Chapter 4 The Multi-Underlying Black-Scholes Model and Correlation So far we have discussed single asset options, the payoff function depended only on one underlying. Now we want to allow multiple underlyings. A standard example would be a simple basket call option with payoff H basket call (S,T,, S m,t max { m S l,t m l S l,0, 0 } (4. Other examples could be options which measure relative performance. For example, a two-underlying option with underlyings S,t and S 2,t could simply pay out the relative performance H(S,T, S 2,T S,T /S,0 S 2,T /S 2,0 (4.2 If S and S 2 have comparable volatilities, it is intuitively cleat that the fair value of the payoff (4.2 should be sensitive to correlation: If S and S 2 move perfectly in sync, that is, if the returns ds /S and ds 2 /S 2 are highly correlated, it is clear that the payoff stays close to. On the other hand, if the returns ds /S and ds 2 /S 2 move in opposite directions, that is, if they are anti-correlated, the payoff can attain a broader range of values. In other words, correlation is a relevant model parameter and we recall some basic definitions and properties in the next section. The multi-underlying Black-Scholes model is then defined in section 4.2. For the payoff (4.2 we will actually find in example 4.7 below that its Black-Scholes time 0 price V 0 is given by V 0 e rt e (σ2 2 σ σ 2 ρt (4.3 if the underlyings S and S 2 have time independent volatilities σ and σ 2 and if underlying returns have the time independent correlation ρ. The fact that a multi asset option has a closed form solution for its price is quite exceptional: already the simple basket call with 07
08 Chapter 4 payoff (4. does not have a closed form solution in the multi-underlying Black-Scholes model. Therefore the standard calculation method for multi asset options is Monte Carlo. 4. Correlation The correlation of two random variable X and Y is defined by Corr(X, Y : where the covariance of X and Y is given by Cov(X, Y { V[X] V[Y ] } /2 (4.4 Cov(X, Y : E [ (X E[X](Y E[Y ] ] E[XY ] E[X]E[Y ] (4.5 In particular, Corr(X, X and Corr(X, X. For random variables X,..., X m, the covariance matrix C (c ij i,j m : ( Cov(X i, X j i,j m (4.6 is symmetric and positive definite (or more pricisely, at least positive semidefinite, since for arbitrary λ,..., λ m d ( m λ i λ j c ij Cov λ i X i, i,j i m λ j X j j [ m ] V λ i X i i 0 (4.7 By substituting λ i λ i / V[X i ], the same conclusion holds for the correlation matrix In terms of the covariance matrix, ρ (ρ ij i,j m : ( Corr(X i, X j i,j m (4.8 ρ ij c ij cii c jj. (4.9 Definition 4.: Let C R m m be a symmetric matrix. Any matrix A R m m which fulfills is called a Cholesky root of the matrix C. AA T C (4.0 Example: Let ρ and C ( ρ ρ (4.
Chapter 4 09 then is a Cholesky root for C. A ( 0 ρ ρ 2 (4.2 Lemma 4.2: Every symmetric and positive semidefinite (that is, eigenvalues 0 can occur matrix C has a Cholesky root A. That is, for every symmetric and positive semidefinite C R m m there is an A R m m such that AA T C (4.3 Proof: Since C is symmetric, it can be diagonalized with an orthogonal matrix V, C V DV T (4.4 with D diag(λ,, λ m and λ i 0 for all i m and V V T. Thus, if we put D diag( λ,, λ m C V DV T V D DV T V DV T V DV T AA T with A V DV T. The A we constructed here is also symmetric, A A T, but as the example above shows, this does not necessarily has to be the case. The following theorem tells us how to construct random variables with a prescribed correlation from uncorrelated random variables. Theorem 4.3: Let Y, Y m be m independent random variables with mean 0 and variance. That is, E[Y i ] 0, V[Y i ], Corr[Y i, Y j ] 0. (4.5 Let C (c ij i,j m R m m be a symmetric positive semidefinite matrix and let ρ (ρ ij i,j m be the matrix with entries ρ ij c ij / c ii c jj. Let A be a Cholesky root for C, Put Y (Y, Y m and let X (X, X m be given by AA T C (4.6 X A Y (4.7
0 Chapter 4 or X. X m a Y.. a m Y m a, Y. a m, Y if we denote the row vectors of A by a i and let, denote the standard scalar product in R m. Then we have: a The X i s have variance c ii and correlation ρ ij, E[X i ] 0, V[X i ] c ii and Corr[X i, X j ] ρ ij. (4.8 That is, C is the covariance matrix for the (X,..., X m and ρ is their correlation matrix. b Suppose that the Y i s are normally distributed. That is, for any function F F (y,..., y m : R m R and with y, y m E[F (Y ] i y2 i, R m F (y e y,y 2 d m y. (4.9 Suppose further that all eigenvalues of C are strictly positive, no zero eigenvalues. Then, with X AY, we have E[F (X] F (x e x,c x 2 d m x (4.20 R m det C c Introduce the notation with C m c,m C m. c m,m c m, c m,m c m,m (4.2 C m (c ij i,j m R (m (m. Then, for x (x,..., x m, x m (x, x m with x (x,, x m R m, we have R det C m e x,c m x 2 dx m det C m e x,c m x 2 (4.22
