Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusion-convection equations, where the stock price, volatility, trend and option price may depend on the time, the stock price or the option price itself. In this chapter we will be concerned with several models from the most relevant class of nonlinear Black-Scholes equations for American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will analytically approach the option price by following the ideas proposed by Ševčovič and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain. Finally, we will present the results of a split-step finite difference schemes for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model

Hedging Contingent Claims in Markets with Jumps

Contrary to the Black-Scholes paradigm, an option-pricing model which incorporates the possibility of jumps more accurately reflects the evolution of stocks in the real world. However, hedging a contingent claim in such a model is a non-trivial issue: in many cases, an infinite number of hedging instruments are required to eliminate the risk of an option position. This thesis develops practical techniques for hedging contingent claims in markets with jumps. Both regime-switching and jump-diffusion models are considered

VALUATION OF OPTIONS IN A HIGH VOLATILE REGIME SWITCHING MARKET

Financial modeling by Stochastic Differential Equations-SDEs with regime-switching has been utilized to allow moving from one economic state to another. The aim of this thesis is to tackle the pricing of European options under a regime-switching model where the volatility is augmented. Regime-switching models are more realistic since they encompass the effect of an external event on the underlying asset prices. But they are challenging, considering in addition increased volatility in the model will for sure make the option pricing problem more complicated and its solution may not exist analytically. Numerical methods for finance could be very helpful in this case. This study proposes a new SDE for the prices of the underlying financial asset under regime-switching with a high volatility model. The suggested model covers the crisis model of [3] and [4] for highly volatile situations and the regime-switching model of [5]. Under these settings, the valuation of European options is investigated based on the works of [5], [7], [8], and [9]. As an application, a study of two states is developed: state 1 when the economy is going well, and state 2 when the economy is under stress

Structural pricing of XVA metrics for energy commodities OTC trades

The aim of the present Chapter is to improve of the structural rst-passage framework built in Chapter 1 along several directions as well as test its robustness. Since typically commodity trades are not clearable under Central Clearing Counterparts (CCPs), it is worthy to assess the eect of bilateral Collateral Support Annex (CSA) agreements on CVA/DVA metrics. Moreover I introduce within my CCR modelling, the impact of state-dependent stochastic recovery rates. Furthermore, in order to stress-test my framework, I investigate the eects on CCR measures of multiplicative shocks to the two major drivers in the game: credit and volatility. Finally I propose an alternative balance-sheet calibration based on hybrid market/accounting data which is well suited in the commodity context in the light of small and medium size of corporations usually operating in the EU commodity derivatives market for risk-management purposes.The global nancial crisis revealed that no economic entity can be considered default-free any more. Because of that, both banks and corporations have to deal with bilateral Counterparty Credit Risk (CCR) in their OTC derivatives trades. Such evidence implies the fair pricing of these risks, namely the Credit Valuation Adjustment (CVA) and its counterpart, the Debt Valuation Adjustment (DVA). Despite the more commonly used reduced-form approach, in this work the random default time is addressed via a structural approach a la Black and Cox (1976), so that the bankruptcy of a given rm is modelled as the rst-passage time of its equity value from a predetermined lower barrier. As in Ballotta et al. (2015), I make use of a time-changed Levy process as underlying source of both market and credit risk. The main advantage of this setup relies on its superior capability to replicate non null short-term default probabilities, unlike pure diusion models. Moreover, a numerical computation of the valuation adjustments for bilateral CCR in the context of energy commodities OTC derivatives contracts has been performed.The global nancial crisis revealed that no economic entity can be considered default-free any more, so that both banks and corporate rms have to cope with bilateral Counterparty Credit Risk (CCR) when negotiating OTC derivatives. Since the mainstream approach typically used in practical settings is to evaluate derivatives in terms of the cost of their respective hedging strategies, the pricing of CCR metrics implicitly relates to the way these strategies are nanced. Within the numerical section of the present work, the valuation adjustments for CCR have been computed. Moreover, the role played by funding costs and their impact in widening bid-ask spreads have been assessed. A similar reasoning has been applied for the investigation of the cost of funding Initial Margins (IM), typically eective on top of Variation Margins (VM) when trading under Central Clearing Counterparties (CCPs). As the Initial Margin Valuation Adjustment (MVA) is concerned, it is here showed that, dierently from what can happen for FVAs, no osetting eect can materialize. As a consequence, in aggregate terms IMs can cause systemic liquidity eects. The computed XVA metrics are relative to energy commodities OTC derivative trades

