efficient mc simulation with stochastic volatility

Efficient Monte Carlo Simulation with Stochastic Volatility

T O R B J Ö R N O D E L M A N

Master of Science Thesis Stockholm, Sweden 2009

Efficient Monte Carlo Simulation with Stochastic Volatility

T O R B J Ö R N O D E L M A N

Master’s Thesis in Numerical Analysis (30 ECTS credits) at the School of Engineering Physics Royal Institute of Technology year 2009 Supervisor at CSC was Anders Szepessy Examiner was Michael Hanke TRITA-CSC-E 2009:020 ISRN-KTH/CSC/E--09/020--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.csc.kth.se

Abstract

Monte Carlo simulation is a powerful aid in many fields.In this thesis it is used for pricing of financial derivatives.Achieving accurate results with Monte Carlo is rather time-consuming due to its slow convergence. However, there areways to improve the accuracy of each simulation, for in-stance by reducing the inevitable discretization bias. Inour financial context, this concerns the discretization of thestochastic processes governing the underlying asset. Specif-ically, we consider assets following Heston’s stochastic volatil-ity model. We examine different discretization schemeswith the aim of an efficient implementation of the Hestonmodel in a generic Monte Carlo engine.

It turns out to be quite demanding to outperform thesimple Euler scheme. In fact, we did not find any reason touse anything but (a slightly modified) Euler.

Referat

Effektiv Monte Carlo-simulering med

Stokastisk Volatilitet

Monte Carlo-simulering är ett kraftfullt hjälpmedel inommånga områden. I det här examensarbetet används detför prissättning av finansiella derivat. Att nå precisa re-sultat med Monte Carlo är tidskrävande på grund av desslångsamma konvergens. Det finns dock metoder för att göravarje simulering noggrannare, till exempel genom att reduc-era det oundvikliga diskretiseringsfelet. I det här samman-hanget berör det diskretiseringen av de stokastiska pro-cesser som styr den underliggande aktien. Vi använder isynnerhet Hestons volatilitetsmodell för den underliggande.Genom att undersöka olika diskretiseringsmetoder strävarvi efter en effektiv implementering av Hestons modell i engenerisk Monte Carlo-motor.

Det visar sig svårt att överträffa den simpla Euler--diskretiseringen. Vi ser ingen anledning att gå ifrån (ennågot modifierad) Euler.

Acknowledgements

First and foremost, I would like to thank my supervisors at SunGard Front Arena,Dr. Jonas Persson and Gustaf Sterner, for their never-ending encouragement andguidance throughout this project. I would also like to thank Annika Mårdfalk andHåkan Norekrans for giving me the opportunity to perform this thesis project atFront Arena.

Finally, I would like to express my gratitude to Prof. Anders Szepessy at theRoyal Institute of Technology for his indispensable expertise in the subject and thevaluable comments on my report.

Contents

1 Introduction 1

2 Stochastic Volatility 3

2.1 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . 3

3 The Heston Stochastic Volatility Model 7

3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . . . . 7

3.1.2 The Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Option Pricing in Heston . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Alternative Computation . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Pricing with Fast Fourier Transform . . . . . . . . . . . . . . 12

4 Monte Carlo Simulation 13

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Discretization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.1 Full Truncation Euler Scheme . . . . . . . . . . . . . . . . . . 15

4.2.2 Milstein Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.3 IM-ĲK Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.4 Exact Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.5 Quadratic Exponential Scheme . . . . . . . . . . . . . . . . . 16

4.2.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Implementation in PRIME 21

5.1 User-Defined Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2 Pricing as in B&S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Efficiency 25

6.1 Performance of Monte Carlo Simulation . . . . . . . . . . . . . . . . 25

6.2 Bias and Variance Results . . . . . . . . . . . . . . . . . . . . . . . . 26

6.2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 RMSE Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.4 Path-dependent Options . . . . . . . . . . . . . . . . . . . . . . . . . 326.4.1 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Price Sensitivity 35

7.1 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.4 Alternative Methods for Greeks . . . . . . . . . . . . . . . . . . . . . 43

7.4.1 Pathwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.4.2 Likelihood Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 Conclusion 45

Bibliography 47

Appendices 50

A Convergence Properties 51

A.1 Efficiency of Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.2 Convergence Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A.2.1 Weak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.2.2 Strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B Additional Price Sensitivity Figures 53

Chapter 1

Introduction

Despite that the drawbacks of the Black and Scholes option pricing model (B&Shenceforth) are well known and empirically supported, it has stood out as themost widely used model for many years. However, there has been an ambitionto replace B&S with a more realistic model. Strong candidates for this purposeare those which model the underlying asset’s volatility as a stochastic process, i.e.stochastic volatility models. This approach has the cost of more intricate pricingand calibration, while B&S excels with straightforward calulations and closed-formsolutions.

In this thesis we will examine the most popular stochastic volatility model, in-troduced by Heston in 1993. The output we are seeking is to implement the Hestonmodel in a generic Monte Carlo engine in Front Arena’s trading application PRIME.Despite the generic structure, the main focus for this thesis is to apply the Hestonmodel to equity options. The Monte Carlo implementation is done in a C++ envi-ronment within the Front Arena software. All other calculations, e.g. all figures,are made in Matlab.

Chapter 2 gives a more detailed survey of the disadvantages of B&S and intro-duces some stochastic volatility models. In Chapter 3 we take a closer look atthe features of the Heston model and its challenges, such as vanilla option pricingand calibration. We then move on to the numerical implementation in Chapter 4,where we investigate various kinds of discretizing methods of the Heston model.The methods we consider most feasible are implemented in PRIME. This is brieflydescribed in Chapter 5, where a few pricing examples are made as well. In Chapter6 these methods undergo some standard performance tests, where we give somesuggestions of efficiency improvements as well. Chapter 7 explores the difficultiesthat comes with estimating sensitivities of option prices with respect to model pa-rameters, especially the underlying asset. Chapter 8 summarizes the thesis andgives some suggestions for future research.

1

Chapter 2

Stochastic Volatility

2.1 The Black-Scholes Model

The B&S model makes the assumption of an underlying asset, S, following a geo-metric Brownian motion under some probability measure P;

dSt = µStdt+ σStdWt, (2.1)

where the asset’s drift µ and volatility σ both are assumed constant. This meansasset price changes are log-normal distributed, which is a well known drawback withB&S as log-returns observed in markets are typically non-Gaussian, with fatter tailsand a high peak. This motivates the desire of modeling volatility as stochastic, see[13]. To further enlighten this problem, B&S shows an inconsistency with marketsin terms of implied volatility. Let CBS(S,K, T, σ, r) denote the B&S price for aEuropean call option with time to maturity T and strike price K. Further, letS be the value of the underlying asset, σ its volatility and r the risk-free interestrate. The implied volatility I is the volatility that makes the B&S price match themarket-observed price,

CBS(S,K, T, I, r) = CMarket(K,T ). (2.2)

By calculating implied volatilities from market data, one observes that it is notconstant but varies with strike price (typically forming a skew or a smile) and timeto maturity. Implied volatilities also move over time in a stochastic manner. Thiscontradicts the assumptions of constant volatility in B&S. Stochastic volatility canresolve this problem and reproduce satisfying implied volatility surfaces.

2.2 Stochastic Volatility Models

The essence with stochastic volatility models is that volatility itself is a stochasticprocess. In the general form, a single underlying asset, S, and the squared volatility

3

CHAPTER 2. STOCHASTIC VOLATILITY

V , is modeled by the following two-dimensional stochastic differential equation:

dSt = µtStdt+√

VtStdW1t (2.3)

dVt = α(S, V, t)dt+ ηβ(S, V, t)√

VtdW2t (2.4)

dW1dW2 = ρdt, ρ ∈ [−1, 1] (2.5)

where ft denotes f(t). There have been several suggestions for the modelling of thevariance process Vt, e.g. the one by Hull and White [18]

dVt = Vt(αdt+ βdW 2t ). (2.6)

Another plausible model is the continuous-time limit of a GARCH series, conse-quently called a GARCH diffusion. It has the following dynamics for the variance:

dVt = κ(θ − Vt)dt+ σVtdW2t . (2.7)

However, the most widely used, and likewise the one we will focus on, is presentedby Heston [16]:

dVt = κ(θ − Vt)dt+ η√

VtdW2t . (2.8)

An extension to Heston is the Bates model [2], which adds a Poisson process toincorporate discontinuous jumps in the market. The asset price process then takesthe form

dSt = (r − λµJ)dt+√

VtdW1t + JtdXt

where Xt ∼ Po(λt) and the jump size Jt follows1

log(1 + Jt) ∼ N(

log(1 + µJ)−σ2J

2, σ2J

)

.

