Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals

Journal of Banking & Finance 53 (2015) 158–178

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Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals Bertrand Tavin ⇑ EMLYON Business School, 23 Avenue Guy de Collongue, 69130 Ecully, France

a r t i c l e

i n f o

Article history: Received 26 July 2013 Accepted 30 December 2014 Available online 9 January 2015 JEL classification: G10 C52 D81 Keywords: Multi-asset derivative Arbitrage Incomplete market Risk-neutral measure Multivariate distribution Copula function

a b s t r a c t In this paper we study the existence of arbitrage opportunities in a multi-asset market when risk-neutral marginal distributions of asset prices are known. We first propose an intuitive characterization of the absence of arbitrage opportunities in terms of copula functions. We then address the problem of detecting the presence of arbitrage by formalizing its resolution in two distinct ways that are both suitable for the use of optimization algorithms. The first method is valid in the general multivariate case and is based on Bernstein copulas that are dense in the set of all copula functions. The second one is easier to work with but is only valid in the bivariate case. It relies on results about improved Fréchet–Hoeffding bounds in presence of additional information. For both methods, details of implementation steps and empirical applications are provided. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The notion of arbitrage is fundamental in economics and finance, as it underpins the setup in which academics and practitioners solve issues about equilibrium, portfolio allocation and contingent claim valuation. In these disciplines, many theoretical developments are thus built on the absence of arbitrage opportunity as a central assumption. For institutions involved in the financial industry, it is a strategic issue to ensure that their systems do not produce such opportunities. Hence, the availability of methods to detect arbitrage is of the utmost interest. In a market with a single underlying asset and a given set of vanilla options, the assessment of the absence of arbitrage is addressed in Carr and Madan (2005), Davis and Hobson (2007) and Cousot (2007). Essentially, the set of option prices is free of arbitrage as soon as butterfly spreads, call spreads and calendar spreads have positive prices. Assessing the absence of arbitrage among a set of derivative prices becomes a much more involved task when the set under scrutiny has some exotic options in addition to vanillas, or in the ⇑ Tel.: +33 (0)478337800; fax: +33 (0)478336169. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jbankfin.2014.12.023 0378-4266/Ó 2015 Elsevier B.V. All rights reserved.

case of a market with multiple underlying assets. The concern of our paper is to address the latter case in a general way that does not rely on the structure of a particular payoff and that is valid beyond the two-dimensional case. To the best of our knowledge, it has not yet been done in the existing literature. Our setup corresponds to a one period multi-asset market with known risk-neutral marginals, in which we obtain a characterization of the absence of arbitrage among a set of derivative prices in terms of copula functions. This characterization allows us to derive two necessary conditions of no-arbitrage, one of which is also sufficient, that both naturally lead to detection methods in the sense that if a condition is not verified then the market is not free of arbitrage. Hence our contribution is twofold. First, from a theoretical standpoint it allows a better understanding of the absence of arbitrage in our market model. Second, with practical perspectives, we detail the detection methods that are deduced from the theoretical part and we apply them to real market situations. For a single risky asset, it is possible to build risk-neutral diffusions that are compatible with a given set of vanilla options. Early references on that topic are Dupire (1993) and Laurent and Leisen (2000). The former considers a local volatility diffusion coefficient and the latter considers the construction of a risk-neutral Markov chain consistent with observed call option prices. It is possible to

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

go further and to obtain no-arbitrage bounds for an additional derivative when some are already available. This question has already been partially addressed. It is closely related, yet different from our concern. An approach to obtain the desired bounds for Asian options is based on the concept of comonotonicity, see Chen et al. (2008) and references therein. Another approach to obtain the desired bounds is via a Skorokhod embedding problem formulation, see Hobson (2010) and references therein. Yet another, more recent, approach is to apply optimal mass transportation theory to obtain the desired bounds, see Beiglbock et al. (2011) and Galichon et al. (2014). In the multi-asset case, bounds on prices of options written on several underlyings are available. When marginals are known, upper and lower bounds for two-asset basket options are initially obtained in Dhaene and Goovaerts (1996), in a context of actuarial analysis of portfolios of dependent risks. The same upper and lower bounds are obtained for other two-asset option payoffs in Rapuch and Roncalli (2001). Tankov (2011) derives improved bounds when some two-asset options are already quoted. For basket options, when single underlying vanillas are quoted, upper and lower bounds are available and the associated replicating strategies are explicit. The lower bound result is only valid in the two-asset case. As for Asian options written on a single asset, the above mentioned comonotonicity approach can be used, see Dhaene et al. (2002a,b) and Vanmaele et al. (2006). Key results for basket options are in Hobson et al. (2005a) and Chen et al. (2008). See also Laurence and Wang (2005) and Hobson et al. (2005b). In d’Aspremont and Ghaoui (2006), the authors work with a linear programming approach and obtain upper and lower bounds on basket option price when other basket options, with different weights, are already available. In Deelstra et al. (2008) the case of Asian basket options is studied in a constant volatility Black–Scholes–Merton framework, these options are path-dependent multi-asset options. Upper and lower bounds are also available for spread options. The respective bounds are obtained in Laurence and Wang (2008, 2009). The case of spread options is particular because arbitrage opportunities did exist during year 2009 among such options written on Constant Maturity Swap rates. This occurrence is documented in McCloud (2011). The pricing of European options written on several underlying assets has been widely studied. This body of research is linked to our problem but, as it is, does not answer it. The classical approach is to postulate a joint distribution for the underlying asset price returns and to calibrate the distribution parameters to available data in order to obtain prices and hedge ratios. For example, with this approach (Margrabe, 1978 and Stulz, 1982) both work in a two-asset extension of the Black–Scholes-Merton model and obtain valuation formulas, respectively, for exchange and rainbow options (also called min–max options). Alexander and Scourse (2004) propose a bivariate distribution built as a mixture for the pricing and hedging of spread options. Dempster et al. (2008) also study spread options and directly model the spread process in a cointegrated two-commodity framework. Nevertheless, in many cases, it is preferable to proceed in two steps by first specifying the marginals and then choosing the dependence structure. This alternative approach relies on the power of copula functions for the modeling of dependence and it allows an easier identification and understanding of the potential sources of risk. See among others (Rapuch and Roncalli, 2001; Coutant et al., 2001; Cherubini and Luciano, 2002 and Rosenberg, 2003). The remainder of the paper is organized as follows. In Section 2 we explain our financial framework and we propose a characterization of the absence of arbitrage in terms of copula functions. In Section 3 we develop a first methodology based on the family of Bernstein copulas. In Section 4 we propose, for the two-asset

159

case, another methodology based on improved Fréchet–Hoeffding bounds. Section 5 concludes. 2. Arbitrage and copulas in a multi-asset market We begin this section by detailing the structure of our market model and formalizing our problem. We then introduce copula functions and deduce a twofold characterization of the absence of arbitrage in our market in terms of such functions. 2.1. Model and assumptions We consider a fundamental probability space ðX; F ; PÞ with P the historical probability measure. Our financial market has one period and n þ 1 non-redundant primary assets (n P 2), t ¼ 0 is the initial time and t ¼ T < þ1 is the final time. The primary   assets are denoted by B; S1 ; . . . ; Sn . Their initial prices   B0 ; S10 ; . . . ; Sn0 2 0; þ1½nþ1 are known (non-random) and their final prices are positive random variables on ðX; F ; PÞ and are   denoted by BT ; S1T ; . . . ; SnT . The 0th asset, B, is a risk-free asset. It earns the risk-free rate r P 0 and its final value is non-random and known at initial time, BT ¼ 1. We suppose vanilla call options of all positive strikes to be available for the n risky assets of our market. For i ¼ 1; . . . ; n, we denote C i ðK i Þ the call option written on Si and struck at K i 2 ½0; þ1½ with the special case C i ð0Þ ¼ Si .  þ and C i0 ðK i Þ denotes Its final payoff is written C iT ðK i Þ ¼ SiT  K i its initial price. Our financial market model departs from reality on two notable characteristics. First, we consider a one-period market ðT ¼ 1Þ where, in reality, trading can almost be done in continuous time. This assumption corresponds to a restriction of trading strategies to only static strategies. By static strategies we mean buy at initial time and hold until final time. Second, we assume the availability of vanilla call prices for a continuum of positive strikes. In reality, vanilla options are traded only at a finite number of strikes hence leaving space for ambiguity in empirical applications. This remaining ambiguity is well documented and can be kept acceptable for underlyings with liquidly traded options such as equity indices or foreign exchange rates. See among others (Jackwerth and Rubinstein, 1996). We now introduce the notion of Risk-Neutral Measure for our financial market. The set of such measures is the cornerstone of the results presented in this paper because it is linked to the existence of arbitrage. Definition 1 (Risk-Neutral Measure). A probability measure Q on ðX; F Þ, equivalent to P, is a Risk-Neutral Measure (RNM) if, for i ¼ 1; . . . ; n

h i C i0 ðK i Þ ¼ B0 EQ C iT ðK i Þ for all K i 2 ½0; þ1½

ð1Þ

We define Q as the set of Risk-Neutral Measures for our basic financial market. The First Fundamental Theorem of Asset Pricing establishes the link between the set of risk-neutral measures and the absence of arbitrage opportunity. It has been first obtained in discrete time in Harrisson and Kreps (1979) and in continuous time in Harrisson and Pliska (1981). This theorem states that there is no arbitrage opportunity in the financial market if and only if Q is non-empty. For proofs, details, further references and extensions see Föllmer and Schied (2002) and Delbaen and Schachermayer (2006).

