WORST-OF OPTIONS AND CORRELATION SKEW UNDER A STOCHASTIC CORRELATION FRAMEWORK
Jacinto Marabel Romo∗ BBVA Vía de los Poblados s/n, 28033, Madrid, Spain email:
[email protected] and University Institute for Economic and Social Analysis, University of Alcalá, Plaza de la Victoria 2, 28802, Alcalá de Henares, Madrid, Spain.
January 2012
Abstract This article considers a multi-asset model based on Wishart processes that accounts for stochastic volatility and for stochastic correlations between the underlying assets, as well as between their volatilities. The model accounts for the existence of correlation term structure and correlation skew. The article shows that the Wishart specification can generate different patterns corresponding to the correlation skew for a wide range of correlation term structures. Another advantage of the model is that it is analytically tractable and, hence, it is possible to obtain semi-closed-form solutions for the prices of plain vanilla options, as well as for the price of exotic derivatives. In this sense, this article develops semi-closed-form formulas for the price of European worst-of options with barriers and/or forward-start features. To motivate the introduction of the Wishart volatility model, the article compares the prices obtained under this model and under a multi-asset stochastic volatility model with constant instantaneous correlations. The results reveal the existence of a stochastic correlation premium and show that the consideration of stochastic correlation is a key element for the valuation of these structures. Keywords: Wishart process, stochastic volatility, stochastic correlation, skew of correlation, worst-of options, forward-start options. ∗ The content of this paper represents the author’s personal opinion and does not reflect the views of BBVA.
1
Introduction
Multi-asset exotic options exhibit sensitivity to the volatilities of the underlying assets, as well as to their correlations. Within the class of multi-asset exotic options the worst-of structures are quite popular in Europe. For instance, one of the most successful types of instruments at the Swiss market for structured financial products is a reverse convertible on multiple assets. It consists of a structure where the particular investor buys a straight bond and at the same time sells a worst-of put option. This option usually includes knock-in and/or knock-out features. Using an adequate model to price the risks associated with this kind of options is still an open question. The fact that volatilities are stochastic is widely recognized. But instantaneous correlations are also stochastic within different market regimes and increase during a crisis when they become highly positive correlated with volatility and highly negative correlated with the stock market. In this sense, Ball and Torous [1] show that the correlations between financial quantities change over time and are notoriously unstable. Moreover, Solnik et al. [24], as well as Loretan and English [20] present evidence for the existence of a positive link between correlation and volatility. Unfortunately, despite the previous considerations multi-asset options have been usually priced assuming constant instantaneous correlations between the underlying assets. But since the 2008 crisis, which generated a huge increase in equity correlations that resulted in large losses for several trading books, it seems necessary to leave the simplistic assumption of constant instantaneous correlations. In the literature there are three main categories of models that accounts for the existence of non-constant correlations between the underlying assets. The first category corresponds to the models based on Wishart processes. These models account for the existence of stochastic correlation between assets returns. The Wishart process was mathematically developed in Bru [4] and was introduced into finance by Gourieroux and Sufana [12]. In their model the Wishart process describes the dynamics of the covariance matrix. They assume that this matrix is independent of the assets noises. On the other hand, da Fonseca et al. [7] introduce a correlation structure between the single asset noise and the volatility factors and da Fonseca et al. [6] consider the pricing of multi-asset structures under a Wishart specification. Branger and Muck [3] rely on Wishart processes in the pricing of quanto derivatives. In the context of Wishart processes, Leippold and Trojani [17] allow for jumps and apply their model to the pricing of single asset derivatives, as well as to a multiasset allocation problem. Finally, Marabel [22] considers a Wishart specification to derive semi-closed-form solutions for the price of outperformance options. The second category of models that accounts for variability in the instantaneous correlations between assets returns is given by the uncertain correlation models. Under these models there is uncertainty about correlation, but it is assumed to lie within a certain range. Using this approach Marabel [21] shows that it can be dangerous to price options, with a cross-gamma that changes sign, assuming a constant instantaneous correlation. Wilmott and Zhou [25] use the 1
