Worst-case optimal investment with a random number of crashes

We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. We then verify that this strategy outperforms every other trading strategy using a direct comparison approach. We conclude with numerical examples and calculating the costs of hedging against crashes

Value Matters: The Long-run Behavior of Stock Index Returns

We present a simple dynamical model of stock index returns grounded on the ability of the Cyclically Adjusted Price Earning valuation ratio devised by Robert Shiller to predict long-horizon performances of the market. Specifically, within the model returns are driven by a fundamental term and an autoregressive component perturbed by external random disturbances. The autoregressive component arises from the agents\u2019 belief that expected returns are higher in bullish markets than in bearish ones. The fundamental value, towards which fundamentalists expect that the current price should revert, varies in time and depends on the initial averaged Price-to- Earnings ratio. We demonstrate both analytically and by means of numerical experiments that the long-run behavior of the stylized dynamics agrees with empirical evidences reported in literature

Mathematical models for financial bubbles

Financial bubbles have been present in the history of financial markets from the early days up to the modern age. An asset is said to exhibit a bubble when its market value exceeds its fundamental valuation. Although this phenomenon has been thoroughly studied in the economic literature, a mathematical martingale theory of bubbles, based on an absence of arbitrage has only recently been developed. In this dissertation, we aim to further contribute to the developement of this theory. In the first part we construct a model that allows us to capture the birth of a financial bubble and to describe its behavior as an initial submartingale in the build-up phase, which then turns into a supermartingale in the collapse phase. To this purpose we construct a flow in the space of equivalent martingale measures and we study the shifting perception of the fundamental value of a given asset. In the second part of the dissertation, we study the formation of financial bubbles in the valuation of defaultable claims in a reduced-form setting. In our model a bubble is born due to investor heterogeneity. Furthermore, our study shows how changes in the dynamics of the defaultable claim's market price may lead to a different selection of the martingale measure used for pricing. In this way we are able to unify the classical martingale theory of bubbles with a constructive approach to the study of bubbles, based on the interactions between investors.Finanz-Blasen sind seit der Entstehung der Finanzmärkte bis zur heutigen Zeit gegenwärtig. Es gilt, dass ein Vermögenswert eine Finanzblase aufweist, sobald dessen Marktwert die fundamentale Bewertung übersteigt. Obwohl dieses Phänomen in der Wirtschaftsliteratur ausgiebig behandelt wurde, ist eine mathematische Martingaltheorie von Blasen, die auf der Abwesenheit von Arbitragemöglichkeiten beruht, erst in letzter Zeit entwickelt worden. Das Ziel dieser Dissertation ist es einen Beitrag zur Weiterentwicklung dieser Theorie zu leisten. Im ersten Abschnitt konstruieren wir ein Model mit Hilfe dessen man die Entstehung einer Finanz-Blase erfassen und deren Verhalten anfänglich als Submartingal in der build-up phase beschrieben werden kann, welches dann in der collapse phase zu einem Supermartingal wird. Zu diesem Zweck entwickeln wir einen Zahlungsstrom im Raum der äquivalenten Martingalmaße und wir untersuchen die zu dem Vermögenswert passende Verschiebung des fundamentalen Werts. Der zweite Teil der Dissertation beschäftigt sich mit der Bildung von Finanz-Blasen bei der Bewertung von Forderungen, die mit Ausfallrisiken behaftet sind, in einer reduzierten Marktumgebung. In unserem Model ist die Entstehung einer Blase die Folge der Heterogenität der Investoren. Des Weiteren zeigen unsere Untersuchungen, inwieweit Veränderungen der Dynamik des Marktpreises einer risikobehafteten Forderung zu einer Veränderung des zur Bewertung verwendeten Martingalmaß es führen kann. Dadurch sind wir in der Lage die klassische Martingaltheorie von Finanz-Blasen mit einem konstruktivem Ansatz zur Untersuchung von Finanz-Blasen zu vereinigen, der auf den Interaktionen zwischen Marktteilnehmern basiert

Volatility Modeling Using High Frequency Trade Data to Identify Cryptocurrency Bubbles

In the light of sudden interest in Bitcoin during 2017, which saw Bitcoin growing multifold in market price, I study blockchain, Bitcoin and few other top cryptocurrencies, and examine whether Bitcoin was in a financial bubble during late 2017 and early 2018, when Bitcoin price had a sudden and dramatic run up. I conduct statistical analysis on the High Frequency Trade (HFT) data, sourced from Bloomberg and other crypto exchanges. The statistical analysis includes filtering in price data using 5%, 7% and 10% daily price jump rules (considered separately), interpolating price points between low and high prices in the time series, estimating price volatility at discrete price points, analyzing the volatility behavior and concluding whether or not price process is a strict local martingale. A bubble is confirmed if the price process is a strict local martingale, and not a true martingale. I run the test for Bitcoin, and find that Bitcoin was in intermittently in a bubble during the years 2017 and 2018. I repeat the test for Ethereum, another top trading cryptocurrency, and find that Ethereum was in a bubble during Nov 2017 - Feb 2018, but infrequently and for lower duration of days as compared to Bitcoin. Though Bitcoin price dramatically increased during 2017, the number of transactions and transactions volume rather fell. I develop a statistical test that can be applied on the High Frequency Trade (HFT) data of any highly traded asset to identify whether or not that asset has been in a bubble during the period of consideration. I also find that Bitcoin has positive correlation with other top cryptocurrencies and almost zero correlation with S&P 500, gold, and the crude oil

Shifting Martingale Measures and the Birth of a Bubble as a Submartingale

In an incomplete financial market model, we study a flow in the space of equivalent martingale measures and the corresponding shifting perception of the fundamental value of a given asset. This allows us to capture the birth of a perceived bubble and to describe it as an initial submartingale which then turns into a supermartingale before it falls back to its initial value zero