Quantitative Finance

The first widely used financial model is linked to dynamical Hamilton jacobi model

Valuation adjustments are nowadays a common practice to include credit and liquidity effects in option pricing. Funding costs arising from collateral procedures, hedging strategies and taxes are added to option prices to take into account the production cost of financial contracts so that a profitability analysis can be reliably assessed. In particular, when dealing with linear products, we need a precise evaluation of such contributions since bid-ask spreads may be very tight. In this paper we start from a general pricing framework inclusive of valuation adjustments to derive simple evaluation formulae for the relevant case of total return equity swaps when stock lending and borrowing is adopted as hedging strategy.

Siegel's paradox is a fundamental question in international finance about exchange rates for futures contracts and has puzzled many scholars for over forty years. The unorthodox approach presented in this article leads to an arbitrage-free solution which is invariant under currency re-denominations and is symmetric, as explained. We will also give a complete classification of all such aggregators in the general case. The formula obtained in this setting therefore describes all the negotiated no-arbitrage forward exchange rates in terms of a reciprocity function. Keywords: Siegel's paradox, forward exchange rates, discount bias.

Over the past few years, the futures market has been successfully developing in the North-West region. Futures markets are one of the most effective and liquid-visible trading mechanisms. A large number of buyers are forced to compete with each other and raise their prices. A large number of sellers make them reduce prices. Thus, the gap between the prices of offers of buyers and sellers is reduced due to high competition, and this is a good criterion for the liquidity of the market. This high degree of liquidity contributed to the fact that futures trading took such an important role in commerce and finance. A multi-step, non-cooperative n persons game is formalized and studied

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

This paper investigates the dependence of functional portfolio generation, introduced by Fernholz (1999), on an extra finite variation process. The framework of Karatzas and Ruf (2017) is used to formulate conditions on trading strategies to be strong arbitrage relative to the market over sufficiently large time horizons. A mollification argument and Komlos theorem yield a general class of potential arbitrage strategies. These theoretical results are complemented by several empirical examples using data from the S&P 500 stocks.

In this short note, we will strengthen the classic Doob's L p inequality for sub-martingale processes. Because this inequality is of fundamental importance to the theory of stochastic process, we believe this generalization will find many interesting applications.

We introduce generalized filtration with which we can represent situations such as some agents forget information at some specific time. The filtration is defined as a functor to a category Prob whose objects are all probability spaces and whose arrows correspond to measurable functions satisfying an absolutely continuous requirement [Adachi and Ryu, 2019]. As an application of a generalized filtration, we develop a binomial asset pricing model, and investigate the valuations of financial claims along this type of non-standard filtrations.

Generalized statistical arbitrage concepts are introduced corresponding to trading strategies which yield positive gains on average in a class of scenarios rather than almost surely. The relevant scenarios or market states are specified via an information system given by a σ -algebra and so this notion contains classical arbitrage as a special case. It also covers the notion of statistical arbitrage introduced in Bondarenko (2003). Relaxing these notions further we introduce generalized profitable strategies which include also static or semi-static strategies. Under standard no-arbitrage there may exist generalized gain strategies yielding positive gains on average under the specified scenarios. In the first part of the paper we characterize these generalized statistical no-arbitrage notions. In the second part of the paper we construct several profitable generalized strategies with respect to various choices of the information system. In particular, we consider several forms of embedded binomial strategies and follow-the-trend strategies as well as partition-type strategies. We study and compare their behaviour on simulated data. Additionally, we find good performance on market data of these simple strategies which makes them profitable candidates for real applications.

To convert standard Brownian motion Z into a positive process, Geometric Brownian motion (GBM) e β Z t ,β>0 is widely used. We generalize this positive process by introducing an asymmetry parameter α≥0 which describes the instantaneous volatility whenever the process reaches a new low. For our new process, β is the instantaneous volatility as prices become arbitrarily high. Our generalization preserves the positivity, constant proportional drift, and tractability of GBM, while expressing the instantaneous volatility as a randomly weighted L 2 mean of α and β . The running minimum and relative drawup of this process are also analytically tractable. Letting α=β , our positive process reduces to Geometric Brownian motion. By adding a jump to default to the new process, we introduce a non-negative martingale with the same tractabilities. Assuming a security's dynamics are driven by these processes in risk neutral measure, we price several derivatives including vanilla, barrier and lookback options.