A Heterogeneous Agent Model for Volatility Smile Dynamics

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A Heterogeneous Agent Model

for Volatility Smile Dynamics

MSc Computational Science

Master Thesis

by

Azer Aras

January 17, 2020

Examiner: Prof. dr. B.D. Kandhai Daily supervisor: Prof. dr. C.G.H. Diks Second Assessor: Mr. I. Anagnostou MSc

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Abstract

The determinants of the volatility smile have long been a topic of discussion in the financial world. In this thesis, we study and extend an options market model by Qiu et al. [1], designed to shed light on the factors that drive the shape and dynamical properties of the implied volatility curve. We introduce a small num-ber of heterogeneous belief types and an evolutionary switching mechanism that determines the popularity of each type. Through this extension, we gain an im-proved understanding of the determinants of the shape of the implied volatility curve, and demonstrate that the dynamical properties are highly dependent on the behavioural parameters. We also show that our model, for a suitable choice of behavioural parameters, can reproduce empirically observed dynamical properties of implied volatility curves.

Title: A Heterogeneous Agent Model for Volatility Smile Dynamics Author: Azer Aras

Examiner: Prof. dr. B.D. Kandhai Daily supervisor: Prof. dr. C.G.H. Diks Second assessor: Mr. I Anagnostou MSc Examination date: January 17, 2020

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1 Introduction

Since the stock market crash of 1987, traders in equity options markets have observed a remarkable phenomenon known as the volatility smile [4]. The origins of this phenomenon have been the topic of much discussion in the financial world. In this thesis, we study and extend a model developed by Qiu et al. [1], which attempts to shed light on the factors that determine the properties of the smile. This chapter introduces the main concepts related to the volatility smile and discusses the motivation for this project.

1.1 Options

Financial institutions trade not only in markets where stocks and bonds are traded, but also in derivatives markets. Derivatives are financial instruments whose value depends on the value of other, more basic, underlying variables [4]. These un-derlying variables are usually the price of some other asset, such as a stock. The derivatives market has become increasingly important and is much bigger than the stock market, when measured in terms of underlying assets [4].

An important class of derivatives are the options. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price, while a put option gives the holder the right to sell the underlying asset by a certain date for a certain price [4]. The price for which the underlying asset can be bought or sold is known as the strike price. The date in the contract is known as the maturity of the option. So called European options can only be exercised at maturity, while American options can be exercised at any time leading up to maturity. Options can be used for speculation, hedging or arbitrage purposes [4].

In contrast to futures and forwards, options give the holder the right, but not the obligation to buy or sell the underlying asset. Consider a European option with strike price K on a stock which has a value of ST per share at the time of

maturity. The payoff from the option to the holder is max(ST − K, 0)

if it is a call option, and

max(K − ST, 0)

if it is a put option.

A call option is said to be in the money (ITM) at time t if the share price exceeds the option’s strike price, i.e. St > K. If St < K, the option is said to be

out of the money (OTM), and if St= K, the option is at the money (ATM). For

a put option, the situation is reversed. It is in the money when St < K and out

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money is called the moneyness of the option. We define the moneyness at time t as K/St [1], so that options are at the money when their moneyness is 1.

1.2 The Black-Scholes model

Over the past decades, researchers have asked what the value or fair price of an option is, and have developed option pricing models to answer this question. The most famous of these models is the Black-Scholes model, introduced in 1973 [6]. Two of its creators where awarded the The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 1997 [14].

The main idea behind the model is that a portfolio, consisting of positions in the underlying stock and bonds, can be set up which replicates the payoff of an option. The value of the option is then equal to the cost of setting up this portfolio [1]. The stock price process in the model is a geometric Brownian motion and can be written as

dS = µSdt + σSdZ.

Here, the stochastic variable S is the stock price, µ is the stock’s expected rate of return and dZ is a Wiener process, or Brownian motion. The variable σ is the volatility, a measure of uncertainty about future stock price movements [4]. This model implies that ST, the stock price at time T , is lognormally distributed, or

log(ST) − log(S0) ∼ Normal

µ − σ 2 2 T, σ2T ,

where the two parameters on the right hand side are the mean and variance of the normal distribution.

The Black-Scholes model gives explicit formulas for the value of European call and put options [4]:

c = S0Φ(d1) − Ke−rTΦ(d2) (1.1)

for calls, and

p = Ke−rTΦ(−d2) − S0Φ(−d1) (1.2)

for puts, where K is the strike price of the option, T the time to maturity, r the risk-free interest rate, Φ is the CDF of the standard normal distribution, and

d1 = log(S0/K) + (r + σ2/2)T σ√T , d2 = d1− σ √ T .

The Black-Scholes price of an option is thus a function of its strike price, its maturity, the risk free rate, the current stock price and the stock volatility. Of these variables, only the volatility is not directly observable. Traders can estimate

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the volatility from the history of the stock price. In option pricing, however, one is interested in the future volatility of the underlying asset. Therefore, historical estimates are not preferred. Instead, traders commonly use observed option prices to compute so called implied volatilities.

Equations (1.1) and (1.2) cannot be inverted in closed form to express σ as a function of the other variables. However, when the other variables are known, a numerical search procedure can be used to compute the unique value of σ that fits the equation. This σ is known as the implied volatility. It is the volatility that is implied by the observed option prices in the market [4].

1.3 The volatility smile phenomenon

Although the Black-Scholes model is celebrated and widely used, it fails to account for a phenomenon known as the volatility smile. Consider a set of options on an asset with identical time to maturity, but different strike prices. The implied volatility of these options can be plotted against the strike prices in a graph. The resulting graph is known as an implied volatility curve (IV curve). In the Black-Scholes framework, implied volatilities are independent of the strike price. Therefore, implied volatility curves should be horizontal, straight lines. In reality, however, implied volatilities exhibit a remarkable dependence on the strike price [1]. The characteristic U-shape of observed IV curves is known as the volatility smile phenomenon.

Moneyness

Implied volatility

IV curve

Rate of return

Probability density

Implied distribution

Implied

Normal

Figure 1.1: Volatility smile and corresponding implied distribution

Figure 1.1 shows the typical shape of a volatility smile. As is common, the implied volatilities are plotted againt moneyness rather than strike price. The figure shows that the implied volatility is relatively low for at-the-money options, and higher for both in-the-money and out-of-the-money options. Also shown is the corresponding implied distribution over the rate of return on the stock. Under Black-Scholes assumptions, the rate of return is normally distributed, as is shown in orange for comparison. The implied distribution has heavier left and right tails than the normal distribution, meaning that large stock price movements in either direction are more likely. Therefore, deeply out-of-the-money calls (high

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moneyness) and deeply out-of-the-money puts (low moneyness) are both more likely to pay off at maturity. Since they are more likely to pay off, their price is higher, which corresponds to a higher implied volatility. A smile with this general shape is typically observed in markets for foreign currency options [4].

Moneyness

Implied volatility

IV curve

Rate of return

Probability density

Implied distribution

Implied

Normal

Figure 1.2: Volatility skew and corresponding implied distribution

Figure 1.2 shows the general form of volatility smiles observed in equity options markets. It is sometimes called a volatility skew [4]. The implied distribution has a heavier left tail and a less heavy right tail than the normal distribution. Large downward movements of the stock price are therefore more likely than large up-ward movements, and more likely than in the Black-Scholes model. Consequently, deeply out-of-the-money puts (low moneyness) are more likely to pay off than under a lognormal distribution, leading to a higher price, which corresponds to a higher implied volatility. Deeply out-of-the-money calls (high moneyness) are less likely to pay off than under a lognormal distribution, leading to a relatively low price, which corresponds to a low implied volatility [4].

