Basket option pricing and implied correlation in a Lévy copula model - PDF

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Basket option pricing and implied correlation in a Lévy copula model Daniël Linders, Wim Schoutens AFI_1494 Basket option pricing and implied correlation in a Lévy copula model Daniël Linders Wim Schoutens

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Basket option pricing and implied correlation in a Lévy copula model Daniël Linders, Wim Schoutens AFI_1494 Basket option pricing and implied correlation in a Lévy copula model Daniël Linders Wim Schoutens Version: July 23, 2014 Abstract In this paper we employ the Lévy copula model to determine basket option prices. More precisely, basket option prices are determined by replacing the real basket with an appropriate approximation. For the approximate basket we determine the underlying characteristic function and hence we can derive the related basket option prices by using the Carr-Madan formula. wo approaches are considered. In the first approach, we replace the arithmetic sum by an appropriate geometric sum, whereas the second approach can be considered as a three-moments-matching method. Numerical examples illustrate the accuracy of our approximations; several Lévy models are calibrated to market data and basket option prices are determined. In a last part we show how our newly designed basket option pricing formula can be used to define implied Lévy correlation by matching model and market prices for basket options. Our main finding is that the implied Lévy correlation smile is flatter than its Gaussian counterpart. Furthermore, if near at-the-money option prices are used, the corresponding implied Gaussian correlation estimate is a good proxy for the implied Lévy correlation. Keywords: basket options, characteristic function, implied correlation, Lévy market, Variance-Gamma. 1 Introduction Nowadays, an increased volume of multi-asset derivatives is traded. An example of such a derivative is a basket option. he basic version of such a multivariate product has the same characteristics as a vanilla option, but now the underlying is a basket of stocks instead of a single stock. he pricing of these derivatives is not a trivial task because it requires a model that jointly describes the stock prices involved. Stock price models based on the lognormal model proposed in Black and Scholes 1973 are popular choices from a computational point of view, however, they are not capable of capturing KU Leuven, Leuven, Belgium. KU Leuven, Leuven, Belgium. 1 the skewness and kurtosis observed for log returns of stocks and indices. he class of Lévy processes provides a much better fit of the observed log returns and, consequently, the pricing of options and other derivatives in a Lévy setting is much more reliable. In this paper we consider the problem of pricing multi-asset derivatives in a multivariate Lévy model. he most straightforward extension of the univariate Black & Scholes model is the Gaussian copula model, also called the multivariate Black & Scholes model. In the Gaussian copula model, the stocks composing the basket are assumed to be lognormal distributed and a Gaussian copula connects the marginals. Even in this simple setting, the price of a basket option is not given in closed form and has to be approximated; see e.g. Hull and White 1993, Brooks et al. 1994, Milevsky and Posner 1998, Rubinstein 1994, Deelstra et al. 2004, Carmona and Durrleman 2006 and Linders 2013, among others. However, the normality assumption for the marginals used in this pricing framework is too restrictive. Indeed, in Linders and Schoutens 2014 it is shown that calibrating the Gaussian copula model to market data can lead to non-meaningful parameter values. his dysfunctioning of the Gaussian copula model is typical observed in distressed periods. In this paper we extend the classical Gaussian pricing framework in order to overcome this problem. Several extensions of the multivariate Black & Scholes model are proposed in order to model the joint dynamics of a number of stocks in a more realistic way. For example, Luciano and Schoutens 2006 introduce a multivariate Variance Gamma model where the dependence is modeled through a common jump component. his model was generalized in Semeraro 2008, Luciano and Semeraro 2010 and Guillaume A framework for modeling dependence in finance using copulas was described in Cherubini et al However, the pricing of basket options in these advanced multivariate stock price models is not a straightforward task. here are several attempts to derive closed form approximations for the price of a basket option in a non-gaussian world. In Linders and Stassen 2014, approximate basket option prices in a multivariate Variance Gamma model are derived, whereas Xu and Zheng 2010, 2014 consider a local volatility jump diffusion model. McWilliams 2011 derives approximations for the basket option price in a stochastic delay model. In this paper we start from the one-factor Lévy model introduced in Albrecher et al to