A Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes Equations
aa r X i v : . [ q -f i n . C P ] J un A Method of Reducing Dimension of Space Variables inMulti-dimensional Black-Scholes Equations Hyong-chol O, Yong-hwa Ro Ning Wan , Faculty of Mathematics,
Kim Il Sung
UniversityPyongyang , D. P. R. of Korea Department of Applied Mathematics, Tong-ji UniversityShanghai, Chinae-mail: [email protected], [email protected], [email protected] Abstract
We study a method of reducing space dimension in multi-dimensionalBlack-Scholes partial differential equations as well as in multi-dimensional parabolicequations. We prove that a multiplicative transformation of space variables in theBlack-Scholes partial differential equation reserves the form of Black-Scholes partialdifferential equation and reduces the space dimension. We show that this transforma-tion can reduce the number of sources of risks by two or more in some cases by givingremarks and several examples of financial pricing problems. We also present that theinvariance of the form of Black-Scholes equations is based on the invariance of theform of parabolic equation under a change of variables with the linear combination ofvariables.
Keywords
Black-Scholes equations; Multi-dimensional; Reducing dimension; Op-tions; Foreign currency strike price; Basket option; Foreign currency option; Zerocoupon bond derivative.
Multi-assets option prices satisfy multi-dimensional Black-Scholes partial differential equa-tions [3]. It is well known that the change of numeraire gives very important computationalsimplification in multi-assets option pricing. See [1]. But using the technique of numerairechange, we can reduce the number of sources of risks by one. See [4].On the other hand, in some pricing problems of financial derivatives, we can see sometransformations of variables reducing the number of sources of risks by two or more. Forexample, in [6], they reduced the number of sources of risks from 3 to 1 in pricing a Europeancall foreign currency option. Their main method is to combine the change of numeraire andthe transformation of multiplication of two variables. Furthermore in [3] Jiang L.S reducedthe number of sources of risks from n to 1 at one try in pricing a basket option with theexpiry payoff of the geometric mean of n underlying assets without any use of change ofnumeraire.The works [3, 6] make us confirm that there must be another general transformation(other than the change of numeraire) reducing the space dimension of multi-dimensionalBlack-Scholes partial differential equations and give us the clue of this article.From the results of [3,6] about pricing options with foreign currency strike price (pricingthe option in Dollars), we have the idea that the transformation of multiplication of twovariables would reserve the form of Black-Scholes partial differential equations. This trans-formation can reduce the dimension of space variable, and furthermore, this transformationcan be applied repeatedly if some condition holds, so in such a case we can reduce the
Hyong-chol O, Yong-hwa Ro, Ning Wan number of sources of risks by two or more. The result of Jiang [3] makes us to find moregeneral transformation.In this article we prove the invariance of the form of multi-dimensional Black-Scholes par-tial differential equations under the multiplicative transformation of variables and providesome examples where we can reduce the space dimension of multi-dimensional Black-Scholespartial differential equations with this transformation.According to our study, in pricing financial derivatives, the possibility to reduce dimen-sion depends on the expiry payoff function structures.The remainder of this article consists as follows. In section 2 we prove the invariance ofthe form of multi-dimensional Black-Scholes partial differential equations under the multi-plicative transformation of variables. In section 3, we give such examples as • Pricing options whose strike price is in a currency different from the stock price, • Pricing a basket option with the expiry payoff of the geometric mean, • Pricing a European call foreign currency option .In section 4, we consider a relationship between the invariance of the form of Black-Scholes partial differential equations and the invariance of the form of parabolic equations.
