Pricing Exchange Options under Stochastic Correlation
PPricing Exchange Options under StochasticCorrelation
Enrique Villamor and Pablo OlivaresJanuary 14, 2020
Abstract
In this paper we study the pricing of exchange options when underlyingassets have stochastic volatility and stochastic correlation. An approxi-mation using a closed-form approximation based on a Taylor expansionof the conditional price is proposed. Numerical results are illustrated forexchanges between WTI and Brent type oil prices.
In this paper we study the pricing of exchange options when the underlyingassets have stochastic volatility and correlation. Its main contribution is theproposal of a approximated closed-form formula under this framework.The exchange of two assets is used to hedge against the changes in price ofunderling assets by betting on the difference between both.The price of these instruments has been first considered in [12] under a stan-dard bivariate Black-Scholes model, where a closed-form formula for the pricingis provided. The results have been extended in [5, 6] to the case of a jump-diffusion model, while in [3] it has been considered the pricing of the derivativeunder stochastic interest rates.It is well known that constant correlation and volatilities assumed in the con-text of a Black-Scholes model are not supported by empirical evidence. In theseminal paper of Heston, see [9], the pricing of option contracts under stochasticvolatility is studied. The idea is extended to stochastic correlation in [1], whilestill considering constant volatilities.As an alternative view to correlation, models for the covariance process havebeen proposed. The pricing of exchanges under stochastic covariance is adoptedin Olivares and Villamor(2018), see [13]. See for example [8] for the Wishartmodel and [14] for an Ornstein-Uhlenbeck Levy type model.We consider a bivariate continuous-time GARCH process to model the cor-relation combined with a pricing method based on a Taylor expansion of theconditional Margrabe price. Continuous-time GARCH processes as limits of theembedded discrete-time counterpart have been proposed, for example, in [7] or[4]. See also [11]. 1 a r X i v : . [ q -f i n . P R ] J a n he organization of the paper is the following:In section 2 we introduce the model, discuss the approximated pricing formulaand compute the first and second order moment of the underlying assets, theirvolatilities and their correlations, whose proofs are deferred to the appendix. Insection 3 we discuss the numerical results for the pricing of exchange optionsbetween WTI and Brent type oil futures. Finaly, we present the conclusions. Let (Ω , F , ( F t ) t ≥ , P ) be a filtered probability space. We denote by Q a risk-neutral equivalent martingale measure(EMM) and E Q the expected value withrespect to the measure Q . For a process ( X t ) t ≥ , the integrated process associ-ated with it is denoted by ( X + t ) t ≥ and defined as: X + t = (cid:90) t X s ds The functions f X ( x ) and f X/Y ( x/y ) are respectively the probability densityfunction (p.d.f.) of the random vector X and the conditional p.d.f. of therandom vector X on the random vector Y .A two-dimensional adapted stochastic process ( S t ) t ≥ = ( S (1) t , S (2) t ) t ≥ , wheretheir components are prices of certain underlying assets, is defined on the filteredprobability space above.We assume that the process of prices has a dynamic under Q given by: dS (1) t = r S (1) t dt + σ (1) t S (1) t dZ (1) t (1) dS (2) t = r S (2) t dt + σ (2) t (cid:113) − ρ (2) t S (2) t dZ (2) t + σ (2) t ρ t S (2) t dZ (1) t (2)where the ( σ t ) t ≥ = ( σ (1) t , σ (2) t ) t ≥ is the volatility process and ρ t is the instan-taneous correlation coefficient, which in our models are going to be stochastic.The payoff of a European exchange option, with maturity at time T > h ( S T ) = ( cS (1) T − mS (2) T ) + (3)where m is the number of assets of type two exchanged against c assets of typeone. To simplify we assume c = m = 1.The volatilities are modeled as an Ornstein-Ulenbeck processes: dσ ( j ) t = − α j σ (1) t + β j dW ( j ) t , j = 1 , W (1) t ) t ≥ and ( W (2) t ) t ≥ have instantaneous correlation ρ V .By Ito formula: dV ( j ) t = c ( V ( j ) L − V ( j ) t ) dt + ξ j σ ( j ) t dW ( j ) t , j = 1 , dρ t = ¯ γ (¯Γ L − ρ t ) dt + ¯ α (cid:113) − ρ t d ¯ W t (6)2here V t = ( V (1) t , V (2) t ) t ≥ , with V ( j ) t = ( σ ( j ) t ) , j = 1 , V L = ( V L , V L ) and c j > L and¯ γ play a similar role in the correlation process.The two components of the Brownian motion ( Z t ) t ≥ = ( Z (1) t , Z (2) t ) t ≥ areassumed to be independent of the second set of Brownian motions ( W t ) t ≥ =( W (1) t , W (2) t ) t ≥ and ¯ W t . Remark 2.1.
