_{1}

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We give a study result to analyze a rather different, semi-analytical numerical algorithms based on splitting-step methods with their applications to mathematical finance. As certain subsistent numerical schemes may fail due to producing negative values for financial variables which require non-negativity preserving. These algorithms which we are analyzing preserve not only the non-negativity, but also the character of boundaries (natural, reflecting, absorbing, etc.). The derivatives of the CIR process and the Heston model are being extensively studied. Beyond plain vanilla European options, we creatively apply our splitting-step methods to a path-dependent option valuation. We compare our algorithms to a class of numerical schemes based on Euler discretization which are prevalent currently. The comparisons are given with respect to both accuracy and computational time for the European call option under the CIR model whereas with respect to convergence rate for the path-dependent option under the CIR model and the European call option under the Heston model.

Stochastic differential equations (SDEs) are fundamental in mathematical finance. Particularly, they serve as models for describing the evolution of certain financial variables, such as the stock price, interest rates or volatility of an asset. Although there are extensive studies on SDEs, explicit solutions of SDEs are rarely known or very difficult to obtain, so numerical approximations are relied on. Actually a wealth of numeric methods have been proposed and tested (see for example [

In some cases, the Itô-type SDEs of the form

d X ( t ) = f ( X ( t ) ) d t = σ ( X ( t ) ) d W ( t ) , X ( 0 ) = X 0 ∈ D (1.1)

are well-defined only with certain boundary conditions. For example, the Cox-Ingersoll-Ross (CIR) [

d V ( t ) = κ ( θ − V ( t ) ) d t − σ V ( t ) d W V ( t ) (1.2)

where W V ( t ) is a Wiener process and κ , θ , σ are positive constants. The Heston model is a two-factor model of the form:

d S ( t ) = μ S ( t ) d t + V ( t ) S ( t ) d W S ( t ) (1.3)

d V ( t ) = κ ( θ − V ( t ) ) d t + σ V ( t ) d W V ( t ) (1.4)

where W S ( t ) , W V ( t ) are two correlated Wiener processes with d W S ( t ) d W V ( t ) = ρ d t , ρ ∈ ( − 1 , 1 ) , and μ , κ , θ , σ are positive parameters. The component S describes the evolution of a financial variable such as stock index or exchange rate, and V describes the stochastic variance of its returns. One can notice that the component V in the Heston model evolves according to the CIR process (1.2).

First let’s mention that the SDE (1.2) for CIR process is not explicitly solvable but its transition probability is known; it can be represented by a non-central chi-square density. Depending on the number of degrees of freedom d : = 4 κ θ / σ 2 , there are significant differences in the boundary behavior of CIR process. According to Feller’s classification [

Second, as the square-root term in CIR process avoids the possibility of negative values, Euler-Maruyama [

order 1 2 − ε , Moro and Schurz [

convergence at least of order 1. Although they are more costly with respect to computational time, they give higher accuracy and convergence rate. With this motivation, we would like to evaluate some important financial instruments using splitting-step methods with comparison to Euler-type numerical schemes. We apply them to option pricing including a path-dependent option valuation comparing both accuracy and computational cost.

Third, exact simulation methods exist for both the CIR process (see Glasserman [

For the CIR process, a number of numerical schemes based on implicit time-stepping integrators have been devised for the case of unattainable boundary condition, see for example Alfonsi [

Other direct approaches that can be applied to both attainable and unattainable zero boundary cases are based on modification of Euler-Maruyama approximations. See for example Deelstra and Delbaen [

The exact simulation method for CIR model exists, see Glasserman [

Anderson [

Further, for pricing financial derivatives under Heston model, closed form semi-analytical formulae for plain vanilla option prices have been derived in [

In the remaining paper, we introduce the general structure and properties of splitting-step algorithm proposed by Moro and Shurtz in [

In Section 3, we present numerical results of applications to CIR model, evaluating a European plain vanilla call option and a path-dependent option with comparisons to the methods based on Euler discretization. The comparisons are with respect to both accuracy and computational time. All of our numerical results are generated on a MacBook Pro with an Intel Core i7 2.3 GHz processor, 16 GB 1600 MHz DDR3 memory, using Xcode 6.4 and R 3.1.0 in a Mac OS X Yosemite environment.

