Bubbles in the Heston Stochastic Local Volatility Model Msc Thesis Financial Econometrics University of Amsterdam

Author: Felix Eikenbroek Supervisor: Peter Boswijk

May 30, 2014

Abstract

When the popular Heston model is extended with a local level dependent volatility part the Heston Stochastic Local Volatility (HSLV) model is created. Through this extension the model is able to capture more dynamics of the stock price movement and even those of an asset bubble. We provide a proof that shows that for certain parameter values the HSLV model is a strict local martingale and hence mathematically able to detect bubbles. By applying the Indirect Inference method we provide an estimation method for non-aﬃne stochastic local volatility models such as the HSLV model and show that the parametric estimation of this work is an improvement of the non-parametric approach used before to detect bubbles of certain assets during the dot-com bubble. Contents

1 Introduction 4

2 History 7 2.1 Tulip Mania ...... 7 2.2 Dot-Com Bubble ...... 8 2.3 Real Estate Bubble ...... 9 2.4 Common Factors ...... 10

3 Framework 11 3.1 Economic Framework ...... 11 3.2 Mathematical Framework ...... 13

4 Models 18 4.1 Geometric Brownian Motion ...... 18 4.2 Heston Model ...... 20 4.2.1 Stochastic Volatility Models ...... 23 4.3 CEV Model ...... 23 4.3.1 Local Volatility Models ...... 26 4.4 HSLV Model ...... 27 4.4.1 HSLV Model vs. Heston Model ...... 30 4.4.2 Stochastic Local Volatility Models ...... 32

5 Martingale Property 34 5.1 Martingale Measure ...... 34 5.2 Proof of Martingale Property ...... 36 5.3 Failure of Martingale Property ...... 38 5.3.1 α > 1&ρ < 0 ...... 39 5.3.2 α > 1&ρ > 0 ...... 42 5.3.3 α < 1 ...... 44

6 Estimation 46 6.1 Indirect Inference ...... 46 6.2 Asymptotic Distribution of the Estimator ...... 49

2 7 Data 51 7.1 InfoSpace ...... 51 7.2 eToys ...... 52 7.3 Geocities ...... 53

8 Results 55 8.1 InfoSpace ...... 55 8.2 eToys ...... 58 8.3 Geocities ...... 59 8.4 Auxiliary Model ...... 60

9 Conclusion 62

A Appendix 66

3 1 Introduction

When we are speaking of an asset bubble a pattern of a sharp price increase followed by a huge decrease is the first thing that comes to mind. Since the 17th century financial bubbles have been documented and to this day they still occur in financial markets. In contrast to the first known bubble (Tulip Mania in 1637) the bursting of recent financial bubbles,the housing-bubble in 2007 and the dot-com bubble around 2000, caused huge losses for financial institutions and affected the global economy. The unusual pattern of the stock price and its corresponding huge impact makes the asset bubble a subject of interest for a lot of researchers. The research on this subject varies from the existence of a bubble in an economic equilibrium to the question regarding the modelling of an asset bubble. The research in the field of economics focusses on the different restrictions on the economy such that a bubble can exist while the field of mathematical finance impose less restrictions on the economy and tries to find the effect on the price of an asset and derivatives when the underlying asset exhibit a bubble, using the local martingale definition. The reports of financial bubbles in the field of mathematical finance have defined that when the price process of an asset is a strict local martingale under the risk neutral measure, the asset exhibits a bubble. This thesis approaches an asset bubble as a strict local martingale as well. The main question that we try to answer in this work is: is it possible to detect an asset bubble by estimating the parameters of the Heston Stochastic Local Volatility model? By the word ’detect’ we mean the possibility to estimate the parameters of the HSLV model such that, according to a predefined definition, the stock price indeed exhibits a bubble. Since the introduction of a stochastic differential equation to drive a stock price process, the process has been extended to a multi-dimensional stochastic differential equation to give a better fit of the true dynamics of the stock price. The addition of a stochastic differential equation generates stochastic volatility models in which a nonzero risk premium for volatility is considered. Mainly, this is useful for pricing derivatives, where the addition of an extra stochastic differential equation give rise to a set of call prices that are close to the observed market prices. The representation of a stock price process also becomes the basis of this thesis where we choose to extend the Heston model. The Heston model is a popular way of describing a stock price process because it is able to price options in closed-form and close to the true price. In addition to the Heston model the volatility can be extended with a local volatility part, which means that the volatility is, besides a random process, also directly dependent on the price of the stock itself. Earlier work showed that the

4 derivative prices based on this model are an even better fit for the market price of the derivatives. This extended Heston model is known as the Heston Stochastic Local Volatility (HSLV) model. Besides the fact that the model is a better fit to the observed derivative prices, the HSLV model is also able to detect an asset bubble by the way we choose to extend the Heston model in this thesis. In other words, the parameter that is related to the local volatility part adjusts such that whenever an asset exhibits a bubble, the HSLV model is a strict local martingale instead of a martingale. Hence in order to find out whether a certain asset exhibited a bubble we have to estimate the parameters of the HSLV and check whether the model is a strict local martingale. Usually, the parameters are estimated by calculating the call prices (that depends on the parameters) and minimize the difference between the call prices from the assumed model and the true observed call prices. This method is called calibrating. However, call prices are not available in this thesis so we have to stick to the historical stock price. It is not possible to find the closed-form likelihood function or moments of the HSLV model (for Heston model as well) and hence a different estimation method should be applied. Since it is easy to simulate a HSLV model it makes sense to apply a simulation based method. A simulation based method makes it possible to estimate parameters of the HSLV model by inferring from the simulated stock prices. The Indirect Inference method is such a type of estimation method and has proven to be success- ful. This method estimates the parameters by taking an auxiliary model that can be estimated by the Maximum Likelihood principle and tries to find the parameters by minimizing the squared distance between the parameters of the auxiliary model based on historical observed stock price and the parameters of the auxiliary model based on the simulated stock price. Although the parameters of the assumed underlying model are not directly estimated, the auxiliary model is based on the observed stock price hence dependent on the true parameters of the model and the name indirect inference. The reason to estimate the HSLV model and check whether a stock exhibits a bubble or not is based on the work of Jarrow et al. [19]. The non-parametric approach of this work had it shortcomings by not being able to conclude whether a stock exhibits a bubble or not in some cases even if the observed price process (high peak followed by quick decline) and the information on the stock (a stock during the dot-com crisis that went bankrupt) was accepted as a bubble. So, this thesis tries to detect bubbles using a parametric approach by estimating the parameters of the HSLV model using the Indirect Inference method and checks whether the stock prices exhibit a bubble according to these estimates. Furthermore, whenever it is found out that the stock price indeed exhibited a bubble, traders could be suspicious of a next price increase of the stock because it may

5 be a bubble as well. The thesis is arranged in the following way: first we discuss some historical financial bubbles and discuss the similar characteristics of these historic events. Second, we provide an economic framework in which a bubble can exist. This framework will be translated to a mathematical finance framework and the definition of a bubble in the context of martingales will be discussed. Subsequently, the HSLV model, which is able to generate an asset bubble, is introduced and an overview of the published literature about this type of model is given where after we prove that the HSLV model is able to detect a bubble. Then the Indirect Inference method for estimation the parameters of this model is introduced. Finally, the results will be discussed.

6 2 History

This section gives an overview of some financial bubbles. We discuss the first documented financial bubble (tulip mania in the 17th century) and the two most recent bubbles which impacted the global economy.

2.1 Tulip Mania

During the second half of the 16th century the tulip was introduced in the Netherlands. The Dutch started to seed the tulip bulbs at the end of the 16th century and found out that the weather conditions in the Netherlands were favorable for the tulips. The beauty and rareness of the flower made the tulip a status symbol for the wealthy and it created a market for the durable bulbs. The bulbs were traded at the end of the season (the spot market) and during the year tulip traders signed a contract which gave the buyer the right to buy the tulips at the end of the season, similar to the future contracts nowadays. The Dutch were able to develop different tulips variants such that the bulbs that created unique patterns had a higher price than the more common tulips. Because of the increasing popularity, not only professional but also speculators joined the tulip bulb trade market (Garber [12]). In the beginning of the market only bulbs were traded that had already become a tulip. The participation of speculators in the market caused a trade in bulbs that did not bloom to a tulip yet (so-called unbroken bulbs) as well. Till 1636 the prices of the bulbs rose steadily. At its peak, a contract would change hands more than ten times in a day and a single bulb was traded for a price of more than ten times the one-year salary of a skilled craftsman. In February 1637, the price decreased drastically; buyers were not able to pay the agreed price and the distrust between the traders caused bulbs to be sold for less than 10 percent of their earlier price. The above described chronicle is known as the tulip mania and is the first documented financial bubble in history. Even though the price correction of the bulb price is extremely large (decrease of more than 90 percent) intervention of the local government prevented damage to the whole Dutch economy. However, recent history has shown that the collapse of a bubble has huge impact on the global economy (dot-com bubble around 2000 and the housing bubble in 2008). The pattern of a bubble is similar when we review all the documented financial bubbles: a huge increase in the price is followed by a sudden and quick drop. Literature about the cause of bubble does not point out one specific cause of the origin of a bubble. Irrational behavior, restriction of short-shelling, excessive leverage, feedback mechanism

7 and heterogeneous believes are diﬀerent causes discussed [32].

2.2 Dot-Com Bubble

During the nineties of the last century, internet became part of households and companies, instead of being a medium strictly used by the government of the United States. The possibilities seemed unlimited and this sentiment became also apparent on the trading floor. In 1995 the initial public offering of Netscape (a web-browser) at a price of 28 dollars shot on the first day to a price of 75 dollars and closed at 58.25 dollars eventually [26]. This event is considered as the start of the financial bubble. A lot of companies that focused their business on internet had huge increases at their first day of going public. The low interest rate at that time caused also a lot of venture capitalism which drove the price up. It is assumed that one of the main reasons for the huge increase during the first day of a stock going public was the informational friction between the different traders. Since internet was a new phenomenon some people interpreted the possibilities differently. Also, some people neglected traditional measures such as the P/E (Price/Earnings) ratio because of the confidence in the possibilities of internet and the companies using this new medium. From 1999 to 2000 the NASDAQ 100 (stock index in the United States consisting of the 100 largest non-financial companies of the NASDAQ) doubled without any fundamental news that could explain this increase [5]. After this peak, early 2000, the Federal Reserve increased the interest rate more than 6 times and the index dropped by more than 75 % in 2.5 years. Again there was no fundamental news that declared this drop.

Figure 1: NASDAQ 100 during the dot-com bubble

The fact that a huge proﬁt and loss was observed within a few years and without any

8 fundamental news is in correspondence with the deﬁnition of a bubble; the fundamental value of the companies did not change much, only the market price was subject to large deviations from the fundamental price. The main reason for this large diﬀerence was the fact that some investors had other information/interpretation of a certain internet stock than others.

2.3 Real Estate Bubble

In the aftermath of the dot-com bubble (mid 2001) the United States of America suffered a mild recession. The Federal Reserve tried to stimulate the spending and investment by lowering the interest rate. People became suspicious towards stocks as an investment after the dot-com bubble and in combination with the low interest rate real estate became an attractive alternative. The mortgage interest rate was relatively low as well: since its peak in 1981 at the rate of more than 18 percent the rate was around 6 percent at the beginning of the 21st century. The low interest rate resulted in a lot of Adjustable Rate Mortgages. This type of mortgage had an interest rate which was determined by the short-term interest rate. This way, the monthly payments were lower compared to a fixed interest rate mortgage. The Adjustable Rate Mortgages had the effect of houses becoming affordable to more people. In advance, different type of mortgages were con- structed with contracts were the monthly payment only had to cover the interest instead of the both the interest and principal. The increase on the demand side increased the house price. Furthermore, the low interest rate made it attractive to invest with bor- rowed money (leveraging) for financial institutions. A popular financial instrument was the Mortgage Backed Security. Rating agencies gave this type of securities a high rating (hence low risk) which made it, combined with the relative high return, an attractive investment. As a result, more financing was available for mortgage lending. The relaxed standard on the monthly payments of the mortgage were a direct effect of the relaxed requirements for having a mortgage. Home ownership was stimulated by the government through tax deductibility and relaxing the standards for obtaining a mortgage in terms of reducing the down payment and the required salary. All the circumstances and policy changes described above were on the assumption that the housing price would increase; since the Great Depression the housing price had not decreased in any year. The combination of relaxing standards of obtaining a mortgage and the assumption that the house price would increase created a huge opportunity for speculators. Mortgage loans were issued with the intention of selling the house even when the buyers could not fulfill the monthly obligation. Investment banks

9 were buying these mortgages to issue more mortgage backed securities and were still rated as low risk by the credit rating agencies even if the mortgage was not affordable by the owner. The house price reached its peak in 2006. After that, the foreclosure rate increased by 46 percent in the second half of 2006 and by 75 percent in 2007 [17]. Mortgage lenders did not receive the expected payments of the mortgages, investment banks made huge losses on the mortgage backed securities and insurance companies who sold insurances with mortgages as underlying as well. The aftermath of the housing bubble was the financial crisis (because of all the financial institutions that were involved in the housing market) that started in 2008 and affected the world economy.

