Euler and Milstein Discretization

8/10/2019 Euler and Milstein Discretization

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Euler and Milstein Discretizationby Fabrice Douglas Rouah

www.FRouah.comwww.Volopta.com

"Monte Carlo simulation" in the context of option pricing refers to a setof techniques to generate underlying valuestypically stock prices or interestratesover time. Typically the dynamics of these stock prices and interest ratesare assumed to be driven by a continuous-time stochastic process. Simulation,however, is done at discrete time steps. Hence, the rst step in any simulationscheme is to nd a way to "discretize" a continuous-time process into a dis-crete time process. In this Note we present two discretization schemes, Eulerand Milstein discretization, and illustrate both with the Black-Scholes and theHeston models.

We assume that the stock price St is driven by the stochastic dierentialequation (SDE)

dSt= (St; t) dt + (St; t) dWt (1)

where Wt is Brownian motion. We simulate St over the time interval [0; T],which we assume to be is discretized as 0 = t1 < t2


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identical fashion, the second integral is approximated as

Z t+dt

t

(Su; u) dWu (St; t)Z t+dt

t

dWu

= (St; t) (Wt+dt Wt)= (St; t)

pdtZ;

sinceWt+dtWt andp

dtZare identical in distribution, where Zis a standardnormal variable. Hence, Euler discretization of (2) is

St+dt = St+ (St; t) dt + (St; t)p

dtZ: (3)

1.1 Euler Scheme for the Black-Scholes Model

The Black-Scholes stock price dynamics under the risk neutral measure are

dSt= rStdt + StdWt: (4)

An application of Equation (3) produces Euler discretization for the Black-Scholes model

St+dt= St+ rStdt + Stp

dtZ: (5)

Alternatively, we can generate log-stock prices, and exponentiate the result. ByItos lemma ln St follows the process

d ln St =

r 1

22

dt + dWt: (6)

Euler discretization via Equation (3) produces

ln St+dt= ln St+

r 1

22

dt + p

dtZ

so that

St+dt= Stexp

r 1

22

dt + p

dtZ

: (7)

where dt= ti ti1.

1.2 Euler Scheme for the Heston Model

The Heston model is described by the bivariate stochastic process for the stock

priceS

t and its variancevt

dSt = rStdt +p

vtStdW1;t (8)

dvt = ( vt) dt + p

vtdW2;t

where E[dW1;tdW2;t] = dt.

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1.2.1 Discretization of vt

The SDE for vt in (8) in integral form is

vt+dt= vt+

Z t+dtt

( vu) du +Z t+dtt

p

vudW2;u: (9)

The Euler discretization approximates the integrals using the left-point rule

Z t+dtt

( vu) du ( vt) dtZ t+dtt

p

vudW2;u pvt(Wt+dt Wt)

=

pvtdtZv

where Zv is a standard normal random variable. The right hand side involves( vt)rather than( vt+dt)since at time t we dont know the value ofvt+dt.This leaves us with

vt+dt= vt+ ( vt) dt + p

vtdtZv:

To avoid negative variances, we can replace vt with v+t = max(0; vt). This is

the full truncation scheme. Thereectionscheme replaces vt with its absolutevaluejvtj.

1.2.2 Process for St

In a similar fashion, the SDE for St in (8) is written in integral form as

St+dt = St+ r

Z t+dtt

Sudu +

Z t+dtt

pvuSudWu:

Euler discretization approximates the integrals with the left-point rule

Z t+dtt

Sudu StdtZ t+dtt

pvuSudW1;u

pvtSt(Wt+dt Wt)

=p

vtdtStZs

where Zs is a standard normal random variable that has correlation with Zv.We end up with

St+dt= St+ rStdt +p

vtdtStZs:

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1.3 Process for lnSt

By Itos lemma ln St follows the diusion

d ln St =

r 1

2vt

dt +

pvtdW1;t

or in integral form

ln St+dt = ln St+

Z t0

r 1

2vu

du +

Z t0

pvudW1;u:

Euler discretization of the process forln St is thus

ln St+dt = ln St+ r 1

2vt

dt +p

vt(W1;t+dt W1;t) (10)

= ln St+

r 1

2vt

dt +

pvtdtZs:

Hence the Euler discretization ofSt is

St+dt = Stexp

r 1

2vt

dt +

pvtdtZs

:

Again, to avoid negative variances we must apply the full truncation or reectionscheme by replacing vt everywhere with v

+t or withjvtj.