Black-Scholes partial differential equations are one of main models in financial mathematics.The following partial differential equation ∂V∂t + 12 n X i,j =0 a ij S i S j ∂ V∂S i ∂S j + n X i =0 ( r − q i ) S i ∂V∂S i − rV = 0 , (1)is called the ( n + 1)-dimensional Black-Scholes equation with risk free rate r . Here r > S i the i -th underlying asset with dividend rate q i ( i = 1 , · · · , n ), A = [ a ij ] ni,j =0 a non-negative definite ( n + 1) × ( n + 1) matrix and V ( S , S , · · · , S n , t ) theprice of an option derived from underlying assets S , S , · · · , S n at time t . The existenceand representation of the solution to equation (1) are described in [3]. The following theo-rem is our main result. Theorem 1
Under the transformation z = S α S α , z i = S i , i = 2 , · · · , n, (2)equation (1) is transformed into an n -dimensional Black-Scholes equation with risk free rate r . In other words, the equation (1) has a solution of the form V ( S , S , · · · , S n , t ) = U ( z , · · · , z n , t ) . (3)where U ( z , · · · , z n , t ) is a solution to the following n dimensional Black-Scholes equationwith risk free rate r : ∂U∂t + 12 n X i,j =1 ¯ a ij z i z j ∂ U∂z i ∂z j + n X i =1 ( r − ¯ q i ) z i ∂U∂z i − rU = 0 . (4) Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes · · · a ij , ¯ q i are provided as follows:¯ a ij = a α + a α α + a α α + a α , i = j = 1 ,a j α + a j α , i = 1 , j = 2 , · · · , n,a i α + a i α , i = 2 , · · · , n, j = 1 ,a ij , i, j = 2 , · · · , n, ¯ q k = (cid:26) r − P i =0 (cid:0) r − q i − a ii (cid:1) α i − P i,j =0 a ij α i α j , k = 1 ,q k , k = 2 , · · · , n. (5) Proof
If we rewrite the derivatives of V on S i ( i = 0 , · · · , n ) in (1) by the derivatives of U on z i ( i = 1 , · · · , n ) using (3), then we have ∂V∂S S = α ∂U∂z z , ∂V∂S S = α ∂U∂z z , ∂V∂S i S i = ∂U∂z i z i , i = 2 , · · · , n,∂ V∂S S = α ∂ U∂z z + α ( α − ∂U∂z z ,∂ V∂S S = α ∂ U∂z z + α ( α − ∂U∂z z ,∂ V∂S ∂S S S = α α ∂ U∂z z + α α ∂U∂z z ,∂ V∂S ∂S j S S j = α ∂ U∂z ∂z j z z j , j = 2 , · · · , n∂ V∂S ∂S j S S j = α ∂ U∂z ∂z j z z j , j = 2 , · · · , n∂ V∂S i ∂S j S i S j = ∂ U∂z i ∂z j z i z j , i, j = 2 , · · · , n. If we expand the terms of second order derivatives in (1) as follows n X i,j =0 b ij = X i,j =0 b ij + X i =0 n X j =2 b ij + n X i =2 1 X j =0 b ij + n X i,j =2 b ij , and substitute the above derivatives into here, then we can easily have (4) and (5). Thenthe n -dimensional matrix ¯ A = [¯ a ij ] ni,j =1 is evidently symmetric and non-negative. In fact, Hyong-chol O, Yong-hwa Ro, Ning Wan for any ξ = ( ξ , · · · , ξ n ) ⊥ ∈ R n (the superscript ” ⊥ ” denotes transpose), we have ξ ⊥ ¯ A ξ = n X i,j =1 ¯ a ij ξ i ξ j = ¯ a ξ + n X j =2 ¯ a j ξ ξ j + n X i =2 ¯ a i ξ i ξ + n X i,j =2 ¯ a ij ξ i ξ j = X i,j =0 a ij α i α j ξ + n X j =2 X i =0 a ij α i ! ξ ξ j + n X i =2 X j =0 a ij α j ξ i ξ + n X i,j =2 a ij ξ i ξ j = X i,j =0 a ij ( α i ξ )( α j ξ ) + X i =0 n X j =2 a ij ( α i ξ ) ξ j + n X i =2 1 X j =0 a ij ξ i ( α j ξ ) + n X i,j =2 a ij ξ i ξ j . So if we let η = ( α ξ , α ξ , ξ , · · · , ξ n ) ⊥ ∈ R n +1 , then from the nonnegativeness of A , wehave ξ ⊥ ¯ A ξ = η ⊥ A η ≥ , which completes the proof of theorem 1. (QED)If we apply the transformation (2) repeatedly, then we can get the following corollary1. Corollary 1