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In the sequel we suppose the above described basic market to be free of arbitrage so that Q is non-empty. Let Q 2 Q. We denote Q by HQ the joint distribution of ðS1T ; . . . ; SnT Þ under Q and F Q 1 ; . . . ; F n its

marginals that are the univariate distributions of S1T ; . . . ; SnT under Q. With the knowledge of initial call prices for all positive strikes, univariate distributions can be recovered independently of Q via the Breeden–Litzenberger formula. See Breeden and Litzenberger (1978). The (univariate) distribution of SiT is the same under all risk-neutral measures in Q, written F i (i ¼ 1; . . . ; n). The Breeden– Litzenberger formula writes, for i ¼ 1; . . . ; n

F i ðxÞ ¼ 1 þ

1 @C i0 ðxÞ x P 0 B0 @K i

ð2Þ

We also suppose that the available call prices are such that the F i : ½0; þ1½!½0; 1 are continuous functions. This assumption is essentially technical and corresponds to the realistic and fairly general case in which no particular value taken by a risky asset is charged with a strictly positive mass. In our setup there are n sub-markets composed of the risk-free asset, one risky asset and all the corresponding call options. The situation of each of these sub-markets can be described as static completeness due to the availability of call options with all positive strikes, written on the risky asset and with maturity T. Hence, within one of these sub-markets, a European contingent claim written on the considered risky asset and with maturity T will admit a single no-arbitrage price. The only relevant information to compute this no-arbitrage price is the univariate distribution of its underlying asset price. And this univariate distribution is shared by all measures in Q. On the contrary, a derivative written on two or more risky assets will, in general, have multiple no-arbitrage prices because our multi-asset financial market stays incomplete. For a discussion of the concepts of static and dynamic completeness we refer to Carr and Madan (2001) who study a financial market with the same characteristics as those of the individual sub-markets considered here. Multi-asset derivatives are now introduced to complement the basic market. A multi-asset derivative Z is a derivative written on up to n risky assets. Its final payoff is a positive random variable on ðX; F ; PÞ written Z T ¼ zðS1T ; . . . ; SnT Þ for some positive payoff function z on ½0; þ1½n . The notion of no-arbitrage price corresponds to a price at which the considered multi-asset derivative can be added to the basic market without creating an arbitrage opportunity. The notion of no-arbitrage price vector extends the one of no-arbitrage price. It corresponds to a vector of prices at   which a finite set of multi-asset derivatives Z 1 ; . . . ; Z q can be sequentially added to the basic market without creating arbitrage opportunity at any step and irrespective of the addition order.   Definition 2 (No-arbitrage Price Vector). Let Z 1 ; . . . ; Z q be a finite set of multi-asset derivatives ðq P 1Þ written on up to n risky   assets (and at least two) and with final payoffs Z 1T ; . . . ; Z qT .

p ¼ ðp1 ; . . . ; pq Þ 2 ½0; þ1½q is a no-arbitrage price vector for   Z1 ; . . . ; Zq

if it is compatible with, at least, one risk-neutral h i measure in Q, that is if there exists Q 2 Q such that pk ¼ B0 EQ Z kT 1

q

for k ¼ 1; . . . ; q. PðZ ; . . . ; Z Þ denotes the set of no-arbitrage price   vectors for Z 1 ; . . . ; Z q , it is a subset of ½0; þ1½q and is written

Our goal is to detect the existence of arbitrage in a multi-asset market. In our setup, the check is straightforward and well documented for single underlying derivatives because the associated sub-markets are complete. Hence our focus in the sequel will be on the q multi-asset derivatives rather than on derivatives written on one of the n risky assets. 2.2. Characterization in the general case A n-dimensional copula function (or copula) is a n-dimensional joint distribution function on ½0; 1n with uniform marginals. We denote by CðnÞ the set of all n-dimensional copula functions. Sklar’s Theorem is the fundamental result for the use of copulas in multivariate probabilistic modeling, see Sklar (1959, 1973) for the original statement and proof. According to this theorem, if H is a n-dimensional joint distribution function on ½0; þ1½n with continuous marginals denoted by F 1 ; . . . ; F n , then there exists a unique copula C H 2 CðnÞ such that, 8x 2 ½0; þ1½n

HðxÞ ¼ C H ðF 1 ðx1 Þ; . . . ; F n ðxn ÞÞ

  1 and, 8u 2 ½0; 1 ; C ðuÞ ¼ H F 1 1 ðu1 Þ; . . . ; F n ðun Þ . n

Conversely, if F 1 ; . . . ; F n are continuous univariate distribution functions on ½0; þ1½, and if C 2 CðnÞ , then HC : ½0; þ1½n !½0; 1 such that HC ðxÞ ¼ C ðF 1 ðx1 Þ; . . . ; F n ðxn ÞÞ is a n-dimensional joint distribution function with marginals F 1 ; . . . ; F n . We consider that a finite set of multi-asset derivatives   Z 1 ; . . . ; Z q is available in the market at initial price vector

p ¼ ðp1 ; . . . ; pq Þ. Our arbitrage detection problem is solved by   assessing whether or not p belongs to P Z 1 ; . . . ; Z q , the associated set of no-arbitrage price vectors. Given the structure of P in (3), we need to explore the set of risk-neutral models Q to check whether p is compatible with, at least, one of them. The Corollary below links Q, the set of risk-neutral measures, and the set CðnÞ of ndimensional copula functions. Following this remark we will work with copula functions because it is easier than working with multivariate distributions, as argued in Sklar (1973). Corollary 3. In the multi-asset market model detailed in Section 2.1, where univariate risk-neutral distributions of asset prices are known and continuous, the set Q is isomorphic to the set CðnÞ of n-dimensional copula functions.

Proof. The isomorphism between Q and CðnÞ is built from Sklar’s theorem and its corollary. The correspondence is written as follows. (from Q to CðnÞ ) Let Q 2 Q and denote by HQ the joint   distribution of S1T ; . . . ; SnT under Q. The corresponding copula C Q is defined for u 2 ½0; 1n as

  1 C Q ðuÞ ¼ HQ F 1 1 ðu1 Þ; . . . ; F n ðun Þ

ð5Þ C

(from CðnÞ to Q) Let C 2 CðnÞ and denote H the corresponding distribution, defined for x 2 ½0; þ1½n as

HC ðxÞ ¼ C ðF 1 ðx1 Þ; . . . ; F n ðxn ÞÞ

ð6Þ n

Let Q be the probability measure on ½0; þ1½ associated with HC .   By construction, the joint distribution of S1T ; . . . ; SnT under QC has C

n h i   PðZ 1 ; .. .; Z q Þ ¼ B0 EQ Z 1T ;. .. ;B0 EQ Z qT jQ 2 Q h i o and EQ Z kT < 1; k ¼ 1; .. .; q

ð4Þ

H

ð3Þ

marginals F 1 ; . . . ; F n , given by the Breeden–Litzenberger formula (2) so that all vanilla calls are properly repriced. Hence QC 2 Q.

h

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

For C 2 CðnÞ we introduce the notation EC denoting the expectation under QC , the probability measure associated with HC given by   (6). Under QC ; S1T ; . . . ; SnT has a dependence structure described by the copula C and marginals F 1 ; . . . ; F n . For C 2 CðnÞ and k ¼ 1; . . . ; q h i the expectation EC Z kT can be written as a multi-dimensional integral on ½0; 1n as

h i h  i EC Z kT ¼ EC zk S1T ; . . . ; SnT Z zk ðs1 ; . . . ; sn ÞdCðF 1 ðs1 Þ; . . . ; F n ðsn ÞÞ ¼ ¼

Z

1 zk ðF 1 1 ðu1 Þ; . . . ; F n ðun ÞÞdCðuÞ

ð7Þ n

C If C is absolutely continuous, it has density c ¼ @u1@...@u and the n integral becomes

h i Z EC Z kT ¼

½0;1n

1 zk ðF 1 1 ðu1 Þ; . . . ; F n ðun ÞÞcðu1 ; . . . ; un Þdu

ð8Þ

Under the same conditions as in Definition (3), the set of noarbitrage price vectors PðZ 1 ; . . . ; Z q Þ straightforwardly writes

PðZ 1 ; . . . ; Z q Þ ¼

n h i o   B0 EC Z 1T ; . . . ; B0 EC Z qT jC 2 CðnÞ

ð9Þ

We denote by q the pricing rule between CðnÞ and PðZ 1 ; . . . ; Z q Þ   so that elements of P are written qðCÞ ¼ q1 ðCÞ; . . . ; qq ðCÞ .

q : C 2 CðnÞ # qðCÞ 2 PðZ 1 ; . . . ; Z q Þ  ½0; þ1½q 1

ð10Þ

and u2 ¼ 1  u1 on ½0; 12 . See Example 2.11 in Nelsen (2006). þ

Z



Z

EC ½Z T  ¼

PðZ 1 ; . . . ; Z q Þ. With (9) the new expression of PðZ 1 ; . . . ; Z q Þ, it can be done by exploring the set CðnÞ . If the search fails to find a copula compatible with p we will deduce that p is out of the set and hence not free of arbitrage. We define the q subsets of CðnÞ composed of n-dimensional copulas compatible with single components of p. These subsets are written, for k ¼ 1; . . . ; q

n h i o  Ck;ðnÞ ¼ C 2 CðnÞ jB0 EC Z kT ¼ pk ¼ C 2 CðnÞ jqk ðC Þ ¼ pk and Ck;ðnÞ is empty if pk R PðZ k Þ. The twofold characterization of no arbitrage in terms of copulas can now be stated. It is directly obtained by examining definitions of P; q and Ck;ðnÞ .

p 2 ½0; þ1½q . The following two characterizations

p 2 PðZ 1 ; . . . ; Z q Þ () 9C 2 CðnÞ such that qðCÞ ¼ p p 2 PðZ 1 ; . . . ; Z q Þ ()

q \ k¼1

Ck;ðnÞ –£

ð11Þ ð12Þ

1

0

q

PðZ Þ  . . .  PðZ Þ. In most cases it has a non-trivial structure. As an example, consider a market with three risky assets on which three multi-asset derivatives are available. These derivatives are calls with identical strikes. The first one is written on the best of the three assets, the second is written on the worst and the third is written on the second to best. The price of any of the three derivatives is fully determined by the prices of the two others and of the three corresponding single asset vanilla calls with identical strikes. Intuitively, when a price is fixed for a first multi-asset derivative, the freedom to choose a copula to obtain a price for a second one is reduced. For the two prices to be free of arbitrage the pricing copula has to be chosen among those compatible with the first price. Our concern is to determine whether a given p belongs or not to

Corollary 4. Let hold

In this subsection we make some comments about the particular case of a market in which a single multi-asset derivative Z with payoff function z, written on exactly n risky assets, is available at initial price p P 0. If n ¼ 2 and z is either 2-increasing or 2decreasing, the Fréchet–Hoeffding bounds C  and C þ directly lead to the bounds of PðZÞ. See Appendix A for the definitions of n-increasing functions and Fréchet–Hoeffding bounds.  If z is 2-increasing then 8C 2 Cð2Þ we have EC ½Z T  6 EC ½Z T    þ   Cþ . If z is 2-decreasing then 6 E ½Z T  and PðZÞ ¼ qðC Þ; q C    þ  8C 2 Cð2Þ we have EC ½Z T  6 EC ½Z T  6 EC ½Z T  and PðZÞ ¼ q C þ ;  qðC Þ. This result is from Rapuch and Roncalli (2001) and is based on results in Tchen (1980) and Müller and Scarsini (2000). An early version of this result for basket options is obtained in Dhaene and Goovaerts (1996). The expectations involving the Fréchet–Hoeffding bounds are computed as one dimensional integrals because C þ and C  respectively have support u2 ¼ u1

q

It is important to note that PðZ ; . . . ; Z Þ is a strict subset of the Cartesian product of individual sets of no-arbitrage prices 1

