0.01% -0.01% -0.01% -0.01% -0.01% -0.02% -0.02%AppChF -0.01% -0.01% -0.01% -0.01% -0.01% -0.01% -0.01% -0.02% -0.02% -0.03%5Y MC 22.69% 22.58% 22.45% 22.30% 22.13% 21.94% 21.74% 21.53% 21.31% 21.08%Exp -0.02% -0.02% -0.02% -0.03% -0.05% -0.08% -0.12% -0.18% -0.25% -0.35%ExpChF -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02%AppChF -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.02% -0.03% -0.05%7Y MC 22.92% 22.83% 22.71% 22.58% 22.43% 22.27% 22.10% 21.92% 21.73% 21.54%Exp -0.00% -0.00% -0.01% -0.02% -0.03% -0.05% -0.08% -0.12% -0.17% -0.24%ExpChF -0.00% -0.00% -0.00% -0.01% -0.01% -0.01% -0.01% -0.01% -0.02% -0.03%AppChF -0.00% -0.00% -0.00% -0.00% -0.00% -0.01% -0.01% -0.02% -0.03% -0.06%10Y MC 23.34% 23.25% 23.14% 23.02% 22.89% 22.76% 22.62% 22.47% 22.31% 22.15%Exp +0.01% +0.01% +0.01% +0.01% +0.00% -0.01% -0.04% -0.07% -0.11% -0.16%ExpChF +0.01% +0.01% +0.02% +0.02% +0.02% +0.01% +0.00% -0.01% -0.02% -0.03%AppChF +0.01% +0.02% +0.02% +0.02% +0.02% +0.01% +0.00% -0.01% -0.04% -0.07%
Table 3: Implied volatilities of ATM put options with vol-of-vol specified on the top row andother parameters unchanged. The Monte-Carlo simulation (MC) means the implied volatilityof it and the differences from MC are displayed for the expansion method (Exp), the expan-sion & characteristic function hybrid method (ExpChF) and the approximate characteristicfunction method (AppChF).
T Method 1% 2% 3% 4% 5%1Y MC 21.94% 21.89% 21.85% 21.82% 21.81%Exp -0.03% -0.03% -0.03% -0.03% -0.03%ExpChF -0.01% -0.01% -0.01% -0.01% -0.01%AppChF -0.01% -0.01% -0.02% -0.02% -0.02%3Y MC 22.16% 22.10% 22.18% 22.39% 22.72%Exp -0.02% -0.02% -0.02% -0.02% -0.01%ExpChF -0.01% -0.01% -0.01% -0.01% +0.00%AppChF -0.01% -0.02% -0.02% -0.02% -0.01%5Y MC 22.41% 22.49% 22.93% 23.69% 24.76%Exp -0.02% -0.02% -0.01% +0.00% +0.02%ExpChF -0.02% -0.01% -0.01% +0.01% +0.02%AppChF -0.02% -0.02% -0.02% -0.01% +0.01%7Y MC 22.68% 23.03% 24.02% 25.59% 27.62%Exp -0.01% +0.00% +0.02% +0.04% +0.06%ExpChF -0.00% +0.01% +0.02% +0.04% +0.07%AppChF -0.00% +0.00% +0.01% +0.03% +0.05%10Y MC 23.17% 24.13% 26.25% 29.28% 32.96%Exp +0.02% +0.03% +0.05% +0.08% +0.11%ExpChF +0.02% +0.04% +0.06% +0.08% +0.11%AppChF +0.02% +0.03% +0.04% +0.06% +0.09% (a) domestic currency
T Method 1% 2% 3% 4% 5%1Y MC 21.94% 22.03% 22.13% 22.25% 22.37%Exp -0.03% -0.02% -0.02% -0.01% -0.00%ExpChF -0.01% -0.01% -0.00% +0.01% +0.01%AppChF -0.01% -0.00% +0.00% +0.01% +0.02%3Y MC 22.13% 22.49% 22.96% 23.53% 24.20%Exp -0.02% -0.02% -0.01% +0.01% +0.02%ExpChF -0.01% -0.01% +0.00% +0.02% +0.03%AppChF -0.01% -0.00% +0.01% +0.03% +0.04%5Y MC 22.33% 23.05% 24.04% 25.25% 26.67%Exp -0.03% -0.02% -0.00% +0.02% +0.04%ExpChF -0.02% -0.01% +0.00% +0.02% +0.04%AppChF -0.02% -0.00% +0.01% +0.03% +0.06%7Y MC 22.52% 23.66% 25.25% 27.21% 29.47%Exp -0.01% +0.00% +0.02% +0.04% +0.06%ExpChF -0.01% +0.00% +0.02% +0.04% +0.06%AppChF -0.00% +0.01% +0.03% +0.06% +0.08%10Y MC 22.84% 24.68% 27.25% 30.36% 33.87%Exp +0.01% +0.03% +0.05% +0.07% +0.10%ExpChF +0.01% +0.03% +0.05% +0.07% +0.10%AppChF +0.02% +0.04% +0.06% +0.09% +0.12% (b) foreign currency