Essays in financial asset pricing

Three essays in financial asset pricing are given; one concerning the partial differential equation (PDE) pricing and hedging of a class of continuous/generalized power mean Asian options, via their (optimal) Lie point symmetry groups, leading to practical pricing formulas. The second presents high-frequency predictions of S&P 500 returns via several machine learning models, statistically significantly demonstrating short-horizon market predictability and economically significantly profitable (beyond transaction costs) trading strategies. The third compares profitability between these [(mean) ensemble] strategies and Asian option Δ-hedging, using results of the first. Interpreting bounds on arithmetic Asian option prices as ask and bid values, hedging profitability depends largely on securing prices closer to the bid, and settling midway between the bid and ask, significant profits are consistently accumulated during the years 2004-2016. Ensemble predictive trading the S&P 500 yields comparatively very small returns, despite trading much more frequently. The pricing and hedging of (arithmetic) Asian options are difficult and have spurred several solution approaches, differing in theoretical insight and practicality. Multiple families of exact solutions to relaxed power mean Asian option pricing boundary-value problems are explicitly established, which approximately satisfy the full pricing problem, and in one case, converge to exact solutions under certain parametric restrictions. Corresponding hedging parameters/ Greeks are derived. This family consists of (optimal) invariant solutions, constructed for the corresponding pricing PDEs. Numerical experiments explore this family behaviorally, achieving reliably accurate pricing. The second chapter studies intraday market return predictability. Regularized linear and nonlinear tree-based models enjoy significant predictability. Ensemble models perform best across time and their return predictability realizes economically significant profits with Sharpe ratios after transaction costs of 0.98. These results strongly evidence that intraday market returns are predictable during short time horizons, beyond that explainable by transaction costs. The lagged constituent returns are shown to hold significant predictive information not contained in lagged market returns or price trend and liquidity characteristics. Consistent with the hypothesis that predictability is driven by slow-moving trader capital, predictability decreased post-decimalization, and market returns are more predictable midday, on days with high volatility or illiquidity, and during financial crises

A consistent framework for valuation under collateralization, credit risk and funding costs

We develop a consistent, arbitrage-free framework for valuing derivative trades with collateral, counterparty credit risk, and funding costs. This is achieved by modifying the payout cash-flows for the trade position. The framework is flexible enough to accommodate actual trading complexities such as asymmetric collateral and funding rates, replacement close-out, and rehypothecation of posted collateral. We show also how the traditional self-financing condition is adjusted to reflect the new market realities. The generalized valuation equation takes the form of a forward-backward SDE or semi-linear PDE. Nevertheless, it may be recast as a set of iterative equations which can be efficiently solved by our proposed least-squares Monte Carlo algorithm. We numerically implement the case of an equity option and show how its valuation changes when including the above effects. We also discuss the financial impact of the proposed valuation framework and of nonlinearity more generally. This is fourfold: Firstly, the valuation equation is only based on observable market rates, leaving the value of a derivatives transaction invariant to any theoretical risk-free rate. Secondly, the presence of funding costs and default close-out makes the valuation problem a recursive and nonlinear one. Thus, credit and funding risks are non-separable in general, and despite common practice in banks, the related CVA, DVA, and FVA cannot be treated as purely additive adjustments without running the risk of double counting. To quantify the valuation error that can be attributed to double counting, we introduce a nonlinearity valuation adjustment (NVA) and show that its magnitude can be significant under asymmetric funding rates and replacement close-out at default. Thirdly, as trading parties cannot observe each others liquidity policies nor their respective funding costs, the bilateral nature of a derivative price breaks down. Finally, valuation becomes aggregation-dependent and portfolio values cannot simply be added up. This has operational consequences for banks, calling for a holistic, consistent approach across trading desks and asset classes.Open Acces

Non-stationary data-driven computational portfolio theory and algorithms

The aim of the dissertation is the development of a data-driven portfolio optimization framework beyond standard assumptions. Investment decisions are either based on the opinion of a human expert, who evaluates information about companies, or on statistical models. The most famous methods based on statistics are the Markowitz portfolio model and utility maximization. All statistical methods assume certain knowledge over the underlying distribution of the returns, either by imposing Gaussianity, by expecting complete knowledge of the distribution or by inferring sufficiently good estimators of parameters. Yet in practice, all methods suffer from incomplete knowledge, small sample sizes and the problem that parameters might be varying over time. A new, model-free approach to the portfolio optimization problem allowing for time-varying dynamics in the price processes is presented. The methods proposed in this work are designed to solve the problem with less a-priori assumptions than standard methods, like assumptions on the distribution of the price processes or assumptions on time-invariant statistical properties. The new approach introduces two new parameters and a method to chose these based on principles of information theory. An analysis of different approaches to incorporate additional information is performed before a straightforward approach to the out-of-sample application is introduced. The structure of the numerical problem is obtained directly from the problem of portfolio optimization, resulting in a system of objective function and constraints known from non-stationary time series analysis. The incorporation of transaction costs allows to naturally obtain regularization that is normally included for numerical reasons. The applicability and the numerical feasibility of the method are demonstrated in a low-dimensional example in-sample and in a high-dimensional example in- and out-of-sample in an environment with mixed transaction costs. The performance of both examples is measured and compared to standard methods, as the Markowitz approach and to methods based on techniques to analyse non- stationary data, like Hidden Markov Models