The Pricing of Quanto Options: An empirical copula approach
TThe Pricing of Quanto Options
An empirical copula approachRafael Felipe Carmargo Prudencio ∗ and Christian D. Jäkel † May 7, 2020
Abstract
The quanto option is a cross-currency derivative in which the pay-off isgiven in foreign currency and then converted to domestic currency, througha constant exchange rate, used for the conversion and determined at con-tract inception. Hence, the dependence relation between the option under-lying asset price and the exchange rate plays an important role in quantooption pricing.In this work, we suggest to use empirical copulas to price quanto op-tions. Numerical illustrations show that the flexibility provided by this ap-proach, concerning the dependence relation of the two underlying stochas-tic processes, results in non-negligible pricing differences when contrastedto other models.
Contents ∗ [email protected], Department of Applied Mathematics, University of SãoPaulo (USP), Brazil. † [email protected], Department of Applied Mathematics, University of São Paulo (USP), Brazil. a r X i v : . [ q -f i n . M F ] M a y Numerical illustration 9 t -student copula, long term option . . . . . . . . . . . . 124.4 Case IV: t -student copula, short term option . . . . . . . . . . . 134.5 Case V: Frank copula, long term option . . . . . . . . . . . . . . 144.6 Case VI: Frank copula, short term option . . . . . . . . . . . . . 15 The quanto option is a cross-currency contract. The payoff is defined with re-spect to an underlying asset or index in one currency, but for payment, thepayoff is converted to another currency. The constant exchange rate is es-tablished at contract inception. Hence, the modelling of the dependence re-lation between the underlying asset and the exchange rate (which are bothmarket observable variables), is mandatory for quanto options pricing. Inthis work, we propose a new approach, based on empirical copula, to pricequanto options. We compare this approach with what is hereafter named thepractitioners model (based on the Black-Scholes framework) and the Dimitroff-Szimayer-Wagner (DSW) framework [1]. Without loss of generality, only calloptions are analysed, with the dividend yield of the underlying asset set tozero.The practitionersâ ˘A ´Z approach is based on the assumptions that “assetprices follow a geometric Brownian motion” and “volatility is constant”. Stochas-tic volatility models, such as the one proposed in [1], relax the “volatility isconstant” assumption. In the quanto option context, the dependences amongthe relevant variables can considerably impact the pricing. Both the practition-ersâ ˘A ´Z approach and the DSW model [1]¢ause a constant correlation in orderto address this issue. However, financial quantities (including the underlyingasset and the exchange rate) can be related in a non-linear way (see, e.g. , Tenget al. [4]). Hence a simple constant correlation cannot fully represent the de-pendence relation between the relevant variables.The copulas framework, which we propose, intends to provide a more flex-ible framework to set the dependence relation between the market variablesused in the pricing of quanto options. Besides, the empirical copula model(just like the DSW model) can adapt to a non-constant volatility smile. Beforewe start our discussion, we would like to note that we are aware of the short-comings of our approach: it is computationally expensive and does not offeranalytical tractability. 2
The quanto process A quanto call option is a financial instrument that gives the holder the right,but not the obligation, to buy an underlying asset S f , quoted in a foreign currency (FOR), at a predetermined price K (given in units of FOR currency), at matu-rity time T . The payoff amount, if positive, is converted to the domestic currency (DOM) at an exchange rate q ( ≡ DOMFOR ) . The latter is predetermined at the con-tract inception. Hence, the payoff, at maturity time T , is C q ( T ) = max (cid:8) q ( S f ( T ) − K ) , 0 (cid:9) . (1) From the risk-neutral pricing formula it follows that the price c q of a quantocall option at time t = c q ( ) = e − rT E Q (cid:104) max { q ( S f ( T ) − K ) , 0 } (cid:105) , (2)where Q is the domestic risk-neutral measure and E Q denotes the associated ex-pectation value.We now derive the stochastic differential equation