Let k ∈ { , , · · · , n } . Under the transformation z k = S α · · · S α k k , z i = S i , i = k + 1 , · · · , n. (6)the equation (1) is transformed into an ( n + 1 − k )-dimensional Black-Scholes equation.Let consider a terminal condition V ( S , S , · · · , S n , T ) = P ( S , S , · · · , S n ) (7)of ( n + 1)-dimensional Black-Scholes equation (1). Then from theorem 1, we have thefollowing corollary 2. Corollary 2
Assume that there exists an n -dimensional function F such that P ( S , S , S , · · · , S n ) = F ( S α , S α , S , · · · , S n ) . (8)Then by the change of variables given by (2) and (3), the terminal value problem given by(1) and (7) is transformed into the n -dimensional Black-Scholes equation’s terminal valueproblem given by (4) and U ( z , · · · , z n , T ) = F ( z , · · · , z n ) . (9) Remark 1
As shown in [4], if the expiry payoff function P ( S , S , · · · , S n ) has thehomogeneity for its variables: P ( aS , aS , · · · , aS n ) = aP ( S , S , · · · , S n ) , ∀ a > , Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes · · · U = VS , z i = S i S , i = 1 , · · · , n, (10)the ( n + 1)-dimensional terminal value problem given by (1) and (7) is transformed into aterminal value problem for an n -dimensional Black-Scholes equation with risk free rate 0.The new transformed expiry payoff function F is given by F ( z , · · · , z n ) , P (1 , z , · · · , z n ) . Then the new expiry payoff function F no more has the homogeneity for its variables. Forexample P ( S , S , S ) = max( S , S , S )has the homogeneity but its new transformed 2 dimensional function F ( z , z ) = max(1 , z , z )has no homogeneity. Thus the change of numeraire only can reduce the number of sourcesof risks by one and it is impossible to use repeatedly. Remark 2
In corollary 2, if the new expiry payoff F has the same property with P in(8), then we can use the transformation (2) repeatedly. For example, if P ( S , S , S ) = max( S S S − K, z = S · S , then the new expiry payoff F is also given by F ( z , z ) = max( z z − K, , thus it is also the function of z · z and we can use the transformation (2) once more, andthen we have one-dimensional function G ( x ) = max( x − K, Remark 3
In our opinion, in multi-dimensional Black-Scholes equations, the principle toreduce dimension is based on its invariance under the transformation (6) and the change ofnumeraire (10). Thus in pricing financial derivatives problems derived to multi-dimensionalBlack-Scholes equations, the possibility to reduce dimension depends on the structures oftheir expiry payoff functions. If the price model of a derivative can be derived to Black-Scholes equation and its expiry payoff is a function of some group of risk source variables suchas (6) and furthermore, if the new expiry payoff function of group variables has homogeneity,then using (6) and (10) we can reduce the dimension of the problem by two or more.
This example is studied in [1] by expectation method and in [4] by PDE method. Fordetails about the financial background, we refer to [1, 4]. Here we only mention the use ofthe transformation given by the equation (2).