build a multivariate stock price model with correlated Lévy marginals. Stock prices are assumed to be driven by an idiosyncratic and a systematic factor. Conditional on the common or market factor, the stock prices are independent. We show that our model generalizes the Gaussian model with single correlation. Indeed, the idiosyncratic and systematic component are constructed from a Lévy process. Employing in that construction a Brownian motion delivers the Gaussian copula model, but other Lévy copulas arise by employing different Lévy processes like VG, NIG, Meixner,... As a result, this new Lévy copula model is more flexible and can capture other types of dependence. From a tractability point of view the copula is still on the basis of a single correlation number. In a first part of this paper, we consider the problem of finding accurate approximations for the price of a basket option in the Lévy copula model. In order to value a basket option, the distribution of this basket has to be determined. However, the basket is a weighted sum of dependent stock prices and its distribution function is unknown or too complex to work with. herefore, we replace the random variable describing the basket price at maturity by a random variable with a more simple structure. Moreover, the characteristic function of the log transfor- 2 mation of this approximate random variable is given in closed form, such that the Carr-Madan formula can be used to determine approximate basket option prices. We propose two different approximations. Both methods are already applied for deriving approximate basket option prices in the multivariate Black & Scholes model. In this paper we show how to generalize the methodologies to the Lévy case. In this paper, two methodologies for pricing basket options in a Gaussian copula model, proposed in Korn and Zeytun 2013 and Brigo et al. 2004, are generalized to our Lévy copula model. A basket is an arithmetic sum of dependent random variables. However, stock prices are modeled as exponentials of stochastic processes and therefore, a geometric average has a lot of computational advantages. he first methodology, proposed in Korn and Zeytun 2013, consists of constructing an approximate basket by replacing the arithmetic sum by a geometric sum. he second valuation formula is based on a moment-matching approximation. o be more precise, the distribution of the basket is replaced by a shifted random variable having the same first three moments than the original basket. his idea was first proposed in Brigo et al Note that the approximations proposed in Korn and Zeytun 2013 and Brigo et al were only worked out in the Gaussian copula model, whereas the approximations introduced in this paper allow for Lévy marginals and a Lévy copula. Furthermore, we determine the approximate basket option price using the Carr-Madan formula, whereas a closed form expression is available in the special situation considered in Korn and Zeytun 2013 and Brigo et al Numerical examples illustrating the accuracy of the approximations are provided. In a second part of the paper we show how the well-established notions of implied volatility and implied correlation can be generalized in our Lévy copula model; see also Corcuera et al We assume that a finite number of options, written on the basket and the components, are traded. he prices of these derivatives are observable and will be used to calibrate the parameters of our stock price model. One main advantage of our Lévy copula model is that each stock is described by a volatility parameter and that the marginal parameters can be calibrated separately from the correlation parameter. We give numerical examples to show how to use the vanilla option curves to determine an implied Lévy volatility for each stock based on a Normal, VG, NIG and Meixner process and determine basket option prices for different choices of the correlation parameter. However, the available market prices for basket options together with our newly designed basket option pricing formula enables us to determine implied Lévy correlation estimates. Indeed, once we have calibrated the model to the vanilla option curves, the only unspecified parameter in our approximate basket option pricing formula is the correlation and an implied Lévy correlation estimate arises when we match the market and the model price of a basket option with given strike. We observe that implied correlation depends on the strike and the so-called implied Lévy correlation smile is flatter than its Gaussian i.e. Black & Scholes counterpart. he standard technique to price non-traded basket options or other multi-asset derivatives, is by interpolating on the implied correlation curve. It is shown in Linders and Schoutens 2014 that in the Gaussian copula model, this can sometimes lead to non-meaningful correlation values. We show that the Lévy version of the implied correlation solves at least to some extent this problem. his paper is organized as follows. In Section 2 we introduce the Lévy copula model as an extension of the classical Gaussian copula model. In Section 3 and Section 4 we propose two different approximating random variables and show how the Carr-Madan formula, discussed 3 in Section 5, can be used to determine approximate basket option prices. We give numerical illustrations in Section 6. Implied Lévy volatility and correlation are defined and investigated in Section 7. 