Chapter 4 Proof: a The equations E[X i ] 0 are an immediate consequence of E[Y i ] 0. Furthermore, by the assumption of independence of the Y i s and because of V[Y i ] E[Yi 2 ], { for i j E[Y i Y j ] 0 for i j δ i,j (4.23 such that m E[X i X j ] a ik a jl E[Y k Y l ] k,l m a ik a jl δ k,l k,l m a ik a jk (AA T ij c ij (4.24 In particular, V[X i ] E[X 2 i ] E[X i ] 2 E[X 2 i ] c ii and Corr[X i, X j ] E[X i X j ] / {V[X i ]V[X j ]} /2 c ij / c ii c jj ρ ij which proves part (a. Part (b is obtained from the substitution of variables x Ay, E[F (X] E[F (AY ] F (Ay e R m ya x F (x e R m F (x e R m F (x e R m y,y 2 d m y A x,a x 2 d m (A x x,(a T A x 2 det(a d m x x,c x 2 d m x det C (4.25 since det(a / (det A 2 / det(aa T / det C. Finally, if we consider equation (4.25 above for some F which depends actually only on x,..., x m, it is clear that part (c should hold. However, let us prove equation (4.22 by explicit calculation: c Abbreviate x (x, x m with x (x,..., x m, x m R m x (x,..., x m R m
2 Chapter 4 Since we do not use the Cholesky root in this part of the proof, we use the notation A for a different matrix: Write the m m matrix Cm as a block matrix with one (m (m block A, one (m block B, a (m block B T and a number D, C m ( A B B T D (4.26 Then the quadratic expression in the exponent on the left hand side of (4.22 can be written as ( A B ( x, Cm x (x x, x m, B T D x m x, Ax + 2(B T x x m + Dx 2 m (4.27 Thus we have to calculate the one dimensional gaussian integral R e 2 ( x,ax +2(B T x x m+dx 2 m dx m e D 2 R e D 2 R R ( x 2 m +2 (BT x D xm 2 x,ax dxm ( x m+ (BT x D 2 ( + D 2 e D 2 x2 dx e (B T x 2 2 D 2 x,ax e 2 y2 dy R 2π D e D e 2 ( x 2,Ax (BT x 2 D (B T x 2 D 2 x,ax dxm ( x,ax (BT x 2 D (4.28 The exponent in (4.28 is again quadratic in x and can be rewritten as follows (we rewrite,, R m to make the dimensionality of the vectors in the scalar product more explicit x, Ax R m D BT x, B T x R x, Ax R m D x, BB T x R m x, (A BD B T x R m (4.29 Because of (4.26 (used in the second equality below, the first equality is just the definition of C m and C m, we have C m c,m C m. c m,m c m, c m,m c m,m ( A B B T (4.30 D
Chapter 4 3 Now use Lemma 0.2.2 of the extremely nice reference [] (okay, okay... It states that ( ( A B E F B T D G H (4.3 where in particular E is given by E (A BD B T (4.32 On the other hand, we have E C m. Thus from (4.30 and (4.29 we get x, Ax R m D BT x, B T x R x, (A BD B T x R m and from (4.28 we obtain det C m R x, E x R m x C m x R m (4.33 e 2 x,c m x dx m 2π det C m D det C m D e ( x 2,Ax (BT x 2 D e 2 x,c m x (4.34 Thus it needs to be checked that D det C m det C m. However, this follows immediately from Cramer s rule since D (Cm m,m and, by Cramer s rule, inverse matrix elements are given by quotients of subdeterminants or Adjunkten, in this case det C m, divided by the determinant of the whole matrix, in this case det C m. This proves part (c and the theorem. The multi-underlying Black-Scholes model is defined in terms of correlated Brownian motions. They are constructed in the following Lemma 4.4: Let y t (y,t,, y m,t be a vector of m uncorrelated Brownian motions given by y t lim t φ k (4.35 t 0 with φ k (φ k,,, φ k,m and the φ k,l being independent (in both dimensions k and l and normally distributed random numbers with mean 0 and variance. Let ρ ρ(t ( ρ i,j (t i,j m Rm m (4.36
4 Chapter 4 be a (eventually time dependent correlation matrix and let A A(t be a Cholesky root for ρ(t, that is, AA T ρ or, if we denote the row vectors of A by a i a i (t, a i (t, a j (t ρ i,j (t (4.37 Then, if we put x t (x,t,, x m,t with and t k k t, we have x i,t lim t a i (t k, φ k (4.38 t 0 dx i,t dx j,t ρ i,j (t dt (4.39 where the above equation (4.39 is just the differential notation of the following more precise statement on the quadratic covariation of the correlated Brownian motions x,t,..., x m,t (with x i,tk x i,tk x i,tk : lim t 0 x i,tk x j,tk t 0 ρ i,j(u du (4.40 Proof: For fixed t we put I t : x i,tk x j,tk t a i (t k, φ k a j (t k, φ k (4.4 We calculate the mean and the variance of I t. The main observation will be again (we used this reasoning already to show that the quadratic variation of a -dimensional Brownian motion is deterministic and equal to t that the variance of I t converges to 0 as t 0 which means that I t itself converges to its mean in probability as a consequence of Tchebyscheff s inequality.