All these methods have their advantages and drawbacks. Equation (2.6) does nothave mean reverting volatility, which is prefered for most purposes, see [13]. Com-paring (2.7) and (2.8), one observes that the Heston model only differs from theGARCH diffusion by a square root factor of the variance. However, as the first oneprovides a closed-form solution for European call option prices (to be introduced inChapter 3), the latter does not. The main reasons for the popularity of Heston are,in brief:

• Volatility is mean-reverting.

• It allows non-Gaussian log-returns.

• It yields implied volatilities reasonably consistent with market values.

1Xt ∼ Po(λt) means that jumps occur randomly and independently of each other. The expectednumber of jumps per time unit is λ.

4

2.2. STOCHASTIC VOLATILITY MODELS

• The existence of a semi-analytical solution for European Options.

For the interested reader, more stochastic volatility models are discussed by A.Lewis [21].

At this point, it should be said that pricing financial derivatives with stochasticvolatility is an incomplete problem. In the B&S framework, the randomness of anoption is solely due to the randomness of the underlying asset, which is tradable.This means that any option can be hedged with the underlying and a risk-free asset,and the market model is said to be complete. In stochastic volatility models, how-ever, with the addition of a stochastic process of the volatility there are two sourcesof randomness. But the volatility is not a tradable asset which makes hedging onlywith the underlying and the risk-free asset impossible. As for the Heston model,this topic will be handled further in Section 3.1.1.

5

Chapter 3

The Heston Stochastic Volatility Model

In this chapter we introduce the Heston model and discuss some of its challenges,including parameter estimation and vanilla option pricing.

3.1 The Model

First, recall from the previous chapter that the underlying asset’s price S and vari-ance V have the following dynamics under some probability measure P:

dSt = µStdt+√

VtStdW1t , (3.1)


VtdW2t . (3.2)

As before, dW 1t dW

2t = ρdt , with ρ ∈ [−1, 1]. Equation (3.2) is a version of the

square root mean reverting process first introduced by Cox et al., [11].

3.1.1 Risk-Neutral Measure

As we mentioned in the end of Section 2.2 Heston is an incomplete model. Com-pleteness means that there exists a unique risk-neutral probability measure Q andonly one fair price of a certain derivative.1 Finding Q is equivalent to determiningthe market price of risk, denoted Λ, which can be seen as a measure of the riskaversion in the market. In a complete market, the value of Λ is constant no matterwhat option or asset we are concerned with. However, dealing with an incompletemodel, Heston assumes that there is a market price of volatility risk proportionalto the volatility (i.e. the square root of variance),2

Λ(St, Vt, t) = k√

Vt for some constant k

=: λ(St, Vt, t).

1see Second Fundamental Theorem of Asset Pricing, e.g. [24] p. 232.2This is justified in [11].

7

CHAPTER 3. THE HESTON STOCHASTIC VOLATILITY MODEL

With λ(St, Vt, t) introduced and assuming a deterministic interest rate r, it is nowpossible to move over to the risk-neutral measure Q with the aid of Girsanov’stheorem3

dW 1t = dW 1

t +µ− r√Vtdt

dW 2t = dW 2

t + λdt

dQ

dP= exp

− 1

2

∫ t

0

(

µ− r√Vs

)2

+ λ(Ss, Vs, s)2ds−

∫ t

0

µ− r√Vs

dW 1s −

∫ t

0λ(Ss, Vs, s)dW

2s

Equations (3.1)-(3.2) then, under Q, become

dSt = rSt +√

VtStdW1t


VtdW2t

dW 1t dW

2t = ρdt,

where

κ = κ+ kη, θ =κθ

κ+ kη.

Thus, λ(St, Vt, t) and µ are "eliminated" under Q and incorporated by the newBrownian motions. Nevertheless, Q is not unique, it depends on the value of λ andhence different λ values produce different option prices. Fortunately, pricing andcalibration, which is done under Q, will not be affected by this problem in practice.This is due to the model’s parametric nature and the existence of a closed-formsolution for vanilla options (to be introduced in Section 3.2). From now on, wewill only work in the risk-neutral world and therefore we simplify the notation bydropping all ˜signs. Thus, under Q, we write

dSt = rSt +√

VtStdW1t (3.3)


VtdW2t (3.4)

dW 1t dW

2t = ρdt. (3.5)

3.1.2 The Parameters

Now that we have established the risk-neutral dynamics of the Heston process, wetake a look at the parameters of the model. As λ(St, Vt, t) and µ have "vanished"under Q we are only left with these 4 parameters,

• θ, the long-term mean variance. Typically the value is in proximity to theimplied volatility for long-term at-the-money options,

3see [24] p. 212.

8

3.1. THE MODEL

• κ, the speed of mean reversion, i.e. how fast V is moving towards θ,

• η, often refered to as the "vol-vol", indicating how volatile the variance is, and

• ρ, the correlation between the two Browninan motions W 1t t≥0 and W 2

t t≥0.

Of the additional values needed, St=0 and r are found in markets, while the initialvariance Vt=0 =: v0 is usually found by calibration, see Section 3.1.3.

As we mentioned earlier, empirical studies, e.g. [10], show that there is a skew-ness in the distribution of stock returns in markets, typically volatility tends toincrease when asset prices go down. This is known as the leverage effect and itmotivates choosing ρ < 0.

The volatility of variance does not change this skewness, but it has an impacton the kurtosis of the stock return distribution. The kurtosis affects the relationbetween near-the-money options and far-from-the-money options while skewnessaffects the relation between in-the-money options and out-of-the-money options.Consequently, both these properties of the Heston model have an impact on theimplied volatility, as illustrated in figure 3.1.

3.1.3 Calibration

As stochastic volatility models are more realistic than the Black-Scholes model, theyare also, unfortunately, more complex. To price any exotic option with Heston, themodel’s parameters (κ, v0, θ, η, ρ) must be found. Bakshi et al. [4] show that time-series estimates of the parameter values rarely coincide with implied parameters(i.e. those parameter values giving correct option prices from the market).

The standard approach is instead to calibrate the model to market data, which typ-ically means minimizing the discrepancy between market prices and model pricesof vanilla options. This can be written as

minΩS(Ω) = min

Ω

N∑

i=1

wi[

CModel(Ki, Ti,Ω)− CMarket(Ki, Ti)]2, (3.6)

where CModel(Ki, Ti,Ω) is the theoretical price of a vanilla call option for some pa-rameter set Ω and with maturity Ti and strikeKi. The observed value CMarket(Ki, Ti)is the corresponding market price. Choosing proper weights wi can be done in vari-ous intricate ways or not done at all depending on the application of the calibration.However, this will not be discussed in greater detail here.

For the actual minimization in (3.6) one has a choice of either applying a local orglobal optimization method. Matlab’s function lsqnonlin (which is a non-linearleast-squares optimizer) and the Solver in Excel are examples of local optimizers.They are both quite fast but on the other hand very sensitive to the initial guess

9


70 80 90 100 110 120 13025

26

27

28

29

30

31

32

33

34

Moneyness (%)

Impl

ied

Vol

(%

)

ρ=−0.5

η=0.2η=0.4η=0.6

70 80 90 100 110 120 13028.5

29

29.5

30

30.5

31

Moneyness (%)

Impl

ied

Vol

(%

)

ρ=0

η=0.2η=0.4η=0.6

70 80 90 100 110 120 13026

27

28

29

30

31

32

33

Moneyness (%)

Impl

ied

Vol

(%

)

ρ=0.5

η=0.2

η=0.4

η=0.6

Figure 3.1. The effect of η and ρ on implied volatility with the other parametersfixed: κ = 2, θ = 0.1, v0 = 0.08, r = 3% and time to maturity T = 1. Moneynessdefined as K/S

of the parameter vector Ω. The objective function of the optimization (3.6) is non-convex and consequently, an obvious drawback with these methods is that one mightend up in a local optimum. To be certain of finding a global optimum (if existing),one has to use a global optimizer, e.g. a simulated annealing method, where theinitial guess should be irrelevant. The most widely known method of this kind isthe Adaptive Simulated Annealing (ASA) presented in [19].