In Section 4 we study the option valuation under the Heston model. The results show that the splitting-step method gives the best convergence rate for pricing a European plain vanilla call option among the five algorithms utilized in our project.

Finally we conclude and point out the future work in Section 5.

The splitting-step algorithm [

d X ( t ) = [ α ( X ( t ) , t ) + β ( X ( t ) , t ) ] d t + σ ( X ( t ) , t ) d W ( t ) , (2.1)

where W ( t ) is a standard Wiener process? Then decompose the above equation into two subsystems

d X 1 ( t ) = β ( X 1 ( t ) , t ) d t + σ ( X 1 ( t ) , t ) d W ( t ) , (2.2)

d X 2 ( t ) = α ( X 2 ( t ) , t ) d t (2.3)

where it’s required that we know the exact strong solution for X 1 ( t ) or the conditional probability P [ X 1 ( t ) | X 1 ( 0 ) ] . Afterwards, approximate the solution of (2.1) along the time interval [ t , Δ t ] using the following two-step algorithm for each Δ t .

Step 1: Knowing the value X t , taking it as an initial data of (2.2), i.e. X t = X 1 ( t ) , we obtain an intermediate value X ˜ t = X 1 ( t + Δ t ) through the exact integration of (2.2), or alternatively, through the conditional transition probability P [ X 1 ( t + Δ t ) | X 1 ( t ) ] .

Step 2: Using X_{t} obtained in step 1 as the initial condition for (2.3), i.e. X ˜ t = X 2 ( t ) , integrate (2.3) by any converging deterministic numerical algorithm to get X 2 ( t + Δ t ) (at least of deterministic order 1). Then X t + Δ t = X 2 ( t + Δ t ) .

Easily speaking, the procedure is as follow:

We adapt the example in [

P [ X 1 ( t ) | X 1 ( 0 ) ] = 2 t ( X 1 ( 0 ) X 1 ( t ) ) 1 2 I 1 ( 4 t X 1 ( t ) X 1 ( 0 ) ) e − 2 t [ X 1 ( t ) + X 1 ( 0 ) ] + e − 4 t X 1 ( 0 ) δ ( X 1 (t) )

where δ ( x ) is the Dirac delta function and I is the modified Bessel function of the first kind with index n which is given by

I ν ( x ) = ∑ n = 0 ∞ 1 n ! ( ν + n + 1 ) ( x 2 ) ν + 2 n , x > 0

with gamma function Γ ( t ) : = ∫ 0 ∞ x t − 1 e − x d x , t > 0 specially Γ ( n ) = ( n − 1 ) ! for n = 0 , 1 , 2 , ⋯ . Then we can sample the conditional probability distribution function using the rejection or inverse methods but it’s computational expensive. There is a more simple way to obtain X 1 ( t ) . Noticing that the variable

Z ( t ) = 4 t X 1 ( t ) follows a non-central c^{2}-distribution, that is

P [ Z | Z 0 ] = ∑ j = 1 ∞ ( λ / 2 ) j e − λ / 2 j ! P χ 2 j 2 ( Z ) + e − t 4 λ δ (Z)

where λ = 4 t Z 0 and P χ 2 j 2 ( x ) is c^{2}-pdf with 2j degrees of freedom. Then

X 1 ( t ) = 1 2 k { 0 if K = 0 , ∑ i = 1 2 K z i 2 if K ≠ 0 ,

where k = 2 t and K is chosen from a Poisson distribution with mean λ / 2 and

z i are independent Gaussian random numbers with zero mean and unit variance.

