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 Volatility (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. 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 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 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.