1.4 Motivation

Because the Black-Scholes model cannot account for the existence of the volatility smile, researchers have developed many alternative option pricing models which assume an underlying process other than a geometric Brownian motion. Some of these models have been found to perform better than the Black-Scholes model in an empirical study [5].

However, these models have also attracted criticism. Cont and Da Fonseca [7] point out that many of these models do not correctly reproduce the profile of empirically observed volatility smiles. In particular, the model dynamics are often inconsistent with those observed in options markets. A model may be well calibrated to IV curves on a given day, but give a bad fit the next day, meaning that the model needs to be constantly recalibrated. Cont an Da Fonseca argue that this inability to describe options markets dynamics is not simply caused by mis-specification of the underlying process. It is also caused by the fact that options markets have become increasingly autonomous, and the price dynamics

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are influenced not just by changes in the underlying, but also by the options market’s internal supply and demand [7].

Pe˜na et al. [10] point out that the models mentioned above do not directly explain the determinants of the smile. In addition, Qiu et al. [1] note that the volatility skew in equity options markets has only been observed since 1987, while the non-normal distribution of price changes has been identified in financial time series since at least the beginning of the twentieth century. Qiu et al. also note that these models can disagree about the value of exotic options [1].

These considerations have led to the development in recent years of heteroge-neous agent models of options markets. One such model was developed by Qiu et al. [1]. Their options market model contains heterogeneous traders and demon-strates how volatility smiles can arise as a natural consequence of heterogeneity in the behaviour and expectations of these traders.

The dynamics in this model are entirely driven by stochastic processes. These processes determine the price of the underlying as well as the expectations of traders and the fractions of different trader types. The model dynamics are thus exogenous and have no direct behavioural interpretation. In recent decades, how-ever, financial models such as the Brock-Hommes asset pricing model [2] have been developed that demonstrate how complex economic dynamics can arise as a result of interaction between boundedly rational, heterogeneous agents, even in the absence of external shocks. The Brock-Hommes model allows for theoreti-cal investigation of the price dynamics and its dependence on behavioural and economic parameters.

In this thesis we study the Qiu-Kandhai model and extend it with elements from the Brock-Hommes model. In particular, we divide the speculators in the Qiu-Kandhai model into different types that are in dynamic competition with each other. The dynamics of the model thus become dependent on behavioural assumptions about the speculators’ beliefs. We can then investigate how the shape and dynamical properties of the IV curves depend on the model’s behavioural parameters.

Chapter 2 will provide an overview of relevant literature, including summaries of the Qiu-Kandhai and Brock-Hommes models. The main model of this thesis is introduced in Chapter 3, followed by the simulation results in Chapter 4 and concluding remarks in Chapter 5.

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2 Literature review

This chapter provides an overview of some of the existing literature relevant to the subject of this thesis. First, we will provide a brief overview of several models designed to overcome the shortcomings of the Black-Scholes model. These include jump-diffusion models, pure jump models and stochastic volatility models, as well as more recently introduced agent-based approaches. Next, we will examine in more detail two models that are central to this thesis; the Qiu-Kandhai options market model and the Brock-Hommes asset pricing model.

2.1 Alternatives to Black-Scholes

As noted in the introduction, the existence of a volatility smile conflicts with the Black-Scholes framework. Since the introduction of the Black-Scholes model in 1973, a large number of models have been designed to overcome these shortcomings [5]. These models are usually based on an underlying price process other than geometric Brownian motion.

Merton [8], for instance, introduced a jump-diffusion model, i.e. a model in which the underlying price process is a combination of continuous changes and jumps. The stock price process in this model is

dS

S = (r − q − λk)dt + σdZ + dP,

where q is the dividend yield, λk is the average growth rate in the asset price from jumps, dZ is a Wiener process and dP is a Poisson process generating the jumps [4]. Particular cases of this model give rise to heavier left and right tails than the Black-Scholes model and can be used to price currency options.

Alternatively, a price process can be considered in which all the price changes that take place are jumps. An example of such a pure jump model is the variance-gamma model [9]. It is based on the variance-gamma process; a jump process in which small jumps occur much more frequently than large jumps. The model tends to produce convex smiles that are not necessarily symmetrical, and can be used for either equity or foreign currency options [4].

Another approach is to consider the volatility σ in the Black-Scholes as a stochastic variable rather than a constant. These stochastic volatility models thus have two stochastic variables instead of one. One such model that has been studied is dS S = (r − q)dt + √ V dZS, dV = a(VL− V )dt + ξVαdZV,

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where V is the asset’s variance rate, a, VL, ξ and α are constants and dZS and dZV

are Wiener processes [4]. Different cases of this model result in IV curves that are similar to those observed for either currency options or equity options. The impact of stochastic volatility on pricing is in particular significant for options with a long time to maturity.

An empirical study of option pricing models, carried out by Bakshi, Cao and Chen [5], concluded that models with stochastic volatility and random jumps are preferable to the Black-Scholes formula, because they perform better and are prac-tically implementable. As explained in the introduction, however, these models have also attracted criticism for failing to identify the determinants of the volatility smile, and researchers have explored alternative approaches to explain the origins of this phenomenon. Pe˜na et al. [10], for instance, carried out an empirical study and concluded that transaction costs play an important role in determining the shape of the IV curve.

In more recent years, agent-based and microsimulation models have become increasingly popular tools to model options markets and IV curves. We will discuss several of these models in the next section.

2.2 Heterogeneous agent models

The models described in the previous section are adaptations of the Black-Scholes model, that assume a different stochastic process for the price dynamics of the underlying asset. An alternative approach to modelling the volatility smile is through a heterogeneous agent model. Examples of this approach include Buraschi & Jiltsov [11], Vagnani [12], Suzuki et al. [13] and Qiu et al. [1]. These models typically include a set of heterogeneous agents and investigate how the IV curve depends on the beliefs and interaction between these agents.

Buraschi & Jiltsov [11] introduce an options market model with two sets of agents, who observe two stochastic processes. One is a dividend process δ(t), which is a geometric Brownian motion with stochastic drift µδ(t). The other

process, z(t), contains a signal for the growth rate of the dividends, and also has a stochastic drift term µz(t). The two sets of agents are aware of the dynamics

of δ(t) and z(t), but do not know the current value of the stochastic drifts µδ(t)

and µz(t). They have different initial beliefs about the value of µδ(0) and they

rationally update their estimates of µδ(t) and µz(t) given the observed history of

δ(t) and z(t).

In relation to the volatility smile, Buraschi & Jiltsov [11] find a significant effect of the difference in beliefs on the slope of the smile. The greater the dispersion of beliefs, the steeper the volatility smile.

Vagnani [12] considers a market where European call options are traded. The market is populated by noise traders in proportion θ and ‘Black-Scholes traders’ in proportion 1−θ. Noise traders perceive the expected prices of the call option as an independent and identically distributed random variable. Black-Scholes traders, on the other hand, assume that the underlying asset price follows a geometric Brownian motion and use the Black-Scholes formula (1.1) to compute their

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ex-pected option prices. However, the traders do not know the value of the stock volatility. Instead, they assume a normal distribution over possible values for the volatility and compute the average option price given this distribution. The mean and standard deviation of this normal distribution may be heterogeneous over the population.