Chapter 4 5 The mean of I t is given by E[ I t ] t E[ a i (t k, φ k a j (t k, φ k ] t t m l,l m l,l a i,l (t k a j,l (t k E[ φ k,l φ k,l ] a i,l (t k a j,l (t k δ l,l t a i (t k, a j (t k (4.37 t ρ i,j (t k t 0 t ρ 0 i,j(u du (4.42 and, since I t is given by a sum of independent random variables, its variance converges to zero: [ V[ I t ] V t a i (t k, φ k a j (t k, φ ] k t 2 V [ a i (t k, φ k a j (t k, φ k ] t 2 V [ a i (t k, φ a j (t k, φ ] t t 0 du V[ a i (u, φ a j (u, φ ] }{{} some constant t 0 0 (4.43 From (4.42 and (4.43 we get the convergence in probability of I t to t 0 ρ i,j(udu. 4.2 The Multi-Underlying Black-Scholes Model The time-dependent multi-underlying Black-Scholes model is given by m underlyings S t (S,t,, S m,t with dynamics ds i,t S i,t µ i,t dt + σ i,t dx i,t (4.44
6 Chapter 4 with m correlated Brownian motions x t (x,t,, x m,t with correlations dx i,t dx j,t ρ i,j (t dt (4.45 As in the single underlying case, equation (4.44 is solved by S i,t S (µ i,t S i,0 e t 0 σ i,udx i,u + t 0 (µ i,u σ 2 i,u /2du (4.46 In terms of some uncorrelated Brownian motions y t (y,t,, y m,t, we can write dx t A(tdy t (4.47 if A(t is a Cholesky root for the correlation matrix ( ρ i,j (t. If we denote the row vectors of A(t by a i (t again, (4.47 reads in components dx i,t a i (tdy t (4.48 with the multiplication on the right hand side of (4.48 being a scalar product of m- dimensional vectors. Thus we have S i,t S (µ i,t S i,0 e t 0 σ i,u a i (udy u+ t 0 (µ i,u σ 2 i,u /2du (4.49 The following theorem is the analog of Theorem 3.4 now for the multi asset case. It states that also in the multi-underlying case, we still can calculate option prices by simply taking expectation values of the payoff function. Again, the drift parameters µ i,t have to be substituted by the interest rate r. Theorem 4.5: Let S t (S,t,, S m,t S (µ t abbreviate m underlyings with correlated time dependent Black-Scholes dynamics given by (4.44,4.45 or (4.49. Let H H({S (µ BSTD MU t } be some multi asset option payoff. Then the price V 0 V0 of this multi asset option is given by where S (r t (y ( S (r,t (y,, S m,t(y (r and [ BSTD MU V0 e rt (r E W H({S t } ] e rt H ({ S (r t (y } dw (y (4.50 S (r i,t (y S i,0 e t 0 σ i,u a i (udy u+ t 0 (r σ2 i,u /2du (4.5
Chapter 4 7 and the Wiener measure dw (y for the m-dimensional uncorrelated Brownian motion y t (y,t,, y m,t is given by ( y tk y tk y tk and the limit t 0, N T to be taken dw (y N T Π [2π(tk t k ] e 2(t k t k yt k, yt k d m y m tk (4.52 The proof is analog to the single asset case and we skip it. Instead of this, we will now consider the question of how the expectation (4.50 can be concretely calculated if the payoff H depends only on the underlying values at maturity, H H(S T. That is, we write down and prove the analog of Theorem 3.5 which reads as follows in the multi asset case: Theorem 4.6: a Let y t (y,t,, y m,t denote an m-dimensional uncorrelated Brownian motion, let σ,t,..., σ m,t be some instantaneous