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arXiv2014

Fair and profitable bilateral prices under funding costs and collateralization

Bielecki and Rutkowski (2014) introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts and margin accounts. In this paper, we examine the pricing and hedging of contract both from the perspective of the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we give the range for either fair bilateral prices or bilaterally profitable prices. We also study the monotonicity of a unilateral price with respect to the initial endowment. Our study hinges on results for BSDE driven by continuous martingales obtained in Nie and Rutkowski (2014), but we also derive the pricing PDEs for path-independent contingent claims of European style in a Markovian framework.

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arXiv2015

Generalized Dynkin game of switching type representation for defaultable claims in presence of contingent CSA

We study the solution's existence for a generalized Dynkin game of switching type which is shown to be the natural representation for general defaultable OTC contract with contingent CSA. This is a theoretical counterparty risk mitigation mechanism that allows the counterparty of a general OTC contract to switch from zero to full/perfect collateralization and switch back whenever she wants until contract maturity paying some switching costs and taking into account the running costs that emerge over time. In this paper we allow for the strategic interaction between the counterparties of the underlying contract, which makes the problem solution much more tough. We are motivated in this research by the importance to show the economic sense - in terms of optimal contract design - of a contingent counterparty risk mitigation mechanism like our one. In particular, we show that the existence of the solution and the game Nash equilibrium is connected with the solution of a system of non-linear reflected BSDE which remains an open problem. We then provide the basic ideas to numerically search the game equilibrium via an iterative optimal stopping approach and we show the existence of the solution for our problem under strong condition, in the so called symmetric case.

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arXiv2014

Fair bilateral prices in Bergman's model

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arXiv2014

Rationality parameter for exercising American put

The main result of this paper is a probabilistic proof of the penalty method for approximating the price of an American put in the Black-Scholes market. The method gives a parametrized family of partial differential equations, and by varying the parameter the corresponding solutions converge to the price of an American put. For each PDE the parameter may be interpreted as a rationality parameter of the holder of the option. The method may be extended to other valuation situations given as an optimal stopping problem with no explicit solution. The method may also be used for valuations where actors do not behave completely rationally but instead have randomness affecting their choices. The rationality parameter is a measure for this randomness.

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arXiv2014

On volatility smile and an investment strategy with out-of-the-money calls

A motivating question in this paper is whether a sensible investment strategy may systematically contain long positions in out-of-the-money European calls with short expiry. Here we consider a very simple trading strategy for calls. The main points of this note are the following. First, the presented trading strategy appears very lucrative in the Black-Scholes-Merton (BSM) framework. In fact, it is such even to the extent that the BSM model turns out to be, in a sense, incompatible with the CAPM. Second, if one wishes to adapt these models together, then the adjustment of the consistent pricing rule (i.e. modifying state price densities) inevitably leads to some form of volatility smile and this is the main point of the paper. Moreover, these observations arise from purely structural considerations.

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arXiv2016

Path Integral and Asset Pricing

We give a pragmatic/pedagogical discussion of using Euclidean path integral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to "less-tractable" models such as the Black-Karasinski model. We give explicit formulas for computing the bond pricing function in such models in the analog of quantum mechanical "semiclassical" approximation. We also outline how to apply perturbative quantum mechanical techniques beyond the "semiclassical" approximation, which are facilitated by Feynman diagrams.

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arXiv2015

Ross Recovery with Recurrent and Transient Processes

Recently, Ross showed that it is possible to recover an objective measure from a risk-neutral measure. His model assumes that there is a finite-state Markov process X that drives the economy in discrete time. Many authors extended his model to a continuous-time setting with a Markov diffusion process X with state space R. Unfortunately, the continuous-time model fails to recover an objective measure from a risk-neutral measure. We determine under which information recovery is possible in the continuous-time model. It was proven that if X is recurrent under the objective measure, then recovery is possible. In this article, when X is transient under the objective measure, we investigate what information is sufficient to recover.