1.3.1 Process for (St; vt) or (ln St; vt)

Start with the initial valuesS0for the stock price andv0 for the variance. Givena value for vt at time t, we rst obtain vt+dt from


vtdtZv

and we obtain St+dt from

St+dt = St+ rStdt +p

vtdtStZs

or from

St+dt = Stexp

r 1

2vt

dt +

pvtdtZs

:

To generate Zv and Zs with correlation , we rst generate two independent

standard normal variable Z1 and Z2, and we set Zv = Z1 and Zs = Z1 +p

1 2Z2.

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2 Milstein Scheme

This scheme is described in Glasserman [2] and in Kloeden and Platen [4] forgeneral processes, and in Kahl and Jackel [3] for stochastic volatility models.The scheme works for SDEs for which the coecients (St) and (St)dependonly on S, and do not depend on t directly. Hence we assume that the stockprice St is driven by the SDE

dSt = (St) dt + (St) dWt (11)

= tdt + tdWt:

In integral form

St+dt = St+ Z t+dt

t

sds + Z t+dt

t

sdWs: (12)

The key to the Milstein scheme is that the accuracy of the discretization isincreased by considering expansions of the coecients t = (St) and t = (St) via Itos lemma. This is sensible since the coecients are functions ofS. Indeed, we can apply Itos Lemma to the functions t and t as we wouldfor any dierentiable function of S. By Itos lemma, then, the SDEs for thecoecients are

dt =

0tt+

1

200t

2t

dt + (0tt) dWt

dt =

0tt+

1

200t

2t

dt + (0tt) dWt

where the prime refers to dierentiation in Sand where the derivatives in t arezero because we assume that t and t have no direct dependence on t. Theintegral form of the coecients at time s (witht < s < t + dt)

s = t+

Z st

0uu+

1

200u

2u

du +

Z st

(0uu) dWu

s = t+

Z st

0uu+

1

200u

2u

du +

Z st

(0uu) dWu:

Substitute for s and s in (12) to produce

St+dt = St+ Z t+dt

t t+ Z

s

t 0uu+

1

200u

2u

du +

Z s

t

(0uu) dWuds

+

Z t+dt

t

t+

Z s

t

0uu+

1

200u

2u

du +

Z s

t

(0uu) dWu

dWs

The terms higher than order one are dsdu= O

(dt)2

and dsdWu = O

(dt)3=2

and are ignored. The term involving dWudWs is retained since dWudWs =

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O (dt) is of order one. This leaves us with

St+dt = St+ t

Z t+dt

t

ds + tZ t+dt

t

dWs+Z t+dt

t

Z s

t

(0uu) dWudWs: (13)

Apply Euler discretization to the last term to obtain

Z t+dtt

Z st

0uudWudWs 0ttZ t+dtt

Z st

dWudWs (14)

= 0tt

Z t+dtt

(Ws Wt) dWs

= 0tt

Z t+dtt

WsdWs WtWt+dt+ W2t!