Generally, a non-central chi-square distribution random variable χ ′ d 2 ( λ ) with d degree of freedom and non-centrality parameter λ whose probability density function is given by

P [ χ ′ d 2 ( λ ) = x ] = p ( x ; d , λ ) = e − ( λ + x ) / 2 2 ( x λ ) ( d − 2 ) / 4 I ( d − 2 ) / 2 ( λ x ) (2.4)

see Johnson et al., 1994 [^{2} variables with Poisson weights, then it can be sample as follow: Choose K from a Poisson distribution with mean = 2, then

χ ′ d 2 ( λ ) = χ d + 2 K 2

And

χ ′ d 2 ( λ ) = { 0 if d + 2 K ≤ 0 χ ′ d + 2 K 2 if d + 2 K > 0 d = 0 , − 2 , − 4 , ⋯

where χ d + 2 K 2 can be sampled using any standard random number generator of the χ d 2 distribution. This sampling is used specially when λ is small. For the case of large λ , other approximations (see e.g. Johnson et al., 1994 [

Let C i , j ( ℝ d × [ 0 , T ] ) denote the vector space of continuous functions f = f ( x , t ) which are i times continuously differentiable with respect to the space

Coordinate x k ∈ ℝ , ( k = 1 , 2 , ⋯ , d ) and j times continuously differentiable with respect to the time coordinate t ∈ [ 0 , T ] . Let

0 < t 0 < t 1 < ⋯ < t n < t n + 1 < ⋯ < t N = T

be any random partition of the given time interval [ 0 , T ] with sufficiently small maximum step size Δ = max i = 1 , 2 , ⋯ , N | t i − t i − 1 | ≤ 1 . Then the time discretized approximation X ˜ Δ of a continuous-time process X, is said to be of general strong order of convergence g to X at time T if there exists a positive constant C, which does not depend on D, and a δ 0 > 0 such that the following strong error ε ( Δ ) satisfies

ε ( Δ ) = E ( | X ( T ) − X ˜ Δ ( T ) | ) ≤ C Δ γ

for each Δ ∈ ( 0 , δ 0 ) .

Along with the strong convergence, the weak convergence can be defined. A discrete-time approximation Y Δ is said to converge with weak order β > 0 to X at time T as Δ → 0 if for each smooth function g of polynomial growth there exists a constant C g , which does not depend on D and Δ 0 ∈ [ 0 , 1 ] such that the following weak error ϵ ( Δ ) satisfies the estimate

ϵ ( Δ ) = | E g ( X ( T ) ) − E g ( Y Δ ( T ) ) | ≤ C g Δ β

for each Δ ∈ ( 0 , Δ 0 ) .

The splitting-step algorithm has been shown that has strong and weak order 1.0 of convergence under certain assumption of the coefficient functions in [

Recall that the original SDE is

d X ( t ) = [ α ( X ( t ) , t ) + β ( X ( t ) , t ) ] d t + σ ( X ( t ) , t ) d W ( t ) . (2.5)

We refer to the splitting

d X 1 ( t ) = β ( X ( t ) , t ) d t + σ ( X ( t ) , t ) d W ( t ) (2.6)

d X 2 ( t ) = α ( X ( t ) , t ) d t (2.7)

Theorem Assume that the coefficient functions α , β ∈ C 2 , 1 ( ℝ d × [ 0 , T ] ) and σ ∈ C 3 , 2 ( ℝ d × [ 0 , T ] ) with exclusively uniformly bounded derivatives are such that

sup 0 ≤ t ≤ T E [ | X ( t ) 2 | + | α ( X ( t ) , t ) | 2 + | β ( X ( t ) , t ) | 2 + | σ ( X ( t ) , t ) | 2 ] < + ∞

for a fixed finite, nonrandom terminal time T > 0 . Then the splitting-step algorithm with step 1 and 2 (see section §2.1) has (global) strong and weak order 1.0 of convergence on the interval [ 0 , T ] (in the worst case).