The main disadvantage with global optimizers is that they are computationallyfar more burdensome than local optimizers, and with an accurate initial guess ofΩ a local optimizer could definetely be good enough. As calibration of the Hestonmodel is not a primary objective with this thesis we leave it with the conclusionthat it is an area of study with profound complexity.

An interesting property of the Heston model is that the variance is allowed toreach zero. Assuming v0 > 0, then this can be prevented by imposing the so-calledFeller condition

2κθ > η2. (3.7)

However, in practical situations, e.g. optimal calibration to market prices, this

10

3.2. OPTION PRICING IN HESTON

condition is most often left unsatisfied. Moreover, for a discrete approximation of(3.4) there is a strictly positive probability that the process can become negative,even if (3.7) holds. This problem is taken in to consideration when discretizing theHeston process in Chapter 4.

3.2 Option Pricing in Heston

One of the reasons for the popularity of the Heston model is that it provides aclosed-form solution for pricing of vanilla options. This is of great benefit whencalibrating against market prices. In [16], the following solution for the price attime t = 0 of a European call option on a non-dividend paying asset, with strike Kand maturity T , is given:4

C(S,K, T, r, v0, θ, κ, η, ρ) = K(exP1(x, v0, T )− P0(x, v0, T )) (3.8)

where

Pj(x, v0, T ) =1

2+

1

π

∫ ∞

0ℜ[ 1

iueCj(u,T )θ+Dj(u,T )v0+iux

]

du,

and x = log KFT (FT = erTS being the forward price of S), and

Cj(u, τ) = κ[

r−τ −2

η2log

(1− ge−dτ1− g

)]

(3.9)

Dj(u, τ) = r−1− e−dτ1− ge−dτ (3.10)

where

g =r−r+, r± =

β ± dη2

d =√

β2 − 4αγ, α = −u2

2− iu

2+ iju

β = κ− ρηj − ρηiu, γ =η2

2, j = 0, 1

3.2.1 Alternative Computation

A simplified method of calculating (3.8) is presented by Gatheral [13], where it isderived that the European call option price can be written as5

C(S,K, T, r, v0, θ, κ, η, ρ) = S −√SKe−rT

π

∫ ∞

0

du14 + u2

ℜ[

e−iuxψT (u− i/2)]

(3.11)

4We have followed the notation in [13] which differs slightly from the one in [16].5Actually, Gatheral makes the proof assuming zero interest rate. This has no fundamental

impact, it just simplifies the appearance of the pricing formula.

11


where ψT (u) = exp(C0(u, T )θ + D0(u, T )v0). Cj(u, τ) and Dj(u, τ) are definedearlier. As (3.11) only consists of one numerical integration it is prefered to (3.8).This will be our method of choice and an implementation of (3.11) in Matlab,with the built-in adaptive Gauss-Lobatto quadrature for the integration, will giveus our reference values for European option prices from now on.

3.2.2 Pricing with Fast Fourier Transform

If avoiding numerical integration is desired, it is worth mentioning a pricing methodbased on Fourier transforms. The Fast Fourier Transform (FFT) is an effectivealgorithm to calculate sums of the following form:

w(k) =N∑

j=1

e−i2πN

(j−1)(k−1)x(j). (3.12)

Carr and Madan [9] show that if the characteristic function of a stochastic process isknown, the price of a European call options can be expressed as a sum in the shapeof (3.12). The characteristic function of logST under the risk-neutral measure Q isdefined by

φT (u) = EQ[eiu logST ].

Gatheral [13] shows that for the Heston model,

φT (u) = eiu(logS0+rT )eC0(u,T )θ+D0(u,T )v0

where we recognize C0(u, T ) and D0(u, T ) from previous sections. The FFT tech-nique, although very fast for calculating a vast amount of prices simultaneously,needs some parameters to be carefully chosen for great accuracy. This can be arather demanding issue and therefore we abandon FFT and stay with numericalintegration of (3.11).

12

Chapter 4

Monte Carlo Simulation

4.1 Introduction

Determining the price of some exotic financial derivative, usually means facing thesituation where one must numerically compute E[f(St)] for some payoff functionf : RN → R, where S is an underlying asset described by a stochastic process,

dSt = a(St, t)dt+ b(St, t)dWt, (4.1)

where Wt is a standard Wiener process. Presuming (4.1) one can, in some cases,with standard non-arbitrage arguments, find a self-financing strategy replicating thespecific derivative. This will imply a partial differential equation (PDE) which theprice of the derivative must satisfy. To find the price, one approach is to numericallysolve the PDE using finite-difference methods.

However, if the replicating strategy demands several underlying assets, this willbe inefficient and time-consuming. Further, concerning pricing of path-dependentderivatives, i.e. payoffs not only depending on the underlying’s terminal value, theconstruction of the PDE may be complicated and solving it can be unpractical andrender complex calculations. For these situations, it is simple and convenient to sim-ulate a discrete-time approximation of (4.1) to obtain a Monte Carlo approximationof E[f(St)]. This means discretizing and simulating a large number of trajectories ofthe underlying process. For every simulated trajectory, a payoff is calculated. Theexpected payoff E[f(St)] is simply the average of these payoffs. The idea is plain,but many questions will be raised along the way.

4.2 Discretization Schemes

Recall the dynamics of the underlying asset’s price in the Heston model:

dSt = rStdt+√

VtStdW1t , (4.2)

where we assume a deterministic interest rate and zero dividends. To facilitate thediscretization it is sensible to work with the underlying asset in terms of the log-price Xt = logSt. Applying Itô’s lemma on (4.2), and together with the variance

13

CHAPTER 4. MONTE CARLO SIMULATION

process introduced earlier, the dynamics of the Heston model that we will consideris:

dXt = (r − Vt2

)dt+√

VtdW1t (4.3)


VtdW2t , (4.4)

where W 1t and W 2

t are standard Wiener-processes with correlation ρ. Now ourmain task begins, to find a proper time-discretization of (4.3) and (4.4). There arenumerous different ways to to this. The most simple method is the Euler scheme,which is easily implemented and straightforward, but gives rise to numerical prob-lems.

A basic Euler discretization of (4.3) and (4.4) would take the form:1

Xt+∆ = Xt +(

r − Vt2

)

∆ +√

Vt∆W1 (4.5)

Vt+∆ = Vt + κ(θ − Vt)∆ + η√

Vt∆W2, (4.6)

with standard ρ-correlated Gaussian variables W1 and W2, and time step ∆.2 Un-fortunately, when moving from continuous to discrete processes a serious problemarises: the V -process can now become negative (even if (3.7) holds). Specifically, ifV (t) > 0, then

P(

V (t+ ∆) < 0)

= Φ

(−(1− κ∆)V (t)− κθ∆η√

V (t)∆

)

(4.7)

where Φ is the standard normal cumulative distribution function. This probabilitywill be positive for any choice of ∆. Negative variance is not only non-intuitive, italso causes the computation of

√

V (t) to fail.

We will look at three (possible) improvements of the Euler scheme, namely FullTruncation Euler, Milstein and IM-ĲK, in search of a better discretization candi-date. Furthermore, we look at a conceptually different approach, methods relyingon the analytical properties of the variance. In [8], Broadie and Kaya present away to simulate from the exact distribution of the variance process. However, theirmethod turns out to be too complex and almost useless in practice. But an approxi-mation of the exact distribution with moment-matched techniques, the QE scheme,is fast enough and still accurate. All these methods just mentioned are presentedbelow.