uncertain correlation model to price collateralized debt obligations (CDOs). The third group of models that leave the constant correlation assumption corresponds to the local correlation models introduced by Langnau [16] and Reghai [23]. These models try to extend the local volatility model of Dupire [10] and Derman and Kani [8] by making correlation a dynamic variable of the market. Langnau [16] shows that if the individual constituent distribution of an equity index are inferred from the single-stock option markets and combined via a Gaussian copula, then it is not possible to explain the steepness of the observed volatility skew for the index. The local correlation model is able to achieve consistency with the constituent and with the index option market generating a correlation skew for the basket of components. But under this model the instantaneous correlation is deterministic and the correlation skew for the basket can provide only little insight into the pricing of worst-of structures. This article considers a multi-asset model based on Wishart processes, introduced by Marabel [22], that accounts for stochastic volatility and for stochastic correlation between the underlying assets returns, between their instantaneous volatilities and between the volatilities and the assets returns. The model is able to generate different patterns corresponding to the term structure of correlation and a positive link between correlation and volatilities that is consistent with the empirical evidence. One of the contributions of this paper consists of providing semi-closed-form solutions for the price of several exotic options which exhibit sensitivity to the existence of correlation skew. In particular, I develop analytical solutions for the price of European worst-of puts with double barrier, as well as for the price of European worst-of options. Moreover, the article also offers semi-closed-form solutions for the price of forward-start multi-asset exotic options, i.e. options with a strike price that will be determined at a later date. The price of these options depends crucially on the evolution of correlations and volatilities. Importantly, the article shows that the model is able to account for the “chewing-gum” effect detailed in Langnau [16] that occurs when the spot dispersion is such that the average basket level barely changes whereas the worst-of performance is deeply altered. In this sense, I provide a numerical example that illustrates the implied correlation skew generated by a worst-of call under different correlation term-structures for different maturities that is consistent with the stylized facts about market correlations. I then provide a numerical illustration that motivates the introduction of the model comparing the prices associated with worst-of options under this model and under a multi-asset stochastic volatility model with constant correlations. The results show the existence of a stochastic correlation premium that should be taken into account in the valuation of this kind of options. This is an important contribution since, to my knowledge, it represents the first analysis of the price discrepancies associated with the existence of correlation skew under a stochastic correlation model and under a constant correlation framework in the valuation of worst-of options. The rest of the paper is organized as follows. Section 2 presents the multiasset Wishart specification and its correlation structure. Section 3 considers the pricing problem and provides semi-closed-form solutions for the price of worst-of 2
options, with and without barriers, as well as with forward-start features. Section 4 provides a numerical illustration that shows different patterns of implied correlation skew that can be generated by the model and applies it to the pricing of forward-start worst-of call options. Section 5 provides a numerical analysis that shows the advantages of the multi-asset Wishart specification with respect to a multi-asset stochastic volatility framework with constant instantaneous correlations. Finally, section 6 concludes.
2 2.1
Model Specification The multi-asset Wishart volatility model
Although the model can be easily generalized to consider an arbitrary number of assets, for simplicity, let us consider n the joint dynamics of assets prices when we o 0 consider two underlying assets. Let St := (S1t , S2t ) ∈ R2 t ≥ 0 be the price 0
process of the assets and let us denote by Yt = (ln S1t , ln S2t ) the log-return vector. For simplicity, I assume that the continuously compounded risk-free rate r and dividend yield qi , for i = 1, 2, are constant. Let Θ denote the probability measure defined on a probability space (Π, z, Θ) such that assets prices expressed in terms of the current account are martingales. We denote this probability measure as the risk-neutral measure. I assume the following dynamics for the return process Yt under Θ: p 1 (2.1) dYt = r1−q − diag (Xt ) dt + Xt dZt , 2 0
where 1 is a 2 × 1 vector of ones, q = (q1 , q2 ) and the vector diag(Xt ) ∈ R2 has the i-th component equal to Xt,ii . The matrix Xt belongs to the set of symmetric 2 × 2 positive semi-definite matrices and satisfies the following stochastic differential equation: h 0 i p 0 0 0p dXt = ΩΩ + M Xt + Xt M dt + Xt dBt Q + Q dBt Xt , (2.2) where Ω, M, Q ∈ R2×2 and Bt is a matrix of standard Brownian motions in R2×2 . Eq. (2.2) characterizes the Wishart process introduced by Bru [4] and represents the matrix analogue of the square root mean-reverting process. In order to grant that Xt is positive semi-definite and the typical mean reverting feature of the volatility, the matrix M is negative semi-definite, while Ω satisfies 0
0
ΩΩ = βQ Q,
(2.3)
with the real parameter β > 1. This condition ensures that X is positive semidefinite at every point in time if X0 is. In Eq. (2.1), Z ∈ R2 is a standard Brownian motion of the form: p Zt = Bt ρ + 1 − ρ0 ρWt , 3
where W ∈ R2 is another standard Brownian motion independent of B and 0 ρ = (ρ1 , ρ2 ) is a fixed correlation vector, with ρi ∈ [−1, 1] , that determines the correlation between the returns and the state variables. This correlation vector satisfies ρ0 ρ < 1. The matrix Q characterizes the randomness of the state variable X. The mean reversion level X, associated with this variable, is given by the following expression: 0 0 (2.4) ΩΩ + M X + XM = 0. Taking into account the previous equation, it is possible to rewrite Eq. (2.2) as follows: h p 0i 0 0p dXt = −M X − Xt − X − Xt M dt + Xt dBt Q + Q dBt Xt . Hence, the matrix M can be considered as the negative of the mean reversion speed corresponding to the volatility process. Note that is matrix is negative semi-definite.
