Once we have studied the convergence properties of Chebyshev polynomial interpola-tion, we can now apply the Chebyshev interpolation to parametric option pricing. For parameterspPRD, whereDPNdenotes the dimensionality of the parameter space, the pricePricepis approximated by tensorized Chebyshev polynomialsTj with pre-computed coefficientscj,jPJ, as follows,
Pricep «ÿ
jPJ
cjTjppq.
4.3.1 Exponential Convergence of Chebyshev Interpolation for POP
In this section we embed the multivariate Chebyshev interpolation into the option pricing framework. We provide sufficient conditions under which option prices depend analyti-cally on the parameters.
Analytic properties of option prices can be conveniently studied in terms of Fourier transforms. First, Fourier representations of option prices are explicitly available for a large class of both option types and asset models. Second, Fourier transformation unveils the analytic properties of both the payoff structure and the distribution of the underlying stochastic quantity in a beautiful way. By contrast, if option prices are represented as expectations, their analyticity in the parameters is hidden. For example the function K ÞÑ pST ´Kq` is not even differentiable, whereas the Fourier transform of the dampened call payoff function evidently is analytic in the strike, compare (Gaß et al., 2016, Table 1).
Conditions for Exponential Convergence
Let us first introduce a general option pricing framework. We consider option prices of the form
Pricep“pp1,p2q“E`
fp1pXp2q˘
(4.57) wherefp1 is a parametrized family of measurable payoff functions fp1 :RdÑ R` with payoff parametersp1 PP1andXp2 is a family ofRd-valued random variables with model parametersp2 PP2. The parameter set
p“ pp1, p2q PP “P1ˆP2ĂRD (4.58)
is again of hyperrectangular structure, i.e. P1“ rp1, p1s ˆ. . .ˆ rpm, pmsand
P2 “ rpm`1, pm`1s ˆ. . .ˆ rpD, pDs for some 1 ď m ď D and real pi ď pi for all i“1, . . . , D.
Typically we are given a parametrizedRd-valued driving stochastic processHp1 withSp1 being the vector of asset price processes modeled as an exponential ofHp1, i.e.
Stp1,i“S0p1,iexppHtp1,iq, 0ďtďT, 1ďiďd, (4.59) and Xp2 is an FT-measurable Rd-valued random variable, possibly depending on the history of theddriving processes, i.e. p2 “ pT, p1q and
Xp2 :“Ψ`
Htp1,0ďtďT˘ , whereΨis anRd-valued measurable functional.
We now focus on the case that the price (4.57) is given in terms of Fourier transforms.
This enables us to provide sufficient conditions under which the parametrized prices have an analytic extension to an appropriate generalized Bernstein ellipse. For most relevant options, the payoff profile fp1 is not integrable and its Fourier transform over the real axis is not well defined. Instead, there exists an exponential dampening factor η P Rd such that exη,¨yfp1 PL1pRdq. We therefore introduce exponential weights in our set of conditions and denote the Fourier transform ofgPL1pRdq by
ˆ gpzq:“
ż
Rd
eixz,xygpxqdx
and we denote the Fourier transform of exη,¨yf P L1pRdq by fˆp¨ ´iηq. The exponential weight of the payoff will be compensated by exponentially weighting the distribution of Xp2 and that weight will reappear in the argument of ϕp2, the characteristic function of Xp2.
Conditions 4.3.1. Let parameter setP “P1ˆP2 ĂRD of hyperrectangular structure as in (4.58). Let % P p1,8qD and denote %1 :“ p%1, . . . , %mq and %2 :“ p%m`1, . . . , %Dq and let weight ηPRd.
(A1) For everyp1 PP1 the mapping xÞÑexη,xyfp1pxqis in L1pRd).
(A2) For every z P Rd the mapping p1 ÞÑ fxp1pz ´iηq is analytic in the general-ized Bernstein ellipse BpP1, %1q and there are constants c1, c2 ą 0 such that supp1PBpP1,%1q|fxp1p´z´iηq| ďc1ec2|z| for allzPRd.
(A3) For everyp2 PP2 the exponential moment condition E`
e´xη,Xp
2y˘
ă 8holds.
(A4) For every z P Rd the mapping p2 ÞÑ ϕp2pz`iηq is analytic in the generalized Bernstein ellipseBpP2, %2q and there are constantsα P p1,2s andc1, c2 ą0 such thatsupp2PBpP2,%2q|ϕp2pz`iηq| ďc1e´c2|z|α for allzPRd.