2.4 Common Factors

This section only discussed three bubbles extensively while the last century more bubbles were documented in the financial market. A common factor in the bubbles discussed is the speculation effect. That is, buying an asset with the intention of selling it at a higher price. The reason for speculation is that speculators expect the price trend to be consistent with the recent historical trend. The enthusiasm of speculators attracts more speculators and hence a further increase in the price, this is called the feedback mechanism. Before the collapse of the bubble this is a profitable strategy. However, the existence of the bubbles means that the market price is different from the fundamental price and a correction can take place any time. During the housing bubble, it was realized that the price increase was not sustainable and that the house price did not reflect it’s ’true value’. After the first price drop, a lot of people were not able to pay their mortgage. As a result the supply of houses increased and the price decreased more. Also during the tulip mania speculators saw an opportunity by buying the tulips and selling it for a higher price. After the market realized that the price increase was not sustainable, the price dropped. Notice that speculators joining the market were not solely responsible for the existence of a bubble in the housing market; policy change that made the requirements for obtaining a mortgage (too) low caused an increase in the housing price which was not sustainable as well. Although speculators also joined the market during the dot-com bubble, one of the main reasons for the large price increase was the heterogeneous beliefs of investors [2] . Internet was a new phenomenon and each investor valuated the company differently. This explains (partly) the price increases at the first days of internet stocks going public.

10 3 Framework

Roughly speaking, a bubble is a deviation between the market price and the fundamental price of an asset. To better understand this concept we have to look how the fundamental price of an asset is determined and how a bubble can exist. First we show how a bubble can exist in a discrete time economy. This leads to an economic framework which will be extended to the existence of a bubble in a continuous time economy by applying the martingale theory of bubbles, which is the main focus of this thesis.

3.1 Economic Framework

We assume that the market is efficient, this means that all information that is known about the asset is incorporated in determining the future returns, hence the price. The efficient market hypothesis requires the agents to have rational expectations. The fundamental price in a complete market is determined by the asset’s discounted expected cash flows using the martingale measure. Since we consider an incomplete market we know by the Second Fundamental Theorem of Asset Pricing there are multiple risk-neutral measures [23]. The market chooses a risk-neutral measure Q that is in correspondence with the measure used for determining the option price with the same underlying asset. Under this measure Q the discounted wealth process obtained by owning a stock and the corresponding dividend is a martingale.

Given the probability space (Ω, F, Ft∈[0,K], P) where F is a sigma algebra, Pt and Dt are adapted to Ft and P is the real probability measure under which the process P moves. Under the risk-neutral probability measure Q and the rationality of expectations we know that : Pt+1 + Dt+1 EQ[Rt+1|Ft] = r where Rt+1 = − 1 (3.1) Pt where r is the risk free rate. Equation (3.1) can be rewritten as:

P + D P = t+1 t+1 |F (3.2) t EQ 1 + r t Using the Law of Iterated Expectations gives the following expression of the price [4]:

" K i # " K # X 1 1 P = D |F + P |F (3.3) t EQ 1 + r t+i t EQ 1 + r t+K t i=1 Usually for K → ∞ the second expectation tends to zero which means that the price

11 equals the sum of the expected discounted dividend payments. This ﬁrst expectation is called the fundamental value. Hence when the second expectation tends to zero the price of the asset equals the fundamental value. However, when we relax the assumption that the second expectation converges to zero there are inﬁnitely many solutions to (3.2). However, Campbell [4] noticed that some solutions (negative bubbles, the start of a bubble within an asset pricing model and a bubble within an asset that has an upper- limit) are excluded from the solution. Nevertheless, all the solutions can be written in the form:

Pt = Pfund + Bt, (3.4) where

B B = t+1 |F (3.5) t EQ 1 + r t

Bt is known as the rational bubble. It refers to a bubble because it causes a difference between the market price and the fundamental price. Since the rationality assumption is not violated in this setting the term rational bubble is used. Camerer [3] noticed that since the current price of the bubble is equal to the discounted expected value of the future rational bubble, the rational bubble is still consistent with the efficient market hypothesis. And since the expected growth rate of the bubbles equals r (the risk-free rate) traders do not have an arbitrage opportunity. The expected growth rate of r for the rational bubble does not mean that we cannot create a setting such that a bubble can burst [4]. Empirically, the rational bubble described above falls short. The trading volume tends to increase during a bubble, something that is neglected in this model and added in the work of Scheinkman and Xiong [32]. They created a market in which a bubble exists by the heterogeneous beliefs in the expected dividend payments (as during the dot-com bubble) and a speculation effect is added. In their context the bubble can exist whenever there is someone who does not own the stock at this moment has a possibility of having a higher reservation price in the future. This implies that the current price should be higher than the reservation price of the current owner and hence a difference between the fundamental value (the reservation price of the owner) and the current price is created, something we consider a bubble. As history has shown, speculation is one of the main reason for the existence of a bubble. Scheinkman and Xiong [31] also gave an overview of the existence of the bubble through the speculation motive. Speculation exist whenever the willingness to pay of an agent is higher than his expected discounted

12 dividend stream. This argument can only exist whenever it is costly to short sell the asset or even impossible. To extend the diﬀerent models explained in Scheinkman [32], a lot of assumptions on the agents and restrictions have to be put on the economy for a bubble to exist. Extending the model to the mathematical ﬁnance framework demands only a few restrictions on the economy and considers bubble in a continuous time.

3.2 Mathematical Framework

In the quantitative/mathematical ﬁnance literature the existence of an asset bubble is quantiﬁed and can be distinguished to three cases (Jarrow et al. [20] [22]):

1. A bubble is a local martingale that is uniformly integrable if it has an inﬁnite lifetime

2. A bubble is a local martingale but not uniformly integrable a ﬁnite lifetime and unbounded stopping time

3. A bubble is under the risk-neutral measure Q a strict local martingale

For this work we evaluate assets with a finite lifetime (t ≤ T ) and for t > T the asset is worthless, this means the first two types of bubbles are excluded. In advance, we assume a frictionless market (i.e. no transaction costs) and a probability space (ω, F, Ft∈[0,T ∗], P) where F is a sigma algebra, St and Dt adapted to Ft and P the probability measure under which the process S moves. This means that the main question asked with regards to the analysis of different asset bubbles is whether the bubble is a strict local martingale under the risk-neutral measure. In earlier work this is called the martingale theory of bubbles (Jarrow and Protter [21]). In the previous subsection we showed in (3.3) that the price equals the expected amount of dividend gained and asset price after K time steps. Similar to this expression, in the mathematical finance context we use the term wealth Wt (for t ∈ [0,T ]) to express a continuous function that is created by owning the stock and receiving the dividend payments. Z t 1 Wt = St + Bt dDu (3.6) 0 Bu

St represents the asset price at time t, Dt the accumulated dividends up to time t and

Bt the value of 1 dollar invested at t = 0 in the bank account. The restrictions on the economy we have to impose are based on the work of Jarrow et al. [22]. They provided the restrictions on the economy for a bubble to exist. In an incomplete market beside

13 the No Free Lunch with Vanishing Risk (NFLVR) condition the No Dominance (ND) condition is imposed as well on the economy. The NFLVR condition is similar to the no arbitrage condition which makes it not possible to make a risk-less profit. The ND condition makes sure that there is no other portfolio that creates the same cash flow as the evaluated assets and has a lower construction price. In this setting a bubble can exist in an incomplete market. The No Dominance condition prevents bubbles to exist in a complete market [20]. The No Free Lunch with Vanishing Risk restriction prevents arbitrage opportunities by traders. Otherwise the arbitrage opportunity that is created by bubbles would quickly remove the bubble from the market. Historical events have shown that bubbles do exist and hence the NFLVR restriction is justified. This NFLVR restriction makes sure that the trading strategies are admissible (has a lower bound) at all time. It is now impossible to sell an asset exhibiting a bubble short (which seems like an easy arbitrage opportunity) because the losses can become too high when the stock reaches its peak. In the mathematical finance field, bubbles are related to the martingale property of a stochastic process. To better understand the concept of martingales and local martingales we give a definition of both properties.

Deﬁnition 1. A stochastic process X : T · Ω → R+ is a martingale with respect to ﬁltration Ft∈[0,T ] on probability space (Ω, F, Ft∈[0,T ], P) whenever for all t E[Xt] < ∞ and for k ≤ t:

EP[Xt|Fk] = Xk

Deﬁnition 2. A stochastic process X : T · Ω → R+ is a local martingale with respect to ﬁltration Ft∈[0,T ] on probability space (Ω, F, Ft∈[0,T ], P) whenever there exist a postive, ∞ increasing and diverging sequence {τn}n=1 such that:

Xmin{t,τn} is a P martingale for all n Under the NFLVR restriction and the First Fundamental Theorem of Asset Pricing there exist at least one risk neutral probability measure Q that is equivalent to the real probability measure . By this fundamental theorem a positive process Wt is a local P Bt martingale under the risk neutral measure. By deﬁnition, nonnegative local martingales are supermartingales. Note that every martingale is local martingale but not every local martingale is a martingale. The stochastic processes which satisfy the latter are called strict local martingales. The concept of a strict local martingale is important because it is a necessary and suﬃcient condition for an asset to exhibit a bubble as we will show in the next theorem.

14 Before we can prove this statement we have to deﬁne a bubble for a bounded asset with a ﬁnite lifetime:

Definition 3. In a finite lifetime, when a bounded asset exhibits a bubble (of type 3) β it is defined by :

∗ βt = St − St

∗ Where St denotes the market price and St the fundamental price of the asset.

The following theorem shows that the existence of a measure Q, such that the fundamental price is a strict local martingale, is necessary and suﬃcient for an asset with a ﬁnite lifetime to exhibits a bubble. The theorem was already stated in [19] but a proof is provided here as well.

Theorem 1. An asset with a ﬁnite lifetime and a bounded stopping time can exhibit a bubble if and only if the wealth created by owning the asset is a strict local martingale measure under the risk neutral measure Q.

Proof. We mentioned before that under the No Free Lunch with Vanishing Risk and the First Fundamental Theorem of Asset Pricing there exist at least on measure Q such that Wt is a local martingale. Bt Since we assume for the asset to have a ﬁnite lifetime the fundamental price of the ∗ asset St is determined by the sum of the expected dividend payments and is similar to the deﬁnition of the fundamental value in the discrete, economic framework (3.3)

Z T ∗ 1 St = EQ[Bt dDu|Ft] (3.7) t Bu ∗ The fundamental wealth process Wt is now deﬁned (using (3.6)) as

Z t Z T Z t Z T ∗ ∗ 1 1 1 1 Wt = St +Bt dDu = EQ[ dDu|Ft]+Bt dDu = EQ[Bt dDu|Ft] 0 Bu t Bu 0 Bu 0 Bu (3.8) The wealth process at time t is, in contrast to the fundamental wealth process (3.8), ∗ determined by the market price St (and not the fundamental price St ) at time t. The diﬀerence in the wealth process and fundamental wealth process at time t equals the

15 diﬀerence in the market price and the fundamental price of the asset.