When beliefs are homogeneous, Vagnani [12] finds that a symmetric U-shaped smile emerges as long as the market is populated by a sufficient proportion of Black-Scholes traders (θ ≤ 0.67). When heterogeneity is introduced, the smile becomes upward sloping.

Suzuki et al. [13] focus on the effect of investors’ loss-aversion on the volatility smile. Their model consists of three types of traders that trade in a risk-free and a risky asset. The first type of traders are ‘smart traders’. These are informed, rational traders that receive private signals containing information about the fun-damental value of the risky asset. The second group consists of loss-averse traders. These traders do not receive private signals and place excessive importance on ex-pected loss when determining their demand. Their investment behaviour becomes excessively passive when they expect that the price may decline. Finally, there are noise traders, who invest at random.

This model can be used to generate price sequences for the risky asset, which in turn can be used to compute option prices using a Monte Carlo procedure. Note that the traders in this model are only active in the market for the underlying asset, not in the options market.

The simulation results of Suzuki et al. [13] show that the IV curve is a straight, horizontal line when there is no loss-aversion. When loss-aversion is included, however, the IV curve is shaped like a smile. Moreover, the convexity of the smile becomes more pronounced when the degree of loss-aversion is increased, suggesting that the loss-aversion feature may be a reason for the occurence of the volatility smile.

The options market model introduced by Qiu et al. [1] is interesting because it allows for a detailed analysis of all the different shape characteristics of the IV curve (level, slope and convexity) and relates them to trader characteristics. The model also allows for an investigation of the dynamical properties of the IV curve, and it highlights the role of arbitrageurs in determining the shape of the smile. The main model of this thesis is an extension of the model by Qiu et al. We will therefore introduce this model in more detail in the next section.

2.3 The Qiu-Kandhai options market model

In their 2012 paper [1], Qiu, Kandhai, Johnson and Sloot provide an explanation of the volatility smile phenomenon using a microscopic simulation model of an options market. In this model, the option prices are determined from the collective demand of individual, heterogeneous traders. The simulation results suggest that the volatility smile is a natural consequence of traders’ heterogeneity.

The model consists of speculators and arbitrageurs, who trade in a range of European call and put options with identical time to maturity, but different strike

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prices. The market contains Nop call options and Nop put options. Let Kndenote

the strike price of the nth option and let φ be 1 for a call and -1 for a put. The interest rate, dividends, transaction costs and taxes are ignored for the sake of simplicity, while the time to maturity of the options that are being traded is kept constant.

2.3.1 Speculators

Let Stdenote the price of the underlying asset at time t. Speculators each have an individual expectation of what the future price of this asset will be. S_SPi,t denotes the price that the underlying asset will have at the option’s maturity, according to speculator i at time t. S_SPi,t varies over time and has a long term mean Si_SP, which is lognormally distributed for the entire group of speculators. The mean of S_SPi,t for the entire group of speculators is denoted by Mt

SP, and the standard deviaton

by Dt

SP. The difference between St and MSPt reflects the expected price movement

by the group of speculators, while Dt_SP reflects their level of disagreement about future prices.

At each timestep, each speculator determines a demand quantity for each op-tion based on his expected profit. Speculator i’s expected payoff at time t from an option with strike price Kn_{is max(φ(S}i,t

SP− Kn, 0)), while the price of that option

at time t is denoted by Vn,φ,t_{. His demand quantity is then}

Qi,n,φ,t= λSP max(φ(S_SPi,t − K, 0)) − Vn,φ,t,

where λSP indicates the activity level of the speculators.

The version of the model with only speculators is called the S model.

2.3.2 Arbitrageurs

The second group of traders are the arbitrageurs. Arbitrageurs trade those options that are found to violate the put-call parity (PCP) or butterfly spread (BFS) arbitrage relations. An arbitrageur’s transaction quantity at time t is given by

Qi,n,φ,t_AR

PCP = −φλARPCP V

n,1,t_{− V}n,−1,t_{− S}t_{+ K}n

(2.1) for the PCP strategy and

    Qi,n−h,φ,t_AR BFS Qi,n,φ,t_AR BFS Qi,n+h,φ,t_AR BFS     =    1 −2 1   λARBFSmax h Vn,φ,t− 1 2(V n−h,φ,t_{+ V}n+h,φ,t_{) +}h_{, 0}i _(2.2)

for the BFS strategy. Here, λARPCP and λARBFS are parameters that indicate the

activity level of the arbitrageurs for each of the strategies. The value of h varies so that the above equation is applied to all sets of options of which Kn_{is the middle}

strike price. h determines the willingness of the arbitrageurs to pay for butterfly spreads and is given by h _{= αh for a parameter α. The role of the}

h-term is

discussed in more detail in Section 3.3.

The version of the model with both speculators and arbitrageurs is called the SA model.

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2.3.3 Difference in liquidity

Qiu et al. note that in real markets, out-of-the-money options are generally more liquid than in-the-money options. This difference in liquidity is included in the model by making the speculators’ activity strike-dependent. The variable λSP

introduced above is defined as

λSP(Kn) = ηSPφ tanh γ(Kn− St) + 1, (2.3)

where ηSP and γ are positive parameters.

The full version of the model, including liquidity unbalancing is called the SAL model.

Now let Qn,φ,t _{denote the total transaction quantity of the option with strike}

price Kn _{at time t, and let N}

tr denote the total number of traders. The price of

the option at time t + 1 is then given by

Vn,φ,t+1= Vn,φ,t+ρQ

n,φ,t

Ntr

, (2.4)

where ρ is a positive parameter. The underlying asset price St, the expected asset prices S_SPi,t and the fraction of traders that is a speculator Ft

SP also vary over time

and are determined by stochastic processes.

2.3.4 Results

One of the main simulation results of Qiu et al. is shown in Figure 2.1. In these simulations, St, S_SPi,t and F_SPt have been kept constant. It can be seen that the model produces implied volatility curves similar to empirical volatility smiles. Moreover, the figure shows how the shape of the curve depends on model variables. Figure 2.1a shows that the smaller M_SPt is with respect to St, the steeper is the skew. Figure 2.1b shows that higher values of Dt

SP correspond with a higher

overall level of the smile. Finally, figure 2.1c shows that a higher fraction Ft SP of

speculators corresponds to a less convex volatility smile.

As explained in the introduction, the dynamics in the Qiu-Kandhai model are entirely driven by external shocks. St_{, S}i,t

SP and FSPt are updated stochastically

at every timestep. Therefore, the dynamical properties do not have a direct be-havioural interpretation. We wish to reduce the model’s dependence on exogenous shocks, and explore the link between the model’s dynamical properties and be-havioural assumptions regarding the traders. We will therefore borrow elements from the Brock-Hommes asset pricing model, a financial market model that is capable of producing complex price dynamics endogenously.

2.4 The Brock-Hommes asset pricing model

The Brock-Hommes asset pricing model [2, 3] is a financial market model intro-duced in 1998. It consists of heterogeneous agents who can invest in a risk free asset and a risky asset, such as a stock. They can choose from a finite set of

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Figure 2.1: Implied volatility curves. Unless otherwise indicated St = 20, M_SPt = 19, Dt

SP= 3.5 and FSPt = 0.5. a) Various values of MSPt , b) Various values of DtSP,

c) Various values of Ft

SP. Figures from Qiu et al. [1].

future price predictors for the risky asset to base their investment decisions on. Despite its simple setup, the model is capable of producing complex price dy-namics, such as cycles, chaos and bubbles and crashes, without the need for a stochastic process driving the price dynamics. The main dynamics of the model are generated through endogenous interactions, although (small) external shocks may be added to allow the solution to switch between different coexisting attrac-tors (see Subsection 2.4.3). We will summarize the main features of the model here.