1When going from continuous to discrete processes, we change the notation from, e.g., W 1t to

W1

2The correlation is accomplished with W1 ∼ N(0, 1) and W2 = ρW1 +√

1− ρ2Z, whereZ ∼ N(0, 1)

14

4.2. DISCRETIZATION SCHEMES

4.2.1 Full Truncation Euler Scheme

Common approaches to handle the variance when it becomes negative are absorption(setting V (t) = 0) and reflection (V (t) = |V (t)|). Lord et al. [22] solves this problemslightly different with their so-called full truncation scheme, which is written as:

Xt+∆ = Xt +(

r − V +t

2

)

∆ +√

V +t ∆W1 (4.8)

Vt+∆ = Vt + κ(θ − V +t )∆ + η

√

V +t ∆W2, (4.9)

where V +t = max(Vt, 0), and this will be the Euler scheme we refer to from now on.

Notice that (4.9) allows the variance to become negative with the consequence thatthe next time step will be deterministic, with drift κθ ≥ 0.

4.2.2 Milstein Scheme

There are several methods named after Milstein, the one we will consider originatesfrom [23] and with the full truncation incorporated:

Xt+∆ = Xt +(

r − V +t

2

)

∆ +√

V +t ∆W1 (4.10)

Vt+∆ = Vt + κ(θ − V +t )∆ + η

√

V +t ∆W2 +

η2

4∆(W 2

2 − 1). (4.11)

Thus, what we are looking at is the Euler scheme with an extra term for the varianceprocess V . This scheme has a higher strong order of convergence than Euler3, atopic which will be handled in Chapter 6.

4.2.3 IM-ĲK Scheme

Kahl and Jäckel [20] propose an implicit Milstein discretization for the varianceprocess together with their so called ĲK discretization for the stock process.

Xt+∆ = Xt +(

r − 1

4(V +t+∆ + V +

t ))

∆ + ρ√

V +t ∆W2

+1

2

(√

V +t+∆ +

√

V +t

)(

W1 − ρW2

)√∆ +

ηρ

4∆(W 2

2 − 1),(4.12)

Vt+∆ = Vt + κ(θ − Vt+∆)∆ + η√

V +t ∆W2 +

η2

4∆(W 2

2 − 1). (4.13)

Here, the variance stays positive provided η2 < 4κθ. Unfortunately, this is rarelysatisfied, and again a truncation is used to avoid negative values of V . This schemeis designed for situations where the correlation between the stock price process andthe variance process is strongly negative. However, it turns out to be very sensitiveto all parameters in the Heston model and shows very erratic behaviour for somesets of parameters, see e.g. [1] or [22]. Since we are not bound to any specificparameter range in this thesis, we leave the IM-ĲK scheme here.

3Convergence orders are defined in Appendix A.2

15


4.2.4 Exact Simulation

Regardless by which method a continuous process is approximated by a discreteone, it will inevitably cause a bias.4 Broadie and Kaya introduce a bias-free methodthat simulates the process from its exact distribution. Their idea is quite simple,beginning by rewriting equations (4.2) and (4.4) as:

St = Ss exp[

r(t− s)− 1

2

∫ t

sVudu+

∫ t

s

√

VudW1u

]

Vt = Vs + κθ(t− s)− κ∫ t

sVudu+ η

∫ t

s

√

VudW2u (4.14)

The algorithm for sampling from the distribution of (St, Vt) is then:

1. Generate Vt given Vs from (4.15)

2. Generate∫ ts Vudu given Vt and Vs

3. Calculate∫ ts

√VudW

2u from (4.14)

4. Generate St given∫ ts

√VudW

2u and

∫ ts Vudu

The main feature which they rely on is that given Vs, for s < t, Vt is up to ascale factor non-central chi-squared:5

Vt ∼η2(1− e−κ(t−s))

4κχ2ν

(

4κe−κ(t−s)Vsη2(1− e−κ(t−s))

)

(4.15)

Although their method is exact and looks tempting, it has limited practical use; itcontains complex and very time-consuming computations. Numerous disadvantagesare mentioned in [15] and [22]. We will not discuss the exact simulation methodfurther here, but it serves as introduction to the Quadratic Exponential scheme.

4.2.5 Quadratic Exponential Scheme

The Quadratic Exponential (QE) scheme is an algorithm specifically developedfor square-root processes such as the Heston model. Due to the complicated andsomewhat non-intuitive nature of this scheme, we will go through it rather brieflybut yet as comprehensible as possible. For all the details the reader is refered toAndersen’s paper, [1]. The main idea, however, is to find distributions that behavesimilarly to the chi-square. With the knowledge that the conditional distribution ofVt+∆ given Vt is non-central chi-squared, Andersen makes use of the two followingfeatures:

4This will be discussed further in Chapter 65see e.g. Cox et al. [11] or Andersen & Piterbarg [3] for deeper analysis of the Heston variance

process (4.4).

16


1. With a moderate or high non-centrality parameter a non-central chi-squarecan be represented by a power function applied to a Gaussian variable. Hence,for sufficiently large values of Vt, one can write

Vt+∆ = a(b+ ZV )2, (4.16)

where a and b are constants depending on Vt and ∆, and ZV is a standardGaussian random variable.

2. For low values of Vt, one can approximate the density of Vt+∆ by

P(

Vt+∆ ∈ [x, x+ dx])

≈ (pδ(0) + β(1− p)e−βx)dx, x ≥ 0, (4.17)

where δ is Dirac’s delta function, and p and β are non-negative constans.Integrating (4.17) and inverting gives

L−1(u) =

0 if 0 ≤ u ≤ p,β−1 log( 1−p

1−u) if p < u ≤ 1.(4.18)

Then, sampling is done by Vt+∆ = L−1(UV ), where UV is a uniform randomnumber.

It now remains to determine the parameters a, b, p and β and when to use which ofthe two Vt+∆ formulas. We do not go into details how Andersen handles these issues.First, to simplify the outline it is convenient to introduce the following variables:

m = θ + (Vt − θ)e−κ∆, (4.19)

s2 =Vtη

2e−κ∆

κ(1− e−κ∆) +

θη2

2κ(1− e−κ∆)2, and (4.20)

ψ =s2

m2. (4.21)

Further, with the aid of moment-matching techniques Andersen shows that:

b2 = 2ψ−1 − 1 + 2√

2ψ−1√

2ψ−1 − 1 ≥ 0, and (4.22)

a =m

1 + b2, for ψ ≤ 2, (4.23)

p =ψ − 1

ψ + 1∈ [0, 1), and (4.24)

β =1− pm

=2

m(1 + ψ)> 0, for ψ ≥ 1 (4.25)

With these constants defined, and assuming some critical level ψc ∈ [1, 2] has beenchosen, we can formulate the QE algorithm for the variance process:

17


1. Given Vt, compute m and s2 from (4.19) and ψ = s2/m2

2. if ψ ≤ ψc :

(a) Compute a and b from (4.22)

(b) Generate a standard Gaussian random number, ZV

(c) Set Vt+∆ = a(b+ ZV )2

3. else, if ψ > ψc :

(a) Compute β and p from (4.22)

(b) Generate a uniform random number, UV

(c) Set Vt+∆ = L−1(UV )

According to Andersen the exact choice of ψc is not of great importance. We lateruse ψc = 1.5 in our numerical tests. Because of the desirable correlation betweenthe asset price and the variance, an Euler-like X-process leads to flaws, if combinedwith the QE scheme for the variance, see [1]. Instead, one is suggested to look atthe properties of the exact representation of X, derived in [8]. The following schemeis proposed:

Xt+∆ = Xt + r∆ +K0 +K1Vt +K2Vt+∆ +√

K3Vt +K4Vt+∆Z, (4.26)

where Z is a standard Gaussian random variable, not correlated with the randomvariables in the V -process, and

K0 = −ρκθη

∆, K1 = γ1∆(κρ

η− 1

2

)

− ρ

η,

K2 = γ2∆(κρ

η− 1

2

)

+ρ

η, K3 = γ1∆(1− ρ2),

K4 = γ2∆(1− ρ2), γ1, γ2 ∈ [0, 1].

Possible Extension to QE

Recall that St = eXt and assume r = 0. As confirmed in [3], a continous-timeprocess of S is a martingale, i.e.