Each characterization leads to a method to detect arbitrage. The first allows us to express the associated problem as a calibration problem. The second is useful when n ¼ 2, because in that particular case best-possible bounds of the Ck;ðnÞ sets can be computed in a quasi-explicit form. Before developing the details of these arbitrage detection methods, we discuss the particular case of a single multi-asset derivative. 2.3. The case of a single multi-asset derivative

½0;þ1½n

½0;1n

161

EC ½Z T  ¼

0

1

  1 z F 1 1 ðuÞ; F 2 ðuÞ du

ð13Þ

  1 z F 1 1 ðuÞ; F 2 ð1  uÞ du

ð14Þ

As it is remarked in Rapuch and Roncalli (2001) and seconded in Tankov (2011), all common two-asset options have a 2-increasing or 2-decreasing payoff function. Hence, in most encountered cases, explicit bounds of PðZÞ are fast to compute and the check for noarbitrage is straightforward to perform. On the contrary, for n P 3 the key property is no longer valid so that the Fréchet–Hoeffding bounds do not necessarily lead to bounds on the multi-asset derivative price. See Rapuch and Roncalli (2001) and Müller and Scarsini (2000). Even though there exists some partial results, such as the upper bound for the basket option price obtained in Hobson et al. (2005a), bounds of PðZÞ cannot usually be directly computed and the check for no-arbitrage must be addressed the same way as in the general case. It is an active field of research. To conclude the section we present two empirical applications in which we compute individual no-arbitrage bounds for two-asset options. In the first application, the underlying assets are the French and German equity market indices CAC40 and DAX30, denominated in EUR and with initial forward values normalized at 100. It is performed with market data as of May 2008. The considered two-asset options have one-year maturity and are a call on the spread, a call on the equally weighted basket, a put on the maximum and a call on the minimum. In the second application, the underlyings are the foreign exchange rates EURUSD and USDJPY. The considered two-asset options have one-month maturity and are six vanilla options on the cross-rate EURJPY, namely two calls struck out of the money, a call and a put struck at the money, two puts struck out of the money. The application is performed with market data as of January 2006. Spot and forward values of EURJPY and USDJPY are divided by 100 for better readability of figures. Further details about market data can be found in Appendix B. The computations of individual no-arbitrage bounds are performed using the Matlab quadrature routine quadgk.

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B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

We consider four alternatives to model the marginal distribution of log-returns of each underlying asset. These marginal distributions correspond to the Normal Inverse Gaussian model (NIG), the CGMY model (CGMY), the Heston stochastic volatility model with jumps (Bates) and the double exponential jump-diffusion model (Kou). Relevant references and comments on the chosen marginal distributions are gathered in Appendix B. For each underlying, these marginal distributions are calibrated to vanilla option prices under the constraint of a forward no-arbitrage condition. Details concerning these calibrations as well as the obtained parameters for NIG, CGMY, Bates and Kou distributions are gathered in Appendix B. Table 1 presents upper and lower bounds of PðZ Þ for various two-asset options written on CAC40 and DAX30 indices, respectively denoted by S1 and S2 . Table 2 presents upper and lower bounds of PðZ Þ for vanilla put and call options on EURJPY, denoted by X, and seen as two-asset options written on EURUSD and USDJPY exchange rates, respectively denoted by S1 and S2 . The relation between the two primary exchange rates and the cross-rate is X T ¼ S1T S2T . Obtained bounds on the option price are expressed according to the convention in use on FX options market that is JPY pips per EUR notional. It corresponds to the amount in JPY to be paid to acquire an option on EURJPY with 10,000 EUR notional. The individual no-arbitrage intervals in Tables 1 and 2 are wide if compared with typical option prices. This is because the deriva-

tives under scrutiny are strongly linked to dependence risk, making their prices very sensitive to the chosen copula function. As said, these intervals can be used to check that a multi-asset option price, taken individually, is free of arbitrage. They cannot be relied upon to assess the absence of arbitrage among a set of two or more multi-asset options. For a given multi-asset option and compared to the width of the no-arbitrage intervals, the choice of marginal distributions has a limited effect on the obtained individual intervals. This remark is valid for options on equity indices and for options on FX rates, as shown in Tables 1 and 2. 3. A method with Bernstein copulas This section begins with definition and properties of Bernstein copulas. We then present the first main result of the paper as a necessary and sufficient condition of no-arbitrage that is valid irrespective of the number of risky assets in the market. The retained approach corresponds to characterization (11) and relies on particular properties of Bernstein copulas within the set of all copulas. The possibility to use such an approach in bivariate foreign exchange markets is briefly sketched out in the conclusions of Salmon and Schleicher (2006) and Hurd et al. (2007). To the best of our knowledge, this approach has not yet been developed further in the literature. We begin the section by considering Bernstein copulas and their properties.

Table 1 No-Arbitrage bounds for individual two-asset derivatives, with maturity T ¼ 1 year and written on CAC40 (S1 ) and DAX30 (S2 ) indices. Upper panel: marginal distributions of logreturns are NIG and CGMY. Lower panel: marginal distributions of log-returns are Bates and Kou. ZT

K

qðC  Þ

0

15.59



q Cþ



PðZ Þ

qðC  Þ

½0:73; 15:59

15.58

NIG marginals  þ S2T  S1T  K  þ 1 2 1 2 ðST þ ST Þ  K  þ K  maxðS1T ; S2T Þ  þ minðS1T ; S2T Þ  K

0.73



PðZ Þ

CGMY marginals 0.73

½0:73; 15:58

100

0.79

7.82

½0:79; 7:82

0.76

7.81

½0:76; 7:81

110

1.32

12.92

½1:32; 12:92

1.36

12.92

½1:36; 12:92

80

7.85

20.53

½7:85; 20:53

7.86

20.52

½7:86; 20:52

Bates marginals  þ S2T  S1T  K  þ 1 2 1 2 ðST þ ST Þ  K  þ K  maxðS1T ; S2T Þ  þ minðS1T ; S2T Þ  K



q Cþ

Kou marginals

0

15.6

0.74

½0:74; 15:60

15.57

0.82

½0:82; 15:57

100

0.84

7.83

½0:84; 7:83

0.81

7.81

½0:81; 7:81

110

1.26

12.88

½1:26; 12:88

1.32

12.81

½1:32; 12:81

80

7.86

20.53

½7:86; 20:53

7.89

20.49

½7:89; 20:49

Table 2 No-Arbitrage bounds for individual vanilla options on EURJPY (X), with maturity T ¼ 1 month and seen as two-asset options written on EURUSD and USDJPY rates. Upper panel: marginal distributions of log-returns are NIG and CGMY. Lower panel: marginal distributions of log-returns are Bates and Kou. ZT

K

qðC  Þ



q Cþ



PðZ Þ

qðC  Þ

NIG marginals ðK  X T Þþ ðK  X T Þþ ðK  X T Þþ ðX T  K Þþ ðX T  K Þþ ðX T  K Þþ

1.3306 1.3571 1.3828 1.3828 1.4081 1.4321

105.78 179.45 285.98 290.97 180.63 108.82

0.08 0.78 13.67 18.67 0.00 0.00

ðK  X T Þ ðK  X T Þþ ðK  X T Þþ ðX T  K Þþ ðX T  K Þþ ðX T  K Þþ

1.3306 1.3571 1.3828 1.3828 1.4081 1.4321

105.06 178.76 285.52 290.51 180.65 109.19

0.00 0.12 12.43 17.42 0.00 0.00



PðZ Þ

CGMY marginals ½0:08; 105:78 ½0:78; 179:45 ½13:67; 285:98 ½18:67; 290:97 ½0:00; 180:63 ½0:00; 108:82

105.06 178.80 285.35 290.35 180.02 108.29

Bates marginals þ



q Cþ

0.04 0.57 13.25 18.24 0.00 0.00

½0:04; 105:06 ½0:57; 178:80 ½13:25; 285:35 ½18:24; 290:35 ½0:00; 180:02 ½0:00; 108:29

Kou marginals ½0:00; 105:06 ½0:12; 178:76 ½12:43; 285:52 ½17:42; 290:51 ½0:00; 180:65 ½0:00; 109:19

106.07 179.87 286.13 291.11 180.26 108.19

0.00 0.09 14.53 19.51 0.02 0.00

½0:00; 106:07 ½0:09; 179:87 ½14:53; 286:13 ½19:51; 291:11 ½0:02; 180:26 ½0:00; 108:19

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B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

3.1. Bernstein copulas and their properties Bernstein copulas and their first properties were introduced in Li et al. (1997) and Li et al. (1998). This family is usually used in the context of the approximation of copulas or non-parametric estimation of dependence structure. The bivariate case is studied in Durrleman et al. (2000a) and the multivariate case is studied in Sancetta and Satchell (2004) and Sancetta (2007). Janssen et al. (2012) study the asymptotic properties of the Bernstein copula estimator. In insurance, Diers et al. (2012) use Bernstein copulas to model the dependence between non-life insurance risks. In finance, Salmon and Schleicher (2006) and Hurd et al. (2007) apply Bernstein copulas to the pricing of bivariate currency derivatives and their parameters are calibrated to vanilla options on the cross exchange rate via a reconstruction of its probability density. Bernstein polynomials are widely used in the theory of approximation and smoothing. They constitute the building blocks of Bernstein copulas. Definition 5 (Bernstein polynomial). For  amfixed degree m 2 N, there are m þ 1 Bernstein polynomials Bi;m i¼0 defined on ½0; 1 as

Bi;m ðxÞ ¼



m



i

xi ð1  xÞmi

ð15Þ



 m m! is the binomial coefficient. ¼ i!ðmiÞ! i Before stating the definition of a Bernstein copula, it is necessary to introduce the notion of discretization of the n-dimensional unit hypercube ½0; 1n . As there is no reason for discretizing dimensions differently and for simplicity, we consider m 2 N a common number of steps for all dimensions. We denote Ln;m such a discretization, formally defined as where

Ln;m ¼

na

1

;...;

m

o an  aj 2 N and 0 6 aj 6 m for j ¼ 1; . . . ; n m

ð16Þ n

n;m

n;m

n

There are ðm þ 1Þ points in L and L  ½0; 1 . For better readability, elements of Ln;m are denoted by v ¼ ðv 1 ; . . . ; v n Þ and elements of ½0; 1n are denoted by u ¼ ðu1 ; . . . ; un Þ. Definition 6 (Bernstein copula). Let n be a real-valued function on n Ln;m and define C m B : ½0; 1 !½0; 1 as

X

Cm B ðuÞ ¼

nðv Þ

v 2Ln;m

¼

m X

an ¼0

n a an Y 1 n ;...; Ba ;m ðui Þ m m i¼1 i

! ð17Þ

1. for 0 6 aj 6 m  1 ðj ¼ 1; . . . ; nÞ and with dn ¼ 0 or 1 whether n is even or odd, respectively.