2 Lévy copula model We consider a market where n stocks are traded. he price level of stock j at some future time t, 0 t is denoted by S j t 1. Dividends are assumed to be paid continuously and the dividend yield of stock j is constant and deterministic over time. We denote this dividend yield by q j. he price level at time t of a basket of stocks is denoted by St and given by n St = w j S j t, j=1 where w j 0 are weights which are fixed upfront. he pay-off of a basket option with strike K and maturity is given by S K +, where x + = maxx, 0. he price of this basket option is denoted by C[K, ]. We assume that the market is arbitrage-free and that there exists a risk-neutral pricing measure Q such that the basket option price C[K, ] can be expressed as the discounted risk-neutral expected value. In this pricing formula, discounting is performed using the risk-free interest rate r, which is, for simplicity, assumed to be deterministic and constant over time. hroughout the paper, we always assume that all expectations we encounter are well-defined and finite. Since the seminal papers of Bachelier 1900 and Black and Scholes 1973, various models are proposed to capture the dynamics of the stocks and their dependence relations. We first revisit the Gaussian copula model. Next, we generalize this popular multidimensional model by allowing more flexibility when fitting the marginals. 2.1 Gaussian copula model We show how the Gaussian copula model can be constructed using standard Brownian motions. Consider the independent standard Brownian motions W = {W t t 0} and W j = {W j t t 0}, for j = 1, 2,..., n. Let ρ [0, 1]. he log returns of each stock j are driven by the r.v. Z j. We assume that these log returns consist of a systematic component and a stockspecific, idiosyncratic, component. herefore, the r.v. Z j is given by: Z j = W ρ + W j 1 ρ, j = 1, 2,..., n. 1 Because the Brownian motions W and W j are independent, we find that Z d j = N0, 1. Furthermore, for i j, the correlation between Z i and Z j is equal to ρ. Indeed, we have that Corr [Z i, Z j ] = Var [W ρ] = ρ. 1 We use the common approach to describe the financial market via a filtered probability space Ω, F, F t 0 t, P. Furthermore, F t 0 t is the natural filtration and we assume that F = F. 4 he log returns of the different stocks are assumed to be described by the correlated r.v. s Z j, j = 1, 2,..., n. Each r.v. Z j is standard normal distributed. In order to adjust the mean and the variance of the time- stock price S j, we add a stock specific drift parameter µ j R and a volatility parameter σ j 0. he stock prices S j, j = 1, 2,..., n at time are then given by S j = S j 0e µ j +σ j Zj, j = 1, 2,... n. 2 he stock price model 2 is also called the multivariate Black & Scholes model or Gaussian copula model, because the log returns are modeled by Normal marginals and a Gaussian copula connects these marginals. For a detailed description of the Black & Scholes model and its extensions, we refer to Black and Scholes 1973, Björk 1998, Carmona and Durrleman 2006 and Dhaene et al Generalization: the Lévy copula model A crucial and simplifying assumption in the Gaussian copula model 2 is the normality assumption for the risk factors Z j. Indeed, the r.v. s Z j are driven by a systematic factor W ρ and a stock specific idiosyncratic factor W j 1 ρ, where W and W j are standard Brownian motions. However, it is well-known that log returns do not pass the test for normality. Indeed, log returns exhibit a skewed and leptokurtic distribution which cannot be captured by a normal distribution; see e.g. Schoutens We generalize the Gaussian copula model outlined in the previous section, by allowing the risk factors to be distributed according to any infinitely divisible distribution with known characteristic function. his larger class of distributions increases the flexibility to find a more realistic distribution for the log returns. In Albrecher et al. 2007, a similar framework was considered for pricing CDO tranches. he Variance Gamma case was considered in Moosbrucker 2006, whereas Guillaume et al consider the pricing of CDO-squared tranches in this one-factor Lévy model. Consider an infinitely divisible distribution for which the characteristic function is denoted by φ. A stochastic process X can be build using this distribution. Such a process is called a Lévy process with mother distribution having characteristic function φ. he Lévy process X = {Xt t 0} based on this infinitely distribution starts at zero and has independent and stationary increments. Furthermore, for t, s 0 the characteristic function of the increment Xt + s Xt is φ s. For more details on Lévy processes, we refer