The proof and a general theorem on L^{2}-convergence based on variation-of-constants formula (VOP) see Moro and Schurz [

The famous Cox-Ingersoll-Ross (CIR) model was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross in [

d V ( t ) = − κ ( V ( t ) − θ ) d t + σ V ( t ) d W ( t ) , ∀ t ∈ ℝ + (3.1)

with k, θ and σ strictly positive parameters and W ( t ) a Wiener process. The parameter k determines the speed of adjustment, θ is the long-run mean and σ is the so-called volatility to volatility. The drift factor, κ ( θ − V ( t ) ) is exactly the same as in the Vasicek model. It ensures mean reversion of the interest rate towards the long-term value θ, with speed of adjustment governed by the strictly positive parameter k. According to Feller’s boundary criteria, see Feller [

・ If κ θ ≥ σ 2 / 2 , the upward drift is sufficiently large to make the boundary unattainable, i.e., the solution is always positive V ( t ) > 0 if V 0 > 0.

・ If κ θ < σ 2 / 2 , there are infinite many values of t > 0 for which V ( t ) = 0 . The boundary becomes attainable, but it is strongly reflecting. That is, when a sample path reaches 0, then it returns immediately to the positive domain in a reflecting manner.

Since for all positive values of k and θ, the standard deviation factor σ V ( t ) rejects the possibility of negative interest rates, integration strategies must preserve non-negativity. Otherwise, it not only lacks any possible interpretation in the context of finance but also could induce severe errors in option valuation.

We now turn to the simulation of CIR model (12). The exact simulation method for CIR model exists, see Glasserman [

Here we outline the numerical schemes based on Euler discretization which we mentioned above. Let

0 < t 0 < t 1 < ⋯ < t n < t n + 1 < ⋯ < t N = T

be any random partition of the given time interval [ 0 , T ] .

・ Higham and Mao [

V ( t n + 1 ) = V ( t n ) − κ Δ t ( V ( t n ) − θ ) + σ | V ( t n ) | Δ W n , (3.2)

V ( t 0 ) = V ( 0 ) , (3.3)

where Δ W n = W ( t n + 1 ) − W ( t n ) .

Although it can be applied to the simulation, it leads to negative values of V ( t ) in practice as we can see from

・ In Lord et al., 2008 [

V ˜ ( t n + 1 ) = V ˜ ( t n ) − κ Δ t ( V ˜ ( t n ) − θ ) + σ | V ˜ ( t n ) | Δ W n , (3.4)

V ( t n + 1 ) = | V ˜ ( t n + 1 ) | , (3.5)

V ( t 0 ) = V ( 0 ) . (3.6)

・ Deelstra and Delbaen [

V ˜ ( t n + 1 ) = V ˜ ( t n ) − κ Δ t ( V ˜ ( t n ) − θ ) + σ ( V ˜ ( t n ) ) + Δ W n (3.7)

V ˜ ( t n + 1 ) = ( V ˜ ( t n + 1 ) ) + (3.8)

V ( t 0 ) = V ( 0 ) (3.9)

・ Full truncation was devised by Lord, Koekkoek & Van Dijk in [

V ˜ ( t n + 1 ) = V ˜ ( t n ) − κ Δ t ( V ˜ ( t n ) + − θ ) + σ ( V ˜ ( t n ) ) + Δ W n (3.10)

V ˜ ( t n + 1 ) = ( V ˜ ( t n + 1 ) ) + (3.11)

V ( t 0 ) = V ( 0 ) (3.12)

We refer the above four numerical schemes as Higham and Mao, Higham and Mao complemented, partial truncation and full truncation respectively.

According to the general structure of splitting-step algorithm in Section 2, we split the SDE (3.1) into two equations as follows.

d V 1 ( t ) = κ θ d t + σ V 1 ( t ) d W ( t ) (3.13)

d V 2 ( t ) = − κ V 2 ( t ) d t (3.14)