...

l1 ¼0

2. for

 ð1Þ

ln ¼0

Pn 

 l a 1 þ l1 a n þ ln j¼1 j P0 n ;...; m m

dn þ

n;m

v2L max

n X

!

v j  n þ 1; 0

j¼1

m

;...;

an 

6 nðv 1 ; . . . ; v n Þ 6 min ðv j Þ j¼1;...;n

1 X

¼

m

...

l1 ¼0

1 X

 ð1Þ

Pn 

 l a1 þ l1 a n þ ln j¼1 j ;...; n m m

dn þ

ln ¼0

ð18Þ Bernstein copulas are absolutely continuous and, as such, always have a proper density. It is a first important property. Definition 7 (Bernstein copula density). Let C m B be a n-dimensional Bernstein copula and n its associated parameter function. We m denote by cm B the density of C B defined as

@nCm B ðuÞ @u1 . . . @un ! m1 m1 n a X X an Y 1 ... Dn n ;...; Bai ;m1 ðui Þ ¼ m m i¼1 a ¼0 a ¼0

cm B ðuÞ ¼

ð19Þ

n

1

Within the set of copulas, the family of Bernstein copulas have two additional and closely related properties. The first one is an approximation property and the second one is a denseness property in CðnÞ . The denseness property is the one relevant in our arbitrage detection context. We recall both below. For details and proofs, see Li et al. (1997), Durrleman et al. (2000a) and Sancetta and Satchell (2004). Let C be a copula. The order m Bernstein copula approximation n of C is denoted by C m B ðCÞ and defined, for u 2 ½0; 1 , as

X

Cm B ðCÞðuÞ ¼

Cðv Þ

v 2Ln;m m X

a1 ¼0

i¼1

1 X

1

¼

If n is n-increasing and verifies the Fréchet–Hoeffding bounds, then Cm B is a copula, named Bernstein copula. It corresponds to the fulfillment of the two conditions below.

1 X

Dn n

a

Bai ;m ðui Þ

m X

...

a1 ¼0

n Y

The summation in (17) involves ðm þ 1Þn terms as it is done over the elements of the grid Ln;m and in each term there is a product involving n terms. For C m B to be fully determined, it is necessary to know the values taken by n on the ðm þ 1Þn points of the grid. The function n is called parameter function because its values are the parameters of the Bernstein copula. As said, the family of multivariate Bernstein copulas is defined and studied in Sancetta and Satchell (2004) and Sancetta (2007). Before stating the expression of the density of Bernstein copulas, we introduce Dn n the n-dimensional volume operator applied to the parameter function n and involving a multiple summation with 2n terms. It is defined, for 0 6 aj 6 m  1 ðj ¼ 1; . . . ; nÞ, as

...

n Y

Bai ;m ðui Þ

i¼1 m X

an ¼0

C

a

1

m

;...;

n an Y

m

Bai ;m ðui Þ

! ð20Þ

i¼1

We denote by nðCÞ the associated parameter function, written nðCÞðv Þ ¼ C ðv 1 ; . . . ; v n Þ, for v 2 Ln;m . nðCÞ satisfies the two conditions in Definition 6 so that C m B ðCÞ is always a proper copula. The Bernstein copula approximation of C uniformly converges to C as the approximation order m grows to infinity

Cm B ðCÞ ! C The other property of the family of Bernstein copulas is their denseness within CðnÞ . This property means we can always find a Bernstein copula as close as desired to a given copula in CðnÞ . It is the key to their application to our arbitrage detection problem. In particular, we have



8C 2 CðnÞ ; 9 C m B

 m2N

such that C m B ! C

where the convergence is the uniform convergence on ½0; 1n . We denote by ðnm Þm2N the corresponding sequence of parameter functions. Obviously, one of the converging sequences is the Bernstein copula approximation of C. We denote by ðnm ðCÞÞm2N the associated series of parameter functions.

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B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

In the development of this paper we will use theoretical properties of the family of Bernstein copulas. However, the empirical relevance of this family of copulas deserves to be discussed. In non-life insurance, the question is addressed in Diers et al. (2012) in which authors consider the modeling of a portfolio of dependent claims and compare various copulas by means of goodness-of-fit tests. They found that the Bernstein copula provides a proper, although not always best, fit to data in various situations. The same kind of question should be addressed in finance as it could confirm that the use of Bernstein copulas is meaningful from an empirical standpoint in addition to being appropriate from a theoretical one. This development is left for future research. It could have implications in the measurement of risks and on decisions based on risk measures. In the next subsection we express the arbitrage detection problem as a calibration problem and we take advantage of the properties of Bernstein copulas reviewed in the current subsection.

Z

½0;1n

m k k k That is jqk ðC m B Þ  q ðCÞj ¼ jq ðC B Þ  p j!0. Hence, the pricing error converges to zero as m grows to infinity. That is

2 kqðC m B Þ  pk !0

  Again, one of the converging series is C m B ðCÞ m2N , the Bernstein copula approximation of C with growing order. Series converging faster may exist here.   2 m ð(Þ Suppose there exists C m B m2N such that kqðC B Þ  pk !0. From the compactness of CðnÞ , there exists a sequence extracted     uðmÞ ðnÞ from C m the B m2N that converges within C . Denote by C B m2N

extracted sequence and C its limit. Hence we have the uniform convergence

3.2. Arbitrage detection as a calibration problem

CB

The setup in this subsection is the same as in Section 2.2. For a given price vector p 2 ½0; þ1½q , we recall the characterization (11)

and

1

q

p 2 PðZ ; . . . ; Z Þ () 9C 2 C

ðnÞ

uðmÞ

such that qðCÞ ¼ p

Hence assessing whether or not p belongs to P can be done by searching a copula in CðnÞ that perfectly reproduces p. If this search fails then p is not an arbitrage free price vector for the considered set of derivatives. Unfortunately the search over CðnÞ cannot realistically be performed because there exists too many copulas to explore them all. As Bernstein copulas are dense in CðnÞ , if there exists a copula reproducing p, then there is also a sequence of Bernstein copulas converging to that particular copula. This property allows us to obtain a necessary and sufficient condition of no-arbitrage based on the above characterization. That is a necessary and sufficient condition for p to belong to P. This theoretical result is formally stated below. It is one of the main results of this paper. Proposition 8 (Necessary and sufficient condition for no-arbitrage in the general case). The necessary and sufficient condition for noarbitrage with Bernstein copulas writes



p 2 P () 9 C mB

 m2N

 

2

! 0 such that q C m B p

ð21Þ

where k:k is the Euclidean norm on Rq and measures the pricing error   as the discrepancy between q C m and the observed price vector p. B That is

kqðCÞ  pk2 ¼

q X 

2

qk ðCÞ  pk :

k¼1

Here the Euclidean norm is a common choice. Note that other choices are also possible.

p 2 P then there exists   C 2 CðnÞ such that qðCÞ ¼ p, where qðCÞ ¼ q1 ðCÞ; . . . ; qq ðCÞ ¼ h i     B0 EC Z 1T ; . . . ; B0 EC Z qT . From their denseness property, there   m exists a sequence of Bernstein copulas C m B m2N such that C B uniformly converges to C.   The convergence of C m B m2N to C implies the weak convergence of the associated probability measures on ½0; 1n . We refer to Deheuvels (1981) for a proof. Thus, for non-pathological payoff functions zk ; k ¼ 1; . . . ; q, we have Proof. ð)Þ From characterization (11), if

m

1 zk ðF 1 1 ðu1 Þ; . . . ; F n ðun ÞÞdC B ðuÞ Z 1 zk ðF 1 ! 1 ðu1 Þ; . . . ; F n ðun ÞÞdCðuÞ

½0;1n

!C 2 CðnÞ

uðmÞ

kqðC B

Þ  pk2 !0

So that, by continuity of the pricing rule q from CðnÞ to ½0; þ1½n , we obtain qðCÞ ¼ p. h From the necessary and sufficient condition (21) we deduce an arbitrage detection method that has the form of a calibration task for a Bernstein copula. For that purpose we need to fix a finite value for the Bernstein copula order m. This value has to be chosen large enough to have an acceptable precision while keeping the computational load manageable. Consider the following minimization problem

8 2 m > > > min kqðC B ðnÞÞn  pk > < for n 2 Rðmþ1Þ and under constraints P1 : > > nðC  Þ 6 n 6 nðC þ Þ > > : Dn n P 0

ð22Þ

where the inequality constraints ensure that C m B ðnÞ is a proper n

n

copula and are taken element wise in Rðmþ1Þ for n and in Rm for Dn n. P 1 has the typical form of a calibration task where parameters of a model, here n, are fitted to a set of observed prices, here p. The first constraint, corresponding to the Fréchet–Hoeffding bounds, imposes the values of the first and last elements of n in each dimension. Hence it reduces the dimensionality of the searched solution to ðm  1Þn . The particular structure of P 1 makes it difficult to solve. To overcome this difficulty, a possibility is to regularize the problem by adding a penalty term to the objective function. The idea is to solve a problem that is close to the initial one but has a better behavior. For details and a formal approach to the regularization of the model calibration problem, see Crépey (2003), Cont and Tankov (2004) and references therein. We choose the penalty term to be the squared Euclidean distance to a prior set of parameters. This is a common choice in the spirit of Tikhonov regularization. Note that other choices are also possible. The regularized calibration task that is, in fact, solved is written

  8 > min kqðC m ðnÞÞ  pk2 þ akn  n0 k2 > B > > > < ðmþ1Þn 0 and under constraints P 1 : for n 2 R >  > > nðC Þ 6 n 6 nðC þ Þ > > : Dn n P 0

ð23Þ

165

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178 n

where a > 0 is the regularization parameter and n0 2 Rðmþ1Þ is an a priori guess. n0 is also the seed of the minimization algorithm (initial start point). a controls the magnitude of the regularization term when compared to the pricing error term in the objective function. a has to be chosen large enough so that the regularization is useful while keeping the loss of precision acceptable. We denote by n a solution of the regularized calibration problem P 01 and by b the associated final value of the squared pricing error. An alternative way to measure the final pricing error is the average absolute relative error denoted by b0 . b and b0 are written 2  b ¼ kqðC m B ðn ÞÞ  pk q m  k 1 X jq ðC B ðn ÞÞ  pk j b0 ¼ q k¼1 pk

