Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation

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108 4 Numerical Issues of Stochastic Volatility Models

of spot volatility to be the average of some ATM short-term volatilities for a given

smile surface. Figure (4.1) and Figure (4.2) give two pictures about how large the

calibrated spot volatilities (variances) deviate from the corresponding ATM short-

term volatilities (variances) in the Sch

obel-Zhu model (the Heston model) between

October 2008 and March 2009. We can observe that in most time the calibrated spot

volatilities and variances are rather close to their ATM short-term counterparts ex-

cept for the time period between the middle December 2008 and the middle January

2009. The test time period in Figure (4.1) and Figure (4.2) begins directly after the

Lehman Brothers crisis and undergoes a strong turbulence in the FX markets, which

is reﬂected in the erratic jumps of the calibrated volatilities and the ATM short-term

volatilities.

4.8 Markovian Projection

Above we have discussed the calibration methods based on parameter optimiza-

tion, and model parameters are optimized in terms of mean square price errors over

all strikes and maturities. In recent literature, the so-called Markovian projection

method (see Antonov and Misirpashaev (2006), Piterbarg (2006)) is applied to cali-

brate stochastic volatility model, particularly the Heston-like models. The basic idea

of the Markovian projection method is to mimic the marginal distribution at cer-

tain time t of an It

o process with a simple and deterministic distribution by match-

ing their ﬁrst and second moments. The resulting new mimicking process is often

Markovian, therefore, this mimicking approach is referred to as the Markovian pro-

jection. The Gy

ongy theorem (1986) delivers the essential theoretical background

of the Markovian projection.

Theorem 4.8.1. Gy

ongy theorem: Assume an It

o process x(t) of the following form

dx(t)=

(x,t)dt +

(x,t)dW (t). (4.33)

Deﬁne a(t, x) and b(t,x) by

a(t, x)=E[

(x,t)|x(t)=x]

and

(t, x)=E[

(x,t)|x(t)=x],

then the new process given by

dy(t)=a(t, y)dt + b(t, y)dW(t), y

= x

, (4.34)

is a week solution to x(t) so that both processes have the same one-dimensional

distributions.

The new process of y(t) is known as the local volatility process in ﬁnancial liter-

ature, and b(t, x) as local volatility or effective volatility. It is also known, as shown

4.8 Markovian Projection 109

in Dupire (1997), that European-style option prices are uniquely determined by the

terminal distribution at maturity T. Therefore, if the terminal distribution at time T

of the stock price process x(T) in a stochastic volatility model is accurately mim-

icked by another simple one y(T), the European-style option prices with maturity

T of stochastic volatility model is then also accurately approximated by the (Black-

Scholes) prices using the new simple (Gaussian) distribution. The process y(t) is

then labeled as the Markovnian projection of the process x(t).

In the following, we ﬁrst consider the Heston model that is often used as a

volatility-driving process, to illustrate the Markovian projection in literature. Re-

call the Heston model as follows:

dx(t)=



r −

V(t)



dt +



V(t)dW

(t),

dV(t)=

(

−V(t))dt +



V(t)dW

(t).

These two SEDs admit obviously unique solution. Deﬁne a(t,x) and b(t, x) by

a(t, x)=E[r −

V(t)|x(t)=x]

and

(t, x)=E[V(t)|x(t)=x],

the new process of the following form:

dy(t)=a(t,y)dt + b(t, y)dW

(t), y

= x

, (4.35)

has a consistent one-dimensional distribution at time t as x(t) in the sense of ﬁrst and

second conditional moments. In this special case, since r is constant, the expected

value of E[r −

V(t)|x(t)=x

] can be simpliﬁed to

a(t, x)=r −

E[V(t)|x(t)=x]=r −

(t, x).

It becomes clear that mimicking the log return process x(t) in the stochastic volatil-

ity model is essentially equivalent to calculating expected value of V (t) conditional

on x(t).

Generally, computing the conditional expected ﬁrst and second moments is not

trivial, and in practice, raises a big numerical challenge, especially in more com-

plicated stochastic volatility models, for example, local-stochastic volatility mod-

els where local volatility and stochastic volatility are combined. One method is

to compute this conditional expected values in a Kolmogorov’s forward PDE for

(x(t),V(t)), and is numerically extensive. Piterbarg (2006) proposed various meth-

ods such as the Gaussian approximation, parameter averaging as well as local

volatility approximation, to calculate the conditional expected ﬁrst and second mo-

ments. Due to its analytical tractability, the Gaussian approximation is simple and

may be easily implemented. We take again the Heston model for example. Since W

110 4 Numerical Issues of Stochastic Volatility Models

and W

are both Gaussian, it follows

(t, x)=E[V(t)| x(t)=x]