for S f ( T ) under Q . Weassume that, under the domestic risk-neutral measure,d S f ( t ) = µ S f d t + (cid:112) V S ( t ) d W Q ( t ) , (3)where µ S f is the (unknown) drift of S f ( t ) and W Q represents a Brownian mo-tion. The volatility is denoted by √ V .The stochastic process of the exchange rate Q ( t )( ≡ DOMFOR ) under the do-mestic risk neutral measure isd Q ( t ) = Q ( t ) (cid:104) ( r − r f ) d t + (cid:112) V d W Q ( t ) (cid:105) , (4)with W Q ( t ) = ρ ( S f , Q ) W Q ( t ) + (cid:113) − ρ ( S f , Q ) W Q ( t ) , (5)a second Brownian motion, correlated with the Brownian motion W Q . On theother hand, W Q is a Brownian motion, which is independent from W Q . Ascan be read off from (5), the infinitesimal correlation between the increments of S f and Q is denoted by ρ ( S f , Q ) .In order to derive the drift µ S f , we express S f ( t ) in the domestic currency:we multiply S f ( t ) by Q ( t ) , setting S d ( t ) . = Q ( t ) S f ( t ) .From Itøâ ˘A ´Zs product rule it now follows thatd (cid:0) S d ( t ) (cid:1) = Q ( t ) S f ( t ) (cid:104) µ S f + r − r f + ρ ( S f , Q ) (cid:112) V V (cid:105) d t + (cid:112) V d W Q ( t ) + (cid:112) V d W Q ( t ) .3nder the domestic risk neutral measure, the drift of S d ( t ) is equal to r . Thus,it follows that µ S f = r − (cid:104) r − r f + ρ ( S f , Q ) (cid:112) V V (cid:105) .Inserting this expression into (3), we findd S f ( t ) = r − (cid:104) r − r f + ρ ( S f , Q ) (cid:112) V V (cid:105) d t + (cid:112) V S ( t ) d W Q ( t ) .Since we know the dynamics of S f ( t ) , we are now able to compute the expecta-tion (2). In fact, the diffusion of S f is of the same form as the diffusion processfor a dividend paying stock, with dividend rate q = r − r f + ρ ( S f , Q ) (cid:112) V V .Whence, the computation of expectation (2) gives the price of the vanilla calloption on a dividend-paying stock: c q ( ) = q · BS (cid:16) S f ( ) e −( r − r f + ρ ( S f , Q ) √ V V ) T , K , (cid:112) V , T , r (cid:17) .Here BS ( a , b , c , d , e ) stands for the traditional Black-Scholes formula, with a the underlying asset spot price, b the strike value, c the volatility, d the time tomaturity, and e the risk-free interest rate.The final step in the practitioners approach is to replace the constant volatil-ities, √ V and √ V , by at the money or at the strike values : c qp ( ) = q · BS (cid:18) S f ( ) e − T (cid:16) r − r f + ρ ( S f , Q ) √ V atm V atm (cid:17) , K , (cid:113) V strike , T , r (cid:19) . (6)Equation (6) is the V dblack approximation from Le Floc’h [3]; in fact, it is oneof the three approximations studied within [3]. Note that V atmi , i =
1, 2, in ρ ( S f , Q ) (cid:112) V atm V atm , must be the at-the-money value (not the at the strikevalue V strikei , i =
1, 2), as otherwise the price of the quanto forward contractwould depend on the option strike (an exogenous factor).
The DSW approach consists in the use of the following diffusion processes tosimulate values of S f ( t ) (named S ( T ) in their work) and Q − ( T ) (named C ( T ) in their work), in order to compute expectation (7) below and to obtain the4uanto option price value: d S f ( t ) d V ( t ) d Q − ( t ) d V ( t ) = ( r f ( t ) − d ( t )) S f ( t ) κ ( V − V ( t ))( r f ( t ) − r ( t )) Q − ( t ) κ ( V − V ( t )) d t + (cid:112) V ( t ) S f ( t ) η (cid:112) V ( t ) (cid:112) V ( t ) Q − ( t )
00 0 0 η (cid:112) V ( t ) × ρ (cid:112) − ρ ρ (cid:112) − ρ ρρ ρ (cid:112) − ρ (cid:112) − ρ d W ( t ) d W ( t ) d W ( t ) d W ( t ) where ( S f ( t ) , V ( t )) models the stock price and its variance, and ( Q − ( t ) , V ( t )) the foreign exchange rate and its variance with correlation ρ and ρ , respec-tively. The correlation between the Brownian motions of the S f ( t ) and Q − ( t ) diffusions is denoted by ρ ≡ ρ ( S f , Q − ) . The domestic risk-free interest rate isdenoted by r ( t ) , the foreign risk free interest rate by r f ( t ) , and the continuousdividend yield of the stock by d ( t ) . As the Heston model is one of the mainbuilding blocks of the DSW approach, the constants V i , κ i and η i have the tra-ditional meaning, i.e. , V i is the long run variance, κ i is the rate at which V i ( t ) reverts to V i , and η i determines the variance of the process V i ( t ) , i =