Parameters in models (4) and (5) are related by c j = 2 α j , V jL = β j α j and ξ j = 2 β j . Next, we find an expression for the price of the exchange contract. Noticethat the price of this contract at time t , 0 ≤ t ≤ T with maturity at T is givenby: C t = e − r ( T − t ) E Q [ h ( S T )] (7)Its terminal value is C T = h ( S T ).The price of the exchange contract at time t , t < T , depends on the behavior ofthe processes ( V s , ρ s ) t ≤ s ≤ T described by equations (5)-(6) and integrated on theinterval [ t, T ]. It depends also on the spot prices, volatilities and correlation attime t . For simplicity in the notations we explicitly drop this last dependence.For the same reason, we analyze only the case t = 0.Hence: C = e − rT (cid:90) R h ( x ) f S T ,V + T ,ρ + T ( x ) dx = e − rT (cid:90) R (cid:20)(cid:90) R C T ( x (cid:48) , x (cid:48)(cid:48) ) f S T /V + T ,ρ + T ( x (cid:48) /x (cid:48)(cid:48) ) (cid:21) f V + T ,ρ + T ( x (cid:48)(cid:48) ) dx (cid:48)(cid:48) = (cid:90) R C M ( x (cid:48)(cid:48) ) f V + T ,ρ + T ( x (cid:48)(cid:48) ) dx (cid:48)(cid:48) (8)where x = ( x (cid:48) , x (cid:48)(cid:48) ) ∈ R .The function C M ( x (cid:48)(cid:48) ) = e − r ( T − t ) (cid:82) R C T ( x (cid:48) , x (cid:48)(cid:48) ) f S T /V + T ,ρ + T ( x (cid:48) /x (cid:48)(cid:48) ) dx (cid:48) is theMargrabe price conditionally on ( V + T , ρ + T ) = x (cid:48)(cid:48) . After conditioning it equalsthe Margrabe price, see [12]. A closed-form for the latter is given by: C M ( V + T , ρ + T ) = e − rT S (1) t N ( d ( v + T )) − e − rT S (2) t N ( d ( v + T )) (9)with: d ( v + T ) = log (cid:18) S (1) t S (2) t (cid:19) + v + T (cid:113) v + T d ( v + T ) = log (cid:18) S (1) t S (2) t (cid:19) − v + T (cid:113) v + T = d ( v + T ) − (cid:113) v + T v + T = V , + T + V , + T − (cid:113) V , + T V , + T ρ + T and ( V + t ) t ≥ = ( V , + t , V , + t ) t ≥ .Next, to approximate the price in equation (7) we consider a second order Taylorexpansion of the conditional Margrabe price C M ( x ) , x ∈ R around the averagevalues given by x = ( E Q ( V , + T ) , E Q ( V , + T ) , E Q ( ρ + T )). It leads to:ˆ C M ( x ) = C M ( x ) + ∂C M ( x ) ∂x ( x − x , ) + ∂C M ( x ) ∂x ( x − x , ) + ∂C M ( x ) ∂x ( x , − x )+ 12 ∂ C M ( x ) ∂x ( x − x , ) + 12 ∂ C M ( x ) ∂x ( x − x , ) + 12 ∂ C M ( x ) ∂x ( x − x , ) + ∂ C M ( x ) ∂x x ( x − x , )( x − x , )+ ∂ C M ( x ) ∂x x ( x − x , )( x − x , ) + ∂ C M ∂x x ( x )( x − x , )( x − x , ) (10)Combining equations (8) and (10) we have the price C is approximated by:ˆ C = C M ( x ) + 12 ∂ C M ( x ) ∂x V ar Q ( V , + T ) + 12 ∂ C M ( x ) ∂x V ar Q ( V , + T )+ 12 ∂ C M ( x ) ∂x V ar Q ( ρ + T ) + ∂ C M ( x ) ∂x x cov Q ( V , + T , V , + T ) (11)Notice that the Margrabe price C M ( x ) ∈ C ∞ ( R ) except in a set of zeroLebesgue measure.We substitute equation (10) into (8). Noticing that: (cid:90) R ( x − x , ) f V + T ,ρ + T ( x ) dx = (cid:90) R ( x − x , ) (cid:20)(cid:90) R f V + T ,ρ + T ( x ) dx x (cid:21) x = (cid:90) R ( x − x , ) f V , + T ( x ) dx = E Q ( V , + T − E Q ( V , + T )) = 0 (cid:90) R ( x − x , ) f V + T ,ρ + T ( x ) dx = (cid:90) R ( x − x , ) (cid:20)(cid:90) R f V + T ,ρ + T ( x ) dx x (cid:21) x = (cid:90) R ( x − x , ) f V , + T ( x ) dx = E Q ( V , + T − E Q ( V , + T )) = 0 (cid:90) R ( x − x , ) f V + T ,ρ + T ( x ) dx = (cid:90) R ( x − x , ) (cid:20)(cid:90) R f V + T ,ρ + T ( x ) dx x (cid:21) x = (cid:90) R ( x − x , ) f ρ + T ( x ) dx = E Q ( ρ + T − E Q ( ρ + T )) = 0Hence, we have equation (11). 4 emark 2.2. Sensitivities with respect to the parameters in the contract canbe computed in a similar way. For example, an approximation of the deltas inthe exchange contract are obtaining by differentiating equation (11) with respectto the price of the underlying assets.
Computing derivatives of the Margrabe price, given by equation (9), withrespect to the volatilities and correlation is straightforward. This aspect isaddressed in appendix B.In order to estimate the option pricing function above we need to compute themoments of ( V , + T , V + , T , ρ + T ). To this end we introduce the following notations: mr j ( t ) = E [ ρ jt ] , mr + j ( t ) = E [( ρ + t ) j ] , j = 1 , mv j,k ( t ) = E [( V ( k ) t ) j ] , mv + j,k ( t ) = E [( V k, + t ) j ] j, k = 1 , mv ( t ) = E [ V (1) t V (2) t ] , mv +12 ( t ) = E [ V , + t V , + t ]Results are given in the propositions below, while proofs are deferred to ap-pendix A. Proposition 2.3.
