Bubbles in the Heston stochastic local volatility model.

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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 volatil-ity part the Heston Stochastic Local Volatilvolatil-ity (HSLV) model is created. Through this extension the model is able to capture more dynamics of the stock price move-ment 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.

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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

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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 pa-rameters 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

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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.

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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 docu-mented 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

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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 compa-nies, 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 trad-ing floor. In 1995 the initial public offertrad-ing 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 funda-mental 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

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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 be-came 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

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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.

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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 funda-mental 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 :

EQ[Rt+1|Ft] = r where Rt+1=

Pt+1+ Dt+1

Pt

− 1 (3.1) where r is the risk free rate. Equation (3.1) can be rewritten as:

Pt= EQ

Pt+1+ Dt+1

1 + r |Ft

(3.2) Using the Law of Iterated Expectations gives the following expression of the price [4]:

Pt= EQ "_K X i=1 1 1 + r i Dt+i|Ft # + EQ " 1 1 + r K Pt+K|Ft # (3.3) Usually for K → ∞ the second expectation tends to zero which means that the price

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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= Pf und+ Bt, (3.4) where Bt= EQ Bt+1 1 + r|Ft (3.5) Btis 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

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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, Stand Dtadapted to Ftand 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. Wt= St+ Bt Z t 0 1 Bu dDu (3.6)

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

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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 F_{t∈[0,T ]} on probability space (Ω, F , F_{t∈[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 F_{t∈[0,T ]} on probability space (Ω, F , F_{t∈[0,T ]}, P) whenever there exist a postive, increasing and diverging sequence {τn}∞n=1 such that:

X_min{t,τ_n} 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 P. By this fundamental theorem a positive process Wt

Bt is a local

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.

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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 fun-damental 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

Bt is a local martingale.

Since we assume for the asset to have a finite lifetime the fundamental price of the asset S_t∗ 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)

S_t∗= EQ[Bt Z T t 1 Bu dDu|Ft] (3.7)

The fundamental wealth process W_t∗ is now defined (using (3.6)) as

W_t∗= S_t∗+Bt Z t 0 1 Bu dDu= EQ[ Z T t 1 Bu dDu|Ft]+Bt Z t 0 1 Bu dDu= EQ[Bt Z T 0 1 Bu dDu|Ft] (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

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difference in the market price and the fundamental price of the asset. Wt− Wt∗= Wt− EQ[Bt Z T 0 1 Bu dDu|Ft] = St+ Bt Z t 0 1 Bu dDu− EQ[Bt Z T 0 1 Bu dDu|Ft] = St− EQ[Bt Z T t 1 Bu dDu|Ft] = St− St∗ (3.9)

In the early assumptions we mentioned that at the maturity of the asset, there is no liquidation value of the asset and hence S_T∗ = 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 W_t∗ and the wealth process Wt, their

wealth at maturity T is equal:

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

We can show that Wt∗

Bt is a martingale by applying equation (3.8) and the tower

property: EQ[ W_t∗ Bt |Ft−k] = EQ[[ Bt Bt Z T 0 1 Bu dDu|Ft|Ft−k]] = EQ[ Z T 0 1 Bu dDu|Ft−k] = W_t−1∗ Bt−k for k = 1, ..., t (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 W_t∗ and wealth Wt. This leads to the existence of a bubble if and only if

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

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

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complete the theorem EQ[ WT BT |Ft] < Wt Bt → βt> 0 Since EQ[ Z T 0 1 Bu dDu|Ft] = Z t 0 1 Bu dDu+ EQ[ Z T t 1 Bu dDu|Ft] = Z t 0 1 Bu dDu+ St∗ Z t 0 1 Bu dDu+ St∗ < Wt Bt → Z t 0 1 Bu dDu+ St∗< Z t 0 1 Bu dDu+ St→ St∗ < St (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 βtis a

strict local martingale under the measure Q. This can be shown in the following way: EQ[βT|Ft] = EQ[ST − S ∗ T|Ft] = EQ[ WT BT −W ∗ T BT |F_t_{] = E}_Q[WT BT |F_t_{] − E}_Q[W ∗ T BT |F_t] = EQ[ WT BT |Ft] − W_t∗ Bt < βt↔ EQ[ WT BT |Ft] < Wt Bt (3.14) Hence for the bubble βt to be a strict local martingale, W_B_tt should be a strict local

martingale as well.

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

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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:

dXt= (µ −

1 2σ

2_{)dt + σdW}

t (4.2)

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 for µ and σ. This means that the discounted stock price process e−rtSt 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).