= E[V (t)] +

Cov < x(t),V (t) >

Var(x(t))

(x(t) −E[x(t)]),

therefore it is straightforward to obtain the expected value of V (t) conditional on

x(t),

(t, x)=A

(t)+A

(t)[x(t) −x

−rt +

E[V(t)]] (4.36)

with

(t)=



(

−V(s))]ds =

κθ

t −



E[V(s))]ds

κθ

t −



[

+(V (0)−

−

]ds

−V(0))(1 −e

−

)

and

(t)=

ρσ





V(s)]A

(s)ds



(s)ds

Another simple example of the Markovian projection is to mimic some skew

models with the displaced diffusions. Since the displaced diffusion model admit

an analytical pricing formula for European-style options, the Markovian projection

with the displaced diffusion allows for an potential convenience in estimating pa-

rameters. Consider the following process

dx(t)=

(t)dW (t), (4.37)

where

(t) is some function to capturing volatility skew (not smile). The new mim-

icking process y(t) is proposed to have displaced diffusion

y(t)=[1 +

y(t)q(t)]

(t)dW(t), (4.38)

where

y(t)=y(t) −y(0) and

(t) is a deterministic function to ﬁt possible term

structure of volatility in x(t). q(t) controls the degree of skew. By the Gy

ongy theo-

rem, the matching condition is given by

(t)

[1 +

y(t)q(t)]

≈ E[

(t)

|x(t)=x]=b

(t, x).

Minimizing the L

distance between

(t)

[1 +

y(t)q(t)]

and b

(t, x) yields

(t)

= E[

(t)

] (4.39)

and

4.8 Markovian Projection 111

q(t)=

(t)

(t)]

2E[

(t)

]E[

(t)]

. (4.40)

The quality of the Markovian projection via a displaced diffusion depends on the

form of

(t) and get worse for highly heterogeneous cases. Furthermore, it can not

capture volatility smile. We will discuss an application of the Markovian projection

to a more complicated local-stochastic model of Piterbarg (2003) in the context of

Libor Market Model in Chapter 11.

The main drawback of the Markovian projection, however, is its limitation to

a ﬁxed time. In other words, with the Markovian projection method, we can only

approximate the terminal distribution of the original process by other projected pro-

cess, and estimate the model parameters of the projected model for a ﬁxed maturity.

As a result, it is not clear if the mimicking process with estimated parameters ex-

hibits the same (or at least similar) dynamics at any time as the original process.

The answer seems to be a no. As mentioned by Piterbarg (2006), “the prices of

securities dependent on sample of S(t) at multiple times, such as barriers, Ameri-

can options and so on, are different between the original model and the projected

model.” As discussed above, the goal of a calibration is not only to target the prices

of European-style options, but also allows us to value exotic derivatives soundly

with the calibrated model. The potential discrepancy between the original and pro-

jected processes is difﬁcult to gauge, and therefore restricts the Markovian projec-

tion to wide applications in practice.

Chapter 5

Simulating Stochastic Volatility Models

In this chapter we address how to simulate stochastic volatility model. After a

stochastic volatility model is calibrated to market smile surface, we can use the

calibrated stochastic volatility parameters to simulate the prices of the underlying

asset, and then to value exotic products.

The mean-reverting square root process is a very popular process and widely used

to model stochastic interest rates, for instance, by Cox, Ingorsoll and Ross (1985),

Dufﬁe and Kan (1996), and to model stochastic variances by Heston (1993). There-

fore, an efﬁcient and accurate simulation of the mean-reverting square root process

is of a great importance in ﬁnance. On the other hand, due to the non-trivial distri-

bution and the negative values in the simulation of the mean-reverting square root

process, its robust simulation remains the middle point of computational ﬁnance

since long time. Hence we will pay more attentions on the numerical issues of the

mean-reverting square root process. In particular, the negative simulated values and

the related calculation of the square root of negative values are a big challenge for

practitioners. In this chapter, some approaches for an efﬁcient and robust simulation

of the mean-reverting square root process are addressed. According to numerical

experiments, the so-called QE scheme proposed by Andersen (2007) and the trans-

formed volatility scheme proposed by Zhu (2008) perform efﬁciently and robustly.

Both schemes can be easily implemented. Additionally, from the viewpoint of prac-

tical applications, the simulation of the maximum and minimum of asset prices is

often necessary in valuing exotic options, for example, barrier options and look-

back options. Basket options and correlation options are a class of actively traded

exotic options. To value them, we need to set up a multi-asset model. In the last

chapter, we consider a multi-asset model with stochastic volatilities, and propose an

intuitive positive semi-deﬁnite correlation matrix for the multi-asset model. A cor-

responding simulation method for the multi-asset case with stochastic volatilities is

also given.