2.4.1 Price equation

The risk free asset pays a fixed rate of return r, while the risky asset pays an uncertain dividend. Let pt be the price per share of the risky asset at time t, yt

its stochastic dividend process, and zt the number of shares purchased at time t.

The wealth dynamics are then given by

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Here, (1 + r)Wt represents the interest gains from the wealth that was already

acquired at time t, while pt+1+ yt+1− (1 + r)ptztrepresents the net profit or loss

resulting from the price change and dividend yield of the newly acquired shares. Now let Eht and Vht denote the beliefs of trader type h about the conditional

expectation and conditional variance, respectively, based on information such as past prices and past dividends. The traders in the model are myopic mean-variance maximizers, meaning that their demand zht is obtained by solving

max zt n Eht[Wt+1] − a 2Vht[Wt+1] o ,

where a is a risk aversion parameter. We can solve this by setting the derivative equal to zero: 0 = d dzt h Eht[Wt+1] − a 2Vht[Wt+1] i = d dzt h (1 + r)Wt+ Eht[pt+1+ yt+1− (1 + r)pt]zt− a 2 Vht[pt+1+ yt+1]z 2 t i = Eht[pt+1+ yt+1− (1 + r)pt] − aVht[pt+1+ yt+1]zt,

so that the demand is given by zht= E

ht[pt+1+ yt+1− (1 + r)pt]

aσ2 ,

where σ2 _{is the conditional variance V}

ht, which is assumed to be constant and

equal for all types. The supply of outside shares is assumed to be zero, so that equilibrium of demand and supply gives

H X h=1 nhtE ht[pt+1+ yt+1− (1 + r)pt] aσ2 = 0,

and the price equation becomes

(1 + r)pt= H

X

h=1

nhtEht[pt+1+ yt+1],

where H is the number of agent types and nht is the fraction of agents of type h.

2.4.2 Heterogeneous beliefs and switching

All traders in the Brock-Hommes model have the same beliefs about the cond-tional variance, as well as the future dividends. Crucially, however, they are heterogeneous in their expectations about prices. Beliefs about the future price are expressed as a function of xt = pt− p∗t. Here, p

∗

t is the fundamental rational

expectations price, the price that would prevail in an efficient market populated by a homogeneous group of rational traders. xt is thus the deviation from the

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All trader types have a belief about the future price pt+1 of the form

Eht[pt+1] = Et[p∗t+1] + fh(xt−1, ..., xt−L),

where fh is a forecasting rule that represents a belief about the way that prices will

deviate from the fundamental price. Brock and Hommes [2] studied competition between simple linear forecasting rules of the form

fht= ghxt−1+ bh,

where gh and bh are model parameters. For instance, the belief fht = 0 represents

fundamentalist traders, who always expect the asset price to equal the fundamental price, while fht = ghxt−1 with gh > 0 represent trend followers, who attempt to

extrapolate a perceived trend.

nht is the fraction of traders of type h at time t. This fraction is updated

over time as a result of competition between types, whose market performance is measured using an evolutionary fitness or performance measure Uh,t. A natural

performance measure used by Brock and Hommes is accumulated realized profits: Uh,t = (xt− Rxt−1)

f_h,t−1− Rx_t−1 aσ2

− Ch+ wUh,t−1.

Here, R = (1 + r), Ch is the cost of obtaining forecasting strategy h and 0 ≤

w ≤ 1 is a memory parameter that determines the weight of past performance for strategy selection. Using this performance measure, the fractions are given by the multinomial discrete choice model:

nht =

eβUh,t

P

ieβUi,t

.

With this equation, strategies that perform better attract a higher fraction of traders. The parameter β ≥ 0 is called the intensity of choice and determines how quickly traders will switch to more successful strategies. When β = 0, no switching takes place, while in the limit case of β = +∞ all traders will switch to the most successful strategy at each timestep.

2.4.3 Dynamics

Competition between many different belief types has been studied using the above framework [2, 3]. Here, we will only briefly look at one simple case to illustrate the price dynamics that the model can produce. Consider the case of costly fundamentalists versus trend followers, i.e.

f1,t = 0, C1 > 0,

f2,t = gxt−1, C2 = 0.

Here, fundamentalist traders pay a positive amount C1 at every timestep,

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The intensity of choice parameter plays an important role in the resulting dy-namics. Brock and Hommes have shown, through bifurcation diagrams and Lya-punov exponent plots, that as the parameter β increases, the dynamical behaviour of the model goes from a stable steady state, to complicated chaotic dynamics.

Figure 2.2 shows the dynamics for the case of β = 3.6, with and without a small stochastic noise term added to the price equation. Without noise, the system settles into an attractor with temporary bubbles when trend followers dominate the market, followed by crashes when fundamentalists dominate market. When noise is added, the system switches back and forth between two coexisting attractors, with crashes and bubbles above and below the fundamental value [3].

0 100 200 300 400 500 t 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 xt Price dynamics (a) 0 100 200 300 400 500 t 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 xt Price dynamics (b) 0 100 200 300 400 500 t 0.0 0.2 0.4 0.6 0.8 1.0 n1,t Fraction dynamics (c) 0 100 200 300 400 500 t 0.0 0.2 0.4 0.6 0.8 1.0 n1,t Fraction dynamics (d)

Figure 2.2: Time series for xt and n1,t. Figures on the left result from the model

without noise. For the figures on the right, a small noise term has been added to the price equation. Parameters are β = 3.6, g = 1.2, R = 1.1 and C1 = 1.

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3 Model description

The main model of this thesis takes the Qiu-Kandhai model as a starting point. However, as explained in the introduction, we wish to adapt the model such that the dynamics become less dependent on exogenous shocks, and more dependent on behavioural assumptions, so that we can explore the link between the behaviour of traders and the dynamical properties of the IV curve. We will therefore disregard the stochastic processes of St, S_SPi,t and F_SPt and replace them by the dynamics described below.

The main addition to the Qiu-Kandhai model is that we will no longer consider each speculator individually, but will instead introduce different speculator types, that are in dynamic competition with each other. In analogy to the Brock-Hommes asset pricing model, the performance of each strategy will determine its popularity, which in turn influences the realized option prices.

The stock price process S is a geometric Brownian motion, as in the Black-Scholes model, and the unit of time is 1 year. Let µ and σ be the expected return and volatility per year and let ∆t be the duration of one timestep. We thus write t + 1 for one year after time t, and t + ∆t for one timestep after time t. Then µ = µ∆t and σ = σ√∆t are the expected return and volatility per timestep. Furthermore,

log(St+1) − log(St) ∼ Normal

µ − σ 2 2 , σ2 . (3.1)

This is a different stock process from that in the Qiu-Kandhai model, which uses a mean-reverting process, which keeps fluctuating around its mean, even on long timescales. We adopt a geometric Brownian motion to stay closer to the Black-Scholes framework.

The options market consists of speculators and arbitrageurs, who trade in a range of European call and put options. The moneyness of the options being traded, defined at time t as Kn_/S

t, is kept constant, as is their time to maturity.

The strike prices that are being traded thus vary over time. The fraction F_SPt of traders that is a speculator will also be kept constant. Like Qiu et al. we will ignore the interest rate, dividends, transaction costs and taxes for the sake of simplicity.