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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

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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: dSt= µStdt + St p VtdWt1 dVt= κ(θ − Vt)dt + p VtdZt dWtdZt= ρdt (4.3)

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

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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:

dXt= (µ − 1 2Vt)dt + p VtdWt dVt= κ(θ − Vt)dt + p VtdZt dWtZt= ρdt (4.4)

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) .

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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 dZtbecause 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

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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:

min

µ,θ,κ,,ρ

X

k

|C_{M odel}(Ki, Ti, µ, θ, κ, , ρ) − CM arket(Ki, Ti)|

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

(4.5)

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:

dSt= µStdt + σSαtdWt (4.6)

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dXt= (µ − 1 2σ 2_S2(α−1) t )dt + σS α−1 t dWt (4.7)

The most important distinction compared to the GBM is that the volatility of the returns is no longer equal to a constant but involves an extra term S_tα−1. 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 charac-teristics 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 St will increase the volatility Stα−1σ. 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.

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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

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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:}

σ2(K, t) = ∂C(K,t) δt + rK ∂C(K,t) ∂K 1 2K2 ∂ 2_C(K,t) ∂K2 (4.9)

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 ∞

α

x

σ2_(x)dx < ∞ (4.10)

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

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

In our case with the local part extended just as the CEV model to dSt= µStdt + Stα p VtdWt dVt= κ(θ − Vt)dt + p VtdZt dWtdZt= ρdt (4.12)

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

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dXt= (µ − 1 2VtS 2(α−1) t )dt + p VtStα−1dWt (4.13)

The interpretation of the parameters (µ, α, κ, θ, , ρ) are still equal to the interpreta-tion 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

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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 of (α, ρ). In Figure 9 there is still a leverage effect which means corr(dXt, dS2α−2t 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

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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

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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 re-turns 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 pro-cess 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.

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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,

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the local volatility part fills this gap such that the mismatch of (4.5) reduces signifi-cantly. 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]:

σLV(St, t) = σ(St, t)

p

E[f (Vt)2|St= s] = σ(St, t)

p

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 op-tion 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 param-eters). 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 func-tion 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.

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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 , F_{t∈[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: dP dQ|Ft = exp − Z t 0 Λ1_udWu+ Z t 0 Λ2_udZu − 1 2 Z t 0 (Λ1_u)2du + Z t 0 (Λ2_u)2du (5.1)

Were Λ1 _{is known as the market price of stochastic volatility and Λ}2 _{the market}

price of the stock price. Similair to the Heston model we assume that the market price of volatility is proportional to the volatility, hence: Λ1(t) = λ√Vt. By Girsanov’s

theorem under the risk-neutral probability measure P the Brownian motions change: dW∗(t) = dW (t) − Λ1(t) = dW (t) − λpVtdt

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

µ − r = S_tα−1pVt(ρΛ1(t) +

p

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This automatically implies that for the second market price of risk Λ2(t) = p 1 1 − ρ2 µ − r S_tα−1√Vt − λρpVt (5.4)

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: dSt= rStdt + Stα p Vt(ρdWt∗+ p 1 − ρ2_dZ∗ t) dVt= κ(θ − Vt)dt + p Vt(dWt∗+ λ p Vtdt) (5.5) Hence the stochastic variance part also changes under the risk-neutral probability measure (Ω, F , Ft∈[0,T ], P) dVt= (κ + λ)( κθ κ + λ− Vt)dt + p Vt(dW∗t) dVt= κ∗(θ∗− Vt)dt + p Vt(dW∗t) (5.6)

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.

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5.2 Proof of Martingale Property

The driftless HSLV model is a stochastic integral of the integrable process √VtStα 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:

E[St|Fk] = E[ lim

n→∞inf St∧τn|Fk]

≤ lim

n→∞inf E[St∧τn|Fk](by Fatou’s lemma)

= Sk

(5.7)

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}.