Hyong-chol O, Yong-hwa Ro, Ning Wan
Problem:
The underlying stock is traded in UK pounds and the option exercise priceis in US dollars. At t = 0 the option is an at-the-money option (that is, the strike price isthe same with the underlying stock price [3, 5]) when the strike price is expressed in UKpounds. This pound strike price is converted into dollars at t = 0. The dollar strike pricecomputed like this is kept constant during the life of the option. At the expiry date t = T ,the option holder can pay the fixed dollar strike price to buy the underlying stock. Findthe fair price of this option. Mathematical model
Let denote by S ( t ) the stock price (in UK pounds), r p the short rate in UK pound market, r d the short rate in US dollar, X ( t ) dollar/pound exchange rate (then Y ( t ) = X ( t ) − ispound /dollar exchange rate), K d the strike price expressed in US dollar and K p ( t ) thestrike price expressed in UK pound. Assumption
1) The stock price S ( t ) satisfies geometric Brown motion: dS ( t ) = α S S ( t ) dt + σ S S ( t ) dW S ( t ) (objective measure) . r p and r d are deterministic constants.3) Dollar/pound exchange rate X ( t ) satisfies Garman-Kohlhagen model [2]: dX ( t ) = α X X ( t ) dt + σ X X ( t ) dW X ( t ) (objective measure) . Here W S ( t ) and W X ( t ) are the scalar Wiener processes and have the following relations: dW S ( t ) · dW X ( t ) = ρdt, | ρ | < . Explanation about strike price : K p (0) = S (0) , K d = K p (0) · X (0) = S (0) · X (0) ≡ constant. The dollar strike price is constant but the pound strike price randomly varies asa result of varying exchange rate: K p ( t ) = K d · X ( t ) − = S (0) · X (0) · X ( t ) − . Explanation about maturity payoff : The US dollar price of the option at maturityis F d = max( S ( T ) · X ( T ) − K d , . Then the UK pound price of the option at maturity is F p = max( S ( T ) − K p ( T ) , S ( T ) − S (0) · Y (0) − · Y ( T ) , . Now we derive the PDE model for pricing the option in dollars. Let V d = V ( S, X, t ) bethe price of the option in US dollars. By ∆-hedging, construct a portfolio Π asΠ = V − ∆ SX − ∆ X. (the dollar price of this portfolio consists of an option, ∆ shares of stocks and ∆ UKpounds.) Choose ∆ , ∆ such that Π is risk-free in ( t, t + dt ), i.e. d Π = r d Π dt. Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes · · · ∂V∂t + 12 (cid:20) σ S S ∂ V∂S + 2 ρσ S σ X SX ∂ V∂S∂X + σ X X ∂ V∂X (cid:21) +( r p − ρσ S σ X ) S ∂V∂S + ( r d − r p ) X ∂V∂X − r d V = 0 , (11) V ( S, X, T ) = max( S · X − K d , . (12)Here we remind that S is the variable representing stock price, X the variable representingdollar/pound exchange rate, T the maturity and V the option price.The equation (11) is a two-dimensional Black-Scholes equation and T -payoff function(12) does not satisfy homogeneity, but it is a function of the group variable z = SX. (13)(This change of variable transforms the stock price expressed in UK pound into UD dollarprice.) Thus from corollary 2 of our theorem 1, by the transformation (13) our problem istransformed into the following one dimensional problem: ∂V∂t + 12 ( σ S + 2 ρσ S σ X + σ X ) z ∂ V∂z + r d z ∂V∂z − r d V = 0 ,V ( z, T ) = max( z − K d , . (14)The problem (14) can be seen as an ordinary UK dollar call option pricing problem. Bythe standard Black-Scholes formula, V ( z, t ) = zN ( d ) − K d e − r d ( T − t ) N ( d ) , where d = ln zK d + (cid:0) r d + σ S,X (cid:1) ( T − t ) σ S,X √ T − t , d = d − σ S,X √ T − t,σ S,X = σ S + 2 ρσ S σ X + σ X . If we return to the original variables (
S, X ), then we have the dollar price of the option: V d ( S, X, t ) =
SXN ( d ) − K d e − r d ( T − t ) N ( d ) , where d = ln SXK d + (cid:0) r d + σ S,X (cid:1) ( T − t ) σ S,X √ T − t , d = d − σ S,X √ T − t. Considering V p = V d · X − and K d = S (0) · X (0), we have the pound price of option: V p ( S, X, t ) = SN ( d ) − S X X e − r d ( T − t ) N ( d ) , where d = ln SXS X + (cid:0) r d + σ S,X (cid:1) ( T − t ) σ S,X √ T − t , d = d − σ S,X √ T − t. Hyong-chol O, Yong-hwa Ro, Ning Wan