We note that the process defined by (3.13) is a kθ-dimensional squared Bessel

process (BESQ) with index of the process = ν = 2 κ θ σ 2 − 1 (see Makarov and

Glew [

P ( V 1 ( t + Δ t ) | V 1 ( t ) ) = ( V 1 ( t + Δ t ) V 1 ( t ) ) ν 2 e − 2 ( V 1 ( t + Δ t ) + V 1 ( t ) ) / σ 2 Δ t σ 2 Δ t / 2 I ν ( 4 V 1 ( t + Δ t ) V 1 ( t ) σ 2 t ) (3.15)

in the case of a regular boundary, where I ν is the modified Bessel function of the first kind with index n. Then we notice that the transition density P [ V 1 ( t + Δ t ) | V 1 ( t ) ] can be written in terms of a non-central χ 2 distribution. Recalling the probability density function (PDF) of a non-central chi-square distribution random variable with d degree of freedom and non-centrality parameter λ:

P [ χ ′ d 2 ( λ ) = x ] = p ( x ; d , λ ) = e − ( λ + x ) / 2 2 ( x λ ) ( d − 2 ) / 4 I ( d − 2 ) / 2 ( λ x ) , x > 0

Comparing the equation above and (3.15), we have

P ( V 1 ( t + Δ t ) | V 1 ( t ) ) = 4 σ 2 Δ t p ( 4 V 1 ( t + Δ t ) σ 2 Δ t ; 4 κ θ σ 2 , 4 V 1 ( t ) σ 2 Δ t ) , (3.16)

as ν = d 2 − 1 . The Equation (3.16) shows that the random process 4 V 1 ( t + Δ t ) σ 2 Δ t can be represented by a non-central chi-square distribution with d = 4 κ θ σ 2 degree of freedom and non-centrality parameter λ = 4 V 1 ( t ) σ 2 Δ t . Hence we have

V 1 ( t + Δ t ) = σ 2 Δ t 4 χ ′ d 2 ( λ ) (3.17)

where

λ = 4 V 1 ( t ) σ 2 Δ t , d = 4 κ θ σ 2 (3.18)

Then along the time interval [ t , t + Δ t ] , given V ( t ) , first we take it as an initial data of (3.13) and integrate the SDE though the transition conditional probability (3.15) to obtain V 1 ( t + Δ t ) which can be done by sampling a χ ′ d 2 ( λ ) random number in (3.17). Second we regard V 1 ( t + Δ t ) has the initial data of (3.14) and integrate it by any deterministic numerical method of at least order 1. In our project, we practically sample the χ ′ d 2 ( λ ) random number by the aid of the function rnchisq() inside the Rmath library from R and integrate (3.14) with deterministic Euler method.

In

The figure depicts that the splitting-step algorithm preserves non-negativity for positive initial data while Higham-Mao produces negative values occasionally. Thus splitting method is preferable as the CIR process is not defined for negative values.

In this section we verify that the splitting-step algorithm has weak convergence of order 1.0 on the interval [0,T] for a fixed finite, nonrandom terminal time T > 0 .

We take g the identity function and use the model

d V ( t ) = ( a + b V ( t ) ) d t + σ V ( t ) d W ( t ) (3.19)

Then split it in this way:

d V 1 ( t ) = a d t + σ V 1 ( t ) d W ( t ) (3.20)

d V 2 ( t ) = b V 2 ( t ) d t (3.21)

Similarly to (3.17) (3.18), we have

V 1 ( t + Δ t ) = σ 2 Δ t 4 χ ′ d 2 ( λ ) (3.22)

where

λ = 4 V 1 ( t ) σ 2 Δ t , d = 4 a σ 2 (3.23)

We use (3.22) (3.23) to obtain V 1 ( t + Δ t ) which is corresponding to (3.20) and deterministic Euler method to integrate (3.21).

Now, we calculate the expected value of exact solution of (3.19) with the initial value V ( 0 ) = V 0 ≥ 0 .