2.2
Variance-covariance structure
Let us define by Vt the instantaneous covariance matrix of dYt . This covariance matrix is given by: 1 Vt := V ar (dYt ) = Xt . dt The stochastic correlation between the assets returns is: ρ(t)S1 S2 := Corr (dY1t , dY2t ) = p
Xt,12 . Xt,11 Xt,22
On the other hand, the correlation between each asset return and its own instantaneous variance is constant and it can be expressed as follows: ρ1 Q1i + ρ2 Q2i i = 1, 2. ρSi Vii := Corr (dYit , dViit ) = p 2 Q1i + Q22i This correlation depends on the correlation vector ρ, as well as on the matrix Q that characterizes the randomness of the state variable X. In this sense, like in the Heston [16] model, a negative correlation ρ between the process associated with the assets returns and the process corresponding to the variance matrix X leads to a negative implied volatility skew. In the particular case where the matrix Q is diagonal, we have the standard correlation between each asset return and its instantaneous volatility observed in the Heston [13] model. Note that it is easy to make this correlation stochastic by introducing an additional Wishart process as in Branger and Muck [3]. Nevertheless, this effect has the cost of increasing considerably the number of parameters making more difficult the calibration of the model to real market data.
4
The correlation between each variance process is stochastic and it is given by:
Xt,11 Q212
ρ(t)V11 V22 := Corr (dV11t , dV22t ) = ρ(t)S S (Q11 Q12 + Q21 Q22 ) + p 1 22 . 2 (Q11 + Q221 ) (Q212 + Q222 )
+ Q222 + Xt,22 Q211 + Q221 p 4 Xt,11 Xt,22 (Q211 + Q221 ) (Q212 + Q222 )
This correlation is a function of the state variable X as well as of the elements of Q and it is linked to the instantaneous correlation between assets returns ρ(t)S1 S2 . Finally, the cross-asset-variance correlations are also stochastic and can be expressed as: ρ(t)S1 V22
:
= Corr (dY1t , dV22t ) = ρ(t)S1 S2 ρS2 V22 ,
ρ (t)S2 V11
:
= Corr (dY2t , dV11t ) = ρ(t)S1 S2 ρS1 V11 .
These correlations are proportional to the instantaneous correlation between assets returns and between assets returns and their instantaneous variances. Note that the expressions corresponding to the cross-asset-variance correlations are fully consistent with the method proposed by Jäckel and Kahl [15] to obtain positive semi-definite correlation matrices in a multi-asset framework with stochastic volatility.