By definition of a local martingale the process Sn

t := 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 A Pn(dω) = Z A S_Tn S0P(dω) or Pn(A) = EP Sn T S0 1A = 1 S0P(S n T1A) A ∈ FT (5.8)

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

St= S0exp Z t 0 S_uα−1pVudWu− 1 2 Z t 0 S_u2(α−1)Vudu (5.9) 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 ]:

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P(ST1{W ∈Γ}) = Z Γ ST(ω)P(dω) = lim n→∞ Z Γ

S_Tn1τn>TP(dω) Dominated Convergence Theorem

= S0 lim n→∞ Z Γ 1τn>TPn(dω) applying (5.8) = S0 lim n→∞ Z Γ 1τˆn>TP(dω) (5.10)

Where ˆτn= {inf t ∈ (0, ∞) : ˆSt≥ n}, ˆSt and ˆVt are defined as the solution to:

d ˆSt= ˆSt2α−1Vˆtdt + ˆStα q ˆ VtdWt d ˆVt= κ∗(θ∗− ˆVt)dt + ˆVtSˆtα−1dt + q ˆ Vt(ρdWt+ p 1 − ρ2_dZ t) (5.11)

With ˆWt defined as:

d ˆW_tn= dWt+

q ˆ

VtSˆtα−1dt (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:

dPn dP |Ft = S_tn S0 = exp Z t 0 1t≤τn∧TS α−1 u p VudWu− 1 2 Z t 0 1t≤τn∧TS 2(α−1) u Vudu (5.13) Under the new probability measure Pnwe have, the Brownian motion Wtn, Stand Vt

as the solution to the differential equations dW_tn= dWt− 1t≤τn∧TS α−1 t p Vtdt dSt= 1t≤τn∧TS 2(α−1) t Vtdt + Stα p VtdWtn dVt= κ∗(θ∗− Vt)dt + 1t≤τn∧TρVtS α−1 t dt + 1t≤τn∧T p Vt(ρdWtn+ p 1 − ρ2_dZ t) (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 dWnis a Brownian motion . If we take Γ = B(C[0, T ]) we return to the calculation in (5.10)

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P(ST) = S0 lim n→∞

Z

Ω

1ˆτn>TP(dω)

= S0P(Sˆtdoes not explode on [0, T ])

(5.15)

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 ˆSt depends also on ˆVt 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 ˆSt does not explode on [0, T ] (hence τn> T ):

Pn(Ω) = 1 = EP Sn T S0 1Ω → S₀ _{= E}_P(S_Tn) = EP(ST) since τn> T (5.16)

Hence in order to verify whether our stock price process is a martingale we have to check whether ˆSthas 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 ˆStdefined 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.

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When we assume that St is a martingale we apply Girsanov’s Theorem and define a

new probability measure Pn as:

dPn dP|Ft = St S0 = exp Z t 0 S_uα−1pVudWu− 1 2 Z t 0 S_u2(α−1)Vudu (5.17) This Radon-Nikodym derivative is the solution of the drift less HSLV process. Under this new measure Pn the process dWt changes:

dW_tn= dWt−

p

VtStα−1dt (5.18)

Hence the price process changes under the measure Pn to

dSt= St2α−1Vtdt + Stα p VtdWt dVt= κ∗(θ∗− Vt) + ρStα−1Vtdt + (ρdWtn+ p 1 − ρ2_dZ t) (5.19) By Girsanov’s Theorem under this new probability measure Pnis equivalent to P which

means that for all A ∈ Ft P (A) > 0 ↔ Pn(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.

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(a) Vtunder 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

dXt= 0.5S2α−2t Vtndt + Stα−1pVtndWtn

dV_tn= κ∗(θ∗− V_tn)dt + ρS_tα−1V_tndt + pV_tn(ρdW_tn+p1 − ρ2_dZ t)

(5.20)

While under the old measure P

dXt= −0.5St2α−2Vtdt + Stα−1 p VtdWt dVt= κ∗(θ∗− Vt)dt + p Vt(ρdWt+ p 1 − ρ2_dZ t) (5.21)

Since the drift part of the log returns is 0.5S_t2α−2Vtdt hence positive under Pn while

under measure P the drift part equals −0.5St2α−2Vtdt Stn tends to become higher under

Pnas Figure 16 shows. Hence under the new measure Pnthe log returns have a drift part

which increases as the stock price increases and a local part of volatility which increases as well when α > 1. As Figure 17b shows the volatility of the log returns S_tα−1√Vt is

strictly larger than the volatility of the stock price process under P. Figure 17a shows the marginal effect of the stochastic variance Vtwhen the negative drift part ρStα−1Vtdt

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

also to a lowerpV_tn when ρ < 0 (Figure 17a), the positive effect of the local volatility effect S_tα−1 is larger than the downward effect on V_tn (Figure 17b).

For some A ∈ Ft the Brownian motion behaves such that stock price process Stn

reaches extreme high values. This still does not imply the nonequivalence between Pnand

Bubbles in the Heston stochastic local volatility model.