Table 4: Implied volatilities of ATM put options with interest rate volatility specified on thetop row. The left table means varying domestic currency’s interest rate volatility with foreignone and other parameters unchanged. The right table is the opposite case.
As the foreign exchange rate model with the stochastic volatility and the stochastic interestrates, the Heston-Hull-White model is considered. By using the expansion method of option13remium with respect to the volatility of volatility (vol-of-vol), we obtain the new approxi-mation formula for pricing options up to the 2nd order in the Heston-Hull-White model. It isinspired by [3] in which the deterministic interest rates are used, and we partially extends itto the case with stochastic interest rates. The approximation accuracy is numerically studiedagainst the Monte-Carlo simulation. If the volatility of volatility is not so high, the expansionbased formula shows the comparable accuracy to the other method of [5], which uses the ap-proximation of characteristic function and is primarily applicable for low volatility of volatility.In addition to the pure expansion method, we also present the hybrid method that incorpo-rates the expansion method and the characteristic function method in [6]. In the numericalstudies, the hybrid method shows the equivalent or higher accuracy than the pure expansionmethod and retains the good accuracy even for the high vol-of-vol cases.
Appendix A
We summarize the results of representative calculations that is needed for evaluating theexpansion terms. They are obtained by using the Itˆo calculus and the lemma (23) based onthe Malliavin calculus. As explained before, G is a smooth function used to denote the BSformula or the derivatives of it, and the derivatives of G are equivalent to those with respectto x . The expectation value of G or its derivatives are explicitly evaluated by the equation(22). f is a deterministic function. Assuming v = θ v leads to g = ρ sv √ v . v i,t , i = 1 , f, g, h , (cid:18)(cid:90) T dtf ( t ) (cid:90) Tt dug ( u ) (cid:19) (cid:18)(cid:90) T dsf ( s ) (cid:90) Ts drh ( r ) (cid:19) = (cid:90) T dtf ( t ) (cid:90) Tt dug ( u ) (cid:90) Tu dsf ( s ) (cid:90) Ts drh ( r ) + (cid:90) T dtf ( t ) (cid:90) Tt duh ( u ) (cid:90) Tu dsf ( s ) (cid:90) Ts drg ( r )+ 2 (cid:18)(cid:90) T dtf ( t ) (cid:90) Tt duf ( u ) (cid:90) Tu dsg ( s ) (cid:90) Ts drh ( r ) + (cid:90) T dtf ( t ) (cid:90) Tt duf ( u ) (cid:90) Tu dsh ( s ) (cid:90) Ts drg ( r ) (cid:19) . (61)In what follows, we express the equations under the risk neutral measure Q , but the sameequations also hold under the T forward measure Q T if the expectation and the Brownianmotion are replaced with those defined under Q T . E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T f ( t ) v ,t dt (cid:21) = E (cid:20) G (1) (cid:18)(cid:90) T g dW v ( t ) (cid:19)(cid:21) γρ sv v (cid:90) T dte k v t (cid:90) Tt duf ( u ) e − k v u . (62)14 (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T f ( t ) v ,t dW v,t (cid:21) = E (cid:20) G (2) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γρ sv v / (cid:90) T dte k v t (cid:90) Tt duf ( u ) e − k