3
The Pricing Problem
Under the specification corresponding to the multi-asset Wishart volatility model it is possible to obtain semi-closed-form solutions for the price of European options on each underlying asset, as well as for the price of multi-asset options, such as worst-of options. With regard to the price of plain vanilla options, I follow Duffie et al. [9] to calculate option prices via Fourier inversion. In this case, it is possible to obtain the pricing formulas by performing a numerical integral in the real plane. Regarding worst-of options it is necessary to resort to generalized Fourier transforms along the lines of Lewis [18], Lewis [19] and da Fonseca et al. [6]. In this sense, the price corresponding to worst-of options on two underlying assets is obtained through a double numerical integral in the complex plane. Following Duffie et al. [9], given some (Y, T, a, b) ∈ R2 × [0, ∞) × R2 × R2 , let Ga,b (y; Y0, T ) : R → R+ denote the price of a security that pays eaYT at time t = T in the event that bYT ≤ y: Ga,b (y; Y0, T ) := EΘ e−rT eaYT 1(bYT ≤y) , (3.1) where 1(bYT ≤y) is the Heaviside step function or unit step function. The FourierStieltjes transform of Ga,b (y; Y0, T ) is given by: ˆ ∞ h i 0 Λa,b (z; Y0, T ) = eizy dGa,b (y; Y0, T ) = EΘ e−rT eγ YT , −∞
5
where z ∈ R, i2 = −1 and γ = a + bzi. Duffie et al. [9] show that the function Ga,b (y; Y0, T ) can be calculated via the following inversion formula: ˆ Λa,b (0; Y0, T ) 1 ∞ Im Λa,b (v; Y0, T ) e−ivy Ga,b (y; Y0, T ) = − dv, (3.2) 2 π 0 v where Im (c) denotes the imaginary part of c ∈ C. Marabel [22] shows that under the risk-neutral measure Θ, the Fourier transform Λa,b (z; Y0, T ) is given by: 0 Λa,b (z; Y0, T ) = eB(γ,T )+tr[A(γ,T )X0 ]+γ Y0 . The matrix A (γ, T ) can be expressed as follows: −1 A (γ, T ) = C22 (γ, T ) C21 (γ, T ) ,
where: C11 (γ, T ) C12 (γ, T ) = C21 (γ, T ) C22 (γ, T ) d1 (γ)
=
exp
M + Q0 ργ 0 d1 (γ)
−2Q0 Q −M 0 − γρ0 Q
T
1 [γγ 0 − diag (γ)] . 2
On the other hand, B (γ, T ) is given by: B (γ, T ) = d0 (γ) T −
β tr [ln [C22 (γ, T )] + T (M + Q0 ργ 0 )] , 2
where: d0 (γ) =
2 X
γi (r − qi ) − r.
i=1
Taking into account the previous expressions, it is possible to obtain semi-closedform solutions for European options based on the numerical integration of Eq. (3.2). Let us consider a European call on asset 1 with strike K and payoff (S1T − K)+ and let ei , i = 1, 2, denotes the i-th unit vector in R2 so that e1 = (1, 0) and e2 = (0, 1). It is possible to express the payoff associated with the European call as follows: (S1T − K)+ = eY1T 1(−Y1T ≤− ln K) − K1(−Y1T ≤− ln K) . Taking into account the definition of Ga,b (y; Y0, T ) in Eq. (3.1), the time t = 0 price of the European call on asset 1 with maturity t = T is given by: C (S10 , T, K) = Ge1 ,−e1 (− ln K; Y0, T ) − KG0,−e1 (− ln K; Y0, T ) . Analogously, the time t = 0 price of the European call on asset 2 is given by: C (S20 , T, K) = Ge2 ,−e2 (− ln K; Y0, T ) − KG0,−e2 (− ln K; Y0, T ) .
6
3.1
Pricing worst-of options
Regarding worst-of options, I follow the methodology of Lewis [18], Lewis [19] and da Fonseca et al. [6] to calculate the option prices in terms of the generalized Fourier transform associated with the payoff function and with the assets returns. In this sense, let us consider the payoff associated with a worst-of call on asset 1 and asset 2 with strike K and maturity t = T : + + S1T S2T = emin(Y1T −Y10 ,Y2T −Y20 ) − eln K , , −K min S10 S20 where the strike is expressed in percentage terms. For simplicity, let us assume that Si0 = 1 (for i = 1, 2) so that Yi0 = 0 and let us consider a generic payoff function that depends on the terminal assets prices w (Y1T , Y2T ) . In this case, the payoff corresponding to the previous worst-of call is: + w (Y1T , Y2T ) = emin(Y1T ,Y2T ) − eln K . The Fourier transform corresponding to w (Y1T , Y2T ) is given by: ˆ w b (z) := w b (z1 , z2 ) = eihz,YT i w (Y1T , Y2T ) dYT ,
(3.3)
R2
0