3.1 Speculators

Like in the Qiu-Kandhai model, the speculators in our model make an assumption about the behaviour of the stock price process and base their demand for each option on this assumption. However, we will drop the unrealistic assumption that the speculators individually have one specific future price in mind. Instead,

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they are now divided into k ≥ 2 types, denoted by i, which make an assumption about the underlying price process. All groups of speculators assume that the stock price process is a geometric Brownian motion, which is a correct assumption in our model. However, they have different opinions on the mean and standard deviation. Type i assumes that S has a drift of µi and a volatility of σi, so that

the stock price process is given by

log(S_i,tt+1) − log(St) ∼ Normal

µi− σ2 i 2 , σ2_i ,

where S_i,tt+1 denotes the stock price at time t + 1, as expected by a speculator of type i at time t. These beliefs imply a distribution for the price of the stock at the option’s maturity T:

log(S_i,tT ) − log(St) ∼ Normal

µi− σ2 i 2 (T − t), σ_i2(T − t) .

As in the Qiu-Kandhai model, let φ be 1 for a call option and -1 for a put option. Furthermore, let Vn,φ,t _{be the price of the option with strike price K}n _at

time t. Then the demand of a speculator of type i for an option with strike price Kn at time t is

Qi,n,φ,t = λSPEipayoff − cost

= λSPEi max(φ(ST − Kn, 0)) − Vn,φ,t = λSP Z ∞ 0 max(φ(ST − Kn, 0))fi(ST)dST − Vn,φ,t ,

where fi is the probability density function of ST as estimated by speculators of

type i. The above integral equals the Black-Scholes price of an option assuming µi is the risk-free rate and σi is the volatility. So the demand becomes

Qi,n,φ,t = λSP

V_BSφ,t(St, Kn, T − t, µi, σi) − Vn,φ,t

,

where V_BSφ,tis the Black-Scholes price of the option. The total transaction quantity of the speculators is then Ntr· FSP·Pk_i=1ni,tQi,n,φ,t, where ni,t is the fraction of

speculators of type i at time t. As in the Qiu-Kandhai model, the demand from the arbitrageurs is added to the demand of the speculators to get the total transaction quantity Qn,φ,T_{. The price of each option is then given by Equation (2.4).}

3.2 Switching between speculator types

The fraction of speculators of type i at time t is denoted by ni,t. This fraction

can be kept constant over time, but, in analogy to the Brock-Hommes model [2], we will also introduce switching between the available speculator strategies, based on the past performance of each strategy. The fractions ni,t are then updated at

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The performance measure of type i at timestep t is denoted by πi,t. The

defi-nition of πi,t is discussed in Subsection 3.2.1. We add memory to the performance

by introducing a memory parameter w ∈ [0, 1] and defining: Ui,t = w · Ui,t−∆t+ (1 − w) · πi,t.

The fractions ni,t are then given by

ni,t =

eβUi,t

Pk

j=1eβUj,t

.

For the case k = 2 of two speculator types, it is sufficient to compute the difference in performance, π1,t− π2,t. The difference in performance with memory

is then

U1,t− U2,t = w · U1,t−∆t+ (1 − w) · π1,t− w · U2,t−∆t+ (1 − w) · π2,t

= w(U1,t−∆t− U2,t−∆t) + (1 − w)(π1,t− π2,t).

The fractions are then given by n1,t = eβU1,t eβU1,t_{+ e}βU2,t = eβ(U1,t−U2,t) 1 + eβ(U1,t−U2,t), n2,t = 1 − n1,t.

3.2.1 Performance measure

To compute the fractions of each speculator type using the above equations, we need to choose a performance measure πi,t. The performance measure reflects

how well each of the available strategies has worked in the last timestep. We will define three different performance measures, denoted by πa

i,t, πi,tb , πi,tc , and

investigate their suitability in the next chapter.

• The first performance measure we will consider is based on how well each speculator type predicted the IV curve in the last timestep. The value of an option at time t, according to a speculator of type i, is V_BSφ,t(St, Kn, T −

t, µi, σi). The market price of that option is Vn,φ,t. The difference between

the two, summed over all options in the market is thus a measure of how well strategy i predicted the option prices. We define πa

i,t as π_i,ta = −X K,φ V_BSφ,t(St, Kn, T − t, µi, σi) − Vn,φ,t 2 .

The square ensures that over- and underestimates of the option prices do not cancel each other out.

• The second performance measure we will consider is based on how well belief type i fits the realized stock price process. Define u = log(St) − log(St−∆t)

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and mi = µi − σi 2

2 . We define performance measure π b

i,t as the log of the

relative likelihood of the last stock movement under the assumption made by speculators of type i. That is,

πb_i,t = log(φb_i(u)) = − log(√2π) − log(σi) −

(u − mi)2

2σi2

,

where φb_i is the PDF of the normal distribution with mean mi and variance

σi2.

• Speculators make assumptions about the annual drift and volatility of the stock price. For the previous performance measure, these annual assump-tions were scaled to assumpassump-tions on the ∆t-timescale and compared to the most recent stock price movement u. Alternatively, we can scale u to the annual level by comparing u/√∆t to the distributions assumed for annual stock price movements. We thus define πc

i,t as π_i,tc = log φc_i u/ √ ∆t ,

where φc_i is the pdf of the normal distribution with mean mi = µi − σ2

i 2

and variance σ2

i. This performance measure is illustrated in Figure 3.1 for a

2-type case with (µ1, σ1) = (0.1, 0.3) and (µ2, σ2) = (−0.1, 0.4).

Figure 3.1: Illustration of performance measure πc i,t.

3.3 Arbitrageurs

The arbitrageurs in our model behave in the same way as in the Qiu-Kandhai model, except for a minor adaptation we will discuss below. That means they

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seek to profit from violations of the put-call parity and butterfly spread arbitrage relations.

Put-call parity (PCP) is a relation between a European call and put option with the same strike price K and time to maturity T . Disregarding the interest rate r, it states that [1]

Vn,1,t+ Kn = Vn,−1,t+ St.

It follows from the fact that a portfolio consisting of a European call and zero-coupon bond that pays off K at time T always has the same payoff as a portfolio consisting of a European put and a share of the stock [4]. Therefore, a violation of this relation gives rise to arbitrage opportunities.

The demand of the arbitrageurs for the PCP strategy is given by Equation (2.1). It is non-zero when PCP is violated. PCP ensures that the volatility smile is the same for calls and puts [4]. By acting on violations of this relationship, the arbitrageurs thus cause the IV curves for calls and puts to be the same.

A butterfly spread is a set of positions in European options of the same type (put or call) with three different strike prices. It can be created by buying one option with strike price K1, buying one option with strike price K3 and selling two

options with strike price K2, which is the average of K1 and K3. A butterfly spread

has a positive payoff at maturity if K1 < ST < K3 and a payoff of 0 otherwise

[4]. Since its payoff is non-negative and may be strictly positive, its price at time t must be positive for there to be no arbitrage opportunities. This is reflected in the arbitrage relationship [1]

Vn−h,φ,t− 2Vn,φ,t_{+ V}n+h,φ,t _{> 0,}

or

Vn,φ,t−1 2 V

n−h,φ,t_{+ V}n+h,φ,t_{< 0.}

Here, h is any positive integer for which Kn−h _{and K}n+h _{exist in the model, so}

that the rule applies to every possible butterfly spread.