Z T ∗ 1 Wt − Wt = Wt − EQ[Bt dDu|Ft] = 0 Bu Z t 1 Z T 1 St + Bt dDu − EQ[Bt dDu|Ft] = (3.9) 0 Bu 0 Bu Z T 1 ∗ St − EQ[Bt dDu|Ft] = St − St t Bu

In the early assumptions we mentioned that at the maturity of the asset, there is ∗ no liquidation value of the asset and hence ST = ST = 0. This assumption can also be replaced by a terminal payoﬀ at t = T . Whether or not there is a terminal payoﬀ, when ∗ we consider both the fundamental wealth process Wt and the wealth process Wt, their wealth at maturity T is equal:

Z T Z T ∗ 1 1 WT = ST + BT dDu = BT dDu = WT (3.10) 0 Bu 0 Bu

∗ We can show that Wt is a martingale by applying equation (3.8) and the tower Bt property:

∗ Z T Z T ∗ Wt Bt 1 1 Wt−1 EQ[ |Ft−k] = EQ[[ dDu|Ft|Ft−k]] = EQ[ dDu|Ft−k] = for k = 1, ..., t Bt Bt 0 Bu 0 Bu Bt−k (3.11) In (3.9) we have shown that a bubble exists whenever there is a diﬀerence between the fundamental price and the market price there is also a diﬀerence between the fundamental ∗ wealth Wt and wealth Wt. This leads to the existence of a bubble if and only if

∗ ∗ Wt Wt βt > 0 → St − St > 0 → − > 0 Bt Bt ∗ (3.12) Wt WT Wt WT WT Wt = − EQ[ |Ft] = − EQ[ |Ft] > 0 ↔ EQ[ |Ft] < Bt BT Bt BT BT Bt

Were we applied the martingale property of (3.11) and the fact that the terminal wealth is equal (3.10). Now we know that Wt is a strict local martingale under the measure (3.12). To Bt Q

16 complete the theorem

WT Wt EQ[ |Ft] < → βt > 0 BT Bt Z T Z t Z T Z t 1 1 1 1 ∗ Since EQ[ dDu|Ft] = dDu + EQ[ dDu|Ft] = dDu + St 0 Bu 0 Bu t Bu 0 Bu Z t Z t Z t 1 ∗ Wt 1 ∗ 1 ∗ dDu + St < → dDu + St < dDu + St → St < St 0 Bu Bt 0 Bu 0 Bu (3.13) This completes the proof that whenever there exist a bubble under the risk neutral ∗ measure Q the fundamental price St is smaller than the market price St. In addition, we mentioned based on earlier literature that a bubble of type 3 βt is a strict local martingale under the measure Q. This can be shown in the following way:

From now on we assume that wealth is only obtained by owning the stock and there are no dividend payments.

17 4 Models

The model of the price process we choose is a set of stochastic differential equations. In the early days of the expression of a stock price process as a stochastic differential equation, the volatility was only expressed as a constant. Empirical work with regards to the volatility smile shows that this was not sufficient to reflect a realistic view of a stock price process hence the properties of the model had to be extended. Over the years the one-dimensional stochastic differential equation have been extended to multi- dimensional equations. The different stochastic differential equations to describe the dynamics are discussed in this section. Note that the simulated stocks that are used in the graphs use the same independent simulated Brownian motions dWt and dZt . After an introduction of the models we give a literature overview of the different models discussed.

4.1 Geometric Brownian Motion

Over the years, diﬀerent models have been used to model the dynamics of a stock price. The most famous one represents the stock price movements by the following stochastic diﬀerential equation [10].

dSt = µStdt + σStdWt (4.1)

When we take Xt = log St the log returns are described by:

1 dX = (µ − σ2)dt + σdW (4.2) t 2 t

Hence the volatility of the returns only depends on σ. Besides the empirically shown shortfalls of this type of underlying model, the martingale property is always satisﬁed −rt for µ and σ. This means that the discounted stock price process e St is a martingale under the risk neutral measure Q. Since under the risk neutral measure the stock price process is always a martingale for σ > 0, it is not possible to detect an asset bubble by this model. With the term ’detect’ we mean that the parameters can adjust such that whenever the stock exhibits a bubble, the martingale property is no longer satisﬁed and satisfy the strict local martingale property under the risk neutral measure. St follows a geometric Brownian motion and hence Xt follows a normal distribution. In general, the log returns of a stock price process are non-normally distributed: excess kurtosis, asymmetry are some of the observed properties of the distribution of the log returns. Especially for bubbles, the log returns can be extreme which is shown by the fatter tails of the observed distribution of the log returns (section 7).

18 Furthermore, a constant volatility parameters σ is not a good representative of the ’true’ volatility. While volatility is an unobserved process, the implied volatility can be shown by finding the σ such that the observed derivate prices matches the prices calculated by the Black-Scholes model. It is shown that for different strike prices and maturities the implied volatility differs. In conclusion, assuming for a stock price process to follow a geometric Brownian motion is an assumption that lacks a lot of properties which are observed in real life.

Figure 2: 20 simulations of a stock price following a geometric Brownian motion

Figure 3: The log returns of the 20 simulated stock prices following a geometric Brownian motion

The simulations of the geometric Brownian motion show that none of the simulated stock prices have a pattern that is in correspondence with the described deﬁnition of a bubble (except volatility clustering). Of course some stock price processes have an

19 increase followed by a decrease but, as Figure 3 shows, the magnitude of the log returns are limited with a highest increase and decrease of 6 percent, something that is not irregular for an asset not exhibiting a bubble. Whether the increase of the asset price following a geometric Brownian motion is due to an increase of the fundamental price or a result of enthusiasm of investors cannot be said by only evaluating the asset price. Hence theoretically speaking the price increase of an asset following a GBM could be a discrepancy between the market price and fundamental value of the asset and hence imply a bubble. However, the dynamics of the increase and decrease is such that it is not considered a bubble. This means that under these dynamics and the NFLVR hypothesis we assume, the price found by risk neutral valuation is always equal to the market price and hence never a bubble. In other words, the log returns are not extreme either negative or positive. We could say that the geometric Brownian motion is robust against small bubbles, which means a small distinction between asset price and fundamental price can be captured by the geometric Brownian motion and none of the patterns similar to ones of Figure 2 will be recognized as a bubble.

4.2 Heston Model

To overcome the drawbacks of the above described geometric Brownian motion. The stock price process is extended to a multi-dimensional stochastic diﬀerential equation.

One of the most popular extensions is the Heston model [16] where under (Ω, F, Ft∈[0,T ], Q) (St,Vt) satisfy:

p 1 dSt = µStdt + St VtdWt p dVt = κ(θ − Vt)dt + VtdZt (4.3)

dWtdZt = ρdt We impose the condition 2κθ > 2 (known as the Feller property) on the Heston model to prevent the stochastic variance Vt becoming negative. Its popularity is due to the availability of a closed-form solution of the European call prices. Moreover, some of the properties of the returns that were not captured by the geometric Brownian motion are now included in the Heston model. Due to the stochastic volatility part the log returns are now non-normal (in correspondence with the observed log returns) distributed. The diﬀerential equation of the stochastic volatility is also known as Cox-Ingersoll-Ross (CIR) process. The parameter κ controls the level of mean-reverting of the variance Vt, θ the long run variance of Vt and the volatility of the stochastic volatility (volatility of volatility) process. Furthermore, µ reﬂects the

20 rate of return and ρ determines the correlation between the Brownian motions dWt and dZt. Since the dynamics of the log returns (dXt = log St+1 − log St) are:

1 p dX = (µ − V )dt + V dW t 2 t t t p (4.4) dVt = κ(θ − Vt)dt + VtdZt

dWtZt = ρdt the volatility of the log returns does not equal a constant but is a stochastic process itself. For = 0 the volatility process is deterministic and hence the distribution of the log returns dXt follows a normal distribution. However, for > 0 the distribution is non normal. Heston [16] showed that the higher the magnitude of , the more the kurtosis of the log returns increases which results in fatter tails. Besides the excess kurtosis, a skewed distribution of log returns can also be created by the parameter ρ. For ρ > 0 the distribution is skewed to the right while for a negative ρ it is skewed to the left. Again the higher the value of ρ (in absolute values) the more the distribution is skewed. Besides of the advantages discussed above, the Heston model is also able to control for volatility clustering. Volatility clustering is known as the dynamics of large price movements followed by large price movements and small price movements followed by small price movements.

The correlation between the Brownian motion dWt and dZt implies are relation √ between the volatility Vt and the price St. For a Heston model ρ = corr(dXt, dVt) where and is usually negative. A negative correlation between these two processes is known as the leverage eﬀect. In economic literature the intuition is that whenever a stock is subject to a large negative shock, the equity value of a stock decreases which increases the debt-to-equity ratio and leads to larger returns (in absolute values) on equity. Although the Heston model is able to capture way more properties than the geometric Brownian motion, the martingale property is again always satisﬁed under the risk neutral measure [1] (the discounted stock price with drift equal to the risk free rate) .

21 Figure 4: 20 simulations of a stock price following a Heston model

Figure 5: The log returns of the 20 simulated stock prices following a Heston model

Compared to the geometric Brownian motion, the dynamics of the simulated asset prices following the Heston model are much wider; some simulated stocks behave more volatile while others move steadier. Note that we used the same simulated Brownian motion as for the GBM model, where the Heston model has an extra Brownian motion dZt because of the stochastic variance process. Also when we check the log returns of the simulated stock prices in Figure 5 we notice a few diﬀerences compared with the GBM: there are more large negative returns than positive returns which implies skewness of the Heston model and the log returns are larger (in absolute terms) and is in correspondence with the suggested fatter tails of the distribution of the log returns ( is positive). As we can see by Figure 4, none of the simulated stock prices has the pattern similar to a bubble. Hence, the dynamics of the log returns are wider compared to the geometric

22 Brownian motion but they cannot capture the dynamics of an asset bubble.

4.2.1 Stochastic Volatility Models

Extension of the geometric Brownian motion by adding stochastic volatility is a widely applied method. Hull and White [18] gave an overview of several stochastic volatility models and provide a partial diﬀerential equation to determine the option price in an incomplete market for a stochastic volatility model when volatility is a non-tradable asset. The volatility smile can be adjusted by the parameters in this model and hence gives a more realistic view than the ﬂat volatility smile of the GBM. The Heston model is one of the most used stochastic volatility models because of the possibility to derive a closed-form option price via the Fast-Fourier Transform. This makes it possible to calibrate very fast. Calibration is a method where the parameters of a model are found by:

X min |CModel(Ki,Ti, µ, θ, κ, , ρ) − CMarket(Ki,Ti)| µ,θ,κ,,ρ k (4.5)

where Ki,Ti is a range of k diﬀerent strike prices and maturities respectivily.

All the literature discussed here ﬁrst determined the parameters of the Heston model through calibrating. Even though the parameters are found by minimizing the diﬀerence between the market price and the calculated price, the mismatch is still large for large out-of -the money options and options with short maturity in the equity markets [9]. Since the above describe models were not able to detect bubbles, we now introduce models that are able to detect bubbles.

4.3 CEV Model

As noticed before, the common models to describe the dynamics of a stock cannot detect a bubble. A simple extension of the stochastic diﬀerential equation that drives the geometric Brownian motion is the Constant Elasticity of Variance model or the CEV model [8]. In this case the stock price satisfy:

α dSt = µStdt + σSt dWt (4.6)

And the log returns

23 1 dX = (µ − σ2S2(α−1))dt + σSα−1dW (4.7) t 2 t t t The most important distinction compared to the GBM is that the volatility of the α−1 returns is no longer equal to a constant but involves an extra term St . These types of models (where the volatility depends on the level of the stock price) are mathematically known as a non- aﬃne stock price process or local volatility models. For any value of α simulation of the model shows that the CEV model still contains the limited characteristics of the geometric Brownian motion: the model is still symmetric and no excess kurtosis is observed. However, the satisfying property of the CEV model is that it is able to detect a bubble [27]. For α > 1 the stock price following a CEV process will exhibit a bubble:

Figure 6: 20 simulations of a stock price following a CEV model with α = 1.2

While we discussed before that economic theory suggests that an increase of the stock price will decrease the volatility, α > 1 implies the opposite. A higher value of α−1 St will increase the volatility St σ. This is called the inverse leverage effect. For commodities the inverse leverage effect is a common occurence [25]. It turns out that the implied volatility curve is skewed to the right for commodities. An explanation is that an increase in the commodity price causes a panic reaction in the market. Because, in contrast to other assets, an increase in commodities has a bad effect on the economy and the prices for out-of-the-money put options are increasing which results in a higher implied volatility. Although not backed by literature, a similar cause could be assigned to the inverse leverage effect for stocks exhibiting a bubble. A possible reason could be that whenever the stock price increases during a bubble, the crash will be potentially bigger and hence the implied volatility will increase because of the increase in demand of out-of-the-money put options to compensate for the possible crash.