E[St+dt|St] = St. (4.27)

However, the discrete-time process (4.26), which can be rewritten as

St+∆ = SteK0+K1Vt+K2Vt+∆+

√K3Vt+K4Vt+∆Z , (4.28)

18


does not fulfill this criterion. Even if the practical relevance of this usually isinsignificant, a martingale-corrected QE-M scheme for the stock price process isnevertheless suggested by Andersen. In the correction the term K0 is replaced by

K∗0 = − logM − (K1 +K3

2)Vt,

where

M = E[

eAVt+∆ |Vt]

> 0, A = K2 +K4

2.

For the proof that this leads to that (4.27) holds, see [1]. Despite this ambitiousmodification, the benefits of QE-M compared to regular QE are reported to beminor (but observable) and, as we see it, not worth the effort implementing thealgorithm. Hence, QE-M will not be implemented.

4.2.6 Comments

Of the discretization schemes introduced in this chapter, three will be used in theforthcoming chapters: Euler, Milstein and QE. The first two are similar and easilyimplemented. QE, on the other hand, is more complex in terms of both understand-ing as well as implementing. But it is reported, [1], to perform well and thereforewe choose to investigate it further.

There are numerous ways to discretize the Heston model, more schemes can befound in e.g. [1] and [15]. Most of them, e.g. QE-M just mentioned, are mereextensions to known schemes with increasing complexity and minor gains.

19

Chapter 5

Implementation in PRIME

5.1 User-Defined Monte Carlo

Front Arena’s trading application PRIME contains a User-Defined Monte Carloengine (UDMC), which provides an opportunity to specify any desired payoff ex-pression, or use built-in expressions of common types, e.g. Asian options, cliquets,barriers. The payoff function and all necessary parameters, such as dividends andcorrelation, are specified in a simple programming language. An example of thisset-up is shown in Figure 5.1. Then, the UDMC simulates trajectories of the un-derlying asset and calculates payoffs and expected price of the specific derivative.

One of the main objectives with this thesis is to extend UDMC to support stochas-tic volatility, more specifically the Heston model. Euler, Milstein and QE are allimplemented but Euler will be considered as the primary choice, mainly due toits simplicity and its reliability found in literature (e.g. [15],[22]) and forthcomingtests. We will see more of this in Chapter 6, where we make exhaustive tests of ourimplementation, with focus on computational performance. However, a few simplepricing examples are shown here.

21

CHAPTER 5. IMPLEMENTATION IN PRIME

Figure 5.1. Example of set-up of pay-off function (Asian option)

5.2 Pricing as in B&S

As an easy way to verify that our implementation is not done completely wrong,we set η = 0 and θ = v0 in equation (3.4). This means that volatility is keptconstant and we expect the process to behave as in a B&S framework. To examinethis we make a few examples, Figure 5.2, where we show the bias for European calloptions prices between B&S and Monte Carlo-simulated Heston price and betweenB&S and the analytical Heston solution (3.11) with the parameter assumptions justmentioned.1 The Monte Carlo simulations are done with all three discretizations(Euler, Milstein and QE) for n = 64 time steps per year and N = 105 simulatedpaths. Note that, equations (3.9) and (3.10) does not allow η = 0, instead it weset η = 10−4, which is hopefully sufficiently small. As seen, the analytical solutionworks fine, but our Monte Carlo estimates tend to diverge slightly for out-of-the-money options. However, these deviations are of such a small magnitude that theyare in the range of the variance of the simulation, see Section 6.1. Thus, we concludethat this simulation is accurate enough.

1The B&S analytical solution for European call options is:

Ct(St,K, r, σ, T ) = StΦ(d1)−Ke−r(T−t)Φ(d2), where

d1 =log(St/K) + (r + σ2/2)T

σ√T

, d2 = d1 − σ√T .

Φ(x) is the cumulative distribution function of a standard Gaussian random variable.

22

5.2. PRICING AS IN B&S

50 60 70 80 90 100 110 120 130 140 150−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Strike

Pric

e di

ffere

nce

(%)

T=1, r=3%

AnalyticalEulerMilsteinQE

50 60 70 80 90 100 110 120 130 140 150−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Strike

Pric

e di

ffere

nce

(%)

T=4, r=3%

50 60 70 80 90 100 110 120 130 140 150−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Strike

Pric

e di

ffere

nce

(%)

T=1, r=10%

50 60 70 80 90 100 110 120 130 140 150−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Strike

Pric

e di

ffere

nce

(%)

T=4, r=10%

Figure 5.2. Difference from B&S price for vanilla calls when volatility is constant

23

Chapter 6

Efficiency

In this chapter we make thorough efficiency tests of our three chosen discretizationschemes for European call options. The reason for using European calls is theexistence of the semi-analytical solution introduced in Section 3.2.

6.1 Performance of Monte Carlo Simulation

When dealing with Monte Carlo estimates one encounters two separate errors. First,there is an error due to the time-discretization of the stochastic process. Let CNdenote the Monte Carlo estimated price of a European call option price. We thendefine

bias(CN ) = E[

CN − C]

(6.1)

where C is the exact option price. The bias is determined by the choice of discretiza-tion scheme and the size h of the time step in the discretization. With a scheme ofweak order of convergence β, we expect the bias to be of order O(hβ). AlthoughMilstein has a higher strong order of convergence than Euler, they are reported toboth have weak order 1, see [14].1 Even though this applies for a scheme with notruncation, equations (4.5)-(4.6), Lord et al. [22] and Higham and Mao [17] showthat modified schemes such as ours also converge. Presuming the equality of weakorder convergence between Euler and Milstein is preserved, the consequence of thisis that we actually do not expect Milstein to perform better than Euler.

Secondly, all Monte Carlo estimates are associated with a standard error oforder O(N−1/2), where N is the number of simulated paths. An equivalent way ofexpressing this is that the estimate CN has a variance of order O(N−1), where thevariance of the estimate CN is written as2

Var(CN ) = E[

(CN − E[CN ])2]

Generally, h affects the variance as well. However, assuming a sufficiently small hensures this influence to be insignificant. Unfortunately, with these two properties

1See Appendix A.2 for the definition of strong and weak order convergence.2This derives from the central limit theorem, see Appendix A.1.

25

CHAPTER 6. EFFICIENCY

affecting the performance of the Monte Carlo estimate, for a given computationalbudget, one faces a tradeoff: increasing the number of simulated paths (to minimizevariance) requires more computing time, as does decreasing the size of the time step(to minimize bias). As a consequence, it is convenient to introduce the root meansquare error, which is defined as

RMSE(CN ) =√

bias2(CN ) + Var(CN ).

We are interested in how the RMSE varies with increased computational time. Aproblem with this might be how to choose the number of simulation paths N andthe step size h. According to Duffie and Glynn [12] it is optimal to increase thenumber of time steps (decrease h), proportional to the square-root of the numberof simulation paths for a scheme with weak convergence of order 1.

Bearing in mind the convergence orders (of standard error and variance) men-tioned earlier, this procedure seems reasonable, at least for Euler and Milstein.Numerical tests of bias, standard error and root mean square error for Euler, Mil-stein and QE are done in the following sections.

6.2 Bias and Variance Results

We focus our efficiency tests on three cases of European ATM call options, each ofthem with parameters roughly suited for equity option markets, see e.g. [1]. Allparamters are listed in table 6.1. The additional parameters for the QE scheme areset to ψc = 1.5, γ1 = γ2 = 0.5 as in section 5.2.3

κ θ v0 η ρ T r

Case 1 2 0.10 0.08 0.5 -0.7 1y 3%Case 2 1 0.09 0.09 1.0 -0.3 5y 0%Case 3 1.5 0.08 0.09 0.76 -0.3 1y 3%

Table 6.1. Heston parameters for numerical tests

For the calculation of bias we fix the number of simulated paths to a large value,N = 106, to keep the standard error low. Figure 6.1 illustrates the results for cases1, 2 and 3. To exemplify the standard error, i.e. the square root of the variance,on the other hand, the time-discretization is kept fixed, 64 steps per year for Eulerand Milstein and 8 steps per year for QE. See figure 6.2.

3See [1] for a justification of these values.