1, 2.Besides, it is necessary to set V i ( ) , which is the initial variance, in orderto get the full representation of the DSW approach in the risk-neutral format.Finally, the parameters in the equations above can be compiled in the Hestonvector of parameters ϕ S f and ϕ Q − , with ϕ S f = (cid:0) ρ , κ , V , V ( ) , η (cid:1) and ϕ Q − = (cid:0) ρ , κ , V , V ( ) , η (cid:1) .These Heston vectors of parameters are calibrated with market data in order totake into account the respective market volatility smiles. Our new framework (as well as the DSW model) bases the quanto option pric-ing on the diffusion processes of S f ( t ) and Q − ( t ) , 0 (cid:54) t (cid:54) T , under the foreignrisk neutral measure Q f . From a foreign investor’s perspective, the payoff, givenin FOR currency, is C fq ( T ) = Q − ( T ) max (cid:8) q ( S f ( T ) − K ) , 0 (cid:9) .Here Q − ( t )( ≡ FORDOM ) is the exchange rate quoted as foreign currency per unitof domestic currency. 5rom the risk neutral pricing formula, the option value (in FOR currency)is given by c fq ( ) = e − r f T E Q (cid:104) Q − ( T ) max (cid:8) q ( S ( T ) − K ) , 0 (cid:9) (cid:105) .A non-arbitrage argument can be used to value the option in DOM currency: c q ( ) = Q ( ) e − r f T E Q f (cid:104) Q − ( T ) max (cid:8) q ( S f ( T ) − K ) , 0 (cid:9) (cid:105) . (7)Equation (7) sets a starting point for quanto option pricing. A variety of methodologies can be used in order to compute the expectationin (7). We like to make the pricing of quanto options as adaptable as possibleto the dependence relation between S f ( T ) and Q − ( T ) . At the same time, ourapproach is capable to adapt to the market volatility smiles.The expectation in equation (7) involves two random variables, namely S f ( T ) and Q − ( T ) , hence, one approach to solve it, is to estimate the bi-variate cumulative distribution function (CDF) of these random variables, under the prob-ability measure Q f , and to compute the expectation based on simulations ofthis CDF. We denote the CDF by H ( s f ( T ) , q − ( T )) in this text, where s f ( T ) and q − ( T ) are the possible outcomes of the random variables S f ( T ) and Q − ( T ) ,respectively. The main ingredient in our analysis is Sklarâ ˘A ´Zs Theorem. It en-sures the existence of a copula , i.e. , a function C : [
0, 1 ] d → R + with the followingproperties [2]: i . ) if at least one coordinates u j =
0, then C ( u ) = ii . ) C is d -increasing, i.e. , for every a = ( a , . . . , a d ) and b = ( b , . . . , b d ) in [
0, 1 ] d such that a i (cid:54) b i , i =
1, . . . , d , the C -volume V C ([ a , b ]) of the box [ a , b ] = [ a , b ] × · × [ a d , b d ] is positive. iii . ) if u j = j (cid:54) = k for some fixed k , then C ( u ) = u k .We can now state Sklarâ ˘A ´Zs result. Theorem 3.1 (Sklarâ ˘A ´Zs Theorem) . Every multivariate cumulative distributionfunction (CDF), H ( x , . . . , x d ) = P (cid:8) X (cid:54) x , . . . , X d (cid:54) x d (cid:9) , can be expressed in terms of its marginals F i ( x i ) = P (cid:8) X i (cid:54) x i (cid:9) , i = {
1, . . . , d } , and acopula C , such that H ( x , . . . , x d ) = C (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) .6sing this result, the problem of estimating a bivariate distribution function H ( s f ( T ) , q − ( T )) can be divided into two independent problems: i . ) Estimating the marginal distributions. The marginals are the market im-plied cumulative distribution functions of S f ( T ) and Q − ( T ) . We denotethem by F S f ( T ) and F Q − ( T ) ; and ii . ) estimating a copula C = C (cid:16) F S f ( T ) (cid:0) s f ( T ) (cid:1) , F Q − ( T ) (cid:0) q − ( T ) (cid:1)(cid:17) ,which specifies the dependence relation between S f ( T ) and Q − ( T ) . Theexistence of such a copula is guaranteed by Sklarâ ˘A ´Zs Theorem.It follows from point i . ) that, as the market implied cumulative distributionfunctions are used, our model duly adapts to the observed volatility smile.We will address item i . ) in Section 3.1 and item ii . ) in Section 3.2. In order to estimate the marginal distributions, the strategy adopted by DSWis to calibrate the parameters of a single Heston model on the market data ofplain vanilla option prices, for both S f and Q − . The vectors of parameters foreach Heston model are denoted by ϕ S f and ϕ Q − , for S f and Q − , respectively.We will simply take over this first step from DSW and consider it as part of ourown approach.However, for the purpose of illustration only, we will use hypothetical data in Section 4 and the parameters of the ϕ S f and ϕ Q − vectors will be set directly, i.e. , without a calibration to real market data. According to Theorem 3.1, an estimate for H (cid:0) s f ( T ) , q − ( T ) (cid:1) can be providedonce a copula C linking the random variables S f ( T ) and Q − ( T ) is identified.Our approach is to calibrate the copula C using data provided by an expert.The data are represented by a ( N × ) matrix A , the first column contains data of S f ( T ) , and the second column contains data of Q − ( T ) . By A ( n ) , n = {
