Efficient Pricing and Calibration of High-Dimensional Basket Options

AbstractThis paper studies equity basket options – i.e. multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks – and develops a new and innovative approach to ensure consistency between options on individual stocks and on the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket. To address this “insufficient skewness”, we proceed in two steps. First, we propose an “effective” local volatility model by mapping the general multi-dimensional basket onto a collection of marginal distributions. Second, we build a multivariate dependence structure between all the marginal distributions assuming a jump-diffusion model for the effective projection parameters, and show how to calibrate the basket to the index smile. Numerical tests and calibration exercises demonstrate an excellent fit for a basket of as many as 30 stocks with fast calculation time.Keywords: Basket OptionsIndex SkewMonte CarloLocal VolatilityStochastic VolatilityCollocation MethodsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. Notes1 Cf. the prospectus availible in the online records of the U.S. Securities and Exchange Commission at: https://www.sec.gov/Archives/edgar/data/19617/000089109221003578/e13291-424b2.htm2 For example, in the Bloomberg basket options pricing template correlations are, by default, estimated over a 5 year period, whereby to eliminate noise, a given percentile of rolling 6-month cross-correlation estimates is chosen in the parameterization of the full correlation matrix.3 We define the skew here loosely as the difference in implied volatilities between the 85-120% ATM levels.4 As an alternative to [27] one could consider Kou's jump-diffusion model [18] which has the additional benefit of separating the upside and downside skew. However, in this case, we opt for the simplicity and parsimony of Merton's approach5 Without loss of generality, we shall henceforth think of the underlying assets as stocks, however the method developed below is obviously general and, mutatis mutandis, applies to other instruments as well.6 The proposed framework can also be extended with a stochastic volatility process. Such an extension is trivial and will, for simplicity, be omitted.7 The respective dynamics are given by (j=1,2): dSj(t)=rSj(t)dt+vj1/2(t)Sj(t)dWj,1(t), dvj(t)=κj(v¯j−vj(t))dt+γjvj1/2(t)dWj,2(t) with correlations dWj,1(t)dWj,2(t)=ρjdt, dW1,1(t)dW2,1(t)=ρ1,2dt and dWj,2(t)dWk,2(t)=0⋅dt. For reference, we set S1(t0)=1, S2(t0)=2.5, r=0, κ1=1, κ2=0.5, γ1=1, γ2=0.6, ρS1,v1=−0.5, ρS2,v2=−0.7, v1,0=0.1, v2,0=0.05, v¯1=0.1 and v¯2=0.05.8 The Feller's condition is a direct consequence of the so-called Fichera [11] condition for the uniqueness of solutions to elliptic and parabolic equations having diffusion coefficients vanishing on the boundary of the computational domain. It gives necessary and sufficient conditions for advection terms guaranteeing the unicity of solutions.9 The reason why we choose a standard normal distribution in the alternative approach is twofold. First, even for a fundamental distribution as the standard normal results are highly accurate – this is also the case in e.g. [12]. By choosing a different distribution, results may be further enhanced. Secondly, as mentioned in [14], choosing the normal distribution is also motivated by the Cameron-Martin Theorem [30], which states that polynomial chaos approximations based on the normal distribution converge to any distribution.10 The strategy proposed in this part does not require “re-calibration” of αi,j,k coefficients, but only neglects the coefficients of higher order.11 1) UnitedHealth; 2) Home Depot; 3) Goldman Sachs; 4) Microsoft Corp; 5) salesforce.com Inc12 Results for σ are not presented here as they resembled the impacts of ξp and σJ