to Sato 1999 and Schoutens Assume that the random variable L has an infinitely divisible distribution and denote its characteristic function by. Consider the Lévy process X = {Xt t [0, 1]} based on the distribution L. We assume that the process is standardized, i.e. E[X1] = 0 and Var[X1] = 1. One can then show that Var[Xt] = t, for t 0. Define also a series of independent and standardized processes X j = {X j t t [0, 1]}, for j = 1, 2,..., n which are all build on the same distribution L and also independent from X. ake ρ [0, 1]. he r.v. A j is defined as A j = Xρ + X j 1 ρ, j = 1, 2,... n. 3 5 In this construction, Xρ and X j 1 ρ are random variables having characteristic function φ ρ L and φ1 ρ L, respectively. Because the processes X and X j are independent and standardized, we immediately find that E[A j ] = 0, Var[A j ] = 1 and A j d = L, for j = 1, 2,..., n. 4 he assumption that X and X j are both Lévy processes based on the same mother distribution L is crucial for obtaining the equality A d j = L. However, this assumption can be relaxed and we can take X j a Lévy process based on the mother infinitely distribution L j. In this case, A j has not the same distribution as L, but its characteristic function is given by φ ρ L φ1 ρ L j. For notational convenience, we always consider the situation where all processes are based on the same distribution L in the rest of the paper. he parameter ρ describes the correlation between A i and A j, if i j. Indeed, it was proven in Albrecher et al that in case A j, j = 1, 2,..., n is defined by 3, we have that Corr [A i, A j ] = ρ. 5 We model the stock price levels S j at time for j = 1, 2,..., n as follows S j = S j 0e µ j +σ j Aj, j = 1, 2,..., n, 6 where µ j R and σ j 0. Note that in this setting, each time- stock price is modeled as the exponential of a Lévy process. Furthermore, a drift µ j and a volatility parameter σ j are added to match the characteristics of stock j. Our model, which we will call the Lévy copula model, can be considered as a generalization of the Gaussian copula model 2. Indeed, instead of a normal distribution, we allow for a Lévy distribution, while the Gaussian copula is generalized to a socalled Lévy copula. his Lévy copula model can also, at least to some extent, be considered as a generalization to the multidimensional case of the model proposed in Corcuera et al and the parameter σ j in 6 can then be interpreted as the Lévy space implied volatility of stock j. Another related model was proposed in Kawai 2009, where the dynamics of each stock price are modeled by a linear combination of independent Lévy processes. he idea of building a multivariate asset model by taking a linear combination of a systematic and an idiosyncratic process can also be found in Ballotta and Bonfiglioli 2014 and Ballotta et al he risk-neutral stock price processes In order to obtain a martingale for the stock price processes, we change the drift µ j of stock j such that the following relation holds E[S j ] = e r q j S j 0. Plugging expression 6 in this equation results in E [S j 0e µ ] j +σ j Aj = e r qj S j 0, j = 1, 2,..., n, 6 from which we find that µ j = r q j 1 log iσ j. 7 From expression 7 we conclude that the risk-neutral dynamics of the stocks in the Lévy copula model are given by S j = S j 0e r q j ω j +σ j Aj, j = 1, 2,..., n, 8 where ω j = 1 log φ L iσ j is a mean-correction which puts us directly in a risk-neutral world. We always assume that ω j is finite. he first three moments of S j can be expressed in function of the characteristic function. By the martingale property, we have that E [S j ] = S j 0e r qj. he risk-neutral variance Var [S j ] can be written as follows [ Var [S j ] = S j 0 2 e 2r q j ω j E e 2σ ] j Aj S j 0 2 e 2r qj. Because A j = d L and the characteristic function of L is, we find Var [S j ] = S j 0 2 e 2r q j e 2ωj i2σ j 1. In a similar way, the second and third moment of S j can be expressed in terms of the characteristic function : i2σ j E [ S j 2] = E[S j ] 2 E [ S j 3] = E[S j ] 3 2, iσ j i3σ j iσ j 3. If the process X j does not have the same mother distribution L than X, we can still determine the moments of S j. Indeed, assume that X j has mother infinitely distribution L j. hen we have to replace by φ ρ L φ1 ρ L j in expression 7 and the formulas for E [S j 2 ] and E [S j 3 ]. In the following sections, we propose two methodologies to approximate the risk-neutral distribution of the sum S. In Section 3 we replace the arithmetic sum S by a geometric sum, whereas a three-moments matching approach is considered in Section 4. In both situations, the r.v. S is replaced by an approximate r.v. S for which the characteristic function φ log S is known. Approximate basket option prices can then be derived by using the Carr-Madan formula. 7 3 Approximating the arithmetic sum by a geometric sum 3.1 he geometric sum he random variable S is a weighted sum of the dependent random variables S j, j = 1, 2,..., n. Its distribution function cannot be determine
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