1, . . . , N } ,we denote the n -th line of A . N is the number of ordered pairs provided by theexpert.In order to build a copula based on the matrix A , we make use of kernelestimators , following the methodology proposed by Scaillet and Fermanian[2, Section 3.1]. The role of the kernels is to smoothen the data. In case there We refer to [5] for the theory of kernel density estimation. d -dimensional GaussianKernel functions of the form K ( x ) = ( π ) − d e − x T x , x = ( x , . . . , x d ) .As one may expect, the probability density function related to our empiricalCDF places more probability mass where there are more ordered pairs, andless probability mass where there are less ordered pairs.The estimated bivariate cumulative distribution function (CDF) of the twodependent random variables S f and Q − , denoted by (cid:98) F , is given by (cid:98) F ( s f , q − ) = (cid:90) s f − ∞ d s (cid:90) q − − ∞ d r (cid:98) f ( s , r ) ,with (cid:98) f ( s , r ) = Nh (cid:80) Nn = K (cid:16) ( s , r )− A ( n ) h (cid:17) the Kernel estimator of f ( s , r ) .We are now able to define the copula which will allow us to compute theprice of a quanto option. Definition 3.2.
A copula C is obtained by setting C ( u , u ) ≡ (cid:98) F (cid:0) ( ξ ( u ) , ξ ( u ) (cid:1) , (8) where ξ ( u ) = inf (cid:8) y | (cid:98) F S f ( y ) (cid:62) u (cid:9) and ξ ( u ) = inf (cid:8) y | (cid:98) F Q − ( y ) (cid:62) u (cid:9) . Remark 3.3.
One easily verifies that the greater the number N of ordered pairs pro-vided by the expert, the lower the impact of the choice of the kernel function K and thebandwidth h , on the copula estimation. We now state the relation between ρ ( S f , Q ) (the correlation between theinfinitesimal increments of S f and Q ) and ρ ( S f , Q − ) (the correlation betweenthe infinitesimal increments of S f and Q − ). This information will be used inthe numerical illustration section, in order to allow the three approaches to becompared, as the practitioners approach is based on the relation between S f and Q , while the DSW approach and our approach are based on the relationbetween S f and Q − . Proposition 3.4. ρ ( S f , Q ) = − ρ ( S f , Q − ) .Proof. Without loss of generality, only stochastic terms shall be considered.From (4), it follows that d Q ( t ) = Q ( t ) (cid:112) V d W Q ( t ) .8he difference between the Q ( t ) diffusion, under the domestic and the foreignrisk-neutral measure, lies in the drift term. The format of the Brownian motionpart remains unaltered. Thus, under the foreign risk-neutral measure W Q f ,d Q ( t ) = Q ( t ) (cid:112) V d W Q f ( t ) .We apply Itøâ ˘A ´Zs Lemma to Q − . We findd Q − ( t ) = Q − ( t ) (cid:16) − (cid:112) V (cid:17) d W Q f ( t ) .Inspecting equation (5), we get, under the foreign risk neutral measure,d Q − ( t ) = Q − ( t ) (cid:112) V (cid:18) − ρ ( S f , Q ) d W Q f − (cid:113) − ρ ( S f , Q ) d W Q f (cid:19) ,where W Q f and W Q f are independent Brownian motions. Under the foreignrisk-neutral measure, the stochastic process S f satisfiesd S f ( t ) = r f S f ( t ) d t + (cid:112) V ( t ) S f ( t ) d W Q f ( t ) .Hence, ρ ( S f , Q − ) = Cor (cid:20) d W Q f ( t ) , (cid:18) − ρ ( S f , Q ) d W Q f − (cid:113) − ρ ( S f , Q ) d W Q f (cid:19)(cid:21) = − Cor (cid:20) d W Q f ( t ) , (cid:18) ρ ( S f , Q ) d W Q f + (cid:113) − ρ ( S f , Q ) d W Q f (cid:19)(cid:21) ;thus ρ ( S f , Q − ) = − ρ ( S f , Q ) . In order to analyse the pricing differences among the practitionersâ ˘A ´Z frame-work, the DSW framework, and our approach based on empirical copulas, weproceed as follows: we set numerical values displayed in the following table.They are used in all the cases we will discuss. correlation initial