24 Figure 7: The log returns of the 20 simulated stock prices following the CEV model

As said before temporary bubbles can maybe still exist in the geometric Brownian motion but it is not able to detect real large bubbles. Figure 6 with the simulated CEV model and α = 1.2 shows that the CEV model is able to detect a bubble. The most compelling one in the ﬁgure is the purple line. Its asset price increased from around 150 to above the 400 within 50 trading days and eventually returned to its value before the bubble. It is reasonable to assume that this large movement is not due to a change in the fundamental value, otherwise the stock price would not return close to its value before the bubble. The log returns of the simulated prices in Figure 7 also shows large positive and negative log returns of the simulated CEV model compared to the GBM model.

Figure 8: The price process and normal ﬁt of the log returns of two simulated stocks following the GBM (left) and CEV (right) model with same random seeds

Figure 8 shows that the log returns of both the GBM and the CEV model ﬁt a normal distribution. Since the CEV model exhibits a bubble, its variance is larger and the returns are more extreme. However, both distributions are symmetric and do not

25 have an excess kurtosis, both properties are not in correspondence with the observed data.

4.3.1 Local Volatility Models

As mentioned, the CEV model is a special case of the local volatility model where the stock price St is a solution of the equation:

dSt = µStdt + σ(S, t)dWt (4.8)

Dupire [7] and Derman & Kani [6] showed that if there exist a surface of call prices for all strike prices K and maturities T then it is possible to determine σ(St, t) such that the local volatility model give rise to the call prices that are known from the call surface. When σ2(K, t) is deﬁned as:

∂C(K,t) + rK ∂C(K,t) σ2(K, t) = δt ∂K (4.9) 1 2 ∂2C(K,t) 2 K ∂K2 the prices of the derivative will match the prices given by the call price surface. However, there are a few drawbacks concerning this approach: ﬁrst of all, it is a very bold statement to assume the existence of a call surface for all strike prices and maturities. This problem can be overcome by for example interpolation which brings of course some inaccuracies with it. And second, [28] showed that the future volatility smiles are ﬂat which is in contrast with empirical observations. Just as in the case of the CEV model, local volatility models can exhibit a bubble. Mijatovic and Urusov [27] showed that these types of models exhibit a bubble whenever:

Z ∞ x 2 dx < ∞ (4.10) α σ (x) Whenever this criteria is satisﬁed, the stock price process is a strict local martingale under the risk neutral measure. The idea behind this integral is based on the Feller test for Explosions. By performing this test we check whether the stock price process

St exits the state space (0, ∞) during the lifetime of the asset [0,T ]. In other words, whenever the integral condition is satisfied, the price process has a positive probability of reaching infinity. The paper of Jarrow et al. [19] did not specify a specific model underlying the price movement during a bubble. Their approach was to estimate σ2(x) using a non-parametric method and verify the integral condition (4.10). The drawback of this method is that only σ(x) can be estimated properly for all values that the stock

26 has reached during the bubble. Extrapolation methods are necessary for large values of x. The drawback of this method also became clear when the σ(x) estimation of the bubbles were evaluated. For the bubble of eToys they were not able to determine whether the integral would converge or diverge and it gave an inconclusive result. Hence the non-parametric method of local volatility model estimation does not lead to the desired results. A non-parametric method can be preferred over a parametric method for local volatility models because it does, compared to the CEV model, not restrict the stock price to follow a pre-determined form with its corresponding disadvantages. Nevertheless, their method turns out to be inconclusive for one bubble. Also a model assuming its volatility only to depend on the stock price is not realistic [15]. As a matter of fact, the local volatility models require a complete market. In a complete market any claim in the market can be replicated and hedged perfectly . Em- perically, it is shown that this is not always possible, there are always market frictions. Extending the model to an incomplete market is preferred. We already described the √ Heston model which exist in an incomplete market (since the volatility Vt is not a tradable asset). Extending the Heston model should lead to a model which captures all the favorable elements of the above described models and neglect the disadvantages.

4.4 HSLV Model

After discussing the diﬀerent types of models as we did, it makes sense to extend the Heston model the same way as we extended the geometric Brownian motion. The model we apply here is the Heston Stochastic Local Volatility model where (St,Vt) is a solution to

p dSt = µStdt + σ(St, t) VtdWt p dVt = κ(θ − Vt)dt + VtdZt (4.11)

dWtdZt = ρdt In our case with the local part extended just as the CEV model to

αp dSt = µStdt + St VtdWt p dVt = κ(θ − Vt)dt + VtdZt (4.12)

dWtdZt = ρdt This model is known as the Heston Stochastic Local Volatility model (HSLV model) or the CEV-SV model. For Xt = log St

27 1 2(α−1) p dX = (µ − V S )dt + V Sα−1dW (4.13) t 2 t t t t t The interpretation of the parameters (µ, α, κ, θ, , ρ) are still equal to the interpretation of the parameters of the Heston and CEV model. Note that for α = 1 the local part is eliminated and the model is a Heston model while for = 0 the model is reduced to a local volatility model. All the favorable properties such as: asymmetry, fatter tails and excess kurtosis are now included in the model.

Figure 9: 20 simulations of a stock price following a HSLV model with α = 1.2

Figure 10: The log returns of the 20 simulated stock prices following the HSLV model

While the eﬀect of the parameters of the Vt (κ, θ, ) are signiﬁcant but marginal, ρ plays a bigger role compared to its contribution to the Heston model. While for the

28 Heston model ρ determines the skewness of the distribution, it also has this effect on the log returns of the HSLV model but now more strengthened. The reason is because of the additional local term in the model. For α > 1 the CEV model had an inverse leverage effect. For the HSLV model this inverse leverage effect also holds for certain combinations 2α−2 of (α, ρ). In Figure 9 there is still a leverage effect which means corr(dXt, dSt Vt) is negative. Now the case when ρ and α both are positive there is an inverse leverage effect. This means that higher stock prices leads to higher volatility and can cause higher stock prices and bubbles. The difference in stock price and returns compared to ρ = −0.4 as in Figure 9 is shown in Figures 11 and 12.

Figure 11: 20 simulations of a stock price following a HSLV model with α = 1.2 and ρ = 0.30

29 Figure 12: The log returns of the 20 simulated stock prices following the HSLV model with positive ρ

4.4.1 HSLV Model vs. Heston Model

When we compare the blue line of Figure 9 with blue line of Figure 4 there is some similarity pattern wise while the Figure 9 should simulate a bubble and Figure 4 not (see Figure 13 for both processes in one figure). However, a closer look to both dynamics tells us that there are some notable differences in support of a bubble in the HSLV model and not a bubble in Heston model but just a structural increment. First of all, the peak of figure the HSLV model is higher (almost 150) compared to the peak of the Heston model (around 70). But, as the definition of an asset bubble suggests, the peak of the bubble does not make the distinction between a bubble or a fundamental change in the asset value and hence does not satisfy as a distinction. The correction during the burst of a bubble could be an argument in favor of a bubble in the HSLV model. After the peak the asset price following the HSLV model in Figure 13 almost returns to its value before the huge increase which was around the 20 dollars while after the burst the stock is around the 30 dollars after being around 130 dollars less than 100 trading days earlier. This means that the stock price has dropped almost 80 percent within 100 days! This is in contrast with the price process of the Heston model where after the peak of almost 70 dollars the stock price decreases to 35 dollar. This is a decrease of 50 percent, which is of course also a lot. However after the burst of this possible bubble the price is still

30 higher than before the birth of the bubble and the price stays at this higher level. These are arguments in favor of an increase of the fundamental value of the asset and the asset does not exhibit a bubble but a positive trend.

Figure 13: HSLV vs Heston with α = 1.2

When we compare both stock prices discussed here and their corresponding log returns in Figure 13 and 14 it is clear that the stock price process of the HSLV model has a higher peak in both the positive and negative returns which means it capture dynamics of the market that are observed during a bubble (e.g. enthusiasm and panic by investors, see Figure 13). Furthermore the log returns are much volatile for the HSLV model compared to the Heston model (Figure 14). Whenever we simulate a stock price process following a HSLV or CEV model with α > 1 it is possible for a stock price process to exhibit multiple bubbles. Notice that after the burst of the bubble of the HSLV model the stock price increases again sharply which means that a second bubble has probably risen which did not burst in the time interval we evaluate but eventually will. In conclusion, at ﬁrst sight the pattern of the Heston model might look similar to the pattern of the simulated bubble using the HSLV model, a closer look to the dynamics teaches us a clear distinction between these two stock prices.

31 Figure 14: HSLV vs Heston log retuns

Figure 15: The price process and normal ﬁt of the log returns of two simulated stocks following the Heston (left) and HSLV model (right) with the same random seeds

Figure 15 shows that the log returns for both the Heston model and the HSLV model cannot be ﬁtted by a normal distribution. An excess kurtosis and their corresponding fatter tails are observed in both cases, with the variance of the HSLV bubble is, of course, larger. Furthermore, the log returns are asymmetric which is not captured by the CEV or GBM model but observed empirically. The addition of a local volatility part for α > 1 generates a more volatility stock price process with more extreme returns and the ability to simulate a bubble. In conclusion, all the favorable properties of the HSLV model and the possbility to detect an asset bubbles makes us choose this model.

4.4.2 Stochastic Local Volatility Models

The Heston Stochastic Local Volatility model has been used before but in a diﬀerent context. Van der Stoep et al.[34] showed that in cases of a large mismatch (in terms of implied volatility) between the market price of the call price and the Heston price,

32 the local volatility part fills this gap such that the mismatch of (4.5) reduces significantly. Engelmann [9] & Tian et al.[35] showed this as well and also made it possible in cases when the Feller property is not satisfied (parameter values that preserve negative volatility). Their method to reduce the mismatch based their improved results, from assuming an underlying HSLV model instead of a Heston model, on the relationship between the local volatility model (4.8) and any stochastic local volatility model (4.11) by the so-called leverage function proposed by Ren [29]:

p 2 p σLV(St, t) = σ(St, t) E[f(Vt) |St = s] = σ(St, t) E[(Vt)|St = s] (4.14)

Where σLV(St, t) is the volatility part of the local volatility model introduced by Dupire [7] (4.9). The calculation that leads to equation (4.14) assumed that the stock price process following any Stochastic Local Volatility (e.g. HSLV) model satisﬁes the martingale property and hence this method is not applicable here (besides requiring option data which are not available). Since σLV(St, t) is solely based on the option price they were all be able determine this local volatility part. All the papers calculated

E[(Vt)|St = s] such that σ(St, t) can be determined and all the parameters are known for the Heston Stochastic Local Volatility model (they ﬁrst determined the Heston parameters). Their focus was on a better ﬁt of the volatility smile for the Heston Stochastic Local Volatility model compared to the Heston model and the method to determine

E[(Vt)|St = s] is the part that distinguished the papers from each other. Ren et al. [29] and Engelmann et al. [9] both used the parameters estimated by the Heston model to determine the transition density p(t, log(s), t). It reflects the transition probability function that log(St,Vt) reaches (log(s), v) at t given the initial values (log(S0),V0). They find the transition density by solving a Kolmogorov forward equation such that it is possible to determine E[(Vt)|St = s] with the difference that Ren [29] assumed that the stock and the volatility part are uncorrelated (which is unrealistic) and did not consider the Feller property. In contrast, Van der Stoep [34] used a nonparametric approach to determine the conditional expectation. All the works mentioned here showed that with the improved stochastic local volatility model it was possible to outperform the Heston model with respect to calibrating the model to the given market option prices.

33 5 Martingale Property

Previous documented proofs that checked whether a stock exhibited a bubble, based their test on the finiteness of the integral of (4.10). This condition is based on the Feller test for explosions [27] and only holds for an one dimensional stochastic differential equation. The sufficient condition for a process to be a strict local martingle is that the expected value of the process is decreasing in time. This condition is related to the Feller test for explosions as we will show in this section.

