Johannes Ruf

On the hedging of options on exploding exchange rates

We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a change of numéraire (change of currency) technique when the underlying is only a local martingale, modelling for example an exchange rate. The new pricing operator assigns prices to contingent claims according to the minimal cost for superreplication strategies that succeed with probability one for both currencies as numéraire. Within this context, we interpret the lack of the martingale property of an exchange rate as a reflection of the possibility that the numéraire currency may devalue completely against the asset currency (hyperinflation).

Optimal Trading Strategies Under Arbitrage

Optimal Trading Strategies Under Arbitrage Johannes Karl Dominik Ruf This thesis analyzes models of financial markets that incorporate the possibility of arbitrage opportunities. The first part demonstrates how explicit formulas for optimal trading strategies in terms of minimal required initial capital can be derived in order to replicate a given terminal wealth in a continuous-time Markovian context. Towards this end, only the existence of a square-integrable market price of risk (rather than the existence of an equivalent local martingale measure) is assumed. A new measure under which the dynamics of the stock price processes simplify is constructed. It is shown that delta hedging does not depend on the “no free lunch with vanishing risk” assumption. However, in the presence of arbitrage opportunities, finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. In order to apply these analytic tools, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The phenomenon of “bubbles,” which has been a popular topic in the recent academic literature, appears as a special case of the setting in the first part of this thesis. Several examples at the end of the first part illustrate the techniques contained therein. In the second part, a more general point of view is taken. The stock price processes, which again allow for the possibility of arbitrage, are no longer assumed to be Markovian, but rather only Ito processes. We then prove the Second Fundamental Theorem of Asset Pricing for these markets: A market is complete, meaning that any bounded contingent claim is replicable, if and only if the stochastic discount factor is unique. Conditions under which a contingent claim can be perfectly replicated in an incomplete market are established. Then, precise conditions under which relative arbitrage and strong relative arbitrage with respect to a given trading strategy exist are explicated. In addition, it is shown that if the market is quasi-complete, meaning that any bounded contingent claim measurable with respect to the stock price filtration is replicable, relative arbitrage implies strong relative arbitrage. It is further demonstrated that markets are quasi-complete, subject to the condition that the drift and diffusion coefficients are measurable with respect to the stock price filtration.

A new proof for the conditions of Novikov and Kazamaki

This paper provides a novel proof for the sufficiency of certain well-known criteria that guarantee the martingale property of a continuous, nonnegative local martingale. More precisely, it is shown that generalizations of Novikov’s condition and Kazamaki’s criterion follow directly from the existence of Follmer’s measure. This approach allows to extend well-known criteria of martingality from strictly positive to only nonnegative, continuous local martingales.

Supermartingales as Radon–Nikodym densities and related measure extensions

Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.

Why are quadratic normal volatility models analytically tractable

We discuss the class of quadratic normal volatility (QNV) models, which have drawn much attention in the financial industry due to their analytic tractability and flexibility. We characterize these models as those that can be obtained from stopped Brownian motion by a simple transformation and a change of measure that depends only on the terminal value of the stopped Brownian motion. This explains the existence of explicit analytic formulas for option prices within QNV models in the academic literature. Furthermore, via a different transformation, we connect a certain class of QNV models to the dynamics of geometric Brownian motion and discuss changes of numeraires if the numeraire is modeled as a QNV process.

A weak convergence criterion for constructing changes of measure

Abstract Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes.

Pricing corporate bonds in an arbitrary jump-diffusion model based on an improved Brownian-bridge algorithm

We provide an efficient and unbiased Monte-Carlo simulation for the computation of bond prices in a structural default model with jumps. The algorithm requires the evaluation of integrals with the density of the firstpassage time of a Brownian bridge as the integrand. Metwally and Atiya (2002) suggest an approximation of these integrals. We improve this approximation in terms of precision. From a modelers point of view, we show that a structural model with jumps is able to endogenously generate stochastic recovery rates. It is well known that allowing a sudden default by a jump results in a positive limit of credit spreads at the short end of the term structure. We provide an explicit formula for this limit, depending only on the Levy measure of the logarithm of the firm-value process, the recovery rate, and the distance to default.

Trading strategies generated by Lyapunov functions

Functional portfolio generation, initiated by E.R. Fernholz almost 20 years ago, is a methodology for constructing trading strategies with controlled behavior. It is based on very weak and descriptive assumptions on the covariation structure of the underlying market, and needs no estimation of model parameters. In this paper, the corresponding generating functions G

A Practical Guide to Measuring Social Structure Using Indirectly Observed Network Data

G

Negative call prices

are interpreted as Lyapunov functions for the vector process μ