Taking the expected value of both sides of (3.19) reads:

d E [ V ( t ) | V ( 0 ) = V 0 ] = ( a + b E [ V ( t ) | V ( 0 ) = V 0 ] ) d t + E [ σ V ( t ) d W ( t ) | V ( 0 ) = V 0 ] (3.24)

Notice that the second term on the right side E [ σ V ( t ) d W ( t ) | V ( 0 ) = V 0 ] = 0 as W ( t ) is a standard Wiener process. We define

y ( t ) = E [ V ( t ) | V ( 0 ) = V 0 ]

Then y ( t ) is a deterministic function of t. Equation (3.24) becomes

d y ( t ) = ( a + b y ( t ) ) d t

with y ( 0 ) = V 0 . Solving the above equation we have

y ( t ) = y ( 0 ) e b t + a b ( e b t − 1 )

Therefore,

E [ V ( t ) | V ( 0 ) = V 0 ] = y ( t ) = V 0 e b t + a b ( e b t − 1 ) (3.25)

For a = 1 , b = 1 , we have

E [ V ( t ) | V ( 0 ) = V 0 ] = e t ( 1 + V 0 ) − 1 (3.26)

Then we apply the splitting-step method to obtain the numerical approximation V ˜ ( t ) .

ϵ Δ t = | E [ V ( t ) ] − E [ V ˜ ( t ) ] | (3.27)

Versus decreasing uniform step size Dt, which shows that the method has weak order 1.0 while using constant step sizes Dt. Moreover, we compare to Higham-Mao complemented, partial truncation and full truncation for the same equation.

Among the variety of financial derivatives, the option is one of the most important financial instruments. In current financial markets, there are mainly four kinds of options: American option, European option, Asian option, and Barrier option. In our work, we only focus on pricing European options, which can only be exercised on the maturity date whereas an American option can be exercised at any time before expiration.

A European call option on an asset is a contract that allows the buyer to buy (but not obligate) this asset at the price K at time T. The number K > 0 is called the exercise (or strike) price and T > 0 the maturity (or expiration) time. We pay K to buy the asset at time T if V ( T ) ≥ K and sell it immediately at V ( T ) making a profit of V ( T ) − K . The option is worthless if V ( T ) < K as we can buy it cheaper than K.

Thus at time T, the call option has the value

C T = E [ ( V ( T ) − K ) + ] = E [ max ( V ( T ) − K , 0 ) ] (3.28)

Suppose that C_{T} is the numerical approximation of C_{T}, then the error

ϵ c a l l = | C ˜ T − C T | (3.29)

reveals the precision of an algorithm. Since the exact solution C_{T} no explicitly known, we compute a numerical reference solution with a very small t and regard it as C_{T}. Here we use the numerical approximation of C_{T} with t = 10 4 computed by splitting method as reference.

Apart from the typical European call option in the previous section, more exotic options exist in the market (see, for example [

As we say that an U&O option expires worthless if the asset price touches some barrier L from below, say, at any time prior to expiry, where L is larger than the present asset value V 0 , the price of U&O call option at expiry time T is given by

C U & O ( T ) = E [ ( V ( T ) − K ) + 1 { 0 < V ( t ) < L , 0 ≤ t ≤ T } ] (3.30)

where 1 { x } is one if x is true and zero otherwise.

The exact option price C U & O ( T ) is not available since the exact solution of U&O call option under CIR model is not explicitly known. Then instead of evaluating biases we work on the convergence properties of the algorithms.

Let u ˜ Δ t which depends on the time step Dt be a numerical approximation of an exact value u. The numerical method said to be of order p means that there exists a number C independent of Dt such that

| u ˜ Δ t − u | ≤ C Δ t p

At least for sufficiently small Dt. It’s also said that the convergence rate of the method is Δ t p . Normally the error u ˜ Δ t − u depends smoothly on Dt. Then

u ˜ Δ t − u = C Δ t p + O ( Δ t p + 1 )

i.e.,

u ˜ Δ t = u + C Δ t p + O ( Δ t p + 1 )

An approach of p is to check the relative differences between u ˜ Δ t computed for different Dt. In most cases we compare solutions where Dt is halved successively. Then we get

u ˜ Δ t − u ˜ Δ t / 2 = u + C Δ t p − u − C ( Δ t / 2 ) p + O ( Δ t p + 1 ) = C Δ t p ( 1 − 1 2 p ) + O ( Δ t p + 1 ) = C ′ Δ t p + O ( Δ t p + 1 )

with C ′ = C ( 1 − 1 2 p ) Hence, we can get an estimate of the order of accuracy p after computing u ˜ Δ t and u ˜ Δ t / 2 for different Dt.