5.1 Martingale Measure

First of all, the martingale property is tested under the risk neutral measure. Since the HSLV model has a drift part we need to change the probability measure such that the drift µ can be replaced by the risk free rate r. We assume that the stock price and variance are deﬁned under real probability space (Ω, F, Ft∈[0,T ], Q) . Similar to the way the martingale measure for the Heston model is found by Wong and Heyde [36], we ﬁnd the martingale measure for the HSLV model. By Girsanov’s Theorem we know that the class of risk neutral measures can be expressed in terms of the Radon-Nikodym derivative:

Z t Z t Z t Z t dP 1 2 1 1 2 2 2 |Ft = exp − ΛudWu + ΛudZu − (Λu) du + (Λu) du (5.1) dQ 0 0 2 0 0

Were Λ1 is known as the market price of stochastic volatility and Λ2 the market price of the stock price. Similair to the Heston model we assume that the market 1 √ price of volatility is proportional to the volatility, hence: Λ (t) = λ Vt. By Girsanov’s theorem under the risk-neutral probability measure P the Brownian motions change:

∗ 1 p dW (t) = dW (t) − Λ (t) = dW (t) − λ Vtdt (5.2) dZ∗(t) = dZ(t) − Λ2(t)dt For the HSLV model to be risk neutral after this measure change a necessary condition is:

α−1p 1 p 2 2 µ − r = St Vt(ρΛ (t) + 1 − ρ Λ (t)) (5.3)

34 This automatically implies that for the second market price of risk 2 1 µ − r p Λ (t) = √ − λρ Vt (5.4) p 2 α−1 1 − ρ St Vt

Since for diﬀerent values of the unknown parameter λ the drift part of the stochastic diﬀerential equation will still change from µ to r, the new measure is not unique which implies by the second fundamental theorem of asset pricing that the market is incomplete

(assuming there is no tradable asset solely based on Vt). Now under the risk neutral measure P the dynamics of the HSLV model change to:

αp ∗ p 2 ∗ dSt = rStdt + St Vt(ρdWt + 1 − ρ dZt ) (5.5) p ∗ p dVt = κ(θ − Vt)dt + Vt(dWt + λ Vtdt) Hence the stochastic variance part also changes under the risk-neutral probability measure (Ω, F, Ft∈[0,T ], P)

κθ p ∗ dVt = (κ + λ)( − Vt)dt + Vt(dW t) κ + λ (5.6) ∗ ∗ p ∗ dVt = κ (θ − Vt)dt + Vt(dW t)

Note that κ∗θ∗ = κθ and the Feller property is still satisﬁed after the measure change. When we discuss the martingale property the parameters of the variance process (5.6) should be considered. However, our estimation method is based on the real data and hence the parameters under the real probability measure Q are estimated. For the proof in the next subsection we work under the risk neutral measure. From now on we assume (without loss of generality) that the risk free rate is equal to zero and the diﬀerential equation that drives the stock price is driftless. Wt and Zt

(we remove the ∗) are defined on (Ω, F, Ft∈[0,T ], P) and adapted to the filtration Ft∈[0,T ] and T < ∞ is the maturity of the asset, since we consider different assets with a finite lifetime in this work. In this section we try to proof whether the martingale property is satisfied based on earlier proofs of multi-dimensional stochastic differential equations similar to the Heston Stochastic Local Volatility model. The proof described below is therefore analogous to the work of Sin [33] in which such models were discussed.

35 5.2 Proof of Martingale Property

√ α The driftless HSLV model is a stochastic integral of the integrable process VtSt with respect to a local martingale dWt and hence by deﬁnition a local martingale itself. By applying Fatou’s lemma and the fact that there exists a sequence τn such that St∧τn is martingale we get for k ≤ t:

[St|Fk] = [ lim inf St∧τ |Fk] E E n→∞ n

≤ lim inf [St∧τ |Fk](by Fatou’s lemma) (5.7) n→∞ E n

= Sk And hence the process is a supermartingale. The 0 ≤0 sign is replaced by the 0 <0 sign for a strict supermartingale (a strict local martingale) and by the 0 =0 sign for a martingale.

We define an increasing sequence of stopping times: τn = {inf t ∈ (0, ∞): St ≥ n}. n By definition of a local martingale the process St := St∧τn is a (local) martingale under P. Applying Girsanov’s Theorem by introducing a new probability measure Pn on the measurable space (Ω, FT ) for a fixed T ∈ [0, ∞) defined by

Z Z n ST Pn(dω) = P(dω) or A A S0 n (5.8) ST 1 n Pn(A) = EP 1A = P(ST 1A) A ∈ FT S0 S0

The solution of price process (5.5) can be written as

Z t Z t α−1p 1 2(α−1) St = S0 exp Su VudWu − Su Vudu (5.9) 0 2 0 According to Sin (1998), the Lebesque dominated convergence theorem and Gir- sanov’s Theorem, for every set Γ ∈ B(C[0,T ]) where B(C[0,T ]) denotes the Borel sigma- algebra of continuous paths in [0,T ]:

36 Z P(ST 1{W ∈Γ}) = ST (ω)P(dω) Γ Z n = lim ST 1τn>T P(dω) Dominated Convergence Theorem n→∞ Γ Z (5.10) = S0 lim 1τn>T Pn(dω) applying (5.8) n→∞ Γ Z = S0 lim 1τˆn>T P(dω) n→∞ Γ

Whereτ ˆn = {inf t ∈ (0, ∞): Sˆt ≥ n}, Sˆt and Vˆt are deﬁned as the solution to: q ˆ ˆ2α−1 ˆ ˆα ˆ dSt = St Vtdt + St VtdWt q (5.11) ˆ ∗ ∗ ˆ ˆ ˆα−1 ˆ p 2 dVt = κ (θ − Vt)dt + VtSt dt + Vt(ρdWt + 1 − ρ dZt)

With Wˆ deﬁned as: t q ˆ n ˆ ˆα−1 dWt = dWt + VtSt dt (5.12)

The choice of (5.11) and (5.12) is based on the change of measure deﬁned in (5.8). By Girsanov’s theorem we can deﬁne the Radon-Nikodym derivative:

n Z t Z t dPn St α−1p 1 2(α−1) |Ft = = exp 1t≤τn∧T Su VudWu − 1t≤τn∧T Su Vudu (5.13) dP S0 0 2 0

n Under the new probability measure Pn we have, the Brownian motion Wt ,St and Vt as the solution to the diﬀerential equations

n α−1p dWt = dWt − 1t≤τn∧T St Vtdt 2(α−1) αp n dSt = 1t≤τn∧T St Vtdt + St VtdWt ∗ ∗ α−1 p n p 2 dVt = κ (θ − Vt)dt + 1t≤τn∧T ρVtSt dt + 1t≤τn∧T Vt(ρdWt + 1 − ρ dZt) (5.14)

For t ∈ [0, τn ∧ T ] the processes deﬁned in (5.14) are equal to (5.11) & (5.12) and hence the identity between (5.14) and (5.11) is justiﬁed. In equation (5.11) we move under probability measure P hence dW is a Brownian motion while on (5.14) we move under Pn and dW n is a Brownian motion . If we take Γ = B(C[0,T ]) we return to the calculation in (5.10)

37 Z P(ST ) = S0 lim 1τˆn>T P(dω) n→∞ Ω (5.15) ˆ = S0P(St does not explode on [0,T ]) By the term ’explode’ we mean that process exits its state space. In this case the stock price process St has a state space (0, ∞) and explodes whenever St becomes ∞ or 0. Usually this condition is checked by verifying the ﬁniteness of an integral [24]. However, verifying whether a stochastic diﬀerential equation (SDE) explodes by solving an integral is only possible for a one-dimensional SDE. Since Sˆt depends also on Vˆt we are not allowed to apply this method here.

By definition [27] a nonnegative local martingale St (and hence a supermartingale) is a martingale on the interval [0,T ] if and only if E(ST ) = S0. By equation (5.8) we take A = Ω since all different values of ST has to be considered and hence all different values of dW and dZ when we calculate the expected value. In this case the indicator function will always be one. Whenever Sˆt does not explode on [0,T ] (hence τn > T ):

n ST n Pn(Ω) = 1 = EP 1Ω → S0 = EP(ST ) = EP(ST ) since τn > T (5.16) S0 Hence in order to verify whether our stock price process is a martingale we have to check whether Sˆt has a positive probability of exploding on [0,T ]. Note that the equality between the expected value and the initial value of the stock only has to hold at maturity to satisfy the necessary condition for a nonnegative local martingale to be a martingale.

5.3 Failure of Martingale Property

Now we have proven the necessary and sufficient condition for a local martingale to be a strict local martingale, we have to find the parameters of the HSLV model for which this property holds. Since the CEV model lost its martingale property when α > 1 it makes sense that this holds for the HSLV model as well. We proof the statement of the HSLV model not being a martingale for α > 1 by first assuming that the local martingale St is a martingale for all α and contradicting this statement for α > 1 which for a supermartingale (nonnegative martingale) automatically implies that the process is a strict local martingale. In the previous subsection we showed that the martingale property was not satisfied whenever the stochastic process Sˆt defined in (5.11) under the measure P explodes on the interval [0,T ]. Now we have to show that this happens for α > 1 and not for α < 1. Analytically this is not possible and hence we use a numerical approach.

38 When we assume that St is a martingale we apply Girsanov’s Theorem and deﬁne a new probability measure Pn as:

Z t Z t dPn St α−1p 1 2(α−1) |Ft = = exp Su VudWu − Su Vudu (5.17) dP S0 0 2 0

This Radon-Nikodym derivative is the solution of the drift less HSLV process. Under this new measure Pn the process dWt changes:

n p α−1 dWt = dWt − VtSt dt (5.18)

Hence the price process changes under the measure Pn to

2α−1 αp dSt = S Vtdt + St VtdWt t (5.19) ∗ ∗ α−1 n p 2 dVt = κ (θ − Vt) + ρSt Vtdt + (ρdWt + 1 − ρ dZt)

By Girsanov’s Theorem under this new probability measure Pn is equivalent to P which n means that for all A ∈ Ft P (A) > 0 ↔ P (A) > 0. Now we check whether both probability measures are indeed equivalent for different combinations of ρ and α sine ρ also had a significant effect on the stock price movement.

5.3.1 α > 1&ρ < 0

Under the new probability measure Pn the HSLV model changed. In both the stock price and the volatility dynamics a drift term is added. We review the behavior of the stock price process following (5.19). We review a stock price process with the following parameters (α, κ∗, θ∗, , ρ) = (1.5, 5.3610, 0.0250, 0.2960, −0.4) to illustrate our ﬁndings.