v u . (63) E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T f ( t ) v ,t dt (cid:21) = E (cid:20) G (2) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v (cid:90) T dse k v s (cid:90) Ts du (cid:90) Tu dtf ( t ) e − k v t . (64) E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T f ( t )( v ,t ) dt (cid:21) = E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ v (cid:90) T du e k v u (cid:90) Tu dtf ( t ) e − k v t + 2 E (cid:20) G (2) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v (cid:90) T dse k v s (cid:90) Ts due k v u (cid:90) Tu dtf ( t ) e − k v t . (65) E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T ( v ,t ) dW v,t (cid:21) = E (cid:20) G (1) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:90) T dte k v t (cid:90) Tt due − k v u + 2 E (cid:20) G (3) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:90) T dse k v s (cid:90) Ts due k v u (cid:90) Tu dte − k v t . (66) E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T v ,t dW v,t (cid:21) = E (cid:20) G (3) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:90) T dse k v s (cid:90) Ts du (cid:90) Tu dte − k v t . (67) E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T dtf ( t ) v ,t (cid:90) T duf ( u ) v ,u (cid:21) = E (cid:20) G (2) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v (cid:18)(cid:90) T dte k v t (cid:90) Tt drf ( r ) e − k v r (cid:19) (cid:18)(cid:90) T dse k v s (cid:90) Ts duf ( u ) e − k v u (cid:19) + E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ v (cid:90) T dse k v s (cid:90) Ts dtf ( t ) e − k v t (cid:90) Tt duf ( u ) e − k v u + E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ v (cid:90) T dse k v s (cid:90) Ts dtf ( t ) e − k v t (cid:90) Tt duf ( u ) e − k v u . (68)15 (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:90) T dW v,t v ,t (cid:90) t dW v,s v ,s (cid:21) = 12 E (cid:20) G (4) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v (cid:18)(cid:90) T dre k v r (cid:90) Tr dse − k v s (cid:19) + E (cid:20) G (2) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v (cid:90) T dse k v s (cid:90) Ts dte − k v t (cid:90) Tt due − k v u + E (cid:20) G (2) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s . (69) E (cid:20) G (cid:18)(cid:90) T g dW v,t (cid:19) (cid:18)(cid:90) T f ( t ) v ,t dt (cid:19) (cid:18)(cid:90) T v ,s dW v,s (cid:19)(cid:21) = E (cid:20) G (1) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:90) T dse k v s (cid:90) Ts dte − k v t (cid:90) Tt drf ( r ) e − k v r + E (cid:20) G (1) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:90) T dse k v s (cid:90) Ts dt (cid:90) Tt drf ( r ) e − k v r + E (cid:20) G (1) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:90) T dre k v r (cid:90) Tr dtf ( t ) e − k v t (cid:90) Tt dse − k v s + E (cid:20) G (3) (cid:18)(cid:90) T g dW v,t (cid:19)(cid:21) γ ρ sv v / (cid:18)(cid:90) T due k v u (cid:90) Tu dse − k v s (cid:19) (cid:18)(cid:90) T dte k v t (cid:90) Tt drf ( r ) e − k v r (cid:19) . (70) Appendix B