The results of this minimization program can be represented as the triplet ðn ; b; b0 Þ. It is to be noted that this triplet obtained after solving P 01 is sensitive to the choice of n0 . Said differently, different initial seeds are likely to yield different results for n and for the final error values. In order to immunize the conclusion of our method with respect to the choice of a particular n0 we rely on a randomization of the minimization problem P 01 . Instead of solving it once with an arbitrarily chosen initial seed, we repeat N times the minimization task with a random start point and store the results of each repetition. Hence we obtain a sample of size N for the final pricing errors. N is chosen large enough so that we can derive statistical properties of the pricing errors. This randomization of the minimization task allows us to base our conclusion on the statistical distribution of the final pricing error. A way to obtain random draws of the initial seed is to compute it as the Bernstein   copula approximation of a Gaussian copula, that is n0 ¼ n C GR0 with R0 a random correlation matrix of dimension n  n. C GR0 denotes the n-dimensional Gaussian copula with correlation matrix R0 . To obtain random correlation matrices in dimension two, one can draw a random correlation coefficient as a uniform random variable on ½1; 1. For higher dimensions, one can use Algorithm 1 proposed in Cont and Deguest (2013) to draw random correlation matrices. We refer to this article for more details. The intuition behind this approach is that if p 2 P then the values of the final pricing errors in P 01 should be small. It may not be zero because the degree of the Bernstein copula is fixed and finite so as to obtain a solvable calibration problem. Hence the rationale to decide whether p 2 P can have the form of a statistical test built on the randomization of P 01 . Based on the distribution of pricing errors obtained from the N draws of P 01 it is possible to compute the percentiles of these errors and to compare them to predefined thresholds. We advocate the choice of the 99th percentile of the relative error b0 and denote its value by b099 . Let  > 0 be the chosen

threshold. If b099 is found to be below the threshold we conclude that p 2 P. Alternative rules can be defined in terms of the squared pricing error or using the 95th percentile. The value of the threshold depends on the chosen percentile and on the chosen degree m. In practice the value of the pair ðm; Þ can be chosen by benchmarking some cases known to be arbitrage-free. Hence the steps of the arbitrage detection heuristic are as follows. 1. Fix a degree m 2 N and a threshold  > 0. 2. Generate N random draws of the initial seed n0 . 3. For each draw, solve P 01 and compute b and b0 the final pricing error values. 4. From the obtained sample of error values, compute the percentile b099 . 5. If b099 <  then conclude p 2 P, otherwise conclude p R P. Besides being based on a necessary and sufficient condition, the main strength of this method is to remain valid when the number of risky assets becomes larger than 2. And, even if it is not the initial purpose, the Bernstein copula that achieves the lowest pricing error can be used to price and hedge an additional multi-asset instrument, coherently with all the available prices. We now provide two empirical applications to illustrate how the proposed method operates on practical cases. The two considered basic markets are identical to those of empirical applications presented in Section 2.3, both with two primary assets. We refer to this section for details. The market data and parameters of marginal distributions (NIG, CGMY, Bates and Kou) are gathered in Appendix B. In the first application, we consider a set of three two-asset options with one year to maturity, written on the CAC40 and DAX30 indices, namely two calls on the spread and a put on the maximum. The considered market prices for these two-asset options are quotes observed in the interbank market. In the second application, we consider a set of six vanilla options on EURJPY with one month to maturity, namely two calls struck out of the money, a call and a put struck at the money and two puts struck out of the money. These options are treated as two-asset options on the EURUSD and USDJPY exchange rates and the considered market prices are quotes observed in the interbank market, we refer to Appendix B for more details. The observed premiums of the EURJPY vanilla options are expressed in JPY pips per EUR notional. The observed market prices are gathered in Tables 3 and 4, respectively for the first and second application. Individual no-arbitrage bounds are met and are given as a reference. To implement the proposed method, we generate 1000 random draws of n0 and, for each draw, we solve the minimization problem P 01 with Matlab routine fmincon. For the two applications, this procedure is repeated for the four considered marginal distributions. For the first application

Table 3 Market quotes and calibrated prices for the set of two-asset options on CAC40 and DAX30 considered in the first empirical application of Section 3.2. Upper panel: marginal distributions of log-returns are NIG and CGMY. Lower panel: marginal distributions of log-returns are Bates and Kou. k

Z kT

K

pk



qk C 5B





P Zk





qk C 5B



NIG marginals 1 2 3

 þ S2T  S1T  K  þ S2T  S1T  K  þ K  maxðS1T ; S2T Þ

2 3

 þ S2T  S1T  K  þ S2T  S1T  K  þ K  maxðS1T ; S2T Þ



CGMY marginals

0

8.35

8.354

½0:73; 15:59

8.352

½0:73; 15:58

20

2.43

2.427

½0:05; 8:22

2.432

½0:04; 8:22

110

7.21

7.212

½1:32; 12:92

7.211

½1:36; 12:92

0

8.35

8.352

½0:74; 15:60

8.352

½0:82; 15:57

20

2.43

2.429

½0:04; 8:23

2.429

½0:05; 8:28

110

7.21

7.211

½1:26; 12:88

7.215

½1:32; 12:81

Bates marginals 1



P Zk

Kou marginals

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B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

Table 4 Market quotes and calibrated prices for the set of EURJPY vanilla options considered in the second empirical application of Section 3.2 and seen as two-asset options on EURUSD and USDJPY. Upper panel: marginal distributions of log-returns are NIG and CGMY. Lower panel: marginal distributions of log-returns are Bates and Kou. 



k

Z kT

K

pk

1 2 3 4 5 6

ðK  X T Þþ ðK  X T Þþ ðK  X T Þþ ðX T  K Þþ ðX T  K Þþ ðX T  K Þþ

1.3306 1.3571 1.3828 1.3828 1.4081 1.4321

19.90 58.34 145.61 150.59 55.05 17.79

19.994 58.187 145.639 150.662 55.006 17.814

1 2 3 4 5 6

þ

1.3306 1.3571 1.3828 1.3828 1.4081 1.4321

19.9 58.34 145.61 150.59 55.05 17.79

19.959 58.219 145.640 150.632 55.024 17.814

qk C 5B



P Zk





qk C 5B



NIG marginals



P Zk CGMY marginals

½0:08; 105:78 ½0:78; 179:45 ½13:67; 285:98 ½18:67; 290:97 ½0:00; 180:63 ½0:00; 108:82

19.982 58.196 145.635 150.656 55.014 17.813

Bates marginals ðK  X T Þ ðK  X T Þþ ðK  X T Þþ ðX T  K Þþ ðX T  K Þþ ðX T  K Þþ



½0:00; 105:06 ½0:12; 178:76 ½12:43; 285:52 ½17:42; 290:51 ½0:00; 180:65 ½0:00; 109:19

½0:04; 105:06 ½0:57; 178:80 ½13:25; 285:35 ½18:24; 290:35 ½0:00; 180:02 ½0:00; 108:29 Kou marginals

19.966 58.209 145.630 150.652 55.012 17.841

½0:00; 106:07 ½0:09; 179:87 ½14:53; 286:13 ½19:51; 291:11 ½0:02; 180:26 ½0:00; 108:19

Table 5 Global minimums and percentiles (95th and 99th) of squared and relative final errors obtained from 1000 draws of the minimization problem P 01 with a random initial seed n0 . Upper panel corresponds to the first empirical application with equity indices presented in Section 3.2. Lower panel corresponds to the second empirical application with FX rates presented in Section 3.2. Marginals

Relative error – b0

Squared error – b 95th perc.

99th perc.

Global min.

95th perc.

99th perc.

Empirical application with equity indices NIG 2.84e5 CGMY 1.11e5 Bates 4.72e6 Kou 2.66e5

Global min.

0.00011 0.00017 9.27e5 0.00011

0.00011 0.00020 9.66e5 0.00012

0.00041 0.00047 0.00027 0.00034

0.00114 0.00131 0.00099 0.00122

0.00115 0.00134 0.00119 0.00222

Empirical application with FX NIG 0.00926 CGMY 0.00806 Bates 0.00434 Kou 0.00798

0.01870 0.01805 0.01501 0.02112

0.01908 0.01813 0.02271 0.02122

0.00135 0.00131 0.00096 0.00150

0.00201 0.00180 0.00179 0.00208

0.00215 0.00185 0.00185 0.00213

the chosen degree of the Bernstein copulas is m ¼ 5 and the regularization parameter is a ¼ 0:8. For the second application the chosen degree of the Bernstein copulas is m ¼ 8. For both applications we work with a threshold  ¼ 0:003 applied to the 99th percentile of the relative absolute error b0 . The numerical values of the regularization parameters are of different magnitudes in the considered empirical applications. This is because the calibration problems have different numerical scales. However, the threshold is the same because it is applied to the percentile of the average relative pricing error which is comparable across cases. For the two empirical applications, Table 5 presents the global minimums and the percentiles of the final pricing errors obtained with NIG, CGMY, Bates and Kou marginal distributions. For the first application, the percentiles b099 are found to be below the threshold, irrespective of the chosen marginals. The greatest value is found for Kou marginals but remains below the threshold. Hence we conclude that there is no arbitrage among the observed prices. For the second application, the percentiles b099 are also found to be below the threshold, irrespective of the chosen marginals. The greatest value is found for NIG marginals but remains below the threshold. For the two applications, the four considered marginal distributions yield rather close results. Table 5 also reports the global minimums reach by the pricing error measures. For the first application, the Bernstein copula parameters corresponding to these global minimums are respec-

tively denoted by n1;NIG ; n1;CGMY ; n1;Bates and n1;Kou for NIG, CGMY, Bates and Kou marginals. For the second application, the Bernstein copula parameters corresponding to these global minimums are respectively denoted by n2;NIG ; n2;CGMY ; n2;Bates and n2;Kou for NIG, CGMY, Bates and Kou marginals. These parameters are reproduced in Appendix C. For both applications, the lowest global minimum is reached with Bates marginals. For each empirical applications, the choice of marginal distributions (NIG, CGMY, Bates or Kou) has a limited effect on the 95th and 99th percentiles of the final pricing errors corresponding to minimization task P 01 . Moreover, the global minimums found with the considered marginal distributions have comparable values. According to the these results, the proposed method with Bernstein copulas and its conclusions appear to be little sensitive to the choice of marginal distributions. Tables 3 and 4 presents the two-asset options prices obtained with the Bernstein copulas which yield the global minimums in terms of squared error. Their parameters are denoted by n1;NIG ; n1;CGMY ; n1;Bates and n1;Kou for the first application and by n2;NIG ; n2;CGMY ; n2;Bates and n2;Kou for the second application. We name these copulas the calibrated copulas as they produce the best fits to the observed market prices. In Figs. 1 and 2 we plot the joint densities of log-returns obtained with the four alternative marginal distributions and the calibrated Bernstein copulas. The joint densities appear to be smooth, and for equity indices they appear

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

167

(a)

(b)

(c)

(d)

Fig. 1. Density of log-returns obtained for CAC40-DAX30 with the calibrated Bernstein copulas in the first application presented in Section 3.2. The Bernstein copulas degree is m ¼ 5 and their parameters are n1;NIG ; n1;CGMY ; n1;Bates and n1;Kou , respectively for plots (a), (b), (c) and (d). Marginal distributions of log-returns are NIG (a), CGMY (b), Bates (c) and Kou (d).