The arbitrageurs’ demand for the BFS strategy is given by Equation (2.2). It is non-zero when the BFS relation is violated. The term h _{ensures that the}

arbitrageurs also trade when Vn−h,φ,t−2Vn,φ,t_+Vn+h,φ,t _{= 0. In this case, they can}

obtain a butterfly spread for free, which has a non-negative and possibly positive payoff, so there is still an arbitrage opportunity. h _{also causes the arbitrageurs}

to trade when the butterfly spread has a small positive price (< 2h). It thus causes these traders to go somewhat beyond strict arbitrage. This is a reasonable assumption for small values of h_{, since these trades will usually still carry very}

little risk.

Qiu et al. [1] define h _{= αh for a parameter α ≥ 0. We will make a slight}

adaptation and define it as h _{= αh∆m, where ∆m is the step size in moneyness}

of the options that are being traded. This addition makes the model much less sensitive to the number Nop of strike prices that are being traded.

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3.4 Tools

We now discuss some computational and investigative tools that are used in the simulations. The following subsections will cover how we compute implied volatil-ities from the option prices, how we measure convexity, and how we can use Principal Component Analysis to study the dynamical properties of our model.

3.4.1 Computing implied volatilities

At every timestep, the market price of each option in our model is determined by Equation (2.4). From this price, the implied volatility can be computed. The Black-Scholes formulas (1.1) and (1.2) cannot be inverted in closed form to be written as function of σ, so we will need to compute the implied volatility nu-merically. We will use a simple numerical root finding method known as interval bisection [15].

Algorithm 1 shows an overview of the procedure in pseudocode. The main idea is to define an interval [a, b] in which the IV must lie, and repeatedly divide the interval by two until the Black-Scholes price VBS(σ) differs no more than some

value tol from the market price Vn,φ,t_{. It is based on the fact that V}

BS is an

increasing function of σ [4]. In the simulations, we will set tol = 0.001, and initial values of a = 0.01 and b = 1. These are the upper and lower limit of the IVs that our model can compute.

Algorithm 1: Interval bisection while |error| > tol do

m = (a + b)/2; error = VBS(σ = m) − Vn,φ,t; if error > 0 then b = m; else a = m; end end

3.4.2 Measuring convexity

We wish to investigate how the convexity of the IV curve depends on the model parameters and settings. To do so, we need to define a measure of convexity. The convexity measure we will use is illustrated in Figure 3.2.

The IV curves our model generates consist of Nop points. To determine the

convexity, we consider all points that have distance 2 and draw an imaginary straight line between them. We then compute the difference on the y-axis between the midpoint of our imaginary line and the IV that was computed on that location on the x-axis. If the IV is higher than the midpoint of the line, this distance is

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negative. These distances are called d1, d2, d3, etc. We then define our convexity

measure as

C = 100 Pidi if di ≥ −10−3 for all i, or di ≤ 10−3 for all i

NaN else.

The multiplicative constant 100 is added merely for more convenient scaling. If all the values di are positive, then C is positive and the curve is convex. If all the

values di are negative, then C is negative and the curve is concave. If some of the

distances are positive and others negative, then C is NaN (not a number) and the curve is neither convex nor concave.

Instead of demanding that di ≥ 0 for all i or di ≤ 0 for all i for a curve to

be convex or concave, we allow slight deviations from this by demanding that di ≥ −10−3 for all i or di ≤ 10−3 for all i. This ensures that curves that are

almost entirely convex, but very slightly concave in the tails, do not get the value NaN.

Figure 3.2: Illustration of the convexity measure. The convexity of this IV curve is d1+ d2+ d3, multiplied by 100.

3.4.3 Principal Component Analysis

An important empirical fact about the volatilty smile is that it changes over time [1]. We therefore wish to examine not just the shape, but also the dynamical properties of the IV curves that our model produces. Unfortunately, each com-puted curve consists of Nop data points, leaving us with a high-dimensional vector

to keep track off. A very useful tool to make sense of these high-dimensional dynamics is Principal Component Analysis (PCA).

PCA is a coordinate transformation which seeks to identify the directions of maximum variation in the data [16]. It produces a set of eigenvectors ordered by how much of the variance in the data they explain. When a small number of eigenvectors accounts for almost all of the variance, the others can be disregarded and dimension reduction is achieved.

In relation to the volatility smile, it has been shown that three principal com-ponents can account for nearly all of the observed changes over time. These three principal components reflect the shift in overall level, the change in slope and the change in convexity of the curve, respectively [1]. The eigenvectors, as found by Qiu et al. are shown in Figure 3.3. Qiu et al. [1] report empirical findings of ap-proximately 77%, 20% and 3% for the respective proportions of variance explained by the first three principal components in skew-dominated markets.

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Figure 3.3: PCA eigenvectors. Figure from Qiu et al. [1].

3.5 Model parameters

The model parameters are summarized in Table 3.1. Parameters

Ntr Number of traders

Nop Number of call/put options in the market

FSP Fraction of traders that are speculators

µ Expected annual return of S

σ Annual volatility of S

µi Expected annual return of S assumed by speculator type i

σi Annual volatility of S assumed by speculator type i

∆t Size of one timestep

ρ Sensitivity of option price to excess demand

β Intensity of choice for switching between strategies

γ Determines how λSP depends on moneyness

ηSP Speculators’ activity level

k Number of speculator types

λARPCP Arbitrageurs’ activity level for the PCP strategy

λARBFS Arbitrageurs’ activity level for the BFS strategy

α Determines willingness of arbitrageurs to pay for butterfly spreads

w Memory in performance measure

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4 Results

In this chapter we present our simulation results. We first investigate how the shape of the IV curve depends on our model’s behavioural parameters in Section 4.1, by considering a static version of the model in which the stock price S and the speculator fraction ni,t are kept constant. Then, in Section 4.2, we investigate the

dynamical properties of the IV curve for various parameter sets, and for different choices of performance measure.

Some of the model parameters from Table 3.1 will be kept at the same value for all or most of the simulations. These standard parameter values are shown in Table 4.1. Unless otherwise stated, these are the values used in the simulations. They are largely the same as, or equivalent to, the values used by Qiu et al. [1].

Parameters Ntr 3000 Nop 11 FSP 0.5 ρ 0.1 γ 1.5 ηSP 0.1 λARPCP 6 λARBFS 1.5 α 2

Table 4.1: Standard values of model parameters

Additionally, the time to maturity of the options being traded will be kept constant at T − t = 1, and their moneyness at the time of trading is linearly spaced over a range of [0.75, 1.25]. The initial value of the stock price in the simulations is S0 = 20 and the initial values of the fractions are ni,0 = 1/k.

4.1 Shape of the implied volatility curve

In this section, we will keep the stock price and speculator fractions constant, so that we can investigate the shape of the IV curve that arises from different parameter values without yet considering the dynamical properties. We set ∆t = 1/1000 and consider the cases k = 1, k = 2 and k > 2.

4.1.1 One speculator type

We first investigate the case of a single speculator type with parameters µ1, σ1.

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unbalancing mechanism on the shape of the IV curves. We therefore consider four versions of our model:

a) Only speculators (S model), i.e. λARPCP = λARBFS = γ = 0.

b) Speculators and arbitrageurs, but no liquidity unbalancing (SA model), i.e. γ = 0.

c) Speculators, arbitrageurs and liquidity unbalancing, but no BFS arbitrage, i.e. λARBFS = 0.

d) Speculators, arbitrageurs and liquidity unbalancing (SAL model)

The results are shown in Figure 4.1. Like Qiu et al. [1], we have set λARPCP = 1

in this subsection and consider a downward sloping curve (µ1 = −0.08, σ1 = 0.2).

