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Geodesic Strikes for Composite, Basket, Asian, and Spread Options

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Geodesic Strikes for Composite, Basket, Asian, and Spread Options Geodesic strikes for composite, basket, Asian, and spread options Peter J¨ackel∗ First version: 28th July 2012 This version: 11th February 2017 Abstract In practice, it is not uncommon for X to be the mar- ket price of a standard investment asset such as an We discuss simple methodologies for the selection of equity share, denominated in its domestic currency most relevant or effective strikes for the assessment DOM, and for Y to be an FX rate that converts the of appropriate implied volatilities used for the valu- final investment asset value to a target currency TAR. ation of composite, basket, Asian, and spread options As a consequence of the natural direction of the FX following the spirit of geodesic strikes [ABOBF02]. rate, i.e., DOMTAR, meaning the value of one DOM currency unit expressed in units of TAR, the valu- Introduction ation of a composite option may also incur a quanto effect since the asset X and the FX rate DOMTAR In the context of vanilla, or near-vanilla, derivatives cannot be martingales in the same measure, and this trading, we often also encounter composite, basket, has to be taken into account. For the purpose of this Asian, and spread options. Whilst their respective article, however, we shall assume that all involved payoff rules do in principle warrant treating them underlyings are martingales in the same measure. In as genuinely exotic options, it is often desirable to practice, this may mean that we have to determine use a simple approximation for the sake of tract- an effective quanto forward for the asset X in the tar- ability. A popular approach is to use a relatively get currency TAR by other means of approximation simple valuation methodology based on multivariate prior to being able to commence with our effective geometric Brownian motion, the well-known Black- geodesic strike procedure. We shall return to this Scholes-Merton framework, and to find a suitable im- point at the end of section3. plied volatility (or term structure thereof) for each Basket options are, conventionally, derivatives underlying selected by the concept of a most relev- with a payoff of the form ant, or effective, strike for each observation date. For P (θ · [ wiXi (T ) − K]) : (2.2) a plain vanilla option, clearly, the volatility must be i + taken from the implied volatility surface at the op- This is a vanilla option on a linearly weighted average tion's expiry date and at the option's strike. For com- of a number of underlyings, whence it is also referred posite, basket, Asian, and spread options, the most to as an arithmetic basket option. In contrast, whilst suitable strike for the looking up of implied volatility rarely traded, there is also the geometric basket op- is not necessarily as obvious. In this document, we tion suggest a systematic procedure that addresses this Q wi (θ · [ i Xi (T ) − K])+ : (2.3) question. Asian options are in the framework of multivariate geometric Brownian motion merely a special case of Composite, Basket, Asian, and arithmetic basket options in that the fixings that con- spread options tribute to the average are from the same underlying, but for different observation times: A composite option is a contract of European style P (θ · [ i wiX (Ti) − K])+ : (2.4) which pays at a future payment date Tpay the payoff Spread options usually simply pay according to (θ · [X (T ) · Y (T ) − K])+ ; (2.1) (θ · [(X(T ) − Y (T )) − K]) (2.5) with θ = ±1 for calls and puts, based on two underly- + ings X and Y observed on the expiry date T ≤ Tpay. for two underlyings X and Y . By allowing the subscript i on an observation index ∗Deputy head of Quantitative Research, VTB Capital Key words and phrases. effective strikes, geodesic, compos- Xi to indicate a specific underlying as well as fixing ite options, basket options, Asian options, spread options. time, and by additionally permitting weights to be 1 negative as well as positive, it is clear that all of bas- relating to the dispersion matrix A according to ket, Asian, and spread options take on the form of > an arithmetic average option C = A · A : (3.5) P (θ · [ i wiXi − K])+ : (2.6) The approximation of effective geodesic strikes is now to find a set of logarithmic shift coefficients ξ∗, which Equally, it is evident that both composite and geo- relates to the effective strike for underlying #i as metric basket options appear as a geometric average ∗ option ∗ ^ ξi Ki = Xi · e ; (3.6) Q wi (θ · [ i Xi − K])+ : (2.7) Subsequently, we will therefore concentrate on these such that the multivariate probability