initial asset domestic foreign risk constant ρ ( S f , Q − ) exchange value S f ( ) risk free free interest exchangerate Q ( ) interest r rate r f rate q rate r - 0.7 3.1 2500 0.1 0.01 39e will vary • the Heston vector parameters ϕ S f and ϕ Q − ; and • the time to maturity T .We also set to zero the continuous dividend yield d ( t ) , from the DSW approachdepicted in Section 2.2.These choices allow us to compute the prices of foreign vanilla call optionson both DOM currency and on S f , and to derive the implied volatility smilesof these options: i . ) We compute the quanto option prices in the practitioners’ framework,using equation (6) and ρ ( S f , Q ) = − ρ ( S f , Q − ) ; ii . ) We evaluate the quanto option prices using the DSW framework outlinedin Section 2.2; iii . ) We compute the proposed quanto option prices in our new copula ap-proach: – We numerically derive the marginal cumulative distribution func-tions F S f (cid:0) s f ( T ) (cid:1) and F Q − (cid:0) q − ( T ) (cid:1) , respectively, from the Hestonmodel with parameters ϕ S f and ϕ Q − ; – We compute the copula C ( u , u ) form the matrix A (provided byan external expert), using equation (8). Sampling from the copula C ( u , u ) , we obtain ordered pairs of quantiles ( v , v ) ; – The ordered pairs of quantiles ( v , v ) are transformed into S f and Q − outcomes, by setting (cid:16) s f ( T ) , q − ( T ) (cid:17) = (cid:16) F − S f ( v ) , F − Q − ( v ) (cid:17) . – For each obtained ordered pair (cid:0) s f ( T ) , q − ( T ) (cid:1) , equation (7) yields C q ( ) = Q ( ) e − r f T q − ( T ) max (cid:8) q ( s f ( T ) − K ) , 0 (cid:9) .The average of the numerous obtained values of C q ( ) is the price we pro-pose for of the quanto option in the empirical copula dependence relationframework.We now discuss the outcome of these three procedures for different volatil-ity smiles and dependence relation fashions between S f and Q − , in a case bycase analysis. 10 .1 Case I: Gaussian copula, constant volatility The matrix A is set such that the obtained copula C is Gaussian with correlation ρ ( S f , Q − ) and the parameters ϕ S f and ϕ Q − are set to ϕ S f = ϕ Q − = (
0, 0, 0, 0.2, 0 ) (whence no volatility smile is present for both S f and Q − ). The time to matu-rity is T = S f and Q − diffusions are corre-lated by a simple constant correlation ρ ( S f , Q − ) , the basic assumption of thepractitionersâ ˘A ´Z framework is satisfied. The DSW framework and the em-pirical copula approach are capable of adapting to this condition as well: theDSW model directly uses ρ ( S f , Q − ) to correlate S f and Q − diffusions, andthe empirical copula approach simply reproduces the Gaussian copula depen-dence relation with correlation ρ ( S f , Q − ) from the data given by the matrix A .Hence, no pricing differences are observed amongst the three approaches, despiteminor differences due to simulation imprecisions. The matrix A is set such that the obtained copula is Gaussian, and ϕ S f = ϕ Q − = (cid:0) − (cid:1) ,whence a co-inclining volatility smile is obtained for S f and Q − (as can beseen from their vectors of parameters ϕ S f and ϕ Q − , and Figure 1). The timeto maturity is T = S f and Q − , as a function of strike, subject toa co-inclining volatility smile. 11oth the DSW approach and the empirical copula approach adapt to thevolatility smiles, while the practitionerâ ˘A ´Zs approach does not, because of its“volatility is constant” assumption. Concerning the dependence relation be-tween S f and Q − , the analysis is the same as in case I. Hence, no pricingdifferences should be observed between the DSW and the empirical copulaframeworks. The minor differences observed between these two approachesare due to simulation imprecisions.Figure 2: Pricing differences, Gaussian copula, co-inclining volatility smile. t -student copula, long term option In Cases III and IV, the matrix A is set such that the obtained copula is a t -copulawith 3 degrees of freedom and correlation ρ ( S f , Q − ) , ϕ S f = ϕ Q − = (− ) (whence a co-inclining volatility smile is obtained for S f and Q − , which isdisplayed in Figure 1). The time to maturity is T = not able to adapt to the t -copula between S f and Q − . The latter presents moretail dependence than the Gaussian copula, which is intrinsic to the DSW frame-work. Hence, pricing differences are observed amongst all the three frame-works (see Figure 3). The slight difference between the DSW framework andour framework is attributed to the difference between a t -copula (with 3 de-grees of freedom) and a Gaussian copula, with the same correlation parameter.Figure 3: Pricing differences, t -student copula, long term option. t -student copula, short term option In Case IV, the conditions are exactly the same as in Case III, except that T = T =