For completeness, we show the explicit formulas to evaluate the equation (54).Let the maturity T fixed and write exponential terms as x d = e k d T , x f = e k f T , x v = e k v T , (71)and a function c ( t ) as c ( t ) = c + c d e k d t + c f e k f t . (72)If c ( t ) is α ( t ) in (44), the coefficients are given by c = 1 √ v (cid:18) ρ sd η d k d − ρ sf η f k f (cid:19) , c d = − ρ sd η d k d √ v x d c f = ρ sf η f k f √ v x f . (73)If c ( t ) = 1 + α ( t ), c has an additional term 1.16 T dte k v t (cid:90) Tt due − k v u c ( u ) = c Tk v + c d ( x d − k v k d + c f ( x f − k v k f + c (1 /x v − k v − c d ( x d /x v − k v ( k d − k v ) − c f ( x f /x v − k v ( k f − k v ) (74)The similar integral (cid:82) T dte k v t (cid:82) Tt due − k v u c ( u ) is also evaluated by replacing k v → k v and x v → x v in the above expression. (cid:90) T dte k v t (cid:90) Tt du (cid:90) Tu dse − k v s c ( s )= c (cid:18) Tk v + 1 /x v − k v − − (1 + k v T ) /x v k v (cid:19) + c d (cid:18) x d − k v k d − x d /x v − k v ( k d − k v ) − − (1 − ( k d − k v ) T ) x d /x v k v ( k d − k v ) (cid:19) + c f (cid:18) x f − k v k f − x f /x v − k v ( k f − k v ) − − (1 − ( k f − k v ) T ) x f /x v k v ( k f − k v ) (cid:19) (75) (cid:90) T dte k v t (cid:90) Tt due − k v u c ( u ) (cid:90) Tu dse − k v s c ( s )= c (cid:18) T k v + 1 k v x v − k v x v − k v (cid:19) + c d (cid:18) x d /x v − k d + k v )( k d − k v ) + x d − k d k v ( k d + k v ) − x d /x v − k v ( k d − k v ) (cid:19) + c f (cid:32) x f /x v − k f + k v )( k f − k v ) + x f − k f k v ( k f + k v ) − x f /x v − k v ( k f − k v ) (cid:33) + c c d (cid:18) ( k d + 2 k v )( x d − k v ( k d + k v ) k d + x d /x v − k v ( k d − k v ) + 1 − /x v k v ( k d + k v )( k d − k v ) − x d /x v − k v ( k d − k v ) (cid:19) + c c f (cid:18) ( k f + 2 k v )( x f − k v ( k f + k v ) k f + x f /x v − k v ( k f − k v ) + 1 − /x v k v ( k f + k v )( k f − k v ) − x f /x v − k v ( k f − k v ) (cid:19) + c d c f (cid:32) x d /x v − k d − k v )( k f − k v ) + x f /x v − k f − k v )( k d − k v ) − x d x f /x v − k v ( k d − k v )( k f − k v )+ (cid:18) k d + k v + 1 k f + k v (cid:19) x d x f − k v ( k d + k f ) (cid:19) (76) (cid:90) T dte k v t (cid:90) Tt due k v u (cid:90) Tu dse − k v s c ( s )= c (cid:18) T k v + 1 k v x v − k v x v − k v (cid:19) c d (cid:18) x d − k v k d − x d /x v − k v ( k d − k v ) + x d /x v − k v ( k d − k v ) (cid:19) + c f (cid:18) x f − k v k f − x f /x v − k v ( k f − k v ) + x f /x v − k v ( k f − k v ) (cid:19) (77)Note that the above formulas should be modified with the limit value at the singular point.The following expressions are to evaluate the derivatives of Black-Scholes formula. BS ( x, y ) = D d ( T ) ( K Φ( − d ) − e x Φ( − d )) , d = x − log K + y/ √ y , d = d − √ y (78) ∂BS ( x, y ) ∂y = D d ( T ) Kφ ( d ) 12 √ y (79) ∂ BS ( x, y ) ∂x∂y = − D d ( T ) Kφ ( d ) d y (80) ∂ BS ( x, y ) ∂y = D d ( T ) Kφ ( d ) d d − y / (81) ∂ BS ( x, y ) ∂x ∂y = D d ( T ) Kφ ( d ) d − y / (82) ∂ BS ( x, y ) ∂x ∂y = D d ( T ) Kφ ( d ) d d − d d + d − y / (83) References [1] Al`os, E. (2012), “A decomposition formula for option prices in the Heston model andapplications to option pricing approximation,”
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