168

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

(a)

(b)

(c)

(d)

Fig. 2. Density of log-returns obtained for JPYUSD-EURUSD with the calibrated Bernstein copulas in the second application presented in Section 3.2. The Bernstein copulas degree is m ¼ 8 and their parameters are n2;NIG ; n2;CGMY ; n2;Bates and n2;Kou , respectively for plots (a), (b), (c) and (d). Marginal distributions of log-returns are NIG (a), CGMY (b), Bates (c) and Kou (d).

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

169

(a)

(b)

(c)

(d)

Fig. 3. Density of the calibrated Bernstein copulas for CAC40-DAX30 obtained in the first application presented in Section 3.2. Degree is m ¼ 5 and their parameters are n1;NIG ; n1;CGMY ; n1;Bates and n1;Kou , respectively for plots (a), (b), (c) and (d).

170

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

(a)

(b)

(c)

(d)

Fig. 4. Density of the calibrated Bernstein copulas for JPYUSD-EURUSD obtained in the second application presented in Section 3.2. Degree is m ¼ 8 and their parameters are n2;NIG ; n2;CGMY ; n2;Bates and n2;Kou , respectively for plots (a), (b), (c) and (d).

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B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

skewed to the left in both directions. Figs. 3 and 4 plot the densities of the calibrated Bernstein copulas mentioned above. The calibrated copulas are smooth and slightly asymmetric.

In this section we consider the case of a market with two risky assets and we obtain another main result of the paper that has the form of a necessary condition of no-arbitrage. The two-asset case is particular because the partial order on copulas leads, for derivatives with either 2-increasing or 2decreasing payoff functions, to monotonicity in the price and to the associated upper and lower bounds, as seen in Section 2.3. From a calculation standpoint, the two-asset case is considerably easier to handle than the general multivariate case. Within this particular case, we obtain a condition of no-arbitrage and we detail the associated detection method that improves on the method relying on Bernstein copulas. To do so we take advantage of results in two dimensions obtained in Tankov (2011) and Bernard et al. (2012) about improved Fréchet–Hoeffding bounds when additional information is available. These results are applied to the case of options written on two underlyings in Tankov (2011) and to the case of optimal investment strategies in Bernard et al. (2012). We consider a restricted financial market with only two risky   assets, S1 and S2 . We also consider a finite set Z 1 ; . . . ; Z q of derivatives written on these assets and with payoff functions that are either 2-increasing or 2-decreasing. As already remarked, in the two-asset case, all common payoff functions verify this condition. The dimensionality being fixed at n ¼ 2, we drop, for readability, the explicit dependence on n from our notations. Our approach is to start with characterization (12), rewritten below for a given price vector p 2 ½0; þ1½q .





q \

Ck –£

k¼1

For each Ck set, we introduce pointwise best possible upper and lower bounds, respectively denoted by Ak and Bk , and defined for ðu1 ; u2 Þ 2 ½0; 12 and k ¼ 1; . . . ; q as

n o Ak ðu1 ; u2 Þ ¼ max Cðu1 ; u2 ÞjC 2 Ck n o Bk ðu1 ; u2 Þ ¼ min Cðu1 ; u2 ÞjC 2 Ck

ð24Þ ð25Þ

The bounds Ak and Bk are said pointwise best possible in the sense that for all points ðu1 ; u2 Þ 2 ½0; 12 there exists a copula in Ck that reaches the bound value. Extending results in Nelsen et al. (2001b,a), a quasi-explicit expression for the improved bounds is derived in Tankov (2011). This key result makes easy the computation of values taken by Ak and Bk . These improved bounds are proper copulas under sufficient conditions obtained in Tankov (2011) and weakened in Bernard et al. (2012). We refer to these articles for details and proofs. In our framework, these quasi-explicit expressions write, if Z k has a 2-increasing payoff function ( Ak ðu1 ; u2 Þ ¼

k qk;1 if pk 2 ½qk ðC  Þ; qk ðu1 ; u2 ; C þ ðu1 ; u2 ÞÞ  ðu1 ; u2 ; p Þ þ

( Bk ðu1 ; u2 Þ ¼

C ðu1 ; u2 Þ

otherwise 



k qk;1 if pk 2 ½qkþ ðu1 ; u2 ; C  ðu1 ; u2 ÞÞ; qk C þ  þ ðu1 ; u2 ; p Þ 

C ðu1 ; u2 Þ

otherwise

ð26Þ ð27Þ

and, if Z k has a 2-decreasing payoff function ( Ak ðu1 ; u2 Þ ¼ ( k

B ðu1 ; u2 Þ ¼

k qk;1 if pk 2 ½qk ðu1 ; u2 ; C þ ðu1 ; u2 ÞÞ; qk ðC  Þ  ðu1 ; u2 ; p Þ

C þ ðu1 ; u2 Þ

otherwise 



k qk;1 if pk 2 ½qk C þ ; qkþ ðu1 ; u2 ; C  ðu1 ; u2 ÞÞ þ ðu1 ; u2 ; p Þ

C  ðu1 ; u2 Þ

otherwise









 þ k k k qk;1  ðu1 ; u2 ; p Þ ¼ max h 2 ½C ðu1 ; u2 Þ; C ðu1 ; u2 Þjq ðu1 ; u2 ; hÞ ¼ p  þ k k k qk;1 þ ðu1 ; u2 ; p Þ ¼ min h 2 ½C ðu1 ; u2 Þ; C ðu1 ; u2 Þjqþ ðu1 ; u2 ; hÞ ¼ p

4. The appeal of improved Fréchet–Hoeffding bounds

p 2 P Z 1 ; . . . ; Z q ()

where

ð28Þ

and, for ða; bÞ 2 ½0; 12 and h 2 ½C  ða; bÞ; C þ ða; bÞ

qk ða; b; hÞ ¼

Z

ah 0

þ þ þ

Z Z Z

  1 zk F 1 1 ðuÞ; F 2 ð1  uÞ du

a ah

  1 zk F 1 1 ðuÞ; F 2 ða þ b  h  uÞ du

1bþh a 1 1bþh

qkþ ða; b; hÞ ¼

Z

h 0

þ þ þ

  1 zk F 1 1 ðuÞ; F 2 ð1 þ h  uÞ du

  1 zk F 1 1 ðuÞ; F 2 ð1  uÞ du

  1 zk F 1 1 ðuÞ; F 2 ðuÞ du

Z Z Z

a

  1 zk F 1 1 ðuÞ; F 2 ðu þ b  hÞ du

h aþbh a 1 aþbh

  1 zk F 1 1 ðuÞ; F 2 ðu þ h  aÞ du

  1 zk F 1 1 ðuÞ; F 2 ðuÞ du

k 1 k q1 þ ðu1 ; u2 ; p Þ and q ðu1 ; u2 ; p Þ are values taken by copulas. qþ ða; b; hÞ and q ða; b; hÞ are derivative prices obtained when

pricing, respectively, with the upper and lower bounds of the set of copulas taking specified value h at point ða; bÞ in ½0; 12 and are monotonic functions of h. With formulas (26)–(29) the computation of values taken by Ak and Bk is easy to implement because it involves only one-dimensional quadratures and a root search. T We can now return to our initial problem. For qk¼1 Ck to be non-empty, the lowest upper bound should always stay greater than, or equal to, the largest lower bound. The proposed necessary condition of no-arbitrage is based on this idea. The involved ’’super’’ bounds are denoted by A and B, both take values in ½0; 1 and are defined pointwise on ½0; 12 as

n o Aðu1 ; u2 Þ ¼ min Ak ðu1 ; u2 Þjk ¼ 1; . . . ; q n o Bðu1 ; u2 Þ ¼ max Bk ðu1 ; u2 Þjk ¼ 1; . . . ; q

Proposition 9 (Necessary condition for no-arbitrage in the two-asset   case). For p to be a no-arbitrage price vector for Z 1 ; . . . ; Z q ; A should be greater than, or equal to, B everywhere in the unit square. This necessary condition is formally written

p 2 P ) 8ðu1 ; u2 Þ 2 ½0; 12 Aðu1 ; u2 Þ P Bðu1 ; u2 Þ

   Proof. Suppose there exists u1 ; u2 2 ½0; 12 such that       A u1 ; u2 < B u1 ; u2 . By construction of A and B the minimum and the maximum are always reached. So that, for A, the minimum is attained for k ¼ kA 2 1; . . . ; q and for B, the maximum is attained for k ¼ kB 2 1; . . . ; q with kB –kA . Hence

    AkA u1 ; u2 < BkB u1 ; u2 that is

ð29Þ

ð30Þ

n  o n  o   max C u1 ; u2 jC 2 CkA < min C u1 ; u2 jC 2 CkB

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(a)

(b)

(c)

(d)

Fig. 5. Values and contour plots of the objective functions in P 2 obtained in the first application presented in Section 4. Marginal distributions of log-returns are NIG (a), CGMY (b), Bates (c) and Kou (d).

B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

173

(a)

(b)

(c)

(d)

Fig. 6. Values and contour plots of the objective functions in P 2 obtained in the second application presented in Section 4. Marginal distributions are NIG (a), CGMY (b), Bates (c) and Kou (d).