The IV curves qualitatively strongly resemble those found by Qiu et al. [1]. This is as expected, as our model with only one speculator type and no dynamics is very similar to the Qiu-Kandhai model. We thus find that PCP arbitrage, BFS arbitrage and liquidity unbalancing are necessary elements of the model in order to obtain IV curves with realistic shapes.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves call put (a) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves call put (b) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves call put (c) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves call put (d)

Figure 4.1: IV curves. a) S model, b) SA model, c) SAL model without BFS arbitrage, d) SAL model.

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Next, we consider the effect of the behavioural parameters µ1, σ1 and α on the

IV curve in Figure 4.2. These results too are similar to those found by Qiu et al. [1] (see Figure 2.1), although the interpretation of the parameters being varied is different. The parameters µ1 and σ1 in our model represent the assumptions of

the single type of speculator, whereas the parameters Mt

SP and DtSP in the

Qiu-Kandhai model represent the mean and standard deviation of the prices expected by the entire group of speculators.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves 1= 0.08 1= 0 1= 0.08 (a) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves 1= 0.2 1= 0.3 1= 0.4 (b) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves = 0 = 2 = 4 (c)

Figure 4.2: Shape of the IV curve for various values of µ1 (a), σ1 (b), and α (c).

Figure 4.2 shows that the skew of the curve is determined by the assumed mean µ1of S, whereas the assumed volatility σ1 determines the level of the curve. Qiu et

al. [1] show that a decrease in FSP (i.e. an increase in the fraction of arbitrageurs)

leads to a more convex curve. As shown in Figure 4.2, we can more specifically attribute the increase in convexity to an increase in butterfly spread arbitrage, in particular the parameter α, which controls how much money arbitrageurs are willing to spend on a butterfly spread. The parameter α thus plays a crucial role in determining the convexity of the IV curve. We find in particular that the convexity disappears altogether when α = 0. This is the case for horizontal (µ1 = 0) as well as skewed curves, as illustrated by Figure 4.3. This raises the

question whether the activity of BFS arbitrageurs is the only mechanism that leads to convexity in the IV curve. We will show in the next section that this is

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not the case. Heterogeneity among speculators gives rise to another mechanism for producing convex IV curves.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves 1= 0.08 1= 0 1= 0.08

Figure 4.3: IV curves for α = 0.

4.1.2 Two speculator types

We now turn to the case of two speculator types, with parameters (µ1, σ1) and

(µ2, σ2). In particular, we consider the case α = 0. This case corresponded to

flat IV curves in the previous section, so by setting α = 0 we can investigate the influence of speculator heterogeneity on the curve’s convexity. Like the stock price, we will keep the fractions n1,t and n2,t fixed at this stage.

We first consider speculator types with the same belief about the drift µ, but different beliefs about the volatility σ. Figure 4.4 shows the results for (µ1, σ1) =

(0, 0.1), (µ2, σ2) = (0, 0.6). Figure 4.4a shows the distributions of stock returns

assumed by type 1 and type 2. Figure 4.4b shows the average of these distributions. It can be interpreted as the distribution that is collectively assumed by a market populated in equal numbers by speculators of type 1 and type 2. For comparison, a normal distribution is also plotted. We find that the speculator distribution is symmetrical, but has heavier left and right tails than the normal distribution. We have seen in Figure 1.1 that an implied distribution of this shape corresponds to a non-skewed convex volatility smile, and this is indeed the shape of the IV curve that our model produces, as shown in Figure 4.4c.

Heterogeneous beliefs about volatility are thus a second mechanism by which our model can produce convex IV curves. It should be noted, however, that the convexity in Figure 4.4c is not very strong, despite the rather extreme difference in belief about the volatility. Compared to the effect of the α parameter in the previous section, this mechanism thus seems to be of secondary importance to explain the convexity of the IV curves in our model.

Next, we assume that the speculators have different beliefs about both the drift and the volatility. Specifically, (µ1, σ1) = (0.11, 0.1) and (µ2, σ2) = (−0.09, 0.6).

Compared to the previous case, the distributions assumed by type 1 and type 2 in Figure 4.5a have shifted to the right and to the left, respectively. This leads to a collective speculator distribution in Figure 4.5b that is left-skewed. The corresponding IV curve in Figure 4.5c is convex and downward sloping. This is

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2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Probability density Speculator distributions Type 1 Type 2 (a) 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 Probability density

Collective speculator distribution Speculators Normal (b) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves call put (c)

Figure 4.4: Heterogeneity in assumed volatility: (µ1, σ1) = (0, 0.1), (µ2, σ2) =

(0, 0.6). a) Assumed distributions for return on the stock, b) Average of assumed distributions compared to a normal distribution with parameters (0, 0.35), c) Re-sulting IV curves.

consistent with the IV curve and implied distribution that we’ve seen in Figure 1.2.

In the previous subsection, we saw that, in a market with homogeneous specu-lators, the IV curve is downward sloping when the speculators assume a negative drift; µi < 0. Figure 4.5 demonstrates that heterogeneous beliefs about µ and

σ can also cause a downward sloping curve, even when speculators on average assume a drift of (approximately) zero.

In a similar way, heterogeneous beliefs can cause an upward sloping smile, even when the assumed drift is on average zero. This is demonstrated in Figure 4.6, where we have set (µ1, σ1) = (−0.1, 0.1) and (µ2, σ2) = (0.1, 0.6). These

parameter choices lead to a right-skewed collective distribution and an upward sloping IV curve, shown in Figures 4.6b and 4.6c.

These cases demonstrate that our model can produce convex IV curves even when α = 0. However, this is only the case for some choices of the parameters µ1, σ1, µ2 and σ2. To investigate how the convexity of the IV curve depends on

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parame-2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Probability density Speculator distributions Type 1 Type 2 (a) 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 Probability density

Figure 4.5: Heterogeneity in assumed drift and volatility: (µ1, σ1) =

(0.11, 0.1), (µ2, σ2) = (−0.09, 0.6). a) Assumed distributions for return on the

stock, b) Average of assumed distributions compared to a normal distribution with mean 0.01 and standard deviation 0.35, c) Resulting IV curves.

ter combinations. The convexity of each curve was then determined using the convexity measure defined in Section 3.4 and plotted in a heat map.

Figure 4.7a shows the convexity heat map from combinations of σ1 and σ2.

µ1 and µ2 were kept constant at 0 for these simulations. On the diagonal where

σ1 = σ2, the convexity is (close to) zero. The larger the disagreement about σ,

the stronger the convexity.

Figure 4.7b shows the convexity heat map from combinations of µ1 and µ2.

Now, σ1 and σ2 have been kept constant, at 0.1 and 0.6, respectively. In the

majority of cases, the resulting IV curve is neither convex nor concave, as shown by the white spaces in the heat map. The curve is only convex when µ1 ≈ −µ2,

i.e. when µ1 and µ2 are on average close to zero. Moreover, the curve is no longer

convex when µ1 or µ2 becomes too large, even if µ1 ≈ −µ2.

When α = 0, the model thus only produces convex IV curves for specific values of µ1, σ1, µ2 and σ2. Specifically, the curve is convex when the difference between

σ1 and σ2 is large, µ1 ≈ −µ2, and µ1 and µ2 are approximately between -0.1 and

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2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Probability density Speculator distributions Type 1 Type 2 (a) 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 Probability density

Figure 4.6: Heterogeneity in assumed drift and volatility: (µ1, σ1) =

(−0.1, 0.1), (µ2, σ2) = (0.1, 0.6). a) Assumed distributions for return on the stock,

b) Average of assumed distributions compared to a normal distribution with mean 0 and standard deviation 0.35, c) Resulting IV curves.