Let the correlation process ( ρ t ) t ≥ satisfy equation (6). Then: E Q ( ρ + t ) = ¯Γ L t + (cid:18) ρ − ¯Γ L ¯ γ (cid:19) (1 − e − ¯ γt ) (12) V ar Q ( ρ + t ) = b + (cid:18) ρ − ¯Γ L ¯ γ (cid:19) + (cid:18) b + 2¯Γ L (cid:18) ρ − ¯Γ L ¯ γ (cid:19)(cid:19) t + ( b + ¯Γ L ) t + (cid:18) b − L (cid:18) ρ − ¯Γ L ¯ γ (cid:19)(cid:19) te − ¯ γt + b e − (2¯ γ +¯ α ) t − (cid:32) b + b + 2 (cid:18) ρ − ¯Γ L ¯ γ (cid:19) (cid:33) e − ¯ γt + (cid:18) ρ − ¯Γ L ¯ γ (cid:19) e − γt (13) where: a = 2¯ γ ¯Γ L + ¯ α γ + ¯ α , a = 2¯ γ ¯Γ L ( ρ − ¯Γ L )¯ γ + ¯ α b = 1¯ γ (cid:18) − a + ρ − L + 2¯ γ − ¯ α ¯ γ (cid:18) a ¯ γ + a (cid:19) − α ( ρ − a − a )¯ γ (2¯ γ + ¯ α ) (cid:19) b = 1¯ γ (cid:18) − a + 2¯Γ L − γ + ¯ α ¯ γ − a ¯ α ¯ γ (cid:19) b = 1 , b = a ¯ α ¯ γ b = − ¯ α ¯ γ ( ρ − a − a )(2¯ γ + ¯ α )(¯ γ + ¯ α )5econd order moments and covariance of the integrated squared volatilityare given in the propositions above: Proposition 2.4.
Let the process ( V t ) t ≥ satisfy equations (5)-( ?? ). Then: mv +1 ,j ( t ) = V ( j ) L t + V ( j )0 − V ( j ) L c j (1 − e − c j t ) (14) mv +2 ,j ( t ) = P ( t ) + ce − c j t + g e − c j t + g e − c j t + g te − c j t V ar Q [ V + ,jt ] = mv +2 ,j ( t ) − [ mv +1 ,j ( t )] (15) with: P ( t ) = 1 c j ( V ( j ) L ) t + 1 c j (cid:32) (2 − V ( j ) L c j ) V ( j ) L + ξ j V ( j ) L (cid:33) t + 1 c j (cid:32) ( V ( j )0 ) + 2( V ( j ) L ) c j − ξ j V ( j ) L c j (cid:33) c = 1 c j ( 1 c j ξ j V ( j ) L + d + d − V ( j ) L ) c j ) g = − c j [ d + ( V ( j )0 ) − d ] g = − c j , g = 1 c j (cid:104) ξ j ( V ( j )0 − V ( j ) L ) (cid:105) Proposition 2.5.
Let the process ( V t ) t ≥ satisfy equations (5)-( ?? ). Then: cov ( V (1) t , V (2) t ) = mv +12 − mv +1 , ( t ) mv +1 , ( t ) where: mv +12 ( t ) = E Q [ V + , t V + , t ] = 1 c c (cid:104) P ( t ) − ( V (1)0 + c V (1) L t ) mv , ( t ) − ( V (2)0 + c V (2) L t ) mv , ( t )+ ms ( t ) − ξ ξ ρ V e − c t B ( t ) − ξ ξ ρ V e − c t B ( t ) + ξ ξ ρ V A ( t ) (cid:3) where: P ( t ) = V (1)0 V (2)0 + c V (1)0 V (2) L t + c V (2)0 V (1) L t + c c V (1) L V (2) L t A ( t ) = ξ ξ ρ V c + c ) (cid:18) t − c + c (1 − e − ( c + c ) t ) (cid:19) + 2( σ (1)0 σ (2)0 ) c + c ((1 − e − ( c + c ) t )) B j ( t ) = ξ ξ ρ V c + c ) (cid:18) c j ( e c j t − − − j c − c ( e ( − j ( c − c ) t − (cid:19) + σ (1)0 σ (2)0 ( 2( − j c − c ( e ( − j ( c − c ) t − he functions m +1 ,j ( t ) are given by equation (14) while: ms ( t ) = ξ ξ ρ V c + c (cid:18) − exp( −
12 ( c + c ) t ) (cid:19) + σ (1)0 σ (2)0 exp( −
12 ( c + c ) t ) m ,j ( t ) = V ( j ) L + ( V ( j )0 − V ( j ) L ) e − c j t (a) (b) Figure 1: Left: Fifty days moving window correlation coefficient between WTIand Brent daily future prices . Right: Same window for the log-returns
We consider the series of daily closure prices per barrel in US dollars in NYSEof types WTI(blue) and Brent (red), period Dec 2013 to Jan 2019 and thecorresponding log-returns. Both series of prices exhibit similar patterns and,as it is expected, are highly correlated. The overall correlation of the series ofprices is equal to 98% while the correlation of the log-returns is 3 . S (1)0 =7 a) (b)(c) Figure 2: Counterclockwise from the top left figure a simulated series of prices,while the top right figure shows a realization of the squared volatilities. Theseries in the bottom is a simulated trajectory of the correlation process. (a) (b)
Figure 3: A change in the prices of an exchange contract with respect to squaredvolatilities(left) and the correlation(right).800 , S (2)0 = 100 dollars are taken, initial squared volatilities V = (0 . , . ρ = 0 .