where z = (z1 , z2 ) ∈ C2 and where h., .i represents the scalar product in R2 . The Laplace transform of the assets returns is defined as: h 0 i Ψ (λ; Y0 , T ) := EΘ eλ YT λ ∈ R2 , (3.4) Along the lines of da Fonseca et al. [6] it is possible to express the time t = 0 price corresponding to the exotic European option on both underlying assets, EOP0 , as follows: ˆ e−rT Ψ (−iz; Y0 , T ) w b (z) dz, (3.5) EOP0 = 2 (2π) χ where χ ⊂ C2 is the admissible integration domain in the complex plane corresponding to the generalized Fourier transform associated with the payoff function w b (z). Hence, to obtain a semi-closed-form solution for the option price we have to calculate the Laplace transform of assets returns, as well as the Fourier transform associated with the payoff function. 3.1.1
The Laplace transform of assets returns
A shows that under the risk-neutral measure Θ, the Laplace transform Ψ (λ; Y0 , T ) is given by: 0 Ψ (λ; Y0 , T ) = eR(λ,T )+tr[U (λ,T )X0 ]+λ Y0 , (3.6)
7
The matrix U (λ, T ) can be expressed as follows: −1 U (λ, T ) = U22 (λ, T ) U21 (λ, T ) ,
with: U11 (λ, T ) U12 (λ, T ) U21 (λ, T ) U22 (λ, T ) h1 (λ)
M + Q0 ρλ0 h1 (λ)
=
exp
=
1 [λλ0 − diag (λ)] , 2
−2Q0 Q −M 0 − λρ0 Q
T ,
where R (λ, T ) is given by: R (λ, T )
=
h0 (λ)
=
h0 (λ) T − 2 X
β tr [ln [U22 (λ, T )] + T (M + Q0 ρλ0 )] 2
λi (r − qi ) .
i=1
3.1.2
The Fourier transform associated with the payoff of worst-of options
Let us consider the payoff associated with a worst-of call with strike K and expiry t = T on asset 1 and asset 2: + wC (YT ) := wC (Y1T , Y2T ) = emin(Y1T ,Y2T ) − eln K . It is possible to express this payoff as follows: wC (YT ) = eY1T − eln K
+
1(Y1T ≤Y2T ) + eY2T − eln K
+
1(Y2T ≤Y1T ) := wC1 (YT )+wC2 (YT ) .
Hence, we can obtain the inverse Fourier transform associated with wC in terms of the transforms corresponding to the functions wC1 and wC2 . B shows that the Fourier transforms corresponding to the payoff functions are given by: w bC1 (z) =
eln K[1+(z1 +z2 )i] , z2 (z1 + z2 ) [1 + (z1 + z2 ) i]
where the corresponding integrability domain is given by: χC1 = z ∈ C2 : Im (z2 ) > 0, Im (z1 ) + Im (z2 ) > 1 .
(3.7)
(3.8)
In this case, to solve the price Eq. (3.5) we have to perform a double integral along a horizontal strip in the complex plane. This strip can lie anywhere in the region defined by Im (z2 ) > 0 and Im (z1 ) + Im (z2 ) > 1. For example, we can consider Im (z1 ) = 32 and Im (z2 ) = 1. The inverse Fourier transform associated with wC2 is eln K[1+(z1 +z2 )i] w bC2 (z) = , (3.9) z1 (z1 + z2 ) [1 + (z1 + z2 ) i] 8
with the integrability domain χC2 = z ∈ C2 : Im (z1 ) > 0, Im (z1 ) + Im (z2 ) > 1 .
(3.10)
Multi-asset reverse convertible structures often include barrier features. For instance, worst-of puts with down-and-in barrier on the worst performing stock and/or up-and-out barrier on the best performing stock are usually embedded in this kind of structures as a way of providing some additional protection to particular investors that sell the worst-of put. These options, as the regular worst-of options, exhibit sensitivity to the existence of stochastic correlation. In particular, a worst-of put with down-and-in barrier on the worst performing asset and up-and-out barrier on the best performing asset is quite sensitive to correlation skew. On the market downside, the sensitivity of the product with respect to correlation is quite negative. The reason is that the lower the correlation between assets returns, the higher the probability of the worst performing stock to reach the down-and-in barrier that activates the product. On the other hand, on the market upside, the negative sensitivity with respect to correlation is reduced. The reason is that if the best performing stock reaches the up-andout barrier, then the option disappears. If we consider that the barriers are observed at maturity, i.e. the barriers style is European, we can express the payoff associated with a worst-of put with strike K, down-and-in barrier Bd (Bd ≤ K) on the worst performing asset, up-and-out barrier Bu (Bd < Bu ) on the best performing asset and maturity T as follows: + wDBP (YT ) = eln K − emin(Y1T ,Y2T ) 1(min(Y1T ,Y2T )