Figure 16: First 1900 hours of a stock price process under old and new measure

39 (a) Vt under both old and new measure (b) Volatility of log returns

Figure 17: Stochastic variance and volatility compared for α > 1 and ρ < 0

Figure 16 shows a discrepancy between the drift less HSLV of equation (5.5) with r = 0 under the old measure and the stock price following (5.19). The log returns dXt under the new measure Pn satisfy

2α−2 n α−1p n n dXt = 0.5S Vt dt + S V dWt t t t (5.20) n ∗ ∗ n α−1 n p n n p 2 dVt = κ (θ − Vt )dt + ρSt Vt dt + Vt (ρdWt + 1 − ρ dZt)

While under the old measure P

2α−2 α−1p dXt = −0.5S Vtdt + S VtdWt t t (5.21) ∗ ∗ p p 2 dVt = κ (θ − Vt)dt + Vt(ρdWt + 1 − ρ dZt)

2α−2 Since the drift part of the log returns is 0.5St Vtdt hence positive under Pn while 2α−2 n under measure P the drift part equals −0.5St Vtdt St tends to become higher under Pn as Figure 16 shows. Hence under the new measure Pn the log returns have a drift part which increases as the stock price increases and a local part of volatility which increases α−1√ as well when α > 1. As Figure 17b shows the volatility of the log returns St Vt is strictly larger than the volatility of the stock price process under P. Figure 17a shows α−1 the marginal effect of the stochastic variance Vt when the negative drift part ρSt Vtdt is added to stochastic differential equation. Hence the upward effect of the drift part of the log returns leads to a higher St under Pn and although a higher stock price leads p n also to a lower Vt when ρ < 0 (Figure 17a), the positive effect of the local volatility α−1 n effect St is larger than the downward effect on Vt (Figure 17b). n For some A ∈ Ft the Brownian motion behaves such that stock price process St reaches extreme high values. This still does not imply the nonequivalence between Pn and P since theoretically the drift less HSLV can reach extreme high values as well. However,

40 √ α−1 since the drift and stochastic local volatility (St V t) part of the log returns are level dependent and for α > 1 having an increasing effect, stock prices tend to increase more and more and eventually will also have a relative larger effect on the stochastic variance n n Vt (see Figure 17a). The stochastic variance under the new measure Vt becomes about one and a half times as small as Vt. However, the stock price necessary for this relative n α−1 large discrepancy between Vt and Vt is of such a level that the difference between St α−1 under Pn and St under P is even larger and hence the drift and volatility will increase despite the difference of the stochastic variance processes (Figure 16 and 17b). n n Eventually the stock price process is of a certain level such that dVt = −Vt (as- n n suming 0 is the absorbing barrier for Vt ) and hence Vt+1 = 0 and this will lead to a log return dX = 0. In advance, V n = dV n = κ∗θ∗dt = κθdt and the log return t+1 √ t+2 t+1 2α−2 α−1 n dXt+2 = 0.5(St+2 κθdt)dt + St+2 κθdtdWt (5.20). The stock price process St has n reached such high value to reach Vt+1 = 0, that the stock price will increase despite the random Brownian motion dW n. Hence apparently, S is of such high value that √ t t 2(α−1) α−1 n 0.5St+2 κθdtdt >> St+2 κθdtdWt . This will result in an even higher stock price n and again will cause Vt+3 = 0. Now the procedure will repeat and the price process is n now independent of the random part dWt and dZt. This means that St will always have extreme high returns and eventually will reach infinity.

Figure 18: Last 220 hours of the price, stochastic variance and volatlity under the new measure Pn

n After Vt has reached 0 a few times and becomes κθdt the process St will reach infinity in the end. This contradicts the statement that the probability measure Pn is an equivalent measure of P since the process reaches infinity with a positive probability on the time interval [0,T ], something that does not happen for the stock price process driven by (5.21). In summary, the proof worked in the following way: by definition we

41 knew that the HSLV model was a local martingale. When we assume that the process is a martingale we can deﬁne the Radon-Nikodym derivative for the new measure Pn. According to Girsanov’s Theorem, this new probability measure should be equivalent to the old measure P. However, we showed that (in this case) for α > 1 and ρ < 0 this does not hold. This contradicts the statement that St moving under the HSLV equations is a martingale for α > 1 and ρ < 0. Since the HSLV process is a nonnegative local martingale deﬁned in (5.13) it is not a martingale but a strict local martingale. n We showed that the non-equivalence of the measure Pn is showed by the explosion (St n reaching ∞) of the process St hence connects to equation (5.15).

5.3.2 α > 1&ρ > 0

Whenever α > 1 is combined with ρ > 0 the dynamics of the HSLV process under the measure Pn are different. Since ρ is positive a higher stock price will have a positive n effect on the stochastic variance Vt . Equal to the case of ρ < 0 the drift part is positive and now a higher return and hence a higher stock price will have an increasing effect on the drift, stochastic variance and volatility part of the log returns (Figure 19 and 20). As shown by the introduction of the HSLV model for α > 1 some of the simulated stock prices can reach high values ( Figure 9). Where under the P the stock price will return to the values before the peak (the burst of the bubble) because of the decreasing drift in log returns, the drift of the log returns in (5.20) stay high which lead to a further increase of the stock price and so forth. Eventually this will also have its effect on the n variance Vt .

Figure 19: First 900 hours of a stock price process under old and new measure with ρ postive

42 (a) Vt under both old and new measure with ρ positive (b) Volatility of log returns ρ postitive

Figure 20: Stochastic variance and volatility compared for α > 1 and ρ > 0

As the graph shows (Figure 19), the stock price stays high and the difference between n Vt and Vt becomes larger (Figure 20a) . As mentioned at the introduction of the HSLV model, the fact that ρ is positive will cause more volatile stock prices and hence some higher peaked processes. When we compare the graphs for ρ < 0 and ρ > 0 we note α−1√ that the volatility of the log returns St Vt for ρ < 0 is much lower (and almost equal to the volatility of under P, Figure 17a ) after 900 hours compared to the volatility in the case ρ > 0 (Figure 20b). Figure 20b shows the high volatility of the process which resulted in the extreme volatile price process (see Figure 19). As Figure 19 and 20a n suggest, the high stock price do lead to a higher Vt but it has a marginal effect on the stochastic variance. However, the high stock price will eventually have its effect.

Figure 21: Last 100 hours before explosion of Vt and St

As Figure 21 shows, the high stock price causes extreme volatile movements of the stock process. Eventually the high drift part cannot be compensated by the value of the Brownian motion en the returns can only become positive. This leads to a log return of

43 over 20 and the stochastic variance increases more than 170 times. The stock price is now at a level of order 1013 and will become inﬁnite in the following hours. By the same argument as the case α > 1& ρ < 0, the Radon-Nikodym derivative is not an equivalent measure and hence the HSLV model is for α > 1 and ρ > 0 a strict local martingale as well. For ρ = 0 there is under the new measure no change in the stochastic variance n part but the drift term of St will be equal to the previously discussed cases. Since for n ρ < 0,St explodes eventually on [0,T ], this will also happen when ρ = 0 since there is not a decreasing eﬀect in the stochastic variance whenever the stock price increases. The value of α plays an important role in the failure of the martingale property.

As the stochastic diﬀerential equations under Pn shows (5.19), the drift and volatility n part are determined by α, the higher α the higher the probability of explosion St . The magnitude of the other parameters play a less prominent role in the failure of the martingale property. Higher α leads to a lower expected value of the stock price process at maturity and lead to more compelling bubbles (higher rises and fall).

5.3.3 α < 1

Whenever α < 1 a few properties mentioned in the previous section do not hold any α−1 more. Firstly, the added drift part of the stochastic variance equation VtSt κθdt is marginalized when α < 1; an increase in the price will reduce the eﬀect of the added part in (5.20). This will result in a stochastic volatility being almost equal to the volatility under the old risk neutral measure. Secondly, the behavior of the stock under the measure Pn does now not lead to high prices; the drift part of the log returns for α < 1 2α−2 makes St smaller than 1 and hence the drift part will be smaller than Vt. This will lead to stock returns being a little bit larger than under the measure P. However, larger α−1 values of St will automatically cause a lower local volatility St and drift. Figure 22 show some simulations of the HSLV process for α < 1 and it illustrates the equivalence in the measure for both ρ > 0 and ρ < 0. Hence for α < 1 the new measure Pn is equivalent to P and the driftless HSLV model is a martingale.

44 (a) 20 simulation of the HSLV ρ < 0 (b) 20 simulations of the HSLV ρ > 0

Figure 22: 20 simulations of the HSLV under Pn (blue) and P (red) and ρ < 0

Hence for α < 1 we can conclude that St under Pn does not explode and by (5.15) the stock price process is a martingale.

45 6 Estimation

Since this work focusses on the estimation of the parameters of the Heston Stochastic Volatility model we have to determine which method to apply to estimate the parameters of this HSLV model. Previous literature which answered this question all determined first the parameters of the Heston model and estimated the local volatility part of the HSLV model afterwards. However, the martingale property is a necessary condition in order to apply this method [13] and hence cannot be applied for the HSLV model for α > 1. As mentioned before, calibration of the option price with the market price cannot be applied either because of the lack of option data. Since only the historical daily stock price is given it would make sense to apply the maximum likelihood estimator. However, for stochastic volatility models of the type we discuss it is not possible to obtain the likelihood function in closed-form. Also, obtaining different moments for a stock price process with stochastic volatility is hard and leads to inconsistent estimators because of the discretization bias. One of the advantages of the stock price process defined in equation (4.11) is that it can be simulated easily.

By replacing dSt by St+1 − St we are able to simulate the stock price process. This modiﬁcation still leads to the problem of the model being a nonlinear state space model (since the stochastic volatility is not observed) and hence estimating the parameters is not straightforward. However we can adjust the equation such that although it is not exactly equal to the discretized version of the HSLV model, the estimated parameters are still consistent estimators of the parameters of the HSLV model. This method is called the Indirect Inference method.

6.1 Indirect Inference

The Indirect Inference method was introduced by Gourieroux et al. [14]. It is an estimation method in which the parameters are estimated by an auxiliary function. The idea behind the method is to ﬁnd the parameters of the model, in this case the parameters θ corresponding to (4.12) (θ = (µ, α, κ, θ, , ρ)), by solving:

˜ ˜ bT (θ) = β (6.1)

Where β˜ is estimated by maximizing the criterion function Q corresponding to an auxiliary, approximating model with parameters β:

˜ β = arg max QT (ST , β) β∈B

46 It is important to stress that β and θ have a diﬀerent interpretation although they have the same size in this case. Where θ refers to the parameters of the Heston Stochastic Local Volatility model, β are the parameters corresponding to a yet undeﬁned auxiliary model. From now on we call β the auxiliary parameter. In this context β˜ refers to the parameters corresponding to maximizing the function QT (ST , β) which depends on the observed data S and T denotes the length of the dataset S. This function does not assume the HSLV model as underlying but a simpler auxiliary model. The reason why we are looking for β˜ is because it is assumed by Gourieroux et al. [14] that limT →∞ QT (ST , β) = Q∞(β, θ0,F ) where θ0 is the true parameter vector of interest and F denotes the density of dW and dZ. This equality states that the limit of Q is not stochastic and the limit function Q∞ has a unique maximum β0. Given this assumption

Gourieroux states that β˜ provides a consistent estimator of β0. This is necessary because

β0 is unknown since it depends on the true parameters of the underlying HSLV model

θ0. According to (6.1) we have to ﬁnd θ˜ such that the equality holds. The function ˜ on the left side of the equality, bT (θ), is known as the binding function. The binding function is deﬁned as:

b(θ) = arg max Q∞(θ, β) β∈B This implies

β0 = b(θ0) (6.2)

In general, the binding function is not available in closed-form and hence it should be replaced by a consistent estimator. Renault [30] gave a consistent estimator for b(θ0) using H simulations of the stock price process S˜T (θ):

H 1 X h h ˜h βT,H (θ) = βT (θ) where βT (θ) = arg max QT (ST (θ), β) (6.3) H β∈B h=1

Hence (6.1) is a result of the equality (6.2) where both sides are replaced by consistent ˜ estimates and bT (θ) is replaced by a consistent estimate as well, namely equation (6.3). ˜h Note that βT,H (θ) is a function of θ because the h-simulated data ST depends on the parameter θ. For the proof of the consistency we refer to Gourieux et al. [14]. Now we have to translate this theoretical framework to our case. First of all we have to determine which underlying auxiliary model the criterion QT (ST , β) maximizes. Fiorentini et al. [11] argued that in order to estimate a Heston model with the Indirect Inference method a proper choice for the auxiliary model would be the NAGARCH(1,1)

47 model:

1 2 Rt = µ + at; at = ht t where Rt = log St − log St−1 1 2 2 (6.4) ht = ω + βht−1 + α(at−1 + γht−1)

t ∼ N(0, 1) They argued that the NAGARCH model contains the main characteristics that are also apparent in the Heston model such as: fat-tailness, time-varying conditional variance and asymmetries in the response of volatility (leverage eﬀect). Since we are not dealing with a Heston model but the extended Heston Stochastic Local Volatility model we have to extend the NAGARCH(1,1) model as well because the number of parameters in the auxiliary model (now 5) is smaller than the number of parameters in the HSLV model. As mentioned, the parameters in the auxiliary model do not have to refer to the exact same properties in terms of magnitude, but it should capture all the characteristics that are also in the original model. In contrast to the work of Fiorentini et al. [11], the α HSLV model has a local volatility part (σ(St, t) = St in equation (4.11)). Hence a level dependent part should be added to the NAGARCH(1,1) model. It makes sense to extend the model to a NAGARCH model with a local part:

1 δ 2 Rt = µ + St−1at; at = ht t 1 2 2 (6.5) ht = ω + βht−1 + α(at−1 + γht−1)

t ∼ N(0, 1) The GARCH model and its extended versions such as the NAGARCH model discussed here have a known distribution. Since t ∼ N (0, 1) we know that, conditional on past history, Rt ∼ N (µ, ht) and through the maximum likelihood criterion we are able to estimate the auxiliary parameters β = (µ, ω, β, α, γ). The Local-NAGARCH(1,1) 2δ model in (6.5) changes the distribution of Rt. Since Rt ∼ N(µ, St−1ht) it now has a 2δ level dependent volatility part St−1 which means the auxiliary model is able to capture the level dependent volatility characteristic of the HSLV model. The diﬀerence with the √ √ discretized HSLV model is that the stochastic volatility Vt is replaced by ht and is observed at t. Since the conditional distribution of the log returns is known we use the maximum likelihood method in order to obtain the parameters of the auxiliary model.