8, correlation between the Brownian motions in thevolatility ρ v = 0 .
80, the mean-reverting levels and rates of the volatility pro-cesses are V L = (1 ,
1) respectively while analogous parameters in the correlationprocesses are ¯Γ = 0 . γ = 0 .
8. The annual interest rate is r = 4%, and thesimulation time is one year. Parameters were chosen for illustrative proposes.Other parameters are shown in table 2.Asset WTI sqr. vol. Brent sqr. vol. CorrelationComponentMR level V (1) L = 1 V (2) L = 1 ¯Γ = 0 . c = 1 c = 1 ¯ γ = 0 . xi = 1 xi = 1 ¯ α = 1Initial values V (1)0 = 0 . V (2)0 = 0 . ρ = 0 . realizations. Taylor approximation offers a suitable method to price exchanges contracts be-yond the classic framework developed originally by Margrabe. In the parametricset considered it produces accurate results with less computational effort thana traditional Monte Carlo approach.
Proof of proposition 2.3
Proof.
For the first moment notice that: ρ t = ρ + ¯ γ ¯Γ L t − ¯ γ (cid:90) t ρ s ds + ¯ α (cid:90) t (cid:112) − ρ s d ¯ W s (16)9aking expected value on both sides: mr ( t ) := E Q ( ρ t ) = ρ + ¯ γ ¯Γ L t − ¯ γ (cid:90) t mr ( s ) ds Differentiating we get: mr (cid:48) ( t ) = ¯ γ ¯Γ L − ¯ γmr ( t )whose solution is: mr ( t ) = ¯Γ L + ( ρ − ¯Γ L ) e − ¯ γt Similarly, for the integrated process: E Q ( ρ + t ) = (cid:90) t ¯Γ L + ( ρ − ¯Γ L ) e − ¯ γs ds = ¯Γ L t + (cid:18) ρ − ¯Γ L ¯ γ (cid:19) (1 − e − ¯ γt )To compute the second moment we first apply Ito formula to f ( x ) = x and thecorrelation process. Hence: ρ t = ρ + 2 (cid:90) t ρ s dρ s + < ρ t > = ρ + 2¯ γ ¯Γ L (cid:90) t ρ s ds − γ (cid:90) t ρ s ds + 2¯ α (cid:90) t ρ s (cid:112) − ρ s d ¯ W s + ¯ α (cid:90) t (1 − ρ s ) ds = ρ + ¯ α t + 2¯ γ ¯Γ L (cid:90) t ρ s ds − (¯ α + 2¯ γ ) (cid:90) t ρ s ds + 2¯ α (cid:90) t ρ s (cid:112) − ρ s d ¯ W s Taking expected value: E Q ( ρ t ) = ρ + ¯ α t + 2¯ γ ¯Γ L (cid:90) t E Q ( ρ s ) ds − (¯ α + 2¯ γ ) (cid:90) t E Q ( ρ s ) ds or after differentiating: mr (cid:48) ( t ) + (2¯ γ + ¯ α ) mr ( t ) = 2¯ γ ¯Γ L mr ( t ) + ¯ α mr (0) = ρ its solution is: mr ( t ) = a + a e − ¯ γt + ( ρ − a − a ) e − (2¯ γ +¯ α ) t (17)Next, notice that we have: mr +2 dt = 2 E Q [ ρ + t ρ t ]10rom equation (16): E Q ( ρ t + ¯ γρ + t ) = E Q ( ρ + ¯ γ ¯Γ L t + ¯ α (cid:90) t (cid:112) − ρ s d ¯ W s ) Expanding both sides in the equation above we have:
LHS = E Q ( ρ t + ¯ γρ + t ) = E Q ( ρ t ) + 2¯ γE Q ( ρ t ρ + t ) + ¯ γ E Q ( ρ + t ) = mr ( t ) + ¯ γ mr +2 dt + ¯ γ mr +2 ( t )and RHS = ( ρ + ¯ γ ¯Γ L t ) + 2( ρ + 2¯ γ ¯Γ L t )¯ αE Q ( (cid:90) t (cid:112) − ρ s d ¯ W s )+ ¯ α E Q (cid:18)(cid:90) t (cid:112) − ρ s d ¯ W s (cid:19) = ( ρ + ¯ γ ¯Γ L t ) + ¯ α E Q ( (cid:90) t (cid:112) − ρ s d ¯ W s ) = ( ρ + ¯ γ ¯Γ L t ) + ¯ α E Q (cid:18)(cid:90) t (1 − ρ s ) ds (cid:19) = ( ρ + ¯ γ ¯Γ L t ) + ¯ α ( t − (cid:90) t mr ( s ) ds )From equation (19): (cid:90) t mr ( s ) ds = (cid:90) t ( a + a e − ¯ γs + ( ρ − a − a ) e − (2¯ γ +¯ α ) s ds = a t + a ¯ γ (1 − e − ¯ γt ) + ρ − a − a γ + ¯ α (1 − e − (2¯ γ +¯ α ) t )Hence, mr +2 dt + ¯ γmr +2 ( t ) = b ( t )where: b ( t ) = 1¯ γ (cid:18) − mr ( t ) + ( ρ + ¯ γ ¯Γ L t ) + ¯ α t − ¯ α ¯ γ (cid:90) t mr ( s ) ds (cid:19) = 1¯ γ (cid:0) − mr ( t ) + ( ρ + ¯ γ ¯Γ L t ) + ¯ α t − γ (cid:18) a t + a ¯ γ (1 − e − ¯ γt ) + ρ − a − a γ + ¯ α (1 − e − (2¯ γ +¯ α ) t ) (cid:19)(cid:19) and initial condition mr + , (0) = 0.Using the integrating factor e ¯ γt we find that its solution is: mr +2 ( t ) = e − ¯ γt (cid:90) e ¯ γt b ( t ) dt + ce − ¯ γt (18)11ut: (cid:90) e ¯ γt b ( t ) dt = 1¯ γ (cid:90) e ¯ γt (cid:0) − mr ( t ) + ( ρ + ¯ γ ¯Γ L t ) + ¯ α t (cid:1) − ¯ α ¯ γ (cid:90) e ¯ γt (cid:18) a t + a ¯ γ (1 − e − ¯ γt ) + ρ − a − a γ + ¯ α (1 − e − (2¯ γ +¯ α ) t ) (cid:19) dt Moreover, from equation (19): (cid:90) e ¯ γt mr ( t ) dt = (cid:90) e ¯ γt ( a + a e − ¯ γt + ( ρ − a − a ) e − (2¯ γ +¯ α ) t ) dt = a ¯ γ e ¯ γt + a t − ρ − a − a ¯ γ + ¯ α exp ( − (¯ γ + ¯ α ) t )) (cid:90) ( ρ + ¯ γ ¯Γ L t ) e ¯ γt dt = ρ ¯ γ e ¯ γt + 2¯ γ ¯Γ L ( 1¯ γ e ¯ γt t − γ e ¯ γt )+ ¯ γ ¯Γ L ( 1¯ γ t e ¯ γt − γ te ¯ γt + 2¯ γ e ¯ γt )= [ ρ + 2¯ γ ¯Γ L ( t − γ ) + t − γ t + 2¯ γ ] 1¯ γ e ¯ γt (cid:90) te ¯ γt dt = [ t − γ ] 1¯ γ e ¯ γt Hence: (cid:90) e ¯ γt b ( t ) dt = − γ a e ¯ γt − ( a ¯ γ ) t + (cid:18) ρ − a − a ¯ γ (¯ γ + ¯ α ) (cid:19) e − (¯ γ +¯ α ) t + 1¯ γ ( ρ + 2¯ γ ¯Γ L ( t − γ ) + t − γ t + 2¯ γ ) e ¯ γt + ¯ α ¯ γ (cid:18) t − γ (cid:19) e ¯ γt − ¯ α ¯ γ (cid:18) a ¯ γ ( t − γ ) e ¯ γt + a ¯ γ e ¯ γt − a ¯ γ t + ρ − a − a ¯ γ (2¯ γ + ¯ α ) e ¯ γt + ρ − a − a (2¯ γ + ¯ α )(¯ γ + ¯ α ) e − (¯ γ +¯ α ) t (cid:19) Combining the expressions above into equation (18) we have: mr +2 ( t ) = b + b t + b t + b te − ¯ γt + b e − (2¯ γ +¯ α ) t + ce − ¯ γt From the initial conditions c = − b − b .Combining the first and second moments of ρ + t we obtain the expression for thevariance in equation 13. Proof of proposition 2.4 roof. To compute the first and second moments we proceed similarly to theproof of proposition 2.3. Notice equations for squared volatilities are of mean-reverting square root type s.d.e’s as well.Hence: mv ,j ( t ) = V ( j ) L + ( V ( j )0 − V ( j ) L ) e − c j t mv +1 ,j ( t ) = E Q [ V + ,jt ] = V ( j ) L t + V ( j )0 − V ( j ) L c j (1 − e − c j t )Moreover,( V ( j ) t ) = ( V ( j )0 ) + 2 (cid:90) t V ( j ) s dV ( j ) s + < V ( j ) t > = ( V ( j )0 ) + 2 c j V ( j ) L V j, + t − c j (cid:90) t ( V ( j ) s ) ds + 2 ξ j (cid:90) t V ( j ) s σ ( j ) s dW ( j ) s + ξ j V j, + t = ( V ( j )0 ) + (2 c j V ( j ) L + ξ j ) V j, + t − c j (cid:90) t ( V ( j ) s ) ds + 2 ξ j (cid:90) t V ( j ) s σ ( j ) s dW ( j ) s Taking expected value on both sides: mv ,j ( t ) = ( V ( j )0 ) + (2 c j V ( j ) L + ξ j ) (cid:90) t mv ,j ( s ) ds − c j (cid:90) t mv ,j ( s ) ds or mv (cid:48) ,j ( t ) + 2 c j mv ,j ( t ) = (2 c j V ( j ) L + ξ j ) mv ,j ( t ) mv ,j (0) = ( V ( j )0 ) with c ( t ) = (2 c j V ( j ) L + ξ j ) mv ,j ( t ).Its solution is: mv ,j ( t ) = e − c j t (cid:90) e c j t c ( t ) dt + d e − c j t But: (cid:90) e c j t c ( t ) dt = (2 c j V ( j ) L + ξ j ) (cid:90) e c j t mv ,j ( t ) dt = (2 c j V ( j ) L + ξ j ) (cid:90) e c j t ( V ( j ) L + ( V ( j )0 − V ( j ) L ) e − c j t ) dt = (2 c j V ( j ) L + ξ j )( V ( j ) L