Assume that the prices m underlying assets S , · · · , S m − follow geometric Brownian mo-tions and the expiry payoff of an option is V ( S , · · · , S m − , T ) = ( S α , · · · , S α m − m − − K ) + (15)where P α i = 1 , α i ≥
0. Such an option is called a basket option with the expiry payoff ofthe geometric mean of m underlying assets [3].The mathematical model for this option is the terminal value problem (1) and (15) of m -dimensional Black-Scholes equation when n + 1 = m .Under the transformation z = S α · · · · · S α m − m − (16)we have S i ∂V∂S i = α i z ∂V∂z , i = 0 , · · · , m − ,S i ∂ V∂S i = α i z ∂ V∂z + α i z ∂V∂z − α i z ∂V∂z ,S i S j ∂ V∂S i ∂S j = α i α j z ∂ V∂z + α i α j z ∂V∂z , ( i = j ) . If we substitute the above derivative expressions into the equation (1) when n + 1 = m andconsider P α i = 1, then we have ∂V∂t + 12 m − X i,j =0 a ij α i α j z ∂ V∂z + m − X i,j =0 a ij α i α j − m − X i =0 a ii α i z ∂V∂z + r − m − X i =0 q i α i ! z ∂V∂z − rV = 0Let denote ˆ σ := m − X i,j =0 a ij α i α j , ˆ q := m − X i =0 (cid:18) q i + 12 a ii (cid:19) α i −
12 ˆ σ . Then we have the following terminal value problem of one dimensional Black-Scholes equa-tion (standard call option): ∂V∂t + 12 ˆ σ z ∂ V∂z + ( r − ˆ q ) z ∂V∂z − rV = 0 ,V ( z, T ) = ( z − K ) + . If we use the standard Black-Scholes formula and return to the original variables, then wehave the pricing formula of a basket option with the expiry payoff of the geometric mean: V ( S , · · · , S m − , t ) = e − ˆ q ( T − t ) S α · · · · · S α m − m − N ( d ) − Ke − r ( T − t ) N ( d ) , Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes · · · d = ln S α ····· S αm − m − K + (cid:0) r − ˆ q + ˆ σ (cid:1) ( T − t )ˆ σ √ T − t , d = d − ˆ σ √ T − t. This problem was studied in [6] under a general condition with stochastic short rates andstochastic exchange rates. They consider the option as a interest rate derivative, so thatthe partial differential equations of the prices that they derived are not standard multidi-mensional Black-Scholes equations. Thus our result can not be directly applied to theirequation. Fortunately, as mentioned in [4], the prices of zero coupon bonds under the Va-sicek model, Ho-Lee model or Hull-White model follow geometric Brown motions and thecorresponding short rate is a deterministic function of the price of zero coupon bond. Sohere we consider the option as a zero coupon bond derivative, just as in [4], and then weapply our theorem.
Problem:
A holder of an European call foreign currency option has a right to fix theexchange rate as a strike exchange rate. Find the fair price of this option.We denote by r ( t ) the short rate in domestic currency, r ( t ) the short rate in foreigncurrency, F ( t ) domestic currency / foreign currency exchange rate, K the strike exchangerate (domestic / foreign), and T the expiry date. Under this notation, the expiry pay off ofour European call foreign currency option is given by[ F ( T ) − K ] + . Assumptions:
All discussion is done under the risk neutral measure Q (domesticmartingale measure). In what follows, a , a , b , b are all positive constants, σ , σ , σ arelinear independent constant vectors and { W t ; 0 ≤ t ≤ T } = (cid:8)(cid:0) W t , W t , W t (cid:1) ; 0 ≤ t ≤ T (cid:9) isa standard 3 dimensional Wiener process satisfying the following conditions: E ( dW it ) = 0 , V ar ( dW it ) = dt, Cov ( dW it , dW jt ) = 0( i = j ) , ≤ i, j ≤ .
1) The domestic and foreign short rates follow Vasicek model: dr i = ( b i − a i r i ) dt + σ i · dW ( t ) , i = 1 , .
2) The exchange rate F ( t ) follows Garman-Kohlhagen model [2]: dF ( t ) = F ( t )( r ( t ) − r ( t )) dt + F ( t ) σ · dW ( t ) .