26

6.2. BIAS AND VARIANCE RESULTS

100

101

102

10−2

10−1

100

101

time steps per year

Bia

s

Euler

Milstein

QE

100

101

102

10−2

10−1

100

time steps per year

Bia

s

QE

Milstein

Euler

100

101

102

10−3

10−2

10−1

100

time steps per year

Bia

s

Euler

Milstein

QE

Figure 6.1. Bias for cases 1-3

27


104

105

106

10−2

10−1

Number of simulated paths

stan

dard

err

or

EulerMilsteinQE

104

105

106

10−2

10−1


stan

dard

err

or

EulerMilsteinQE

104

105

106

10−2

10−1


stan

dard

err

or

EulerMilsteinQE

Figure 6.2. Standard error for cases 1-3

28

6.2. BIAS AND VARIANCE RESULTS

6.2.1 Comments

In terms of bias, QE is unquestionably outperforming the other two schemes in case1 for large h (i.e. few time steps per year). As the number of time steps increasesEuler and Milstein "catch up" and show a stable convergence. The alternation ofthe parameters (between cases 1,2 and 3) seems to have less effect on Euler thanMilstein and QE. This must be considered as an advantage for Euler, even thoughit is beaten by the others in some situations.

Moreover, even if QE looks promising with its precision for large values of h,there is a hidden flaw here. The algorithm of the QE scheme is slower than Eulerand Milstein, so using fewer time steps does not yield the (imaginable) correspond-ing gain in computation time. This will be exemplified in Section 6.3.

Concerning the standard error, the stability of all three schemes is obvious. How-ever, using a quasi-Monte Carlo method or applying a variance reduction techniquemay improve the convergence of the variance, but it will not affect the discretizationbias, see [14] or [7]. An example of gains in terms of variance is shown in figure6.3, where we have illustrated the standard error of the Euler scheme for antitheticvariables compared to a standard approach.4 This modification is widely used inpractice.

104

105

106

10−1


Sta

ndar

d er

ror

Standard Monte CarloAntithetic

Figure 6.3. Case 2, standard error for Euler scheme

4The idea of antithetic variables: Suppose we want to estimate E[X] and suppose we have gener-ated X1 and X2. Then Var(X) = Var

(

X1+X2

2

)

= 14Var(X1)+ 1

4Var(X2)+ 1

2Cov(X1, X2). Drawing

X1 and X2 independently gives Cov(X1, X2) = 0, but choosing them such that Cov(X1, X2) < 0will reduce the variance.

29


6.3 RMSE Results

As discussed earlier, the convergence of root mean square error is more interestingsince it, as opposed to bias and variance, captures the computational burden ofthe methods. The RMSE is estimated in the lines of [12]; for each doubling of thenumber of time steps, the number of simulations is quadrupled. Note that, since theQE-algorithm is more time consuming than the other schemes, the starting valuehQE is larger than hEuler and hMilstein. Illustrations of RMSE convergence for cases1-3 of table 6.1 is seen in figures 6.4-6.5. Case 1 shows a small advantage for QE,but not for case 2 and 3.

10−2

10−1

100

101

102

103

10−2

10−1

100

total simulation time (seconds)

RM

SE

EulerMilsteinQE

Figure 6.4. Root mean square error, case 1

30

6.3. RMSE RESULTS

10−1

100

101

102

103

10−1

100


RM

SE

10−2

10−1

100

101

102

103

10−2

10−1

100


RM

SE

MilsteinQEEuler

Figure 6.5. Root mean square error, case 2 and 3

31


6.4 Path-dependent Options

By now, we have only considered the path-independent European options, i.e. op-tions whos prices are determined by the value of the underlying at maturity. Thereare, however, many exotic options (despite the name they are becoming more andmore popular) who are path-dependent. As one might suspect, this path-dependencyhas an impact on the performance of our three discretization schemes. We will makea few tests with barrier options, a common type of path-dependent options.

6.4.1 Barrier Options

As the name indicates, a barrier option is associated with a certain barrier, andthe payoff of the option depends on the underlying’s crossing or not crossing ofthat barrier. We will look at two of the (four) basic types of barrier options, theup-and-out (UO) and the down-and-in (DI) call option. If the underlying value hitsthe barrier B before maturity the OU option is "knocked out", and hence worthless,otherwise it pays like an European call option with strike K < B. And in a similarway, the DI option is "knocked in" when hitting B, where B < K in this case. Thus,the price Vt=0 is:

V OU0 = e−rTEQ[h(T )], where

h(T ) =

max(ST −K, 0) if St < B ∀t ∈ [0, T ]0 otherwise

and

V DI0 = e−rTEQ[g(T )], where

g(T ) =

0 if St > B ∀t ∈ [0, T ]max(ST −K, 0) otherwise

Figure 6.6 shows the price convergence for increasing number of time steps for a1-year OU and a 3-year DI call option with S0 = K = 200, Heston parametersκ, θ, v0, η, ρ, r = 2, 0.1, 0.08, 0.5,−0.7, 3% and barriers BUO = 1.25 · K = 250and BDI = 0.9 ·K = 150 respectively. As seen, all three schemes behave in a similarmanner and convergence is reached for a rather large number of time steps peryear. QE is reported to converge much faster (and thus requiring fewer time steps)than Euler for plain vanilla options, [1]. However, with the path-dependency, anyadvantages seems to be set aside. Hence we do not see any benfit of using QE here.

32

6.4. PATH-DEPENDENT OPTIONS

100

101

102

103

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

time steps per year

Pric

e

Up−and−Out call option

EulerMilsteinQE

101

102

103

16

17

18

19

20

21

22

23

24

25

26

time steps per year

Pric

e

Down−and−In call option

EulerMilsteinQE

Figure 6.6. Price convergence of barrier options

33

Chapter 7

Price Sensitivity

As we have seen, finding accurate option prices with Monte Carlo simulation isachievable without too many difficulties. Prices are of course often found in mar-kets as well. However, traders are usually not satisfied with only the prices or,equivalently, implied volatilities. To get knowledge of the risk exposure of any port-folio, price sensitivities, i.e. an option price’s sensitivity with respect to a certainmodel parameter, are vital. In other words, we are interested in the partial deriva-tives of the option price, often refered to as the Greeks.

Payoff functions are usually not differentiable everywhere, which makes analyti-cal computation of derivatives unpleasant. Therefore, we will mainly study finite-difference approximations of the derivatives and by the end of the chapter mentionsome alternative approaches. We will see that, regardless of the method applied,estimating a second derivative is more intricate than estimating a first derivative.

7.1 Delta

The first order derivative of the price C of an option with respect to the underlyingasset is called the delta, and is denoted:

∆ =∂C(t, St, . . . )

∂St(7.1)

We estimate the delta of a vanilla call option by

∆C =CN (S + ǫ)− CN (S − ǫ)

2ǫ,

where we use the central difference estimation

1

2ǫ(f(x+ ǫ)− f(x− ǫ)) = f ′(x) +O(ǫ2). (7.2)

The bias of the central difference estimator ∆C can then be written as

bias(∆C) = E[

∆C −∆]

= O(ǫ2). (7.3)

35

CHAPTER 7. PRICE SENSITIVITY

ǫ is, for obvious reasons, called the shift and plays a key role when estimating greeks.The conclusion from (7.3) would be to choose ǫ as small as possible. However, wewill look at two different approaches here. This concerns the random numbers inthe simulation procedure. We are interested in estimating ∆ for a given range ofinitial stock prices, Sii=1,...,n. With this target, we distinguish the following twocases:

(i) Simulate all prices CN (Si + ǫ), CN (Si − ǫ) etc, with the same random seed,i.e., with the same random number

(ii) Prices simulated independently, i.e., with different random seeds

The bias, (7.3), is unaffected by the choice of (i) or (ii), but the variance is not. Tobe precise, the variance of the Delta estimate is

Var(

∆(N, ǫ))

=1

4ǫ2Var

(

∆N (S + ǫ)− ∆N (S − ǫ))

=1

4ǫ2Var

(

∆N (S + ǫ))

+1

4ǫ2Var

(

∆N (S − ǫ))

=

O(N−1), Case (i)O(N−1ǫ−2), Case (ii)

Thus, for case (i), the shift ǫ can be taken as small as possible to minimize thebias and this is probably the more favorable approach to approximating delta. Incase (ii), on the other hand, reducing both variance and bias of ∆ will introduce atradeoff for ǫ. This is in the lines of the discussion in Section 6.1.