c j e c j t + V ( j )0 − V ( j ) L c j e c j t )13hen: mv ,j ( t ) = d + d e − c j t + d e − c j t (19)where: d = (2 c j V ( j ) L + ξ j ) V ( j ) L c j d = (2 c j + ξ j ) ( V ( j )0 − V ( j ) L ) c j d = ( V ( j )0 ) − d − d Next, notice that we have: dmv +2 ,j dt = 2 E Q [ V ( j, +) t V ( j ) t ] (20)Now: E Q ( V ( j ) t + c j V j, + t ) = E Q [ V ( j )0 + c j V ( j ) L t + ξ j (cid:90) t σ ( j ) s dW ( j ) s ] = ( V ( j )0 + c j V ( j ) L t ) + 2( V ( j )0 + c j V ( j ) L t ) ξ j E Q (cid:18)(cid:90) t σ ( j ) s dW ( j ) s (cid:19) + ξ j E Q (cid:18)(cid:90) t σ ( j ) s dW ( j ) s (cid:19) = ( V ( j )0 + c j V ( j ) L t ) + ξ j (cid:90) t mv ,j ( s ) ds (21)On the other hand, after expanding the expression above and taking into accountequation (20): E Q ( V ( j ) t + c j V j, + t ) = mv ,j ( t ) + c j dmv +2 ,j dt + c j mv +2 ,j (22)Hence, equating equations (21) and (20) we have that mv +2 ,j satisfies: dmv +2 ,j dt + c j mv +2 ,j ( t ) = d ( t ) (23) mv +2 ,j (0) = 014ith: d ( t ) = ( V ( j )0 + c j V ( j ) L t ) c j + ξ j c j (cid:90) t mv ,j ( s ) ds − c j mv ,j ( t )= ( V ( j )0 + c j V ( j ) L t ) c j + ξ j c j (cid:90) t V ( j ) L + ( V ( j )0 − V ( j ) L ) e − c j t ds − c j ( d + d e − c j t + d e − c j t )= ( V ( j )0 + c j V ( j ) L t ) c j + ξ j c j V ( j ) L t − ξ j c j ( V ( j )0 − V ( j ) L ) e − c j t − c j ( d + d e − c j t + d e − c j t )The solution of equation (23) is: mv +2 ,j ( t ) = e − c j t (cid:90) e c j t d ( t ) dt + ce − c j t = e − c j t (cid:90) e c j t (cid:34) ( V ( j )0 + c j V ( j ) L t ) c j + ξ j c j V ( j ) L t − ξ j c j ( V ( j )0 − V ( j ) L ) e − c j t − c j ( d + d e − c j t + d e − c j t ) (cid:21) dt + ce − c j t = e − c j t c j (cid:90) e c j t (cid:34) ( V ( j )0 + c j V ( j ) L t ) + ξ j V ( j ) L t − ξ j c j ( V ( j )0 − V ( j ) L ) e − c j t − ( d + d e − c j t + d e − c j t ) (cid:3) dt + d e − c j t (24)Moreover: (cid:90) e c j t ( V ( j )0 + c j V ( j ) L t ) dt = ( V ( j )0 ) c j V ( j )0 e c j t + 2 c j V ( j ) L (cid:90) te c j t dt + ( V ( j ) L ) c j (cid:90) t e c j t dt = ( V ( j )0 ) c j e c j t + 2 c j V ( j )0 V ( j ) L ( tc j e c j t − c j e c j t ) + c j ( V ( j ) L ) ( t c j e c j t − tc j e c j t + 2 c j e c j t )= e c j t c j (cid:104) c j ( V ( j ) L ) t + 2( c j V ( j )0 V ( j ) L − c j ( V ( j ) L ) ) t + ( V ( j )0 ) + 2( V ( j ) L ) (cid:105) e c j t ξ j V ( j ) L t dt = ξ j V ( j ) L c j e c j t ( t − c j ) (cid:90) e c j t ξ j c j ( V ( j )0 − V ( j ) L ) e − c j t dt = ξ j c j ( V ( j )0 − V ( j ) L ) t (cid:90) e c j t ( d + d e − c j t + d e − c j t ) dt = d c j e c j t + d t − d c j e − c j t = − e − c j t c j [ d + ( V ( j )0 ) − d + d e − c j t ]Therefore substituting in equation (24): mv +2 ,j ( t ) = 1 c j (cid:34) ( V ( j ) L ) t + (2 c j − V ( j ) L c j ) V ( j ) L t + ( V ( j )0 ) + 2( V ( j ) L ) c j (cid:35) + ξ j V ( j ) L c j ( t − c j ) + ξ j e − c j t c j ( V ( j )0 − V ( j ) L ) t − e − c j t c j [ d + ( V ( j )0 ) − d + d e − c j t ] + d e − c j t Where, from the initial conditions: d = 1 c j [ 1 c j ξ j V ( j ) L + d + d − V ( j ) L ) c j ] Proof of proposition 2.5
Proof.
