Spanning With Binary Options: A Vector Lattice Approach

The issue of portfolio insurance is one of the prime concerns of the investors who want to insure their asset at minimum or appropriate cost. Static hedging with binary options is a popular strategy that has been explored in various option models (see e.g. (2; 3; 4; 7)). In this thesis, we propose a static hedging algorithm for discrete time models. Our algorithm is based on a vector lattice technique. In chapter 1, we give the necessary background on the theory of vector lattices and the theory of options. In chapter 2, we reveal the connection of lattice-subspaces with the minimum-cost portfolio insurance strategy. In chapter3, we outline our algorithm and give applications to binomial and trinomial option models. In chapter 4, we perform simulations and analyze the hedging errors of our algorithm for European, Barrier, Geometric Asian, Arithmetic Asian, and Lookback options. The study has revealed that static hedging could be suitable strategy for the European, Barrier, and Geometric Asian options as these options have shown less inclination to the rollover effect.

Spanning With Binary Options: A Vector Lattice Approach

The issue of portfolio insurance is one of the prime concerns of the investors who want to insure their asset at minimum or appropriate cost. Static hedging with binary options is a popular strategy that has been explored in various option models (see e.g. (2; 3; 4; 7)). In this thesis, we propose a static hedging algorithm for discrete time models. Our algorithm is based on a vector lattice technique. In chapter 1, we give the necessary background on the theory of vector lattices and the theory of options. In chapter 2, we reveal the connection of lattice-subspaces with the minimum-cost portfolio insurance strategy. In chapter3, we outline our algorithm and give applications to binomial and trinomial option models. In chapter 4, we perform simulations and analyze the hedging errors of our algorithm for European, Barrier, Geometric Asian, Arithmetic Asian, and Lookback options. The study has revealed that static hedging could be suitable strategy for the European, Barrier, and Geometric Asian options as these options have shown less inclination to the rollover effect.

Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

An exact and explicit formula for pricing lookback options with regime switching

<p style='text-indent:20px;'>This paper investigates the pricing of European-style lookback options when the price dynamics of the underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of the underlying risky asset depend on states of the economy described by a continuous-time Markov chain process. We derive an exact, explicit and closed-form solution for European-style lookback options in a two-state regime switching model.</p>

Option Pricing for Path-Dependent Options with Assets Exposed to Multiple Defaults Risk

In the present paper, we derive analytical formulas for barrier and lookback options with underlying assets exposed to multiple defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. An original technique is developed to valuate the barrier and lookback options by first conditioning on the predefault and the afterdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with the default-free option model and exogenous counterparty default risk option model.