The relative differences u ˜ Δ t − u ˜ Δ t / 2 indicate the convergence rate of an algorithm. The faster converges a scheme (has higher p value), the faster relative differences reduce to 0 as Dt tends to 0. Now we define the relative difference of U&O call option value as:

ϵ Δ t U & O = | C ˜ Δ t U & O ( T ) − C ˜ Δ t / 2 U & O ( T ) | (3.31)

where C ˜ Δ t U & O ( T ) is the numerical approach of C Δ t U & O ( T ) using time step Dt. Thus ϵ Δ t U & O = C Δ t p + O ( Δ t p + 1 ) , with C a constant independent of Dt. We show the relation between ϵ Δ t U & O and the time step Dt to get an estimate of p for the four algorithms. We also analyze the computational cost with respect to the relative difference ϵ Δ t U & O . For sufficient small Dt, the relative differences ϵ Δ t U & O computed by splitting method are much less than standard error of the mean (SEM) which suggests to increase substantially the number of sample paths for

the sake of validity of these ϵ Δ t U & O . But roughly SEM ∼ 1 N where N is size of

the sample, which means that the SEM will reduce to 10% if we increase N to 100N. As we have already used N = 10 7 sample paths, it’s impossible in practice to increase it to at least N = 10 11 to make ϵ Δ t U & O larger than SEM. Then for sufficient small Dt, we estimate ϵ Δ t U & O computed by splitting by the fit of existing data. This has implied that splitting convergent the fastest as it’s the first one to reach the SEM level. ^{7} sample paths are generated.

The Heston model [

d S ( t ) = μ S ( t ) d t + V ( t ) S ( t ) d W S ( t ) (4.1)

d V ( t ) = κ ( θ − V ( t ) ) d t + σ V ( t ) d W V ( t ) (4.2)

where κ , θ , σ are positive constants, W S ( t ) and W V ( t ) are Wiener processes with correlation, i.e., d W S ( t ) d W V ( t ) = d t , ρ ∈ ( − 1 , 1 ) . The parameters μ is the rate of return of the asset. θ is the long variance, which means as t tends to infinity, the expected value of V ( t ) tends to θ. k is the rate at which V ( t ) reverts to θ σ is the so-called vol of vol, which determines the variance of V ( t ) . We note that the variance (4.2) follows a CIR process.

For V ( t ) it has the same boundary behavior as we mentioned in §3.1.

1) 0 is an attainable boundary when σ 2 > 2 κ θ . The boundary is strongly reflecting;

2) ∞ is an unattainable boundary.

There is an additional condition for S ( t ) which is iii) S ( t ) has an absorbing barrier at 0.

There are plenty of methods can be used to simulate Heston model. Broadie and Kaya [

Comparing to the methods above, Euler discretization and splitting-step are very simple to implement. For simulating the variance process (4.2) we use Splitting-step, Higham-Mao, partial truncation and full truncation methods introduced in §3. Then we switch to logarithms for the asset price S ( t ) , as in Lord et al., 2008 [

ln S ( t + Δ t ) = ln S ( t ) + ( μ − 1 2 V ( t ) ) Δ t + V ( t ) ( ρ Δ W V ( t ) + 1 − ρ 2 Δ W S ( t ) ) (4.3)

with W S ( t ) , W V ( t ) two independent Wiener processes. For Higham-Mao, as it may produce negative values of V ( t ) with substantial probability which we have seen in §3.2.1, we follow their spirit to take the absolute value of V ( t ) in (4.3), i.e.,

ln S ( t + Δ t ) = ln S ( t ) + ( μ − 1 2 | V ( t ) | ) Δ t + | V ( t ) | ( ρ Δ W V ( t ) + 1 − ρ 2 Δ W S ( t ) ) (4.4)

Closed form semi-analytical formulae for plain vanilla option prices have been derived in [

For simplicity and efficiency, in this section we mainly work on the convergence performance of the algorithms we mentioned in §4.2 and demonstrate that besides the well performance in convergence rates, splitting has superiority in computational efficiency.