So in this case the criterion function QT (ST , β) is L(ST , β) which leads to the estimated parameters B˜ = (˜µ, ω,˜ β,˜ α,˜ δ,˜ γ˜). Now we know the auxiliary model and criterion function we have to ﬁnd the param-

48 eter vector θ. We have to solve (6.1), this can be done by calibrating the value of θ such ˜ ˜ that bT (θ) is close to β. In other words, we have to minimize:

H H ˜ ˜ 0 ˜ ˜ 1 X h 0 ˜ 1 X h θ = arg min(β − bT (θ)) (β − bT (θ)) = arg min(β − βT (θ)) (β − βT (θ)) θ θ H H h=1 h=1 h h where βT (θ) = arg min L(ST (θ), β) β (6.6)

When we calibrate to value of θ in order to minimize (6.6) there are a few things we need to take into consideration. The simulated stock price could be subject to discretization bias. This means that when we simulate the continuous stock price process (4.12) we discretize the process which leads to a bias. In order to reduce this bias we take 10 steps for each day and take the 10-th value of the simulated data set to obtain the ˜h simulated ST (θ). Also we simulate the Brownian motion dW and dZ and ﬁx this value for all simulations such that the simulated stock price only diﬀers through θ. Note that we not only changed the method to determine the parameters of the Heston Stochastic Local Volatility model compared to the work of Van der Stoep et al. [34], Engelmann et al. [9] and others. We also determine all the parameters of the HSLV simultaneously. Both the local volatility part and the stochastic volatility part are determined by the Indirect Inference method while the previous works only saw the local part as an adding factor to reduce the estimation error.

6.2 Asymptotic Distribution of the Estimator

Under the assumptions mentioned in the section above and some extra assumptions for which we refer to Gourieroux et al. [14], the asymptotic distribution of the Indirect Inference estimator is known. Since the numbers of the parameters in the HSLV model and in the auxiliary model are equal, the estimator is equal to:

√ ˜H d T (θT − θ0) −−−−→ N [0,W (H)] (6.7) T →∞ ˜H Where θT is found by (6.6) using H simulations of the HSLV model where:

1 ∂2Q (θ , β )−1 ∂2Q (θ , β )−1 W (H) = 1 + ∞ 0 0 (I − K ) ∞ 0 0 (6.8) H ∂β∂θ0 0 0 ∂θ∂β0

49 And

√ ∂QT ˜h I0 = lim V T (ST (θ0), β0) T →∞ ∂β √ √ (6.9) ∂QT ˜h ∂QT ˜k K0 = lim Cov T (ST (θ0), β0), T (ST (θ0), β0) T →∞ ∂β ∂β

Since there are no exogenous variables K0 = 0 and we can, according to [14], I0 consistently estimate by:

K X k Γˆ + (1 + )(Γˆ + Γˆ0 ) (6.10) 0 K + 1 k k k=1 Where

T 1 X ∂Ψt−k ∂Ψt Γˆ = (βˆ ) (βˆ ) (6.11) k T ∂β T ∂β0 T t=k+1

2 1 1 2δ T (rt − µ) Ψt(β) = − log(2π) − log(htSt−1) − 2δ (6.12) 2 2 2 htSt−1 1 PT Since QT = T t=1 Ψt(β) and ht deﬁned in (6.5). The cross derivative of Q∞(θ0, β0) is deﬁned as:

2 2  ∂ Q∞(θ0,β0) ∂ Q∞(θ0,β0)  ∂Q ∂Q ··· 2 ∂ ∞ ··· ∞ θ1β1 θ1β6 ∂ Q (θ , β ) ∂β1 ∂β6 ∞ 0 0  . .. .  0 = =  . . .  (6.13) ∂θ∂β ∂θ  2 2  ∂ Q∞(θ0,β0) ··· ∂ Q∞(θ0,β0) θ6β1 θ6β6

˜ ˜ Since Q∞(θ0, β) is consistently estimated by QT (β, ST ) and QT = L(β, ST ) we esti- ˜ mate (6.13) by numerically determining the derivative of ∂L(β,ST ) which can be deter- ∂β˜ ˜ mined easily since the likelihood depends directly on β. For ∂L(β,ST ) we simulate new ∂θ˜ paths (with the same random seeds) for a small change in θ˜ (which is found in (6.6)) and determine the derivative by

H h ˜ ∂L(β, ST ) 1 X ∂L(β, S (θ)) = T (6.14) ˜ H ˜ ∂θ h=1 ∂θ h ˜ ˜ Because for each simulated path QT (ST (θ), β) is a consistent estimate of Q∞(θ0, β).

50 7 Data

Before we discuss the obtained results, we describe the data and their corresponding characteristics. As mentioned before, the bubbles we discuss are the same bubbles discussed by Jarrow et al. [19]. The bubbles evaluated in this paper were from Lastminute.com, InfoSpace, eToys and Geocities. We were not able to obtain the data of Lasminute.com so we cannot discuss this bubble unfortunately.

7.1 InfoSpace

The company InfoSpace was founded in 1996 and oﬀered an online version of the yellow pages. In December 1998 the company went public and after an initial price of 25 dollars the stock reached a value of 1305 dollars in the beginning of 2000, and dropped below 40 dollars one year later. Eventually the company was bought by Blucora.

Figure 23: Stock price process of InfoSpace from 15-12-1998 till 19-09-2002

Figure 24: The log returns of InfoSpace

When looking at the price process of InfoSpace we see that his is case is the most

51 compelling bubble of the three bubbles discussed. Hopefully, this will also be reﬂected in the results.

7.2 eToys eToys is a retail website for toys founded in 1997 and went public in May 1999. After an initial price oﬀering of 20 dollars the ﬁrst trading day made the stock close at 76.25 dollars. After the burst of the dot-com bubble the company went bankrupt in February 2001.

Figure 25: Stock price process of eToys from 20-05-1999 till 26-02-2001

Figure 26: The log returns of eToys

The price process of eToys has a pattern which we could consider as a bubble. After the ﬁrst trading day of having a value of 76 dollar it decreased to 30 dollars after 56 days.

52 After this drop the stock price increased again to 84.25 dollar where after the stock price reduced to zero. A price process like this which reached a value of 30 and after 50 days 84.25 and 50 days later is again round 30 dollar suggests that the raise of price was not because of the change in the fundamental value. Of course, this could be the case but since we know that the stock went bankrupt after all, we can safely assume that this raise and fall was due to a birth and burst of a bubble. To estimate the parameters of the HSVL model we only use the ﬁrst 300 days. This because there are some extreme returns at the end which took place long after the burst of the bubble. After 300 days the stock price is around 5 dollars after being 84 dollars and hence should be suﬃcient to detect a bubble.

7.3 Geocities

Geocities is a Web hosting service founded in 1994 and went public on the NASDAQ in August 1998. After an Initial Price Oﬀering of 17 dollars the stock more than doubled on the ﬁrst day to 37.31 dollars. On May 1999 Yahoo acquired Geocities and was eventually closed by Yahoo in 2009.

Figure 27: Stock price process of eToys from 11-08-1998 till 28-05-1999

53 Figure 28: The log returns of Geocities

In contrast to the other discussed bubbles, the stock price of Geocities has a few strange movements. After about 105 and 120 days the stock price makes a huge increase. Research on the internet found out that around that time a few announcements were made regarding the takeover of the company by Yahoo. These large returns could be explained by this news. Usually a stock price increases after such an announcement and usually this is justified because of the assumed change in the fundamental value. Nevertheless, the consensus is that Geocities exhibited a bubble and apparently the figure suggest a discrepancy of the fundamental value (hence the bubble birth) of the asset after 100 days. However, as usually observed, after the burst of the bubble the stock price did not returned to its value before the burst. Before the stock price returned to its value before the bubble (as assumed by theory) the stock was no longer on the index. Therefore, it is interesting to check whether the HSLV model can detect a bubble during the burst of the bubble or at least indicate the difference of a larger proportion of large positive returns than large negative returns. In contrast to the process of eToys and InfoSpace the first order autocorrelation turned out to be significant and positive.

54 8 Results

In this section we discuss the results obtained using the Indirect Inference method and check whether the stock prices indeed exhibited a bubble. Note that our estimates are based on the real data and will result in estimates of the parameters under the real measure. Hence we estimate (µ, κ, θ, , ρ, α) and not (r, κ∗, θ∗, , ρ, α) which are the parameters under the risk neutral measure. To estimate the parameters we used MATLAB and the Particle Swarm Optimization algorithm which took about 24 hours to estimate the parameters for each data set. All the estimated parameters turned out to be signiﬁcant, the standard error were calculated by the way described in section 6.2 and can be found in the appendix.

8.1 InfoSpace

InfoSpace µ κ θ ρ α objective function HSLV 0.119 5.361 0.025 0.296 −0.874 1.861 0.0029 10−3· (.243) (0.0672) (0.156) (0.210) (0.00550) (0.0991) Heston 0.0060 6.1570 0.0950 0.1560 −0.1480 1 0.0078 (0.0006) (0.0013) (0.0017) (0.0007) (0.0011)

Table 1: Estimated parameters of the HSLV and Heston model using InfoSpace data

A few things stick out when we evaluate the parameters of InfoSpace with the underlying HSLV model. First of all, the stock exhibited clearly a bubble according to our assumed underlying model.α ˆ = 1.8610 and significantly different from 1 implies a more compelling bubble compared to lower values of α which is reflected in the higher peak and drop of the stock price. The relative high value of ρ (in absolute terms) draws our attention. For a Heston model ρ = corr(log St − log St−1,Vt − Vt−1) while the correlation between the simulated stock returns and the difference in the stochastic variance with parameters θˆ is strictly larger and around −0.7390. While a negative ρ had the economic interpretation of the leverage effect for the Heston model, it cannot be said for the HSLV model. Since the volatility of the returns is extended with a local part, the leverage effect should now be considered as the correlation between the log return α−1√ Xt and the volatility St Vt. It turns out that for the case of InfoSpace (hence this combination of ρ and α) the correlation is positive (around 0.3). This correlation is not as strong as the Heston model or the CEV model were the correlation between returns

55 α−1 and volatility St is −1 for α < 1 and 1 for α > 1. Nevertheless, it is a change from a leverage effect (higher returns lead to lower volatility) to an inverse leverage effect (higher returns leads to a higher volatility). Other properties are the skewness, kurtosis and autocorrelation. When we take a look at the log returns series of InfoSpace there is no indication of volatility clustering. This is confirmed when we test for autocorrelation of the squared returns where only the first order autocorrelation is slightly significant. As mentioned at the introduction of the different models: the GBM, Heston and CEV model do not exhibit all the properties of the evaluated bubbles. One of the most important properties were the fat tails on both sides of the log returns. Only the Heston model was able to have an excess kurtosis but only to a certain extent and certainly not sufficient to show the statistics of a bubble. The observed log returns showed a fatter tail on both sides (left graph of Figure 29). The histogram of the simulated log returns (right graph of Figure 29) shows a similar shape. They do not perfectly match each other as you can see by the higher number of observations around a return of 0 but the stock price process of the simulated HSLV model under the found estimate θˆ still is able to simulate a bubble with similar extreme returns.