density of ξ two generic cases: geometric and arithmetic average under the joint normal law with options. D E ξ · ξ> = C (3.7) Geodesic strikes is maximal at ξ = ξ∗ subject to the constraint The formal derivation of the procedure for the calcu- f(K∗) = K: (3.8) lation geodesic strikes in [ABOBF02] involves con- cepts of projection onto an effective local volat- In other words, we seek ility representation of a basket process for large deviations, as well as Varadhan's geodesic the- ∗ ∗ ξ = arg max (ξ) f(K ) = K (3.9) orem [Var67], and is rather technical. In this doc- ξ ument, we shall attempt to obtain a similar result with with a somewhat less rigorous, though tractable, ar- − 1 ξ>C−1ξ gument, bearing in mind that the sole purpose of e 2 (ξ) := (3.10) the exercise is to arrive at a set of suitably chosen p(2π)n · jCj effective strikes for the lookup of implied volatilit- ies from the underlyings' implied volatility smiles. and K∗ being given elementwise in equation (3.6). These volatilities are then to be used in whatever The log-bilinear form of (3.10) allows us to sim- near-vanilla approximation that is chosen for the re- plify (3.9) to spective target product. We emphasize that it is clear that this process cannot possibly arrive at a sophist- ∗ > ∗ z = arg min z · z f(K ) = K (3.11) icated exotic product pricing framework. Instead, it z merely is intended to give a procedure that suffices with for the simplistic, but very fast, valuation of some near-vanilla products whilst preserving some sensible ξ∗ = A · z∗ : (3.12) consistency conditions. The starting point is that all of the involved As we shall see below, it turns out that the effect- stochastic financial observables X are governed by ive strikes themselves depend on (implied) volatilit- a joint log-normal law, and that there is a critical ies. This makes the task of effective geodesic strike level K for a function f(·) of the financial observ- calculation ultimately an implicit problem, since we ables, identified by the fact that the payout is of the need the effective strikes to be able to look up the form implied volatilities in the first place. In practice, we (θ · (f(X) − K))+ : (3.1) resolve this by the approximation that all volatilit- ies that show up in the effective strike formulæ are As is well known, without loss of generality, we can to be taken as at-the-forward implied volatilities. In transform the vector of financial variables X to a this context, we recall that we mentioned at the end vector of independent standard Gaussian variables z of the first paragraph in section2 that, when some according to of the underlyings require translation into the target 1 P − cii+ aij zj Xi = X^i · e 2 j (3.2) valuation measure, an approximation for this trans- lation of forward and implied volatility smile has to with be employed separately and prior to the invocation of ^ X = hXi ; (3.3) the effective geodesic strike procedure. When choos- the matrix A being the dispersion or factor loading ing the quanto-translation procedure, it is useful to matrix, and the log-covariance matrix C whose ele- bear in mind that the subsequent geodesic strike se- ments are lection is only an approximation for the sake of ana- cij = hln Xi; ln Xji (3.4) lytic tractability for a range of near-vanilla products, 2 and thus judge the required level of sophistication As for the effective geodesic strikes, equation (3.6) for the quanto translation in line with the overall gives us level of the chain of approximations. We emphasize, ∗ ^ ξi however, that whilst the geodesic strike procedure is Ki = Xi · e (3.23) only an approximation, its purpose is to be consistent with and accurate in a certain asymptotic sense, namely P that of the local volatility projection for large de- κ · cijwj ξ := j (3.24) viations, i.e, for out-of-the-money options, and that i w> · C · w of the geodesic distance asymptotics of [ABOBF02] and [Var67]. for geometric average options. For composite options as defined in (2.1), we have wi ≡ 1 and Geodesic strikes for geometric average op- ln K · σX ·(σX +ρXY σY ) ^ ^ σ2 +2σ ρ σ +σ2 ∗ ^ XY X X XY Y Y tions KX = X · e (3.25) ln K · σY ·(σY +ρXY σX ) ^ ^ σ2 +2σ ρ σ +σ2 We consider the payoff f(·) being given by ∗ ^ XY X X XY Y Y KY = Y · e : Y f (X) = Xwi : (3.13) It is worth reflecting on equation (3.25) with respect geometric i average i to a few benchmark cases. First, let us consider the case when σ ! 0. In that case, we have The constraint (3.8) becomes Y ∗ K > ∗ lim K = (3.26) ^ w ·A·z X K = G · e (3.14) σY !0 Y^ with which is consistent with the exact plain vanilla op- Y G^ := X^ wi (3.15) tion we arrive at in this limit for a composite option, i and we obtain of course the symmetric equivalent for i σX ! 0.
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