volatilities and let A(tA(t T (ρ i,j (t or, if we denote the row vectors of A by a i (t, a i (t a j (t ρ i,j (t (4.53 Then we have for any function F : R m R ( { T F σ 0 i,t a i (tdy t dw (y } i m F (x,, x m e 2 x,c T x d m x R m det C T (4.54 F (A T y e 2 y,y dm y R m (4.55 with a covariance matrix C T ( T σ 0 i,tσ j,t ρ i,j (tdt i,j m (4.56 and A T being a Cholesky root for C T, A T A T T C T (with the subscript T denoting maturity T and the superscript T denoting matrix transposition. b Let H H(S,T,, S m,t be the payoff of some non path-dependent multi asset BSTD MU option. Then its fair value V 0 V0 in the time-dependent multi-underlying Black- Scholes model is given by V 0 e rt R m H e rt R m H ( {S i,0 e σimp i,t ( {S i,0 e σimp i,t T xi + (r (σimp i,t 2 T 2 } i m T aimp,i,y + (r (σimp i,t 2 T 2 } e 2 x,ρ imp,t x d m x (2πm det ρ imp,t (4.57 i m e 2 y,y dm y (4.58
8 Chapter 4 with implied volatilities σ imp i,t { /2 T T 0 σ2 i,t dt} (4.59 and an implied correlation matrix ρ imp,t ( ρ imp i,j (T with entries i,j m ρ imp If we let A imp denote a Cholesky root of that matrix, T i,j (T : σ 0 i,tσ j,t ρ i,j (tdt { T 0 σ2 i,t dt T 0 σ2 j,t dt} (4.60 /2 A imp A T imp ρ imp,t (4.6 then the vectors a imp,i showing up on the right hand side of (4.58 in the exponential function are the row vectors of that A imp R m m. Proof: a Let x i : T σ 0 i,t a i (tdy t N T lim σ i,tk a i (t k t φ k t 0 with φ k (φ k,,, φ k,m being independent normal random numbers. In matrix notation, with x (x,, x m and σ tk : diag(σ,tk,, σ m,tk R m m, with Wiener measure N T x lim σ tk A(t k t φ k (4.62 t 0 dw (φ N T Π e φ k,φ k 2 d m φ k (4.63 Now we make the substitution of variables φ k ψ k where which gives ψ k σ tk A(t k φ k φ k A(t k σ t k ψ k (4.64 N T x lim t ψk (4.65 t 0
Chapter 4 9 with Wiener measure where we put dw (ψ N T Π e 2 A(t k σt ψ k,a(t k σ k t ψ k k det A(t k det σ tk which has the matrix entries d m ψ k N T Π e 2 ψ k,σt [A(t k A(t k T ] σ k t ψ k k det[σtk A(t k A(t k T σ tk ] N T Π e 2 ψ k,c(t k ψ k det[c(tk ] d m ψ k d m ψ k (4.66 C(t k : σ tk A(t k A(t k T σ tk R m m (4.67 c i,j (t k σ i,tk σ j,tk ρ i,j (t k (4.68 Now we transform back to the Brownian motion type integration variables by putting x tk : t k ψ j (4.69 j ψ k x t k x tk t and arrive at (the limit t 0 to be taken ( { T F σ 0 i,t a i (tdy t dw (y } i m F (x,t,, x m,t N T Π F ( x (ψ,, x m (ψ N T Π e 2 ψ k,c(t k ψ k det[c(tk ] d m ψ k F (x,t,, x m,t N T Π e 2 xt k xt k,[ t C(t k] (x tk x tk det[ t C(tk ] where we abbreviated which has matrix entries e 2 xt k xt k,c t(t k (x tk x tk d m x tk det[c t (t k ] d m x tk (4.70 C t (t k : t C(t k (4.7 c t i,j (t k σ i,tk σ j,tk ρ i,j (t k t (4.72 Now, for any symmetric and positive definite matrix C (c ij i,j m R m m we introduce the kernel (x, y R m p C (x, y : det C e 2 x y,c (x y (4.73