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arXiv2015

Arbitrage theory without a numéraire

This note develops an arbitrage theory for a discrete-time market model without the assumption of the existence of a numéraire asset. Fundamental theorems of asset pricing are stated and proven in this context. The distinction between the notions of investment-consumption arbitrage and pure-investment arbitrage provide a discrete-time analogue of the distinction between the notions of absolute arbitrage and relative arbitrage in the continuous-time theory. Applications to the modelling of bubbles is discussed.

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arXiv2014

Volatility is rough

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

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arXiv2016

Banach geometry of arbitrage free markets

The article presents a description of geometry of Banach structures forming mathematical base of markets arbitrage absence type phenomena. In this connection the role of reflexive subspaces (replacing classically considered finite-dimensional subspaces) and plasterable cones is uncovered.

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arXiv2015

Robust Fundamental Theorem for Continuous Processes

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family $\mathcal{P}$ of possible physical measures. A robust notion ${\rm NA}_{1}(\mathcal{P})$ of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: ${\rm NA}_{1}(\mathcal{P})$ holds if and only if every $P\in\mathcal{P}$ admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.

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arXiv2016

Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model

We obtain stability estimates and derive analytic expansions for local solutions of multi-dimensional quadratic BSDEs. We apply these results to a financial model where the prices of risky assets are quoted by a representative dealer in such a way that it is optimal to meet an exogenous demand. We show that the prices are stable under the demand process and derive their analytic expansions for small risk aversion coefficients of the dealer.

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arXiv2015

Pricing of European Basket Call Option under Exponential Ornstein-Uhlenbeck Process

Pricing of European basket call option with n-assets and a bond is discussed in this paper, where all prices of n-assets and the bond are driven by Exponential Ornstein-Uhlenbeck processes. The close-form of European basket option pricing formula is derived. Utilizing with 1-order differential approximate numerical solution of stochastic differential equation (Milstein method), a simulation example of European basket option pricing with 3 assets is also given.

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arXiv2014

Visualisation of financial time series by linear principal component analysis and nonlinear principal component analysis

In this dissertation, the main goal is visualisation of financial time series. We expect that visualisation of financial time series will be a useful auxiliary for technical analysis. Firstly, we review the technical analysis methods and test our trading rules, which are built by the essential concepts of technical analysis. Next, we compare the quality of linear principal component analysis and nonlinear principal component analysis in financial market visualisation. We compare different methods of data preprocessing for visualisation purposes. Using visualisation, we demonstrate the difference between normal and crisis time period. Thus, the visualisation of financial market can be a tool to support technical analysis.

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arXiv2016

Pricing and Hedging Long-Term Options

In this article, we investigate the behavior of long-term options. In many cases, option prices follow an exponential decay (or growth) rate for further maturity dates. We determine under what conditions option prices are characterized by this property. To see this, we use the martingale extraction method through which a pricing operator is transformed into a semigroup operator, which is easier to address. We also explore notions of hedging long-term options. Hedging is an attempt to reduce market risks, and we investigate the price sensitivities (Greeks) with respect to such risks, which are typically repre- sented by variations in the underlying process of an option. We combine the Malliavin calculus with the martingale extraction method to analyze Greeks. We see that the ratios between Greeks and the option price are expressed in a simple form in the long term.

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arXiv2015

Super-replication with nonlinear transaction costs and volatility uncertainty

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim's payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [11] and [19] for the case of model uncertainty.

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arXiv2015

Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the long maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

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arXiv2015

Incorporating Views on Market Dynamics in Options Hedging

We examine the possibility of incorporating information or views of market movements during the holding period of a portfolio, in the hedging of European options with respect to the underlying. Given a fixed holding period interval, we explore whether it is possible to adjust the number of shares needed to effectively hedge our position to account for views on market dynamics from present until the end of our interval, to account for the time-dependence of the options' sensitivity to the underlying. We derive an analytical expression for the number of shares needed by adjusting the standard Black-Scholes-Merton $Δ$ quantity, in the case of an arbitrary process for implied volatility, and we present numerical results.

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