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so that a copula cannot be at the same time in CkA and CkB . Thus T CkA CkB ¼ £   T T And because qk¼1 Ck  CkA CkB , we finally have q \

Ck ¼ £:

k¼1

That is equivalent to

p R P.

h

The intuition behind the result is that if the inequality is invalidated somewhere in ½0; 12 there exists no model reproducing the observed set of prices and as a consequence this set is not jointly arbitrage free. It allows us to express our arbitrage detection problem as a minimization task with the conclusion depending on the final value of its objective function. Consider A  B as a function of ðu1 ; u2 Þ and denote by P 2 the following minimization problem

P2 :

min

ðu1 ;u2 Þ2½0;12

fAðu1 ; u2 Þ  Bðu1 ; u2 Þg

ð31Þ

The objective function in P 2 takes values in ½1; þ1 and the mini  mization is done over ½0; 12 , a compact set. Denote u1 ; u2 a solution of P 2 and c the corresponding value of the objective function. The value of c is straightforwardly linked to the detection of arbitrage because if it is negative then the intersection of the Ck sets is empty and the price vector p is not free of arbitrage. This second arbitrage detection heuristic has two steps only. 1. Solve P 2 and keep c the final objective function value. 2. If c P 0 then conclude p 2 P, otherwise conclude p R P. A suitable approach to solve P 2 is to adopt a global optimization algorithm that uses multiple start points and keeps the lowest local minimum found. This global optimization approach has good chances to reach a global minimum because the objective function has a reasonable number of peaks and basins (see Figs. 5 and 6 described below). In comparison to the first method that works in the space of prices, the method proposed in this section works in the space of models (copulas). It has two main strengths: first, it is easy to implement and, second, it flags true arbitrage situations because the algorithm is not truncated as it is the case with Bernstein copulas. The output information of this arbitrage detection method cannot be used for pricing additional derivatives because, by construction, it does not include a calibration step. For this specific purpose a calibration has to be performed separately once market prices have been checked to be jointly arbitrage free. To conclude the section we present two numerical applications to illustrate how the method operates. In both applications we consider the CAC40 and DAX30 indices as primary assets. Here again we work with four alternative marginal distributions (NIG, CGMY, Bates and Kou). The first application is performed with the same market conditions and multi-asset options observed quotes as in Section 3.2, only the detection method differs. The market data and marginal distributions parameters are gathered in Appendix B. In the second application we use the same marginal distributions and consider a set made of two derivatives written on two assets, a call on the spread and a call on the equally weighted basket, the prices of which are artificially built in order to obtain a case with arbitrage. In both applications, the minimization task P 2 is solved with a global optimization routine, namely a multi-start version of the Matlab routine fmincon with twenty start points. In the first application, for NIG, CGMY, Bates and Kou marginals, the minimization task P 2 leads to a minimum value of  the objective function c ¼ 0 and this value is reached for any u1 ; u2 such that u1 or u2 is either 0 or 1. We conclude that the condition is met. Fig. 5 plots the surfaces of values and the associated contour

graphs of the objective functions A  B as functions of ðu1 ; u2 Þ 2 ½0; 12 . For this application, the global minimum found for the objective function in P 2 is not sensitive to the choice of marginal distributions. In addition, this choice has a very limited effect on the shape of the objective function, as illustrated in Fig. 5. According to these remarks, the proposed method with improved Fréchet–Hoeffding bounds appears to be little sensitive to the choice of marginal distributions. Compared to the method with Bernstein copulas, it yields the same conclusion. Table 6 presents the artificial prices for two-asset options considered in the second application. As said, these prices are built to obtain a case with arbitrage. Options under scrutiny have a oneyear maturity and are written on CAC40 (S1 ) and DAX30 (S2 ) indices. Table 6 also presents the individual no-arbitrage bounds for the two considered two-asset options. For this application, the values of c, the global minimum of the objective function, as well as ðu1 ; u2 Þ, the location of the minimum, are gathered in Table 7 for NIG, CGMY, Bates and Kou marginals. For the considered marginal distributions, the global minimum values are all negative and the minimums are reached for ðu1 ; u2 Þ located near the center of the unit square. In this application, for the four considered marginal distributions, we conclude that p is not free of arbitrage even if its components are within the corresponding individual no-arbitrage bounds. The choice of marginal distributions has a limited effect on the value of the global minimum and on the location of ðu1 ; u2 Þ. Fig. 6 plots the surfaces of values and the associated contour graphs of the objective functions A  B as functions of ðu1 ; u2 Þ 2 ½0; 12 . A look at the plots confirms that the objective functions go below zero near the center of the unit square. The choice of marginal distributions has a limited effect on the shape of the objective function, as illustrated in Fig. 6. This last application documents a case in which the proposed method is able to flag a situation of arbitrage between the multiasset derivatives where the usual method with individual no-arbitrage bounds fails to do so. In this application the conclusion of the

Table 6 Prices of two-asset options on CAC40 and DAX30 in the artificial case considered in the second numerical application of Section 4. The options maturity is T ¼ 1 year. The individual no-arbitrage bounds obtained with NIG and CGMY distributions (upper panel) are provided as well as those obtained with Bates and Kou marginals (lower panel). k

Z kT



1 2



1 1 2 ðST



1 2

S2T



S2T

1 1 2 ðST



S1T

K

þ

þ S2T Þ  K



S1T

K

þ

þ

þ S2T Þ  K

þ









K

pk

NIG marginals

CGMY marginals

0

15.00

½0:73; 15:59

½0:73; 15:58

100

7.20

½0:79; 7:82

½0:76; 7:81

Bates marginals

Kou marginals

0

15.00

½0:74; 15:6

½0:82; 15:57

100

7.20

½0:84; 7:83

½0:81; 7:81

P Zk

P Zk

Table 7 Value and location of the global minimum reached by the objective function in P 2 obtained with NIG, CGMY, Bates and Kou marginals in the second application of Section 4 (artificial case with arbitrage). Marginals

NIG CGMY Bates Kou

Global min.

Location of min.

c

ðu1 ; u2 Þ

0:0935 0:0955 0:0909 0:0890

ð0:4805; 0:4802Þ ð0:4840; 0:4859Þ ð0:4775; 0:4781Þ ð0:4820; 0:4846Þ

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B. Tavin / Journal of Banking & Finance 53 (2015) 158–178

proposed method is the same with the four considered marginal distributions. 5. Conclusion

A n-dimensional function H : Rn !R is said n-increasing if 8B n-box 1 X k1 ¼0

In a multi-asset financial market with known risk-neutral marginal distributions, we have proposed a twofold characterization of the absence of arbitrage opportunities in terms of copula functions. In these characterizations, a set of multi-asset option prices is free of arbitrage if there exists, at least, one copula function that is compatible with it. It allows us to better understand, for the considered market, the mathematical structure underpinning the set of riskneutral measures, as it is then isomorphic to the set of copula functions. Two no-arbitrage necessary conditions have then been deduced and the associated detection methods have been detailed. The first, which is also a sufficient condition, relies on special properties of Bernstein copulas, in particular their denseness within the set of all copulas and it remains valid when dimensionality increases. The second is made appealing by the availability of quasi-explicit formulas for improved Fréchet–Hoeffding bounds. It is, however, valid in the bivariate case only. Both detection methods have been formalized in such a way optimization algorithms can solve it. The structure of our market model makes them particularly suitable for applications to markets with several equity indices or exchange rates, which correspond to the empirical applications we have provided. The sensitivity of the proposed methods with respect to the choice of marginal distributions has been studied and appears to be small. The proposed methods can easily be used, within a financial institution, by a risk management or model validation department to control for the overall coherence of the pricing tools used by front office operators to quote, trade and value multi-asset options, and by doing so, to prevent their institution from being arbitraged by other players. Thinking the other way round, hedge funds or trading teams can implement the method to detect and eventually take advantage of inconsistencies between quoted prices of multi-asset options available in the market. Acknowledgements I am grateful to Patrice PONCET for his comments and discussions. I thank Carole BERNARD, Jan DHAENE, Jean-Paul LAURENT, Olivier LE COURTOIS, Jean-Luc PRIGENT, Lorenz SCHNEIDER, Steven VANDUFFEL for their comments and suggestions on earlier versions. I also thank an anonymous referee for comments and suggestions that helped improve the quality and clarity of the paper. All remaining errors are mine. Appendix A. Additional elements about copula functions The set CðnÞ is a convex and compact set under the topology of uniform convergence. In CðnÞ , pointwise and uniform convergences are equivalent. See Deheuvels (1978) and Darsow et al. (1992). There exists a partial order on CðnÞ named concordance ordering. For C 1 ; C 2 2 CðnÞ , we write C 1  C 2 if 8u 2 ½0; 1n C 1 ðuÞ 6 C 2 ðuÞ and  is called concordance order. It is known that CðnÞ admits upper and lower bounds with respect to the concordance order. These bounds are named the Fréchet–Hoeffding bounds and respectively denoted by C þ and C  . C þ and C  are respectively written, for u 2 ½0; 1n þ

C ðu1 ; . . . ; un Þ ¼ minðui Þ i

C  ðu1 ; . . . ; un Þ ¼ max

X i

! ui  n þ 1; 0

...

1 X

Pn   ð1Þ i¼1 ki H x1k1 ; . . . ; xnkn P 0

kn ¼0

with xi1 ¼ ai and xi0 ¼ bi for i ¼ 1; . . . ; n. A n-box in Rn is a set B ¼ ½a; b ¼ ½a1 ; b1   . . .  ½an ; bn   Rn with ai < bi 8i ¼ 1; . . . ; n. Further details and properties of multivariate copulas can be found §2.10 in Nelsen (2006) and in Durante and Sempi (2010). The latter reference also offers a wide literature review on the topic. Appendix B. Market data and marginal distributions In this appendix, we gather and describe the market data sets used for the empirical applications presented in the paper. Table B.8 presents the relevant zero-coupon bond prices. Table B.9 gathers spot and forward prices for the considered underlying assets. Note that in Sections 2.3, 3.2 and 4, forward prices of equity indices are normalized to 100 for better readability of results. Note also that in Sections 2.3 and 3.2, forward exchange rates for EURJPY and USDJPY are divided by 100 for better readability of results. Table B.10 reproduces the implied volatility quotes for CAC40 and DAX30 indices. Table B.11 reproduces the implied volatility quotes for the considered exchange rates following the usual conventions of the FX options market, we refer to Clark (2011) for details on these conventions. These datasets correspond to end of day quotes observed in the interbank market. The equity indices dataset corresponds to May 2008 and the considered options and forwards have a one-year maturity. The foreign exchange dataset corresponds to January 2006 and the considered options and forwards have a one-month maturity. In Sections 3.2 and 4 we consider a set of three two-asset options written on CAC40 an DAX30 indices. These options are two calls on the spread and a put on the maximum. Their market prices correspond to observed quotes in the interbank market and are reproduced in Table 3. In the second application presented in Section 3.2 we work with a set of six vanilla options on EURJPY considered as two-asset options written on EURUSD and USDJPY exchange rates. These options are two calls struck out of the money, a call and a put struck at the money and two puts struck out of the money. Their market prices are reproduced in Table 4 and are computed from observed volatility quotes in Table B.11. In empirical applications, we model each underlying asset log-returns with four alternative univariate distributions. These

Table B.8 Zero-coupon bond prices for EUR, USD and JPY currencies for the considered dates and maturities. Date

Currency

Maturity

Price

May 2008 January 2006 January 2006 January 2006

EUR EUR JPY USD

1 year 1 month 1 month 1 month

0.9520 0.9979 1.0000 0.9962

Table B.9 Spot and forward prices for underlying assets of FX and equity index types. Maturities of forwards are one year for CAC40 and DAX30 and one month for EURUSD, USDJPY and EURJPY. Date

Name

Currency

Spot price

Forward price

May 2008 May 2008 January 2006 January 2006 January 2006

CAC40 DAX30 EURUSD USDJPY EURJPY

EUR EUR USD JPY JPY

5055.0 7740.0 1.2136 114.21 138.61

5200.0 7970.0 1.2158 113.78 138.32

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Table B.10 Implied volatility smile quotes for vanilla options on CAC40 and DAX30 indices. Values are expressed in volatility points. Maturity is one year and date is May 2008. CAC40

rimp ðKÞ

K

2528 32.32

3033 30.36

3539 28.12

4044 25.76

4550 23.34

K rimp ðKÞ

3870 35.91

4644 31.90

5418 28.55

6192 25.80

6966 23.50

5055 20.92

5561 18.64

6066 16.66

6572 15.04

7077 14.51

7583 14.51

7740 21.53

8514 20.09

9288 19.29

10062 18.98

10836 19.05

11610 19.32

DAX30

Table B.11 Implied volatility smile quotes for vanilla options on FX rates EURUSD, USDJPY and EURJPY. Values are expressed in volatility points and the ATM column corresponds to the zero delta straddle strategy, according to the FX options market conventions. Maturity is one month and date is January 2006.