(a) (b)

Figure 4.7: Convexity heat maps for α = 0. Colour bars to the side of each graph show the convexity level associated with each colour. White spaces correspond to an IV curve that is neither convex nor concave.

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For comparison, we have also generated heat maps for the case α = 2, shown in Figure 4.8. Note that the overall levels of convexity are considerably higher than in the case α = 0, and that the diagonal patterns of Figure 4.7 are no longer visible in Figure 4.8. This further indicates that the trading activity of the BFS arbitrageurs is a stronger determinant of the convexity than the heterogeneity of speculators.

(a) (b)

Figure 4.8: Convexity heat maps for α = 2. Colour bars to the side of each graph show the convexity level associated with each colour.

4.1.3 More speculator types

We now investigate the effect of increasing the number k of speculator types, in particular on the convexity of the curve. First, consider the cases k = 2, k = 4 and k = 6 with heterogeneous beliefs about the volatility σ. In each case, we will set µi = 0 for all i. The beliefs about volatility are linearly spaced over the speculator

types from 0.1 to 0.6. That is, • (σ1, σ2) = (0.1, 0.6) for k = 2,

• (σ1, ..., σ4) = (0.1, 0.267, 0.433, 0.6) for k = 4,

• (σ1, ..., σ6) = (0.1, 0.2, 0.3, 0.4, 0.5, 0.6) for k = 6.

The speculators’ collective distribution for each case is shown in Figure 4.9a. A normal distribution is also plotted for comparison. We find that the heavy-tailedness of the speculator distribution is less and less pronounced as k increases. Correspondingly, the IV curves in Figure 4.9b are less and less convex for increas-ing k.

Using the convexity measure defined in Section 3.4, we can quantify the con-vexity of the IV curve for each value of k. These results are shown in Figure 4.10. For each k, we have, as before, set µi = 0 for all i, and the σi are linearly spaced

from 0.1 to 0.6. The graph confirms that there is an inverse relationship between the number of speculator types and the convexity of the curve.

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2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Rate of return 0.0 0.5 1.0 1.5 2.0 Probability density

Collective speculator distributions 2 types 4 types 6 types Normal (a) 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Moneyness 0.1 0.2 0.3 0.4 0.5 0.6 Implied volatility IV curves 2 types 4 types 6 types (b)

Figure 4.9: a) Speculator distributions for 2 types, 4 types and 6 types, com-pared to a normal distribution with mean and standard deviation (0, 0.35). b) Corresponding IV curves. 2 4 6 8 10 k 0.8 1.0 1.2 1.4 1.6 Convexity Convexity as a function of k

Figure 4.10: Convexity of the IV curve as the number of speculator types increases.

4.2 Dynamical properties

In the previous section, the stock price S and the fractions ni,t were kept constant,

meaning that the volatility smile did not evolve over time. We will now consider a dynamic version of the model in which S changes according to the stochastic process described by Equation (3.1) and the fractions evolve according to a per-formance measure. We first consider the case of a single speculator type and then investigate the dynamics under the three performance measures that were defined in Section 3.

We will use PCA and time series of IVs and fractions to analyse the dynamical properties of the IV curves. The simulations run over 20000 timesteps with ∆t = 1/200000. The total simulated time period is thus relatively short (1/10 of a year), in order to get more consistent results for different price paths of S. For the PCA, IV curves were selected after every 150 timesteps, starting after a ‘warm up’ period of 1000 timesteps. For the time series, we plot the IV over time for the

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left most, right most and middle point on the curve, i.e. the IVs corresponding to moneyness 0.75, 1 and 1.25. Only the IV curves for calls are used for the PCA and time series analysis.

4.2.1 One speculator type

In the case of one speculator type, the dynamics of the IV curve are entirely driven by the stock price process S. We saw in Section 4.1 that the level, slope and convexity of the curve are determined by the assumed drift and volatility levels µi and σi, as well as the arbitrage parameter α. When there is only one

speculator type, these variables are all constant. Consequently, we can expect the IV curve to remain largely constant over time.

0.8 0.9 1.0 1.1 1.2 Moneyness 0.4 0.2 0.0 0.2 0.4 Factor loading Eigenvectors PC1 PC2 PC3 (a) PC1 PC2 PC3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Proportion

Proportion of variance explained

(b) 0 2500 5000 7500 10000 12500 15000 17500 Timesteps 0.30 0.35 0.40 0.45 0.50 0.55 Implied volatility IV timeseries mon = 0.75 mon = 1 mon = 1.25 (c)

Figure 4.11: PCA results and IV time series for a single speculator type. Param-eters are µ1 = −0.8 and σ1 = 0.35.

The time series in Figure 4.11c confirm that there is fairly limited change in the IV curve over time. The plotted IVs vary only over a range of approximately 0.02. Moreover, the time series follow approximately the same pattern, so the rel-ative position of each point on the IV curve stays largely the same. The variance in the OTM and ITM IVs is however slightly larger than in the ATM IVs. This indicates that the variation that does take place is mostly in the overall level and

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the convexity of the curve. Indeed, the PCA results in Figures 4.11a and 4.11b show that the variance is explained almost entirely by the first two eigenvectors, which represent (approximately) overall level and convexity. The third eigenvec-tor relates to the slope of the curve, but its proportion of variance explained is negligible (approximately 0.0003).

4.2.2 Performance measure A

We now introduce heterogeneity again by defining more than one speculator type and letting the fraction of each type evolve according to performance measure πa

i,t

defined in Section 3.2.

We have investigated time series for the fractions ni,t for many different

param-eter values, in particular of µi, σi, β, w and ∆t. We have found that the evolution

of the fractions is frequently odd or nonintuitive and very sensitive to some of the aforementioned parameters, as well as to the particular price path of S. This is illustrated by the time series in Figure 4.12. In Figures 4.12a, 4.12b and 4.12c, only the beliefs about σ are heterogeneous. However, wrong estimates of σ do not always lead to lower performance scores than assuming the correct value of σ. Underestimates of σ in particular seem to lead to higher performance scores than correct beliefs about σ. In Figures 4.12b and 4.12c we even see, counterintuitively, that the larger the underestimate of σ, the higher the performance score of this speculator type.

In Figure 4.12d, only beliefs about µ are heterogeneous, and ∆t = 1/200000. We find here that the time series produces a wave-like pattern in which the fraction falls and rises successively. The underlying reason for this is unclear.

Because of the difficulties in calibrating the fractions with this performance measure, attempts to obtain useful PCA results about the dynamics of the IV curve have proven unsuccesful. We thus find performance measure πa

i,t to be

unsuitable for the purposes of this model.

0 1000 2000 3000 4000 5000 Timesteps 0.0 0.2 0.4 0.6 0.8 1.0 n1,t Fraction of type 1 (a) σ1 = 0.3, σ2 = 0.4 0 1000 2000 3000 4000 5000 Timesteps 0.0 0.2 0.4 0.6 0.8 1.0 n1,t Fraction of type 1 (b) σ1 = 0.3, σ2= 0.2

Figure 4.12: Time series of the fraction n1,t for various parameter choices. Unless

otherwise stated, µ1 = µ2 = µ = 0, σ1 = σ2 = σ = 0.3, β = 0.1, w = 0.99 and

A Heterogeneous Agent Model for Volatility Smile Dynamics