To compute the covariance of the integrated squared volatilities we startnoticing that < σ (1) t , σ (2) t > = β β ρ V t . Therefore by integration by parts for-mula: σ (1) t σ (2) t = σ (1)0 σ (2)0 + (cid:90) t σ (1) s dσ (2) s + (cid:90) t σ (2) s dσ (1) t + < σ (1) t , σ (2) t > = σ (1)0 σ (2)0 − α (cid:90) t σ (1) s σ (2) s ds + β (cid:90) t σ (1) s dW (2) t − α (cid:90) t σ (1) s σ (2) s ds + β (cid:90) t σ (2) s dW (1) t + β β ρ V t Taking expected value on both sides: E Q [ σ (1) t σ (2) t ] = σ (1)0 σ (2)0 − ( α + α ) (cid:90) t E Q [ σ (1) s σ (2) s ] ds + β β ρ V t ms (cid:48) ( t ) + ( α + α ) ms ( t ) − β β ρ V = 0with ms ( t ) = E Q [ σ (1) t σ (2) t ].Its solution is: ms ( t ) = β β ρ V α + α (cid:16) − e − ( α + α ) t (cid:17) + σ (1)0 σ (2)0 e − ( α + α ) t With the reparametrization in remark 2.1 it becomes: ms ( t ) = ξ ξ ρ V c + c (cid:18) − exp( −
12 ( c + c ) t ) (cid:19) + σ (1)0 σ (2)0 exp( −
12 ( c + c ) t )(25)Moreover, from equation (5): V (1) t V (2) t = V (1)0 V (2)0 + (cid:90) t V (1) s dV (2) s + (cid:90) t V (2) s dV (1) s + < V (1) t , V (2) t > = V (1)0 V (2)0 + c V (2) L t − c (cid:90) t V (1) s V (2) s ds + ξ (cid:90) t V (1) s σ (2) s dW (2) t + c V (1) L t − c (cid:90) t V (2) s V (1) s ds + ξ (cid:90) t V (2) s σ (1) s dW (1) t + ξ ξ ρ V (cid:90) t σ (1) s σ (2) s ds Again, taking expected value on both sides of the equation above and differen-tiating: mv (cid:48) ( t ) = c V (1) L + c V (2) L − ( c + c ) mv ( t ) + ξ ξ ρ V m ( t )whose solution is given by: mv ( t ) = e − ( c + c ) t ξ ξ ρ V (cid:90) e ( c + c ) s ms ( s ) ds + ce − ( c + c ) t = e − ( c + c ) t ξ ξ ρ V (cid:90) e ( c + c ) s [ ξ ξ ρ V c + c (cid:16) − e − ( c + c ) s (cid:17) ds + σ (1)0 σ (2)0 ξ ξ ρ V e − ( c + c ) t (cid:90) e ( c + c ) s e − ( c + c ) s ds + ce − ( c + c ) t = ( ξ ξ ρ V ) c + c ) e − ( c + c ) t (cid:90) e ( c + c ) s ds − ( ξ ξ ρ V ) c + c ) e − ( c + c ) t (cid:90) exp( 12 ( c + c ) s ) ds + σ (1)0 σ (2)0 ξ ξ ρ V e − ( c + c ) t (cid:90) e ( c + c ) s ds + ce − ( c + c ) t = ( ξ ξ ρ V ) c + c ) − ( ξ ξ ρ V ) ( c + c ) e − ( c + c ) t + 2 σ (1)0 σ (2)0 ξ ξ ρ V c + c e − ( c + c ) t + ce − ( c + c ) t From the initial condition mv (0) = V (1)0 V (2)0 we have that: c = V (1)0 V (2)0 + 12 ( ξ ξ ρ V ) ( c + c ) − σ (1)0 σ (2)0 ξ ξ ρ V c + c
17n the other hand, from equation (5): V j, + t = 1 c j [ V ( j )0 + c V ( j ) L t − V ( j ) t + ξ j σ ( j ) t dW ( j ) t ] V ( j ) t = V ( j )0 e − c j t + V ( j ) L (1 − e − c j t ) + ξ j e − c j t (cid:90) t e c j s σ ( j ) s dW ( j ) t Hence: mv +12 ( t ) := E Q [ V , + t V , + t ]= 1 c c E Q [( V (1)0 + c V (1) L t − V (1) t + ξ (cid:90) t σ (1) s dW (1) s )( V (2)0 + c V (2) L t − V (2) t + ξ (cid:90) t σ (2) s dW (2) s )]= 1 c c (cid:20) V (1)0 V (2)0 + c V (1)0 V (2) L t − V (1)0 E Q [ V (2) t ] + ξ V (1)0 E Q [ (cid:90) t σ (2) s dW (2) s ]+ c V (2)0 V (1) L t + c c V (1) L V (2) L t − c V (1) L tE Q [ V (2) t ] + c ξ V (1) L tE Q [ (cid:90) t σ (2) s dW (2) s ] − V (2)0 E Q [ V (1) t ] − c V (2) L tE Q [ V (1) t ] + E Q [ V (1) t V (2) t ] − ξ E Q [ V (1) t (cid:90) t σ (2) s dW (2) s ]+ ξ V (2)0 E Q [ (cid:90) t σ (1) s dW (1) s ] + c ξ V (2) L tE Q [ (cid:90) t σ (1) s dW (1) s ] − ξ E Q [ V (2) t (cid:90) t σ (1) s dW (1) s ] + ξ ξ E Q [ (cid:90) t σ (1) s dW (1) s (cid:90) t σ (2) s dW (2) s ] (cid:21) Now, we have that: E Q [ (cid:90) t σ ( j ) s dW ( j ) s ] = 0 , j = 1 , E Q [ (cid:90) t