Figure 29: Normal ﬁt of the log returns of observed data (left) compared to a simulated stock price (right) of InfoSpace

The stock price of InfoSpace clearly shows the pattern of a bubble with a distinction of the observed data by the sharp rise and fall of the simulated price. The reason is the fact that the volatility of the returns is level dependent and a high price leads to a high volatility (especially for a high α) and hence a large possible price drop. Economically

56 this interpretation makes sense: when a bubble grows it is likely to burst and hence show a volatile pattern. However the effect of the stock price on volatility seems to be overvalued in the HSLV model whenever the stock price is around the height of the bubble. This leads to too extreme log returns when the stock price reaches its height. To reduce this effect of high volatility whenever the stock price is high, the parameter ρ plays a role. To compensate for the high volatility ρ becomes low to make sure that the stochastic volatility decrease whenever the stock price increase. In this caseρ ˆ = −0.9310 which means a negative correlation between the returns and the stochastic variance Vt (and stochastic volatility) and hence it reduces a little the effect between the correlation α−1√ of the return and the local volatility St Vt. But it is apparently not sufficient to give the simulated stock a more realistic dynamic. According to Heston [16] ρ should also be able to control the skewness of the distribution and the high value (in absolute terms) of ρ here should create a much skewed distribution. Though, Figure 29 shows a quite symmetric distribution. Also the property of autocorrelation cannot be controlled in the same way as the Heston model. It seems that these properties disappear whenever α reaches quite high values such as this case. In conclusion, the relative high value of α shows that the stock price process not only exhibit a bubble but also one with a relative high peak and following fall. Its drawback is that the stock price becomes too volatile at the height of the bubble and not all properties of the Heston model can be controlled. When the estimated parameters of the HSLV are compared with the parameters of the Heston model (table 1) we notice a few differences. Sinceρ ˆ < 0 the Heston model suggests a leverage effect instead of an inverse leverege effect as suggested by the HSLV model. In advance, it turns out that because of the lack of the local volatility parameter the long run variance mean θ increases almost four times and is half of its value under the HSLV model. As the stochastic differential equations that drives this process suggests, it is not able to indicate when a bubble starts and hence the simulated stock returns do not have a bubble at the same time as the bubble of the observed data. That both the simulated and observed bubble of Figure 29 start at the same time is a coincidence.

57 8.2 eToys

eToys: µ κ θ ρ α objective function HSLV 0.0360 4.9200 0.0770 0.5140 −0.3760 1.1780 0.000719 10−2· (.0442) (0.110) (0.058) (0.055) (0.036) (0.110) Heston 0.0390 5.5370 0.0870 0.4260 0.0880 1 0.0004715 (0.0005) (0.0020) (0.0008) (0.0008) (0.0001)

Table 2: Estimated parameters of the HSLV and Heston model using the eToys data

Again for the HSLV model, α > 1 means that according to our model eToys exhibited a bubble. This is an improved result of the work of Jarrow [19] in which this conclusion could not be drawn. In contrast to the estimated parameters of InfoSpace and Geocities, the combination of ρ and α is such that there is a leverage effect. Hence the volatility of the log returns is negatively correlated with the log returns. A possible reason could be the pattern at the beginning of the stock price process: the stock price drops right after its initial value and fluctuates a while before rising again. The simulated HSLV processes did not show such a pattern (Figure 9). This is an interesting observation because the other discussed bubbles both had an inverse leverage effect and the CEV model had one as well whenever the stock price exhibited a bubble. Hence according to these estimates, the inverse leverage effect and existence of a bubble do not always hold at the same time. The unusual pattern (start with decrease and increase afterwards) cannot be captured by the HSLV model as Figure 30 shows, this leads to a poor fit of the log returns. The observed histogram shows a distribution with a large kurtosis and fat tails. The histogram of the simulated log returns is different: it has a smoother distribution. Even if the fit is far from perfect (autocorrelation of squared returns is also captured in the simulated process while not observed), the most important aspect, detecting the bubble, was confirmed by the parameters and the parameters can indicate a bubble and a leverage effect. Where the inverse leverage effect of the HSLV model is changed to a leverage effect for the Heston model for InfoSpace, the leverage effect according the HSLV model is now changed to an inverse leverage effect for the Heston model. All the other parameters for the Heston model are close to the parameters estimated for the HSLV model.

58 Figure 30: Normal ﬁt of the log returns of observed data (left) compared to a simulated stock price (right) of eToys

8.3 Geocities

Geocities: µ κ θ ρ α objective function HSLV 0.0280 3.2860 0.0800 0.4660 0.3370 1.0830 0.0026 10−4· (0.843) (0.584) (0.922) (0.189) (0.221) (0.600) Heston 0.0030 6.9750 0.0410 0.5020 0.1980 1 0.0016 10−3· (0.6931) (0.8054) (0.3654) (0.7778) (0.6316)

Table 3: Estimated parameters of the HSLV and Heston model using the Geocities data

Figure 31: Normal ﬁt of the log returns of observed data (left) compared to a simulated stock price (right) of Geocities

Of all the 3 bubbles the indicator of the existence of a bubble α is the smallest (but still signiﬁcant) for this process while the increase was much higher than the one of eToys (25). When we look at the stock price process of Geocities, the stock price did not

59 return to its value before the growth. However, the fall of the price was started and apparently such that the estimated parameters were able to detect that the process was indeed a bubble. The fact that the bubble did not return to the assumed fundamental value (the value before the quick rise of the price) might be a possible reason for α to be relative small since the simulated HSLV process usually involves a growth and burst of bubble. In this case ρ > 0 which means that the inverse leverage effect still holds and is a little higher thanρ ˆ because of α > 1. As mentioned at the introduction of the data a few jumps of the stock price were noticed. The HSLV process that drives St does not incorporate jumps, something that might explain the poor fit in Figure 31. As the histogram of the log returns of a simulated price shows, its variance is much lower compared to the observed log returns. The extreme returns on the positive side are not observed as well. However, ρ > 0 does besides a leverage effect also determine the skewness of the log returns of the HSLV model (just as in the Heston model). The ρ > 0 suggests a skewed distribution to the right something that can be observed in Figure 31. The effect of ρ is equal to its effect on the Heston model because of the close value of α to 1 (which would make the HSLV model the Heston model). As 29 shows, for α large an even very low value of ρ has not much effect on the skewness of the distribution. In advance, the autocorrelation of the squared log returns was also significant for the first order equal to the observed data. Again, because of the similarity to the Heston model such characteristics can be captured by the HSLV model. The inverse leverage effect as suggested by the HSLV model still holds for the Heston model and hence there is no change in leverage effect although the effect (correlation between volatility and returns) is reduced. Other Heston parameters µ, κ, θ are changed a lot even if α is small. However, the particle swarm optimization algorithm determines the parameters it evaluates randomly hence a better fit could possibly be found close to the parameters of the HSLV model (although not likely) and both fits are not perfect since the objective function is not equal to zero.

8.4 Auxiliary Model

As mentioned, the HSLV model has some drawbacks when it comes to capture other properties of the data besides detecting a bubble. Possibly the drawbacks are related to the choice of the auxiliary model, the Local-NAGARCH model. After estimation the rt−µˆ parameters of this auxiliary model the standardized residuals ˆt = 1 were tested for 2 δˆ ht St−1 independent identically distribution by Engle’s ARCH test. Table 4 shows that in the case of eToys and Geocities the null hypothesis of independent and identically distributed

60 standardized residuals is rejected and hence we should choose another auxiliary model. Furthermore, the fatter tails of the standardized residuals showed that there is room for improvement for the Local-NAGARCH model (e.g. a t-distribution for the errors as Figure 32 shows). Hence a way to improve to estimated parameters is to extend/change the Local-NAGARCH model.

Figure 32: Q-Q plot of the standardized residuals of Geocities versus a standard normal distribution (above) and a t-distribution (below) with 7 degrees of freedom

ARCH test: H0 p-value InfoSpace not reject 0.1075 eToys reject 0.0215 Geocities reject 0.0012

Table 4: ARCH test of the standardized residuals

61 9 Conclusion

The main purpose of this thesis was to find a method to detect an asset bubble. The inspiration behind this thesis was the work of Jarrow et al. [19] who tried to detect asset bubbles by estimating the volatility of the stock price process. Their method involved a non-parametric estimation of the volatility part of the stock price process. Besides the drawbacks of non-parametric estimation they assumed a stock price driven by a one dimensional differential equation. In this work, the stock price process is extended to a multi-dimensional stock price process namely the Heston Stochastic Local Volatility model. This extension of the Heston model makes it possible to extend the volatility of the stock price by making it also dependent on the level of the stock price. These so-called non-affine stock price processes have been used before to describe the dynamics of a stock process. However, as far as I am aware, this work is the first document which tries to estimate the parameters with the intention of detecting bubbles of an observed stock price. In order to estimate the parameters we used the Indirect Inference method. This simulation based approach has the advantage of to be able to estimate all parameters of the HSLV model simultaneously, in contrast to earlier work in which the parameters of the Heston model were estimated first and the extra local part afterwards. All this earlier work assumed the HSLV model being a martingale. The HSLV model as a martingale is the property relaxed in this work: by allowing the HSLV model the specific form we chose we showed that it is possible for the stock price process being a strict local martingale. We showed that a stock price with a finite maturity exhibit a bubble whenever the stock price process is a strict local martingale under the risk neutral measure. Aslo, we proved that for certain parameters the HSLV model is a strict local martingale and relate the necessary condition for a local martingale to be a strict local martingale to the probability of explosion of the HSLV driven process. The three bubbles discussed in the work of Jarrow et al. [19] were also estimated in this work. The estimated parameters using the Indirect Inference method shows that all the three discussed bubbles of Jarrow et al. were indeed an asset bubble. This is an improvement of their work since in one case they could not conclude whether there was indeed a bubble (even it was confirmed by the market as a bubble in hindsight). Based on these three bubbles we can conclude that the HSLV model is better able to detect a bubble than the non-parametric method; the wider dynamics of a stock price during a bubble compared to a regular movement are apparently able to be captured by HSLV model. The HSLV model is also able to test whether the stock price exhibits a leverage

62 eﬀect or not which makes it an improvement of the CEV model. The leverage eﬀect can be useful whenever the stock price is known and the volatility has to be forecasted. However, even if the HSLV model is able to detect an asset bubble, it is not able to control all the properties of the evaluated stocks: some extreme returns, excess kurtosis and autocorrelation of the squared returns are not always in correspondence with the observed data. It turns out that the presence of the local volatility also loses some of the favorable properties of the Heston model especially when α is relatively high. Further research could be including jumps in the stock price process to capture these statistics and/or changing the stochastic variance equation or the auxiliary model.

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65 A Appendix

1.0e − 07· α µ κ θ ρ α 0.0983 −0.2412 0.0666 −0.1545 −0.2080 0.0051 µ −0.2412 0.5914 −0.1634 0.3788 0.5102 −0.0125 κ 0.0666 −0.1634 0.0451 −0.1046 −0.1409 0.0035 θ −0.1545 0.3788 −0.1046 0.2426 0.3267 −0.0080 −0.2080 0.5102 −0.1409 0.3267 0.4401 −0.0108 ρ 0.0051 −0.0125 0.0035 −0.0080 −0.0108 0.0003

Table 5: Variance matrix of the Indirect Inference estimator of InfoSpace

1.0e − 05· α µ κ θ ρ α 0.1158 −0.0475 −0.1193 0.0619 0.0588 −0.0387 µ −0.0475 0.0195 0.0489 −0.0254 −0.0241 0.0159 κ −0.1193 0.0489 0.1229 −0.0638 −0.0605 0.0399 θ 0.0619 −0.0254 −0.0638 0.0331 0.0314 −0.0207 0.0588 −0.0241 −0.0605 0.0314 0.0298 −0.0197 ρ −0.0387 0.0159 0.0399 −0.0207 −0.0197 0.0130

Table 6: Variance matrix of Indirect Inference estimator of eToys

1.0e − 08· α µ κ θ ρ α 0.3600 −0.5055 −0.3505 0.5535 0.1131 −0.1322 µ −0.5055 0.7098 0.4921 −0.7771 −0.1588 0.1857 κ −0.3505 0.4921 0.3412 −0.5388 −0.1101 0.1287 θ 0.5535 −0.7771 −0.5388 0.8508 0.1739 −0.2033 0.1131 −0.1588 −0.1101 0.1739 0.0356 −0.0415 ρ −0.1322 0.1857 0.1287 −0.2033 −0.0415 0.0486

Table 7: Variance matrix of the Indirect Inference estimator of Geocities