7.2 Gamma

Gamma is the second order derivative w.r.t. the asset;

Γ =∂∆

∂St=∂2C(t, St, . . . )

∂S2t

The central difference for the second derivative has the form:

1

ǫ2(f(x+ ǫ)− 2f(x) + f(x− ǫ)) = f ′′(x) +O(ǫ2), (7.4)

and thus we can write

ΓC =CN (S + ǫ)− 2CN (S) + CN (S − ǫ)

ǫ2. (7.5)

The bias in this case is then

bias(ΓC) = E[

ΓC − Γ]

= O(ǫ2). (7.6)

Thus, the bias has the same order of convergence in terms of ǫ for delta and gamma,but the variance has not. Unfortunately, not even in case (i) can we get rid of thedependence of ǫ for the variance, it is typically O(ǫ−1) or even worse, see [14]. Thisimplies that we will have to be more careful when choosing ǫ for Gamma than forDelta.

36

7.3. RESULTS

7.3 Results

To investigate the effect of ǫ we study the bias ((7.3) and (7.6)), where we letcentral approximations of the analytical solution (equation (3.11)) represent thetrue values, ∆ and Γ. Note that when studying Greeks throughout this chapter, welimit ourselves to European call options. To begin with, case 3 in Table 6.1 for ATMoptions is employed. Monte Carlo estimates of delta and gamma are simulated withthe Euler scheme for both case (i) and (ii) and with QE for case (i). Case (ii) isomitted for the QE scheme because of the heavy computational burden.

Figures 7.1-7.2 illustrate bias as a function of ǫ. As seen, choosing small ǫ incase (ii) gives highly erratic results. Figure 7.3 gives a further example of how in-appropriate case (ii) is, where delta is simulated with the Euler scheme for a rangeof spot values. More examples of this can be found in Appendix B, where the de-pendence on N also shows. These studies should verify that case (i) is, withoutany doubt, the only one applicable in practice. The significance of using commonrandom numbers is also stressed in [6].

As mentioned earlier, ǫ has an impact on the variance of gamma even when simu-lating with common random numbers. Choosing a small ǫ (to reduce bias) increasesthe importance of a large N . To illustrate this, Figures 7.5-7.6 show the behaviourof gamma when increasing N for different ǫ. A comparison with Figure 7.4 pointsout the difference between delta and gamma in this sense.

Euler and QE are both used and the "true values" ∆ and Γ are shown when(aesthetically) suitable. In these tests, we have limited ourselves to the Hestonparameters from case 1 in Table 6.1.

With the theory from the previous sections in mind, the result is expected;smooth and accurate delta estimations are obtained with as small N as 103 (Figure7.4). Gamma, however, requires much larger N for satisfying accuracy and smooth-ness, at least for ǫ < 1% of the spot price. Notice as well that QE and Euler behaveidentically here. The Milstein scheme is omitted throughout this chapter becauseof its similarity with Euler.

37


10−2

10−1

100

101

102

10−4

10−3

10−2

10−1

100

ε

Bia

s(∆)

case (ii) Euler

case (i) Euler

case (i) QE

Figure 7.1. Bias of Delta. ǫ in % of spot price

10−2

10−1

100

101

102

10−5

10−4

10−3

10−2

10−1

100

101

102

103

ε

Bia

s(Γ)

case (ii) Euler

case (i) Euler

case (i) QE

Figure 7.2. Bias of Gamma. ǫ in % of spot price

38

7.3. RESULTS

50 60 70 80 90 100 110 120 130 140 1500

0.2

0.4

0.6

0.8

1

1.2

Spot price

∆

case (ii)

case (i)

Figure 7.3. Delta simulated with ǫ = 0.1 and N = 106

39


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot / Strike

∆

N = 103, ε = 0.1%

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot / Strike

∆

N = 103, ε = 0.001%

EulerQE

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot / Strike

∆

EulerQE

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot / Strike

∆

EulerQE

Figure 7.4. Delta simulated with N = 103 and N = 104

40

7.3. RESULTS

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.01

0

0.01

0.02

0.03

0.04

0.05

Spot / Strike

Γ

ε = 0.01%

"True values"EulerQE

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Spot / Strike

Γ

ε = 0.1%

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

ε = 1%

Figure 7.5. Gamma simulated with N = 104

41


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.005

0

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

N = 105, ε = 0.01%

"True values"EulerQE

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

N = 106, ε = 0.01%

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

N = 105, ε = 0.1%

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

N = 106, ε = 0.1%

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Spot / Strike

Γ

N = 105, ε = 1%

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

N = 106, ε = 1%

Figure 7.6. Gamma simulated with N = 105 and N = 106

42

7.4. ALTERNATIVE METHODS FOR GREEKS

7.4 Alternative Methods for Greeks

7.4.1 Pathwise

Until now, we have computed greeks by differentiating the estimated price of an op-tion. However, it can instead be more advantegeous to differentiate each simulatedoutcome with respect to the parameter of interest. Let Y (θ) be a discounted payoffof some option and θ one of its model parameters. If

E

[

d

dθY (θ)

]

=d

dθE[Y (θ)] (7.7)

then E[Y ′(θ)] is unbiased and Y ′(θ) is called a pathwise derivative. This generallyrequires that the discounted payoff is a continuous function of the parameter ofdifferentiation, which of course is not always satisfied. Hence, this will not bean applicable method in general. Path-dependent options, such as barrier optionsface a problem with this method, since events during the time to maturity have aconsiderable affect on the payoff. This means that

E

[

dY

dS

]

6= d

dSE[Y ]. (7.8)

One way to solve this problem is to simulate approximations of, in this case, thedelta. Thus, at the same time as the asset price process is incremented by one timestep, an approximation of the increment in delta is calculated. This will disentanglesuch theoretical problems just mentioned for the pathwise method, but will on theother hand require a greater computational effort. There are, as an alternative, afew analytical extensions that may improve the pathwise method, e.g. smoothing ofan discontinuous payoff function. The interested reader is refered to the great bookby Glasserman [14].

7.4.2 Likelihood Ratio

With the pathwise method we run into trouble when the payoff function is discontin-uous. This continuity problem is avoided with the likelihood ratio method (LRM),which differentiates probabilities rather than payoffs. We let Y = f(X), where Xis a random vector dependent on some model parameter θ and with probabilitydensity gθ. We can then write the expected discounted payoff as

E[Y (θ)] = E[f(X1, . . . , Xm)] =

∫

ℜmf(x)gθ(x)dx,

and the derivative with respect to θ as

d

dθE[Y ] =

∫

ℜmf(x)

d

dθgθ(x)dx =

∫

ℜmf(x)

dgθ(x)

dθ

1

gθ(x)gθ(x)dx =

E

[

f(X)1

gθ(X)

dgθ(X)

dθ

]

= E

[

f(X)d log gθ(X)

dθ

]

,

(7.9)

43


where the first equality holds if we can change the order of integration and differ-entiation. If this cannot be done, then LRM may fail. Assuming (7.9) holds, whichit typically does since densities (as opposed to payoffs) often are smooth functions,then

f(X)d log gθ(X)

dθ(7.10)

is an unbiased estimator of the derivative of E[Y ]. Rather than (7.9) not being satis-fied, impendiments with LRM typically concerns lack of knowledge of the density gθand too large variance produced by (7.10). Combinations of LRM and the pathwisemethod are sometimes suitable for second derivatives, where the two methods canbe used for one differentiation each. Once again, we refer to [14] for further details.

7.5 Comments

We have chosen to put larger focus on the finite-difference approach than the othertwo just mentioned. This is partly because of the easy implementation and alsofor its generality. Even though we have only made numerical tests for vanilla calloptions, the extension to other options is straightforward and obvious which is agreat advantage when considering a general framework for applications of equitycontracts.

The pathwise method is computationally more efficient than the finite-differencemethod and usually has smaller variance than LRM, and is preferred when appli-cable. However, it is generally unapplicable for second derivatives, where instead acombination of the pathwise method and LRM, as already mentioned, is favourable.