Now dene dYt = WtdWt. Using Itos Lemma, it is easy to show1 that Yt hassolutionYt=

1

2W2t 12 t so thatZ t+dt

t

WsdWs = Yt+dt Yt = 12

W2t+dt 1

2W2t

1

2dt: (15)

Substitute back into (14) to obtain

Z t+dtt

Z st

0uudWudWs 1

20uu

h(Wt+dt Wt)2 dt

i= :

1

20uu

h(Wt)

2 dti

:

where Wt = Wt+dt Wt, which is equal in distribution to pdtZ with Zdistributed as standard normal. Combining Equations (13) and (15) the generalform of Milstein discretization is therefore

St+dt= St+ tdt + tp

dtZ+1

20ttdt

Z2 1 : (16)

2.1 Milstein Scheme for the Black-Scholes Model

In the Black-Scholes model Equation (4) we have (St) = rSt and (St) = Stso the Milstein scheme (16) is

St+dt= St+ rStdt + Stp

dtZ+1

22dt Z

2 1which adds the correction term 1

22dt

Z2 1 to the Euler scheme in (5). In

the Black-Scholes model for the log-stock price, Equation (6), we have (St) =

1 Indeed, @Y@t

= 12; @Y@W

= W, and @2Y

@W2 = 1, so that dYt =

1

2+ 0 + 1

2 11

dt+

(Wt 1) dWt = WtdWt.

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r 1

22

and (St) = so that

0

t = 0

t = 0: The Milstein scheme (16) istherefore

ln St+dt= ln St+

r 12

2

dt + pdtZ

which is identical to the Euler scheme in (7). Hence, while the Milstein schemeimproves the discretization ofStin the Black-Scholes model, it does not improvethe discretization ofln St.

2.2 Milstein Scheme for the Heston Model

Recall that this model is given in Equation (8) as

dSt = rStdt +p

vtStdW1;t

dvt = ( vt) dt + pvtdW2;t

2.3 Process for vt

The coecients of the variance process are (vt) = ( vt)and (vt) = pvtso an application of Equation (16) for vt produces


vtdtZv+1

42dt

Z2v 1

(17)

which can be written

vt+dt =

pvt+

1

2p

dtZv

2+ ( vt) dt 1

42dt:

This last equation is also Equation (2.18) of Gatheral [1]. Milstein discretizationof the variance process produces far fewer negative values for the variance thanEuler discretization. Nevertheless, the full truncation scheme or the reectionscheme must be applied to (17) as well.

2.3.1 Process for St and ln St

The coecients of the stock price process are (St) =rSt and (St) =p

vtStso Equation (16) becomes


vtdtStZs+1

4S2t dt

Z2s 1

: (18)

We can also discretize the log-stock process, which by Itos lemma follows the

processd ln St =

r 1

2vt

dt +

pvtdW1;t:

The coecients are (St) =

r 12

vt

and (St) =p

vt so that 0

t = 0

t = 0.Sincevt is known at time t, we can treat it as a constant in the coecients. An

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application of (16) produces

ln St+dt = ln St+

r 12

vt

dt +p

vtdtdZs

which is identical to Equation (10). Hence, as in the Black-Scholes model, thediscretization ofln St rather than St means that there are no higher correctionsto be brought to the Euler discretization. The discretization of the stock priceis

St+dt = Stexp

r 1

2vt

dt +

pvtdtZs

: (19)

Again, it is necessary to apply the full truncation or reections schemes inEquations (18) and (19).

2.4 Process for(St; vt)

or(lnSt; vt)

Given a value for vt at time t, we rst update to vt+dt using (17)


vtdtZv+1

42dt

Z2v 1

and we obtain St+dt using


vtdtStZs+1

4S2t dt

Z2s 1

:

or from

St+dt = Stexp

r 1

2vt

dt +

pvtdtZs

:

To generate Zv and Zs with correlation , we rst generate two independentstandard normal variable Z1 and Z2, and we set Zv = Z1 and Zs = Z1 +p

1 2Z2.

References

[1] Gatheral, J. (2006) The Volatility Surface: A Practitioners Guide. NewYork, NY: John Wiley & Sons.

[2] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering.New York, NY: Springer.

[3] Kahl, C., and P. Jackel (2006). Fast Strong Approximation Monte-Carlo

Schemes for Stochastic Volatility Models. Quantitative Finance, Vol. 6, No.6.

[4] Kloeden, P.E., and E. Platen (1992). Numerical Solution of Stochastic Dif-ferential Equations. New York, NY: Springer.

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Euler and Milstein Discretization