Robust optimal stopping

This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples

Robust optimal stopping

This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem represented by the Snell envelope using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples

Estimation Under Stochastic Differential Equations

Stochastic approaches are used in modern financial analysis to explore the underlying dynamics of securities like stocks and options. Statistical modeling and inferences within this aspect is an important concern because pricing errors could lead to serious economic losses. In this thesis, statistical estimation motivated by real applications are developed for inferences under stochastic diffusion processes using tensor method and kernel smooth method. We consider in Chapter 2 parameter estimation for multi–factor stochastic processes defined by stochastic differential equations. The class of processes considered are multivariate diffusion which are popular processes in modeling the dynamics of financial assets. We quantify the bias and variance by developing theoretical expansions for a large class of estimators which includes as special cases estimators based on the maximum likelihood, approximate likelihood and discretizations. We apply the proposed methods to evaluate bias in estimated contingent claims. We also provide simulation results for a set of popular multi-factor processes to confirm our theory. Our Chapter 3 is dedicated to improve the estimation of the market volatility, specifically the VIX index introduced by Chicago Board Option Exchange (CBOE). This index provides a way to measure the 30–day expected volatility of the S & P 500 index. Among a few ways to estimate it, the CBOE and the Goldman Saches had developed an estimator based on the concept of fair value of future variance. In realizing the discretization error, truncation error, and the approximation error in their estimator, as well as the possible option pricing errors involved, we propose a new method that combines the CBOE method and the kernel smoothing method. We derive the weak convergence property of our estimator. Simulation is run to justify the improvement

Schnelle Löser für partielle Differentialgleichungen

[no abstract available

Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

We develop a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in Cázares and Mijatović (SBG approximation. GitHub repository. Available online at https://github.com/jorgeignaciogc/SBG.jl (2020)), exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes

Stochastic Analysis in Finance and Insurance

[no abstract available

Pricing swing options and other electricity derivatives

The deregulation of regional electricity markets has led to more competitive prices but also higher uncertainty in the future electricity price development. Most markets exhibit high volatilities and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. A good understanding of the stochastic price dynamics is required for the purposes of risk management and pricing derivatives. In this thesis we examine a simple spot price model which is the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of this spot price process at maturity T. With some restrictions on the set of possible martingale measures we show that the risk neutral dynamics remains within the class of considered models and hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulas for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process. Further contributions of this thesis include a short discussion of interpolation methods to generate a continuous forward curve based on the forward contracts with delivery periods observed in the market, and an investigation into optimal martingale measures in incomplete markets. In particular we present known results of q-optimal martingale measures in the setting of a stochastic volatility model and give a first indication of how to determine the q-optimal measure for q=0 in an exponential Ornstein-Uhlenbeck model consistent with a given forward curve

An affine jump-diffusion model in the negative rates environment

This Thesis develops a new affine term structure model providing a short rate that is allowed to take negative values and is bounded from below by a randomly varying level. With this approach we conveniently represent many of the most relevant empirical features of the financial market that raised after the financial crisis of 2007-08, when the spread of negative rates became not only realistic but real

No-Arbitrage Conditions for Storable Commodities and the Models of Futures Term Structures

One distinguishable feature of storable commodities is that they relate to two markets: cash market and storage market. This paper proves that, if no arbitrage exists in the storage-cash dual markets, the commodity convenience yield has to be non-negative. However, classical reduced-form models for futures term structures could allow serious arbitrages due to the high volatility of the convenience yield. To avoid negative convenience yield, this paper proposes a semi-affine arbitrage-free model, which prices futures analytically and fits futures term structures reasonably well. Importantly, our model prices commodity-related contingent claims (such as calendar spread options) quite differently with classical models