44

Chapter 8

Conclusion

A vast amount of advanced discretization methods have been presented since theintroduction of the Heston model in 1993. As we see it none of them outperformthe (full truncation) Euler scheme, at least not so extensively that it should be rec-ommended for a generic Monte Carlo engine. Thus, there is no reason to abandonEuler for something more fancy and perhaps less intuitive.

We have been looking at three different ways to implement the Heston model. Theextra term representing the difference between Milstein and Euler seem to havelittle effect on the convergence of option prices. As discussed in Chapter 6, this isbecause the two schemes are expected to have the same order of weak convergence.A difference in strong order convergence does not have any impact on the optionprices. The QE method, that was reported in [1] to perform much better thanEuler, did not fulfil our expectations although it showed fast convergence for plainvanilla options in at least one case. Concerning the Greeks in Chapter 7, there wasno clear advantage found when using QE.

It is important to point out that since the algorithm in QE is relatively slow, itshould perform equivalent to Euler (or Milstein) with less than half the number oftime steps to be considered beneficial. Even if it succeeds with this for Europeanoptions, it did not when it came to the path-dependent options in Section 6.4, wherethe path-behaviour plays a key role. One should be cautious when applying an ad-vanced model on exotic options based on its performance for vanilla options. Thisis perhaps the prime reason for implementing Euler, a method well documented andthat can be considered "safe". On the whole, the reward in accuracy found in a fewcases is not enough motivation to use QE.

One should keep in mind that no matter how accurate the discretization is, ifthe Heston parameters v0, θ, κ, η, ρ are not properly estimated/calibrated, the fi-nal result will be unreliable. In this thesis we have only considered Monte Carlosimulations with one single underlying stock. When customers employ the User-Defined Monte Carlo engine in reality, they will certainly most often utilize it with

45

CHAPTER 8. CONCLUSION

several underlyings, e.g. for pricing of basket options et cetera. Then, the numberof parameters that need to be handled increases dramatically and hence also thecomplexity of pricing. This might further strengthen the will to keep the simulationprocedure as simple as possible.

46

Bibliography

[1] Andersen, L., (2007)Efficient Simulation of the Heston Stochastic Volatility Model. SSRN eLibrary.

[2] Bates, D., (1996)Jumps and Stochastic Volatility: Exchange Rate Processes Implicit inDeutsche Mark Options The Review of Financial Studies, Vol.9, No.1, 69-107

[3] Andersen, L. & Piterbarg, V., (2007)Moment Explosions in Stochastic Volatility Models. Finance and Stochastics,Vol.11, No.1, 29-50.

[4] Bakshi, G., Cao, C. & Chen, Z., (1997)Empirical Performance of Alternative Option Pricing Models. Journal ofFinance, Vol.52, No.5, 2003-2049.

[5] Black, F. & Scholes, M., (1973)The Pricing of Options and Corporate Liabilities. Journal of Political Econ-omy, Vol.81, No.3, 637-659.

[6] Boyle, P., Broadie, M. & Glasserman, P., (1997)Monte Carlo Methods for Security Pricing. Journal of Economic Dynamics &Control, Vol.21, No.8-9, 1267-1321.

[7] Boyle, P., Joy, C. & Tan, K. S., (1996)Quasi-Monte Carlo Methods in Numerical Finance. Management Science,Vol.42, No.6, 926-938.

[8] Broadie, M. & Kaya, O., (2006)Exact Simulation of Stochastic Volatility and other Affine Jump Diffusion

47

BIBLIOGRAPHY

Processes. Operations Research, Vol.54, No.2, 217-231.

[9] Carr, P. & Madan, D. B., (1999)Option Valuation Using the Fast Fourier Transform. The Journal of Compu-tational Finance, Vol.2, No.4, 61-73.

[10] Cont, R., (2001)Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues.Quantitative Finance, Vol.1, No.2, 223-236.

[11] Cox, J., Ingersoll, J. & Ross, S. A., (1985)A Theory of the Term Structure of Interest Rates. Econometrica, Vol.53, No.2,385-407.

[12] Duffie, D. & Glynn, P., (1995)Efficient Monte Carlo Estimation of Security Prices. The Annals of AppliedProbability, Vol.4, No.5, 897-905.

[13] Gatheral, J., (2006)The Volatility Surface: A Practitioner’s Guide. Wiley.

[14] Glasserman, P., (2004)Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York.

[15] van Haastrecht, A. & Pelsser, A. A. J., (2008)Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model.SSRN eLibrary.

[16] Heston, S., (1993)A Closed-Form Solution for Options with Stochastic Volatility with Applica-tions to Bond and Currency Options. The Review of Financial Studies, Vol.6,No.2, 327-343.

[17] Higham, D. & Mao, X., (2005)Convergence of Monte Carlo Simulations Involving the Mean-reverting SquareRoot Process. Journal of Computational Finance, Vol.8, No.3, 35-61.

48

[18] Hull, J. & White, A., (1987)The Pricing of Options on Assets with Stochastic Volatilities. Journal ofFinance, Vol.42, No.2, 281-300.

[19] Ingber, A. L., (1993)Simulated Annealing: Practice Versus Theory. Journal of MathematicalComputational Modelling, Vol.6, No.2, 29-57.

[20] Kahl, C. & Jäckel, P., (2006)Fast Strong Approximation Monte-Carlo Schemes for Stochastic VolatilityModels. Quantitative Finance, Vol.6, No.6, 513-536.

[21] Lewis, A. L., (2000)Option Valuation under Stochastic Volatility. Finance Press.

[22] Lord, R., Koekkoek, R. & van Dĳk, D., (2008)A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.SSRN eLibrary.

[23] Milstein, G. N., (1974)Approximate Integration of Stochastic Differential Equations. Theory ofProbability and its Application, No.5, 557-562.

[24] Shreve, S. E., (2004)Stochastic Calculus for Finance II (Continuous-Time Models). Springer, NewYork.

49

Appendix A

Convergence Properties

A.1 Efficiency of Estimates

Consider an unbiased estimate Cn of the quantity C. Suppose that

Cn =1

n

n∑

i=1

Ci,

with Ci i.i.d., E[Ci] = C and Var(Ci) = σ2C < ∞. The central limit theorem then

states that, for increasing n

Cn − CσC/√n−→ N(0, 1).

In other words, for large nCn ≈ N(C, σ2

C/n).

A.2 Convergence Orders

A.2.1 Weak

Let X denote the time-discretized approximation of X. Then X has weak order ofconvergence β if

∣

∣

∣E[

f(X(h))]

− E[

f(X)]

∣

∣

∣ ≤ chβ

for some constant c, sufficiently small h and all functions f whose derivatives oforder up to 2β + 2 are polynomially bounded.

A.2.2 Strong

X has strong order of convergence β if

E[

∥

∥X(h)−X∥

∥

]

≤ chβ

for some constant c, all sufficienty small h and some vector norm ‖ · ‖.

51

Appendix B

Additional Price Sensitivity Figures

To complete the results of Section 7.3 and to further emphasize the inadequacy ofcase (ii) in Chapter 7 we give some additional delta and gamma plots.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.2

0.4

0.6

0.8

1

Spot / Strike

∆

N=104

case (i)

case (ii)

Figure B.1. Delta simulated with N = 104. ǫ = 1% of the spot value

53

APPENDIX B. ADDITIONAL PRICE SENSITIVITY FIGURES

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot / Strike

∆

N=105

case (i)

case (ii)

Figure B.2. Delta simulated with 105. ǫ = 1% of the spot value

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot / Strike

∆

N=106

case (i)

case (ii)

Figure B.3. Delta simulated with 106. ǫ = 1% of the spot value

54

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Spot / Strike

Γ

N=104

case (i)case (ii)

Figure B.4. Gamma simulated with N = 104. ǫ = 1% of the spot value

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Spot / Strike

Γ

N=106

case (i)

case (ii)

Figure B.5. Gamma simulated with 106. ǫ = 1% of the spot value

55

APPENDIX B. ADDITIONAL PRICE SENSITIVITY FIGURES

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.005

0

0.005

0.01

0.015

0.02

0.025

Spot / Strike

Γ

case (i)

case (ii)

Figure B.6. Gamma simulated with 106. ǫ = 5% of the spot value

56

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