3. 13igure 4: Pricing differences, t -student copula, short term option.Figure 4 shows that no pricing differences are observed . We conclude that nei-ther the dependence relation between S f and Q − nor the volatility smile playa major role in the pricing of quanto options, if the contract is a short-term calloption. In this case, the same simple conditions as in Case I are imposed, except that A is set such that the obtained copula is a Frank copula with parameter α . Theordered pairs of quantiles generated by this copula, when converted to orderedpairs of normal random variables, induce a correlation ρ ( S f , Q − ) . A Frankcopula is less similar to a Gaussian copula than a t -copula is. Whence, this casestresses the modelling of the dependence relation more than Case III does.The DSW and the practitionersâ ˘A ´Z frameworks yield similar results, as novolatility smile is imposed and both approaches adapt to the imposed Frankcopula dependence relation the same way, i.e. , by considering solely its in-duced correlation. The empirical copula framework gives pricing figures con-siderably different from the other approaches as it takes into account the fulldependence relation provided by the imposed Frank copula. Figure 5 illus-trates these results. 14igure 5: Pricing differences, Frank copula, long term option.
The conditions are the same as in Case V, except that now T =
0, 25. As a con-sequence, major pricing differences among the three models are not identified,even though slight pricing differences for deep out-of-the money options exist.Figure 6: Pricing differences, Frank copula, short term optionFigure 6 illustrates the pricing differences. Whence, even in a stressed de-pendence relation context, the dependence relation does not play a major rolein the pricing of short-term quanto options.15
Summary
We have proposed a framework based on empirical copulas for quanto op-tion pricing. We have given numerical examples in order to illustrate the pric-ing differences among our approach and the practitioners as well as the DSWmodel [1]. Looking at the results, we conclude that: i . ) the quanto option requires explicit modelling for accurate pricing, withthe exception of short duration contracts; ii . ) the flexibility provided by the non-parametric approach results in pricingdifferences when compared to the other two approaches.On the proposed empirical copula dependence relation framework, we con-clude that: iii . ) it provides a flexible framework to define the dependence relation be-tween the market variables used in quanto option pricing, by taking intoaccount non-linear dependence relations, through the matrix A and therelated empirical copula estimation framework; iv . ) it can adapt to the observed volatility smiles from the relevant marketvariables, as the marginals of S f and Q − shall be calibrated based onplain vanilla options market prices; and finally v . ) a drawback of the proposed model is that it is computationally more ex-pensive than the other models it was compared to. References [1] Dimitroff G., Szimayer A. and Wagner A.,
Quanto option pricing in the par-simonious Heston model , Berichte des Fraunhofer ITWM (2009) 1–24; Avail-able at SSRN: http:// dx.doi.org/10.2139/ssrn.1477387.[2] Scaillet, O., and Fermanian, J.-D.,
Non-parametric estimation ofcopulas for time series (November 2002). FAME Research PaperNo. 57. Available at SSRN: https://ssrn.com/abstract=372142 orhttp://dx.doi.org/10.2139/ssrn.372142.[3] Le Floc’h, F.,
On the simulation of a quanto process under local volatility (Sept. 2, 2011). Available at SSRN: https://ssrn.com/abstract=2097921;see also http://dx.doi.org/10.2139/ssrn.2097921.[4] Teng, L., Ehrhardt, M. and Günther, M.,
The pricing of quanto options underdynamic correlation , J. Computational and Applied Math. 275, February(2015) 304–310; https://doi.org/10.1016/j.cam.2014.07.017.r165] Gramacki, A.,