3) The price V of the option in domestic currency is given as a deterministic function V = C ( r , r , F, t ) of domestic short rate, foreign short rate and exchange rate and assume V ∈ C , ( D × [0 , T )) , D = ( −∞ , ∞ ) × ( −∞ , ∞ ) × (0 , ∞ ) . Dynamics of the price of zero coupon bond:
Let denote the price of domestic zerocoupon bond with maturity T by p ( t, r ; T ) (in domestic currency) and denote the price0 Hyong-chol O, Yong-hwa Ro, Ning Wan of foreign zero coupon bond with maturity T by p ( t, r ; T ) (in foreign currency). Then theprice of zero coupon bond p i ( t, r i ; T ) satisfies the following equation [5]: ∂p i ∂t + 12 | σ i | ∂ p i ∂r i + ( b i − a i r i − λ i | σ i | ) ∂p i ∂r i − r i p i = 0 ,p i ( T, r i ; T ) = 1 . Here λ = 0 is the price of domestic market risk (under the domestic martingale measure)and λ = 0 is the price of foreign market risk (under the domestic martingale measure), | σ | denotes the length of vector σ . And its solution is expressed by p i ( t, r i ; T ) = A i ( t, T ) e − B i ( t,T ) r i , ∂p i ∂r i = − B i ( t, T ) p i , B i ( t, T ) = 1 a i (cid:16) − e − a i ( T − t ) (cid:17) , and so the short rate r i ( t ) is a deterministic function of p i = p i ( t, r i , T ): r i = r i ( t, p i ) = − B i ( t ) (ln p i − A i ( t )) = − B i ( t ) ln p i + A i ( t ) B i ( t ) . (17)As shown in [4], the dynamics of the zero coupon bond price follows geometric Brownmotion: In fact dp i = (cid:18) ∂p i ∂t + 12 | σ i | ∂ p i ∂r i (cid:19) dt + ∂p i ∂r i dr i = (cid:18) ∂p i ∂t + 12 | σ i | ∂ p i ∂r i + ( b i − a i r i ) ∂p i ∂r i (cid:19) dt + ∂p i ∂r i σ i · dW ( t )= (cid:18) rp i + λ i | σ i | ∂p i ∂r i (cid:19) dt + ∂p i ∂r i σ i · dW ( t )= ( r i − λ i | σ i | B i ( t )) p i dt + p i B i ( t ) σ i · dW ( t ) . Thus we get dp i = α i ( t ) p i dt + p i Σ i ( t ) · dW ( t ) , i = 1 , , Σ i ( t ) = − B i ( t ) σ i . (18) PDE Model and Solving:
Now we can attack the pricing problem. Denote by p i = p i ( t ; T ) . Since p i ( T, T ) = 1, then the price of our option can be rewritten as V T = max( F p − Kp , . (19)From the assumption 3) and the fact that r ( t ) = r ( t, p ) and r ( t ) = r ( t, p ), the domesticprice of the option at time t can be rewritten as a function V = V ( p , p , F, t )of the zero coupon bond prices. By ∆-hedging, construct a portfolio Π asΠ = V − ∆ p − ∆ p F − ∆ F. (in domestic currency) Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes · · · shares of domestic zero coupon bond, ∆ shares offoreign zero coupon bond and ∆ units of foreign currency. Choose ∆ , ∆ , ∆ such thatΠ is risk-free in ( t, t + dt ), i.e. d Π = r Π dt . This is equivalent to the following equality dV − ∆ dp − ∆ d ( p F ) − ∆ dF − ∆ r dtF = r ( t, p )( V − ∆ p − ∆ p F − ∆ F ) dt. (20)By the three dimensional Itˆo formula, we have dV = ∂V∂p dp + ∂V∂p dp + ∂V∂F dF + n ∂V∂t + 12 h | Σ | p ∂ V∂p + | Σ | p ∂ V∂p + | σ | F ∂ V∂F + 2Σ · Σ p p ∂ V∂p ∂p +2Σ · σ p F ∂ V∂p ∂F + 2Σ · σ p F ∂ V∂p ∂F io dt,d ( p F ) = p dF + F dp + Σ · σ p F dt.