J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer Finance,

DOI 10.1007/978-3-642-01808-4

 Springer-Verlag Berlin Heidelberg 2000, 2010

113

114 5 Simulating Stochastic Volatility Models

5.1 Simulation Scheme

5.1.1 Discretization

The simplest discretization scheme is an Euler scheme. Given a time partition

= 0,t

,··· ,t

H−1

}, according to the Euler scheme, the Heston model and

the Sch

oble-Zhu model can be discretized respectively as follows:

The Heston Model:

S(t

h+1

)=S(t

)[1 + r(t

)



V(t

)

)],

V(t

h+1

)=V(t

(

−V(t

))



V(t

)

). (5.1)

The Sch

oble-Zhu model:

S(t

h+1

)=S(t

)[1 + r(t

)

+ v(t

)



)],

v(t

h+1

)=v(t

(

−v(t

))



). (5.2)

Here

is the time step between t

h+1

and t

. Z

) and Z

) are two correlated

standard normally distributed random variables. They may be drawn at time t

ac-

cording to the following methods,

)=Ran

, Ran

∼ N(0,1),

Ran



1 −

Ran

, Ran

∼ N(0,1).

The random variables Ran

and Ran

are two independent standard normally dis-

tritbuted and may be created subsequently by a same random generator. Usually, the

most efﬁcient random generating algorithms produce only uniform random vari-

ables for interval [0,1]. Hence, we need to produce the Gaussian random variables

from the uniform random variables. To this end, some efﬁcient and quick algorithms

are available including the Box-Mueller algorithm and the inverse-transform algo-

rithm. For detailed treatments on this topic, see Glasserman (2004).

To avoid possible negative values of S(t

) in a simulation, we may simulate

X(t)=lnS(t) instead of S(t). The corresponding discretization of X(t) is given

by for the Heston model,

X(t

h+1

)=X(t

)+[r(t

) −

V(t

)]



V(t

)

and for the Sch

oble-Zhu model,

X(t

h+1

)=X(t

)+[r(t

) −

)]

+ v(t

)



The simulated value of S(t

h+1

) is then equal to exp(X(t

h+1

5.1 Simulation Scheme 115

If we simulate S(t) directly, another discretization scheme called the Milstein

scheme is available and improves the Euler’s scheme by adding a term

Di f f (S(t),t)Di f f



(S(t(,t)[Z

(t)

−1]

where Di f f and Di f f



denote the diffusion term and its derivation to S(t) of the

process S(t). The mean-reverting square-root process V(t) has also a Milstein dis-

cretization. The corresponding discretization for the Heston model and the Sch

oble-

Zhu model are given as follows:

The Heston Model:

S(t

h+1

)=S(t

)[1 + r(t

)



V(t

)

V(t

)(Z

) −1)

V(t

h+1

)=V(t

(

−V(t

))



V(t

)

) −1)

. (5.3)

The Sch

oble-Zhu model:

S(t

h+1

)=S(t

)[1 + r(t

)

+ v(t

)



)(Z

) −1)

v(t

h+1

)=v(t

(

−v(t

))



). (5.4)

However, the above deﬁned Milstein scheme has a convergence of the same week

order as the Euler scheme, namely a week order of one.

5.1.2 Moment-Matching

Note the stochastic volatility v(t) follows a mean-reverting Ornstein-Uhlenbeck pro-

cess and is Gaussian, we can utilize this property to arrive at a more efﬁcient dis-

cretization. According to (3.29), conditional on v(t

), v(t

h+1

) is distributed by a

Gaussian law,

v(t

h+1

) ∼ N



),m

)



(5.5)

with

+(v(t

) −

−

κΔ

(1 −e

−2

κΔ

Therefore, v(t) can be simulated as follows

v(t

h+1

)=m



). (5.6)

116 5 Simulating Stochastic Volatility Models

This simulation of v(t) does not have a discretization error. The only discrepancy

from the true process is due to the random variables Z

). Additionally, it allows

for large time steps

without losing simulation quality. Hence, we need only a few

number of time steps to simulate v(t) and achieve remarkable efﬁciency.

5.2 Problems in the Heston Model

This section focuses on a special problem of the Heston model: how to achieve a

robust and accurate simulation for a mean-reverting square root process. More pre-

cisely, the problem is that the possible negative samples for V (t) and the related

squared root



V(t) break down the simulation. This is a problem that many ﬁ-

nancial engineers have tried to solve since more than a decade. Meanwhile, many

approaches are proposed to deal with this problem with some successes. In the fol-

lowing, we ﬁrst discuss the reasons why the negative values of V(t) in a simulation

occur. Then we present some simulation schemes which are especially tailored to do

an accurate, efﬁcient and robust simulation of a mean-reverting square root process.

According to my own experiences, the scheme labeled to QE suggested by Ander-

sen (2007) and the transformed volatility scheme suggested by Zhu (2008) are two

of the most efﬁcient simulation algorithms for the Heston model. Of course, these

schemes can be applied to the CIR (1985) interest rate model without any restriction.

5.2.1 Negative Values in Paths

As discussed early, Feller’s conditions of the Heston model for the positivity of

variance V(t) are the following restrictions of parameters,

κθ

, V

> 0.

An unconstrained calibration of the Heston model does not guarantee the above con-

dition and leads often to a prior possible negative values in the simulation of V(t).

This makes the calculation of the square root



V(t

)

impossible and breaks down

the simulation. Therefore, a constrained calibration with an embedded parameter re-

striction is necessary, and however, sacriﬁces some ﬁtting quality.

But even the required parameter restriction is satisﬁed in the Heston model, it is

still possible to generate some negative values of V(t

) in paths in practical simula-

tions. If the generated Z

) is negative enough, so that

(

−V(t

))



V(t

)

) < 0,

we obtain V(t

h+1

) < 0. More mathematically, even when V(t

) > 0 and the Feller

condition is satisﬁed, the probability of V (t

h+1

) becoming negative in a simple Euler

scheme is given by

5.2 Problems in the Heston Model 117

P(V(t

h+1

) < 0)=N



−(1−

κΔ

)V(t

) −

κθΔ



V(t

)



since the Euler scheme practically replaces the non-central chi-square distribution

of V (t) by a normal distribution. This ﬁnding is sometimes more frustrating than the

parameter constraints. This simple and naive approach to avoiding the computation



V(t

)

is to truncate V(t

) to V

)=max[V(t

),0]. The standard Euler

scheme then becomes

S(t

h+1

)=S(t

)[1 + r(t

)



)

)],

V(t

h+1

)=V(t

(

−V(t

))



)

), (5.7)

which is called the (partial) truncation scheme. Numerical examples show that the

truncation scheme is not robust and generates some largely biased paths. Therefore,

the truncation scheme is not applicable for pricing options.

Approximating a mean-reverting squared-root process with a process or a distri-

bution without the term



V(t

) seems to be the only way to overcome this problem

encountered in the Heston model.

5.2.2 Log-normal Scheme

The idea of the log-normal scheme is straightforward and approximates the condi-

tional non-central chi-square distribution of V(t) by a log-normal distribution, and

the resulting paths are then always positive. The approximation is performed by

matching ﬁrst two moments of non-central chi-square distribution and log-normal

distribution. Let y(t)=lnY (t) follow a Brownian motion with drift,

dy(t)=

dt +

dW(t).

Y(t) is then log-normally distributed with the following mean and variance respec-

tively,

(s)=E[Y(t)|Y (s)] = Y(s)e

(

)(t−s)

Var

(s)=Var[Y (t)|Y(s)]

= Y

(s)



(t−s)

−1



)(t−s)



(t−s)

−1



Setting E

and Var

equal to the true conditional mean and variance of V (t) yields

V(s)e

(

)(t−s)

= E

(s)=

+(V (s)−

−

(t−s)

118 5 Simulating Stochastic Volatility Models



(t−s)

−1



(s)=Var

(s)=



1 −e

(t−s)



+ V (s)

−

(t−s)



1 −e

(t−s)



which can be rearranged to

(s)=V(s)e

(

)(t−s)

Var

(s)=



(t−s)

−1



We can solve

and

from the above equations and obtain

(s)=

t −s



Var

(s)

+ 1



, (5.8)

(s)=

t −s

(s)

V(s)

−

. (5.9)

Based on the approximated log-normal distribution, a new discretization scheme is

given by

V(t

h+1

)=V(t

)exp



)



)



. (5.10)

5.2.3 Transformed Volatility Scheme

As indicated above, the negative values in paths are caused by two reasons. First

one is the unfortunate parameter restriction that allows a negative variance V(t

Second one lies in the nature of the simulation that can not exclude the negative

variance V(t

). Both problems break down the calculation of



V(t

).Iftheterm



V(t

) vanishes, we do not need to care the negative values of V(t

) any more. For

this reason, it is natural to consider v(t)=



V(t

). As mentioned in Chapter 3, we

can derive the process v(t) by It

o’s lemma, and recall it,

dv(t)=



(

−

−1

(t) −v(t)



dt +

(t).

It is straightforward to discretize v(t) and obtain a total new Euler scheme for the

Heston model based on v(t),

X(t

h+1

)=X(t

)+[r(t

) −

)]

+ v(t

)



v(t

h+1

)=v(t



(

−

−1

) −v(t

)





(5.11)