Solution of Stochastic Differential Equations in Finance

Fig. 7. Estimation errors for European call using control variates. Error is proportional to the square root of the number of Monte-Carlo realizations. Compare absolute levels of error with Fig. 5.

∂X
Let C₀ be the option value at time t = 0, which is the goal of the calcula- tion. At the time t = 0, the seller of the option hedges by purchasing ∆ = ^∂C shares of the underlying asset. Thus the cash account, valued forward to time T , holds

⁰ _∂X ^{0 t}0
[C − ^∂C (t )X ]e
^{r(T −t}0 ⁾_.

∂X
At time step t = t₁, the seller needs to hold ∆ = ^∂C (t₁) shares, so after purchasing ^∂C (t₁) − ^∂C (t₀) shares, the cash account (valued forward) drops

∂X ∂X
by
∂C
∂X


(t₁) −
∂C ^l
_∂X (t₀)]X<sub>t1


e</sub>^{r(T −t}1 ⁾_.

Continuing in this way, the cash account of the replicating portfolio at time
T , which must be C_T , equals

^N ∂C
∂C ^l

₀ −
_{C e}^{r(T −t}0 ^{) J}
∂X
(t_k ) −
_∂X (t_k
−1⁾
X_tk
er(T −t_k )

k=0
N −1

∂C
_{= C0e}^{r(T −t}0 ⁾ ₊ ^J
∂X
(t_k )(X_t


k+1



k
_{— Xt e}r∆t_)er(T −t_k+1 )

and so

_{C0 = e}^{−r(T −t}0 ⁾
k=0

T ₋ J
N −1 _∂C
C
∂X
(t_k )(X_t
k+1

l

k
_{— Xt e}r∆t_)er(T −t_k+1 )

k=0
= e^{−r(T −t0 )} [C_T − cv]
where cv denotes the control variate. Estimating the expected value of this expression yields fast convergence, as demonstrated in Fig. 7. Compared to Fig. 5, the errors in pricing of the European call are lower by an order of magnitude for a similar number of realizations. However, the calculation of the control variate adds significantly to the computational load, and depending on the form of the derivative, may add more overhead than is gained from the reduced variance in some cases.

6. Multifactor models

Financial derivatives that depend on a variety of factors should be modeled as a stochastic process that is driven by a multidimensional Wiener process. The various random factors may be independent, but more realistically, there is often correlation between the random inputs.

For multifactor Wiener processes (W ¹, . . . , W ^k ), the generalization of Ito’s
t t
Formula requires that (3) is replaced with


dt dt = 0
dt dW ⁱ = dW ⁱ dt = 0
t t
dW ⁱ dW ^j = ρ_ij dt (22)
t t
where ρ_ij represents the statistical correlation between W ⁱ and W ^j . As usual,
t t
correlation ρ of two random variables X₁ and X₂ is defined as

1 2 ^.
_{ρ(X , X ) =} cov(X₁, X₂)
V (X ) V (X )
1 2

Note that ρ(X₁, X₁) = 1, and X₁ and X₂ are uncorrelated if ρ(X₁, X₂) = 0.

To construct discretized correlated Wiener processes for use in SDE solvers, we begin with a desired correlation matrix
^ ρ₁₁ · · · ρ_1k ^

R = ^

._..
._. ^

.


ρ_k1 · · · ρ_kk
that we would like to specify for Wiener processes W ¹, . . . , W ^k . The matrix R is symmetric with units on the main diagonal. A straightforward way to create noise processes with a specified correlation is through the singular value decomposition (SVD) (see Sauer (2006) for a description). The SVD of R is
R = Γ ΛΓ ^T

where Γ is an orthogonal matrix (Γ ⁻¹ = Γ ^T), and Λ is a diagonal matrix with nonzero entries on the main diagonal.

Begin with k independent, uncorrelated Wiener processes Z₁, . . . , Z_k , sat- isfying dZ_idZ_i = dt, dZ_idZ_j = 0 for i /= j. Define the column vector dW = Γ Λ^1/2dZ, and check that the covariance matrix, and therefore the correlation matrix, of dW is
dWdW^T = Γ Λ^1/2dZ(Γ Λ^1/2dZ)^T
= Γ Λ^1/2dZdZ^TΛ^1/2Γ ^T
= Γ ΛΓ ^Tdt = R dt


For example, a two-asset market has correlation matrix

_{1 ρ} l
R = _{ρ 1}
corr(W ¹, W ¹) corr(W ¹, W ²) ^l
= corr(W ², W ¹) corr(W ², W ²) ^.

Since the SVD of this 2 × 2 correlation matrix is

_{1
1 ρ} l = _√2

√
_{ρ 1} 1
2

1
√₂

1
₋ √₂
l 1 + ρ 0 ^{l 1}

√₂

√₂
0 1 − ρ ¹
₁ l

,
√₂
1
₋ √₂

we calculate
dW ¹ =


dW ² =


^√1 + ρ
√₂
^√1 + ρ
√₂
dZ¹ +


dZ¹ −
^√1 − ρ
√₂
^√1 − ρ
√₂
dZ²


dZ². (23)

2.5
2

value
1.5
1
0.5
0
−1 −0.5 0 0.5 1
correlation 


Fig. 8. European spread call value as a function of correlation. The Euler- Maruyama solver was used with multifactor correlated Wiener processes. The initial ^{values of the underlying assets were X}1^{(0) = 10, X}2^{(0) = 8, the interest rate was} r = 0.05, strike price K = 2, and expiration time T = 0.5.
With a change of variables, the correlation ρ can be generated alternatively as
dW ¹ = dZ¹
dW ² = ρ dZ¹ + 1 − ρ² dZ². (24)
As a simple example, we calculate the value of a European spread call us- ing Monte-Carlo estimation of noise-coupled stochastic differential equations using a two-factor model. Assume there are two assets X₁ and X₂ satisfying arbitrage-free SDE’s of form
dX₁ = rX₁ dt + σ₁X₁ dW ¹
dX₁ = rX₂ dt + σ₂X₃ dW ² (25)
where dW ¹dW ² = ρ dt, and that the payout at expiration time T is max{X₁(T ) − X₂(T ) − K, 0} for a strike price K. The Monte-Carlo approach means estimating the expected value
E(e^−rT max{X₁(T ) − X₂(T ) − K, 0}).
Using either form (23) or (24) for the coupled Wiener increments in the Euler- Maruyama paths, the correct price can be calculated. Fig. 8 shows the depen- dence of the price on the two-market correlation ρ. As can be expected, the more the assets move in an anticorrelated fashion, the more probable the spread call will land in the money.

7.