If we substitute above two expressions into (20), then we get (cid:18) ∂V∂p − ∆ (cid:19) dp + (cid:18) ∂V∂p − ∆ F (cid:19) dp + (cid:18) ∂V∂F − ∆ p − ∆ (cid:19) dF + n ∂V∂t + 12 h | Σ | p ∂ V∂p + | Σ | p ∂ V∂p + | σ | F ∂ V∂F +2Σ · Σ p p ∂ V∂p ∂p + 2Σ · σ p F ∂ V∂p ∂F + 2Σ · σ p F ∂ V∂p ∂F io dt − ∆ Σ · σ p F dt − ∆ r ( t, p ) dt · F = r ( t, p )( V − ∆ p − ∆ p F − ∆ F ) dt. Here we choose ∆ and ∆ such that ∂V∂p − ∆ = 0 , ∂V∂p − ∆ F = 0 , ∂V∂F − ∆ p − ∆ = 0 , equivalently, ∆ = ∂V∂p , ∆ = 1 F ∂V∂p , ∆ = ∂V∂F − p F ∂V∂p . Then we have ∂V∂t + r ( t, p ) ∂V∂p p + [ r ( t, p ) − Σ ( t ) · σ ] ∂V∂p p +[ r ( t, p ) − r ( t, p )] F ∂V∂F + 12 h | Σ ( t ) | p ∂ V∂p + | Σ ( t ) | p ∂ V∂p + | σ | F ∂ V∂F + 2Σ ( t ) · Σ ( t ) p p ∂ V∂p ∂p (21)+2Σ ( t ) · σ p F ∂ V∂p ∂F + 2Σ ( t ) · σ p F ∂ V∂p ∂F i − r ( t, p ) V = 0 . Hyong-chol O, Yong-hwa Ro, Ning Wan
The problem (21) and (19) is the pricing model for European call foreign currency optionwhich is considered as a derivative of two countrys zero coupon bonds.The equation (21) has a simillar form of Black-Scholes equation but the terms of firstorder derivatives and itself of unknown function have strongly varying coefficients whichdepend on space variables. Although our theorem deals with constant coefficient Black-Scholes equation, but the change of variables z = p · F does work well. This change of variables composes the price of foreign zero coupon bondand the exchange rate to the domestic price of foreign zero coupon bond. By this change ofvariables, the space 3-dimensional problem given by (21) and (19) is transformed into thefollowing space 2-dimensional problem: ∂V∂t + r ( t, p ) ∂V∂p p + r ( t, p ) ∂V∂z z + 12 h | Σ ( t ) | p ∂ V∂p + | Σ ( t ) + σ | z ∂ V∂z + 2Σ ( t ) · (Σ ( t ) + σ ) p z ∂ V∂p ∂z i − r ( t, p ) V = 0 , (22) V T = max( z − Kp , . The original expiry payoff function (19) has no homogeneity on its variables ( p , p , F ) butthe changed expiry payoff function of the problem (22) has homogeneity on its new variables( z, p ), and thus by theorem 1 of [4] we can use the standard change of numeraire U = Vp , y = zp (= p Fp ) . This change of variables transforms the bond price and domestic price of foreign zero couponbond into relative price with respect to the zero coupon bond price and we have the followingterminal value problem of 1-dimensional Black-Scholes equation with risk free rate 0: ∂U∂t + 12 | Σ ( t ) − Σ ( t ) − σ | ∂ U∂y y = 0 , (23) U ( y, T ) = max( y − K, . We can easily solve (23) using standard method of [3]. The solution of (23) is U ( y, t ) = yN ( ¯ d ) − KN ( ¯ d ) . Here ¯ d = ln yK + σ ( t, T ) σ ( t, T ) , ¯ d = ¯ d − σ ( t, T ) ,σ ( t, T ) = Z Tt | Σ ( u ) − Σ ( u ) − σ | du. Considering (18), then we have σ ( t, T ) = Z Tt | B ( u, T ) σ − B ( u, T ) σ + σ | du. Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes · · ·
V, p ( t, T ) , p ( t, T ) , F , then we have the price of Europeancall foreign currency option : V ( p , p , F, t ) = p ( t, r , T ) F N ( d ) − Kp ( t, r , T ) N ( d ) , (24)where d = ln p ( t,r ( t ) ,T ) · F ( t ) p ( t,r ( t ) ,T ) · K + σ ( t, T ) σ ( t, T ) , d = d − σ ( t, T ) . Note: The formula (24) coincides with the pricing formula in [6].