EURUSD USDJPY EURJPY

ATM

25DRR

25DBFLY

50DRR

50DBFLY

8.95 9.15 9.30

0:18 1:05 0:70

0.15 0.20 0.20

0:28 1:75 1:20

0.40 0.80 0.65

Table B.12 Values of NIG model parameters used in empirical applications as risk-neutral marginal distributions of log-returns. These parameters are calibrated to vanilla options market quotes corresponding to Tables B.10 and B.11. The obtained RMSE on observed quotes is also reported. NIG parameters

CAC40 DAX30 EURUSD JPYUSD

RMSE

a

b

l

d

21.050 6.530 75.572 54.789

17:467 3:510 1:461 15.343

0.278 0.124 0.001 0:011

0.204 0.237 0.054 0.038

min

hM 2HM

0.085 0.065 0.624 0.616

Table B.13 Values of CGMY model parameters used in empirical applications as risk-neutral marginal distributions of log-returns. These parameters are calibrated to vanilla options market quotes corresponding to Tables B.10 and B.11. The obtained RMSE on observed quotes is also reported. CGMY parameters

CAC40 DAX30 EURUSD JPYUSD

which a jump component is added. It is introduced in Bates (1996) and Bakshi et al. (1997). The Kou model corresponds to a jump-diffusion Lévy process based on two exponential jump components. It is introduced in Kou (2002). NIG and CGMY processes are pure jump Lévy processes with infinite jump activity. The Bates process is a jump-diffusion process with stochastic volatility and finite jump activity. The Kou process is a jump-diffusion Lévy process with finite jump activity. Hence the four considered marginal distributions span various univariate dynamics for the underlying asset prices. Running the proposed methods using these different alternatives allows us to assess the sensitivity of the obtained results with respect to the choice of marginal distributions. For each underlying asset, the four distributions are calibrated to vanilla options data sets. In order to do so, we solve, for each asset and distribution, the minimization task that is written

RMSE

C

G

M

Y

0.155 0.086 0.135 0.087

2.473 1.755 78.202 73.839

63.537 9.368 81.185 42.411

0.915 1.056 0.668 0.663

0.148 0.178 0.664 0.891

marginal distributions correspond to the Normal Inverse Gaussian model (NIG), the CGMY model (CGMY), the Heston stochastic volatility model with jumps (Bates) and the double exponential jumpdiffusion model (Kou). The NIG distribution is a special case of the generalized hyperbolic distribution introduced in BarndorffNielsen (1978), see also Barndorff-Nielsen (1997). The CGMY process is a pure jump Lévy process introduced in Carr et al. (2002). The Bates process is built as a stochastic volatility diffusion to

NK  2 X OðK j ; hM Þ  OObs ðK j Þ j¼1

where hM is a vector of parameters for distribution M; HM is the set of feasible parameters for distribution M and N K is the number of strikes for the considered maturity. OðK; hM Þ denotes the price of the strike K option obtained using the chosen model with parameter hM and OObs ðKÞ denotes the corresponding observed market price. This minimization is solved using a global minimization routine. A way to assess the quality of the fit is by means of the root mean squared error (RMSE) between model and market prices. For a given distribution and underlying asset, hM is the vector of calibrated parameters and the RMSE is written

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u NK  2 u1 X RMSE ¼ t OðK j ; hM Þ  OObs ðK j Þ NK j¼1 The calibrated parameters are reproduced in Tables B.12, B.13, B.14 and B.15. These tables also present the obtained RMSE. Appendix C. Bernstein copulas calibration outputs In this appendix we reproduce the Bernstein copula parameters corresponding to the global minimums in terms of squared error for the calibration task P 01 . n1;NIG ; n1;CGMY ; n1;Bates and n1;Kou are those obtained for the first empirical application of Section 3.2. n2;NIG ; n2;CGMY ; n2;Bates and n2;Kou are those obtained for the second empirical application of Section 3.2.

Table B.14 Values of Bates model parameters used in empirical applications as risk-neutral marginal distributions of log-returns. These parameters are calibrated to vanilla options market quotes corresponding to Tables B.10 and B.11. The obtained RMSE on observed quotes is also reported. Bates parameters

CAC40 DAX30 EURUSD JPYUSD

RMSE

v0

r

q

k

v

j

l

pffiffiffiffi sj

0.029 0.029 0.007 0.006

0.374 0.547 0.259 0.249

0:774 0:488 0:018 0.027

0.822 1.087 1.065 1.981

0.078 0.088 0.028 0.012

0.077 0.113 0.144 1.106

0:155 0:102 0.013 0.047

0.180 0.092 0.012 0.007

0.160 0.157 0.867 1.095

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0

Table B.15 Values of Kou model parameters used in empirical applications as risk-neutral marginal distributions of log-returns. These parameters are calibrated to vanilla options market quotes corresponding to Tables B.10 and B.11. The obtained RMSE on observed quotes is also reported. Kou parameters

CAC40 DAX30 EURUSD JPYUSD

0

0

RMSE

r

d

p

g1

g2

0.093 0.143 0.020 0.016

1.298 0.641 1.177 1.499

0.002 0.080 0.402 0.797

5.433 7.401 83.741 79.103

7.566 5.577 93.772 70.193

0

0

0

0

0

0.235 0.152 0.710 0.938

1

C 0:12 0:16 0:18 0:19 0:2 C C 0:17 0:28 0:36 0:38 0:4 C C  n1;NIG C 0:18 0:36 0:49 0:57 0:6 C C 0:19 0:38 0:57 0:74 0:8 C A 0:2 0:4 0:6 0:8 1 1 0 0 0 0 0 0 C B B 0 0:12 0:16 0:17 0:18 0:2 C C B B 0 0:16 0:29 0:35 0:38 0:4 C C B n1;CGMY ¼ B C B 0 0:18 0:35 0:50 0:57 0:6 C C B B 0 0:19 0:39 0:58 0:75 0:8 C A @ 0 0:2 0:4 0:6 0:8 1 1 0 0 0 0 0 0 0 C B B 0 0:14 0:16 0:17 0:18 0:2 C C B B 0 0:16 0:30 0:35 0:37 0:4 C C B  n1;Bates ¼ B C B 0 0:18 0:35 0:50 0:56 0:6 C C B B 0 0:19 0:38 0:57 0:74 0:8 C A @ 0 0:2 0:4 0:6 0:8 1 1 0 0 0 0 0 0 0 C B B 0 0:14 0:16 0:17 0:18 0:2 C C B B 0 0:16 0:30 0:35 0:37 0:4 C C B  n1;Kou ¼ B C B 0 0:17 0:34 0:50 0:56 0:6 C C B C B @ 0 0:18 0:37 0:56 0:75 0:8 A 0 0:2 0:4 0:6 0:8 1 1 0 0 0 0 0 0 0 0 0 0 C B B 0 0:07 0:09 0:09 0:11 0:12 0:12 0:12 0:13 C C B B 0 0:10 0:15 0:19 0:21 0:23 0:24 0:25 0:25 C C B C B B 0 0:11 0:19 0:26 0:30 0:33 0:35 0:37 0:38 C C B C B n2;NIG ¼ B 0 0:11 0:22 0:31 0:37 0:42 0:46 0:49 0:50 C C B B 0 0:12 0:23 0:34 0:43 0:51 0:56 0:60 0:63 C C B C B B 0 0:12 0:24 0:36 0:47 0:57 0:65 0:71 0:75 C C B B 0 0:12 0:25 0:37 0:49 0:61 0:72 0:82 0:88 C A @ 0 0:13 0:25 0:38 0:50 0:63 0:75 0:88 1 1 0 0 0 0 0 0 0 0 0 0 C B B 0 0:07 0:08 0:09 0:11 0:12 0:12 0:12 0:13 C C B B 0 0:10 0:15 0:19 0:21 0:22 0:24 0:25 0:25 C C B C B B 0 0:11 0:19 0:26 0:30 0:32 0:35 0:37 0:38 C C B C B n2;CGMY ¼ B 0 0:11 0:22 0:30 0:37 0:42 0:46 0:49 0:50 C C B B 0 0:12 0:23 0:34 0:43 0:51 0:56 0:60 0:63 C C B C B B 0 0:12 0:24 0:36 0:47 0:57 0:65 0:71 0:75 C C B B 0 0:12 0:25 0:37 0:49 0:61 0:72 0:82 0:88 C A @ 0 0:13 0:25 0:38 0:50 0:63 0:75 0:88 1 B B0 B B0 B ¼B B0 B B0 @ 0 0

n2;Bates

n2;Kou

0

B0 B B B0 B B B0 B ¼B B0 B B0 B B0 B B @0

0 0 0 B0 B B B0 B B B0 B ¼B B0 B B0 B B0 B B @0 0

0

0

0:07 0:09

0 0:09

0:10 0:15 0:19 0:11 0:19 0:26 0:11 0:22 0:30 0:12 0:23 0:33 0:12 0:24 0:35 0:12 0:25 0:37 0:13 0:25 0:38 0 0 0 0:07

0:09 0:09

0:10 0:15 0:19 0:11 0:20 0:26 0:12 0:22 0:31 0:12 0:24 0:34 0:12 0:25 0:36 0:12 0:25 0:37 0:13 0:25 0:38

0

0

0

0

0

1

0:10 0:12 0:12 0:12 0:13 C C C 0:21 0:22 0:23 0:25 0:25 C C C 0:30 0:32 0:34 0:37 0:38 C C 0:37 0:42 0:46 0:48 0:50 C C C 0:43 0:51 0:56 0:60 0:63 C C 0:47 0:57 0:65 0:71 0:75 C C C 0:49 0:61 0:72 0:82 0:88 A 0:50 0:63 0:75 0:88 1 1 0 0 0 0 0 0:11 0:12 0:12 0:12 0:13 C C C 0:21 0:23 0:24 0:25 0:25 C C C 0:30 0:33 0:35 0:37 0:38 C C 0:38 0:42 0:46 0:49 0:50 C C C 0:43 0:51 0:56 0:60 0:63 C C 0:47 0:57 0:65 0:71 0:75 C C C 0:49 0:61 0:72 0:82 0:88 A 0:50 0:63 0:75 0:88 1

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