20 Chapter 4 which has the following basic property which is proven in Lemma 4.8 below: R m p C (x, yp C2 (y, z d m y p C +C 2 (x, z (4.74 With this, (4.70 can be rewritten as follows: ( { T F σ 0 i,t a i (tdy t dw (y } i m F (x,t,, x m,t N T Π F (x,t,, x m,t N T Π p C t (t k (x tk, x tk d m x tk (4.74 e 2 xt k xt k,c t(t k (x tk x tk d m x tk det[c t (t k ] R m F (x,t,, x m,t p sum C t (t k (x 0, x T d m x T (4.75 where the matrix sum C t (t k is given by sum C t (t k N T C t (t k : C T (4.76 and has matrix entries c ij,t t 0 N T N T t C(t k t σ i,tk σ j,tk ρ i,j (t k T 0 σ i,tσ j,t ρ i,j (t dt (4.77 Since p CT (x 0, x T p CT (0, x T det C T e 2 x T,C T x T (4.78 part (a follows. To obtain part (b, we apply the formula of part (a to the expectation value in Theorem 4.5 to obtain ( V 0 e R rt H {S i,0 e x i,t + } t 0 (r σ2 i,u /2du m i m Recall that e 2 x T,C T x T d m x T (4.79 (2πm det C T T 0 σ2 i,tdt σ 2 imp,t T (4.80
Chapter 4 2 such that, since ρ ii, Thus, with the substitution of variables c ii,t T 0 σ2 i,tdt (σ imp i,t 2 T (4.8 x i,t σ imp i,t T yi c ii,t y i (4.82 and abbreviating D diag( c,t,, c mm,t R m m, equation (4.79 becomes, using x Dy, V 0 e rt R m H e rt R m H e rt R m H which proves the theorem.. ( {S i,0 e σimp i,t T yi + } t 0 (r σ2 i,u /2du i m ( {S i,0 e σimp i,t T yi + } t 0 (r σ2 i,u /2du i m ( {S i,0 e σimp i,t T yi + } t 0 (r σ2 i,u /2du i m e 2 Dy,C T Dy det D d m y (2πm det C T e 2 y,[d C T D ] y d m y (2πm det[d C T D ] e 2 y,ρ imp y d m y (2πm det[ρ imp ] (4.83 Example 4.7: Consider 2 underlyings with dynamics ds,t /S,t µ dt + σ dx,t (4.84 ds 2,t /S 2,t µ 2 dt + σ 2 dx 2,t (4.85 with constant volatilities and constant correlation ρ, dx,t dx 2,t ρ dt (4.86 Then the Black-Scholes fair value V 0 of the two-asset option H with payoff (4.2, is given by H(S,T, S 2,T S,T /S,0 S 2,T /S 2,0 V 0 e rt e (σ2 2 σ σ 2 ρt (4.87 In particular, for σ σ 2 σ, V 0 e rt e ( ρσ2 T (4.88
22 Chapter 4 Proof: We use the pricing formula (4.58 with Cholesky root ( ( ( 0 ρ ρ ρ 2 0 ρ ρ 2 ρ Thus, ( x x 2 ( ( 0 ρ y ρ 2 y 2 ( y ρ y + ρ 2 y 2 (4.89 with uncorrelated y and y 2. Since S,T /S,0 S 2,T /S 2,0 e σ T x +(r σ 2/2T e σ 2 T x2 +(r σ2 2/2T e (σ2 2 σ2 T 2 e T (σ x σ 2 x 2 (4.89 e (σ2 2 σ2 T 2 e T (σ y σ 2 [ρy + ρ 2 y 2 ] e (σ2 2 σ2 T 2 e T ([σ σ 2 ρ]y +σ 2 ρ 2 y 2 ] we have to calculate the 2-dimensional integral ρ 2 y 2 ] e 2 (y2 +y2 2 dy dy 2 e [σ σ 2 ρ] 2 T/2+σ2 2( ρ2 T/2 2π R 2 e T ([σ σ 2 ρ]y +σ 2 such that we end up with V 0 e rt e (σ2 2 σ2 T 2 e (σ 2 2σ σ 2 ρ+σ 2 2 T/2 e rt e (σ2 2 σ σ 2 ρt which is identical to formula (4.88. e (σ2 2σ σ 2 ρ+σ 2 2 T/2 (4.90 Finally we have to prove the following lemma which we used in the proof of Theorem 4.6 where we had to calculate an expectation value with respect to Wiener measure with m correlated Brownian motions. Lemma 4.8: Let C and C 2 be symmetric and positive definite matrices in R m m. For x, y R m, define the kernel p C (x, y : det C e 2 x y,c (x y (4.9
Chapter 4 23 for a symmetric and positive definite C. Then there is the following identity: R m p C (x, yp C2 (y, z d m y p C +C 2 (x, z (4.92 Proof:..to be included..