Theorem 4.3.2. Let % P p1,8qD and weight η P Rd. Under conditions (A1)–(A4), P Q p ÞÑ Pricep has an analytic extension to the generalized Bernstein ellipse BpP, %q andsuppPBpP,%q|Pricep| ďV, and thus,
maxpPP
ˇ
ˇPricep´INpPricep¨qqppqˇ
ˇďmintap%, N, Dq, bp%, N, Dqu, where, denoting by SD the symmetric group on D elements,
ap%, N, Dq “ min we therefore can apply (Eberlein et al., 2010, Theorem 3.2). This gives the following Fourier representation of the option prices,
Pricep“ 1 p2πqd
ż
Rd`iη
fxp1p´zqϕp2pzqdz.
Due to assumptions (A2) and (A4) the mapping
p“ pp1, p2q ÞÑfxp1p´zqϕp2pzq
Moreover, thanks to assumptions (A2) and (A4), dominated convergence shows continu-ity ofpÞÑPricep inBpP, %qwhich yields the analyticity ofpÞÑPricepinBpP, %qthanks to a version of Morera’s theorem provided in (Jänich, 2004, Satz 8).
Similar to Corollary 4.2.14 in the setting of Theorem 4.3.2 additionally the according derivatives are approximated as well by the Chebyshev interpolation. A very interesting
application of this result in finance is the computation of sensitivities like delta or vega of an option price for risk assessment purposes. Theorem 4.3.2 together with Corollary 4.2.14 yield the following corollary.
Corollary 4.3.3. Under the assumptions of Theorem 4.3.2, for all lPN,µ andσ with σą D2, 0ďµďσ and µ´lą D2 there exist a constant C, such that
}Pricep´INpPricep¨qqppq}ClpPqďCN2µ´σ}Pricep}W2σpPq, where the spaces and norms are defined in Section 4.2.1.
In Gaß et al. (2016) it is shown that Conditions (A1)–(A4) are satisfied for a large class of payoff functions and asset models. Here, we focus on basket options in affine models.
Let Xπ1 be a parametric family of affine processes with state space DĂRd for π1 PΠ1 such that for every π1 PΠ1 there exists a complex-valued function νπ1 and a Cd-valued function φπ1 such that
ϕp2“pt,x,π1qpzq “E` eixz,Xπ
1 t yˇ
ˇX0π1 “x˘
“eνπ
1pt,izq`xφπ1pt,izq,xy, (4.60) for every tě0,z PRd and x PD. Under mild regularity conditions, the functions νπ1 and φπ1 are determined as solutions to generalized Riccati equations. We refer to Duffie et al. (2003) for a detailed exposition. The rich class of affine processes comprises the class of Lévy processes, for which νπ1pt, izq “ tψπ1pzq with ψπ1 given as some exponent in the Lévy-Khintchine formula and φπ1pt, izq ” 0. Moreover, many popular stochastic volatility models such as the Heston model as well as stochastic volatility models with jumps, e.g. the model of Barndorff-Nielsen and Shephard (2001) and time-changed Lévy models, see Carr et al. (2003) and Kallsen (2006), are driven by affine processes.
Consider option prices of the form
PricepK,T,x,π1q “E`
fKpXTπ1q|X0π1 “x˘
(4.61) where fK is a parametrized family of measurable payoff functions fK : Rd Ñ R` for K PP1.
Corollary 4.3.4. Under the conditions (A1)–(A3) for weight η PRd, % P p1,8qD and P “P1ˆP2 ĂRD of hyperrectangular structure assume
(i) for every parameter p2 “ pt, x, π1q P P2 Ă RD´m that the validity of the affine property (4.60)extends to z“R`iη, i.e. for every zPR`iη,
ϕp2“pt,x,π1qpzq “E` eixz,Xπ
1 t yˇ
ˇX0π1 “x˘
“eνπ
1pt,izq`xφπ1pt,izq,xy,
(ii) for every zPRdthat the mappingspt, π1q ÞÑνπ1pt, iz´ηqandpt, π1q ÞÑφπ1pt, iz´ηq have an analytic extension to the Bernstein ellipse BpΠ1, %1q for some parameter
%1 P p1,8qD´m´1,
(iii) there existαP p1,2sand constantsC1, C2 ą0such that uniformly in the parameters p2 “ pt, x, π1q PBpP2,%r2q for a generalized Bernstein ellipse with %r2 P p1,8qD´m
<`
νπ1pt, iz´ηq ` xφπ1pt, iz´ηq, xy˘
ďC1´C2|z|α for all zPR. Then there exist constants Cą0,%ą1 such that
pPPmax1ˆP2
ˇ
ˇPricep´INpPricep¨qqppqˇ
ˇďC%´N.
Proof. Thanks to Theorem 4.3.2 and Corollary 4.2.4 and in view of the assumed validity of Conditions (A1)–(A3), it suffices to verify Condition (A4). While assumptions (i) and (ii) together yield the analyticity condition in (A4), part (iii) provides the upper bound in (A4).