In fact, the invariance of the form of Black-Scholes equations is based on the invarianceof the form in parabolic equation under a change of variables with the linear combinationof variables. Using the theorem 1 and the change of variable x i = ln S i , we can easilyget a transformation under which the form of parabolic equation is not changed and thedimension is reduced.As shown in [3], the change of variable x i = ln S i ( i = 0 , , · · · , n ) transforms the equation(1) into a parabolic equation and we have the diagram:(1) ←→ x =ln S ∂V∂t + 12 n X i,j =0 a ij ∂ V∂x i ∂x j + n X i =0 (cid:16) r − q i − a ii (cid:17) ∂V∂x i − rV = 0 l (2) l T x = y (4) ←→ y =ln z ∂V∂t + 12 n X i,j =1 ¯ a ij ∂ V∂y i ∂y j + n X i =1 (cid:16) r − ¯ q i − ¯ a ii (cid:17) ∂V∂y i − rV = 0where new change of variables y = T x is given by y = α x + α x ,y i = x i , i = 2 , · · · i = 2 , · · · , n. This change of variables reserve the form of parabolic equation and reduce the number ofspace variables.
Multi-dimensional Black-Scholes equations have the form invariance under the change ofvariables product (2) and its space dimension is reduced under the change of variables.In the pricing problems of financial derivatives described as a terminal value problemfor multi-dimensional Black-Scholes equation, if its expiry payoff has the combination ofvariables such as (2), then the space dimension can be reduced.In some pricing problems of interest rate derivative that have three or more risk resourcesand are not described by Black-Scholes equations (for example, [6]), the space dimensioncan be reduced by two or more; the main reason is that the pricing problem is described asa simillar form of multi-dimensional Black-Scholes equation when we consider the interest4
Hyong-chol O, Yong-hwa Ro, Ning Wan rate derivative as a risk free zero coupon bond derivative and its expiry payoff function hasnot only a combination of variables but also homogeneity on the new group variables.The method of considering interest rate derivatives as no coupon bond derivatives is stilleffective in any interest rate models satisfied the assumptions ”(i) the volatility of short rate r does not depend on r , (ii) the price of zero coupon bond follows geometric Brown motion(see (18)), (iii) short rate r is a deterministic function of the price of zero coupon bond (see(17))”. For examples, Vasicek, Ho-Lee and Hull-White models satisfy all these properties.For counter examples, HJM model satisfies (i) and (ii) but does not satisfy (iii); CIR modelsatisfies (ii) and (iii) but does not satisfy (i), so the resulting equations (22) and (23) afterchange of variables have the coefficients Σ , Σ that still depend on r or p and p .If an asset F that depends on the short rate satisfies the above three assumptions, thenwe can consider interest rate derivatives as F derivatives and use Black-Scholes equations.For a counter example, if r is stochastic and satisfies (i), B is bank account, that is B ( t ) = exp Z t r ( u ) du, then B satisfies (ii) but it does not satisfy (iii), so we cannot consider interest rate derivativesas B derivatives. References [1] Benninga, S., Bjrk, T. and Wiener, Z. On the Use of Numeraires in Option Pricing.
The Journal of Derivatives . Winter 2002. 10(2): 1-16.[2] Garman, M. and Kohlhagen, S. Foreign currency option values.
Journal of internationalmoney and Finance . 1983. 2(3): 231-237.[3] Jiang, L.
Mathematical modeling and methods of option pricing . Beijing: Higher edu-cation press. 2003. chap. 7 (in Chinese), English translation: Jiang, Li-shang.
Mathe-matical modeling and methods of option pricing . Singapore: World Scientific. 2005.[4] O, H., Ro, Y. and Wan, N. The Use of Numeraires in Multi-dimensional Black-ScholesPartial Differential Equations. Working Paper. Tong-ji University, Department of Ap-plied Mathematics. 2005. http://ssrn.com/abstract=731544. DOI: 10.2139/ssrn.731544(in Chinese), English translation: arXiv 1310.8296[q-fin-PR].[5] Wilmott, P. Derivatives,
The theory and practice of financial engineering . New York:John Wiley & Sons, Inc.1998.[6] Xu, G. Analysis of Pricing European Call Foreign Currency Option Under the VasicekInterest Rate Model.