σ (1) s dW (1) s (cid:90) t σ (2) s dW (2) s ] = E Q (cid:104) (cid:90) t σ (1) s dW (1) s (cid:90) t σ (2) s dW (2) s (cid:105) = ρ V (cid:90) t m ( s ) dsE Q [ V (1) t (cid:90) t σ (2) s dW (2) s ] = E Q [( V (1)0 e − c t + V (1) L (1 − e − c t )+ ξ e − c t (cid:90) t e c s σ (1) s dW (1) s ) (cid:90) t σ (2) s dW (2) s ]= ( V (1)0 e − c t + V (1) L (1 − e − c t )) E Q [ (cid:90) t σ (2) s dW (2) s ]+ ξ e − c t E Q [ (cid:90) t e c s σ (1) s dW (1) t (cid:90) t σ (2) s dW (2) s ]= ξ e − c t E Q (cid:104) (cid:90) t e c s σ (1) s dW (1) s , (cid:90) t σ (2) s dW (2) s (cid:105) = ξ ρ V e − c t (cid:90) t e c s m ( s ) ds E Q [ V (2) t (cid:90) t σ (1) s dW (1) s ] = ξ ρ V e − c t (cid:90) t e c s m ( s ) ds Therefore: mv +12 ( t ) := 1 c c (cid:104) P ( t ) − ( V (1)0 + c V (1) L t ) mv , ( t ) − ( V (2)0 + c V (2) L t ) mv , ( t )+ ms ( t ) − ξ ξ ρ V e − c t B ( t ) − ξ ξ ρ V e − c t B ( t ) + ξ ξ ρ V A ( t ) (cid:3) where: P ( t ) = V (1)0 V (2)0 + c V (1)0 V (2) L t + c V (2)0 V (1) L t + c c V (1) L V (2) L t Moreover, from equation (25) A ( t ) = (cid:90) t ms ( s ) ds = ξ ξ ρ V c + c (cid:18) t − (cid:90) t e − ( c + c ) s ds (cid:19) + σ (1)0 σ (2)0 (cid:90) t e − ( c + c ) s ds = ξ ξ ρ V c + c ) (cid:18) t − c + c (1 − e − ( c + c ) t ) (cid:19) + 2 σ (1)0 σ (2)0 c + c (1 − e − ( c + c ) t ) B j ( t ) = (cid:90) t e c j s ms ( s ) ds = ξ ξ ρ V c + c ) (cid:18) c j ( e c j t − − (cid:90) t e c j − ( c + c ) s ds (cid:19) + σ (1)0 σ (2)0 (cid:90) t e c j − ( c + c ) s ds = ξ ξ ρ V c + c ) (cid:18) c j ( e c j t − − − j c − c ( e ( − j ( c − c ) t − (cid:19) + σ (1)0 σ (2)0 − j c − c ( e ( − j ( c − c ) t − Derivatives of the Margrabe price are computed by elementary differentiation.Indeed, for the function: M ( x ) = x x − √ x √ x x We see that: ∂M ( x ) ∂x = x − √ x x √ x , ∂M ( x ) ∂x = x − √ x x √ x ∂M ( x ) ∂x = − √ x √ x M ( x ) are: ∂ M ( x ) ∂x = √ x x x / , ∂ M ( x ) ∂x ∂x = 1 − x √ x √ x ∂ M ( x ) ∂x ∂x = − √ x √ x , ∂ M ( x ) ∂ x = √ x x x / ∂ M ( x ) ∂x ∂x = − √ x √ x , ∂ M ( x ) ∂x = 0Regarding the function: d ( x ) = M M − ( x ) − M ( x )= M (cid:112) x x − √ x √ x x − (cid:112) x x − √ x √ x x M = log (cid:18) S (1) t S (2) t (cid:19) , the first and second derivatives of d ( x ) are: ∂d ( x ) ∂x j = − M M − ( x ) ∂M ( x ) ∂x j − M ( x ) ∂M ( x ) ∂x j , j = 1 , , ∂ d ( x ) ∂x j ∂x k = 34 M M − ( x ) ∂M ( x ) ∂x j ∂M ( x ) ∂x − M M − ( x ) ∂ M ( x ) ∂x j ∂x k + 18 M − ( x ) ∂M ( x ) ∂x j ∂M ( x ) ∂x k − M − ( x ) ∂ M ( x ) ∂x j ∂x k , j, k = 1 , , ∂C M ( x ) ∂x j = M f Z ( d ( x )) ∂d ( x ) ∂x j − M f Z ( d ( x )) ∂d ( x ) ∂x j , j = 1 , , ∂ C M ( x ) ∂x j ∂x k = M (cid:18) ∂f Z ( d ( x )) ∂x k ∂d ( x ) ∂x j + f Z ( d ( x )) ∂ d ( x ) ∂x j ∂x k (cid:19) − M (cid:18) ∂f Z ( d ( x )) ∂x k ∂d ( x ) ∂x j + f Z ( d ( x )) ∂ d ( x ) ∂x j ∂x k (cid:19) = M (cid:18) − d ( x ) f Z ( d ( x )) ∂d ( x ) ∂x j ∂d ( x ) ∂x k + f Z ( d ( x )) ∂ d ( x ) ∂x j ∂x k (cid:19) − M (cid:18) − d ( x ) f Z ( d ( x )) ∂d ( x ) ∂x j ∂d ( x ) ∂x k + f Z ( d ( x )) ∂ d ( x ) ∂x j ∂x k (cid:19) References [1] Alvarez , A., Escobar, M., Olivares, P. (2012) Pricing two dimensionalderivatives under stochastic correlation. International Journal of FinancialMarkets and Derivatives Volume 2, Number 4/2011, pg.265-287.202] O. Barndoff-Nielsen and N.J. Shephard. Non-gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics.
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