We define C H e s ( T ) to be the exact European call option price at maturity time T based on Heston model, i.e.

C H e s ( T ) = E [ ( S ( T ) − K ) + ] = E [ max ( S ( T ) − K ) , 0 ]

and C ˜ Δ t H e s is an numerical approximation of C H e s ( T ) with time step Dt.

Then the relative difference is defined as

ϵ Δ t H e s = | C ˜ Δ t H e s − C ˜ Δ t / 2 H e s | (4.5)

The same technique with §3.3.2 gives that ϵ Δ t H e s = C Δ t p + O ( Δ t p + 1 ) , with C a constant independent of Dt, where p is the order of accuracy defined in §3.3.2. We estimate p by computing ϵ Δ t H e s for different Dt. The slope of the plots of ϵ Δ t H e s with respect to Dt gives the approximation of p.

^{7} sample paths. Obviously the splitting-step method has the highest order of accuracy p. Although the splitting method is more time-consuming than the others for each Dt, and with respect to relative differences, it costs the most at the beginning as for the case ϵ Δ t H e s > 2 illustrated in

More precisely, we analyze the computational cost by taking the case ϵ Δ t H e s = 0.065 as an example. As partial truncation and full truncation give fluctuant points (see ^{4} seconds » 2.8 hours while Higham-Mao needs much more than 10^{6} seconds » 277.8 hours which is more than 100 times than splitting.

A number of existing numerical algorithms for integrating SDEs cannot be used to simulate certain financial models which require non-negativity, such as the CIR model, the Heston model that we have seen. A slight modification of Euler-Maruyama [

Less precision is compared to semi-analytical numerical schemes such as the splitting-step methods which we are analyzing in our project. Since the exact simulation schemes are thoroughly too slow for simulating a process along a time-grid, splitting-step methods preponderate them with respect to efficiency.

The splitting algorithms heavily rely on the exploitation of the specific structure of original system (2.1). One can decompose the original system into two subsystems appropriately for which either one knows the explicit solution or the conditional transition probability of one of the subsystems and integrate the remaining one by deterministic numerical methods of at least order 1. In this way, it preserves the non-negativity and a maximum of convergence order 1.0 both in strong and weak sense. For the analytical solution of SDEs, an extensive list of known one can be found in textbooks such as Kloeden and Platen [

Among the numerical schemes in §3, splitting-step method gives the best convergence rate and normally highest accuracy. We observe that it doesn’t perform better than other four schemes on call option valuation with CIR model. Despite it costs more computational time to generate sample paths, it may cost even less for certain × errors. For example, if one requires that the relative difference of A European call option under the Heston model should be less than 0.10 with the parameter setting in ×4.3, the splitting method costs the least computational time than others.

Since derivatives are one of the three main categories of financial instruments where the other two being stocks (i.e., equities or shares) and debt (i.e., bonds and mortgages), and as we said that the splitting methods are much more efficient but has less accuracy than exact simulation methods which are not suitable for pathwise simulations, then one direction of our interest is applying splitting-step method to other path-dependent options, probably with comparison to other efficient algorithms like what we have done in §3.3.2.

Another direction is the application of splitting-step models to portfolio problems which are aiming to find the optimal investment strategy of an investor. Following the optimal portfolio strategy leads to the maximum expected utility of the terminal wealth. For example the classical Merton’s problem [

As Moro and Schurz [

I would take the opportunity to thank my tutor Esteban Moro Egido who gives the support during our project.

Yuan, Y. and Egido, E.M. (2018) Non-Negativity Preserving Numerical Algorithms for Problems in Mathematical Finance. Applied Mathematics, 9, 313-335. https://doi.org/10.4236/am.2018.93024