Based on the dual formulation by Rogers (2002), Monte Carlo algorithms to estimate the high-biased and low-biased estimates for American option prices are proposed. Bounds for pricing errors and the variance of biased estimators are shown to be dependent on hedging martingales. These martingales are applied to (1) simultaneously reduce the error bound and the variance of the high-biased estimat...

This paper derives optimal perfect hedging portfolios in the presence of transaction costs within the binomial model of stock returns, for a market maker that establishes bid and ask prices for American call options on stocks paying dividends prior to expiration. It is shown that, while the option holder's optimal exercise policy at the ex-dividend date varies according to the stock price, ther...

In this paper we distinguish between operational risks depending on whether the operational risk naturally arises in the context of model risk. As the pricing model exposes itself to operational errors whenever it updates and improves its investment model and other related parameters. In this case, it is no longer optimal to implement the best model. Generally, an option is exercised in a jump-...

In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. I further test the truncated Levy paradigm using high freq...

When an insurance company sells a mutual fund with death and maturity guarantees to its client, it may consider allowing the client to extend the guarantee for some more years. If the renewal only happens once, a so-called rollover option is implied in the contract. In this paper, we show how the generalized Bermudan option can be applied to the special case of the rollover option. By avoiding ...

Since the ideas of arbitrage free pricing were born, finance has changed radically both in theory and practice. Derivatives markets have evolved and options serve nowadays as underlyings and as hedging instruments. In this thesis, we consider some markets for equity derivatives. We start by statistical analysis of the markets for European options and variance swaps because these products are im...

In this paper, we provide an option pricing formula based on an arbitrarily given stock distribution, where the problem of optimally hedging the payoo on a European call option is considered through a self-nancing trading strategy. An optimal hedging problem is solved on a trinomial lattice by assigning suitable probabilities on the lattice, where the underlying stock price distribution is deri...

Option pricing and hedging in a complete market are well-studied with nice results using martingale theories. However, they remain as open questions in incomplete markets. In particular, when the underlying processes involve jumps, there could be infinitely many martingale measures which give an interval of no-arbitrage prices instead of a unique one. Consequently, there is no martingale repres...

We consider the problem of option pricing and hedging when stock returns are correlated in time. Within a quadratic–risk minimisation scheme, we obtain a general formula, valid for weakly correlated non–Gaussian processes. We show that for Gaussian price increments, the correlations are irrelevant, and the Black–Scholes formula holds with the volatility of the price increments on the scale of t...

We consider a large investor hedging a long or short options position, whose trades generate adverse market impact. Unlike the complete-market or proportional transaction cases, the agent no longer finds it tenable to be perfectly hedged or even within a fixed distance of being hedged. Instead, he may find himself arbitrarily mishedged and optimally trades towards the classical Black-Scholes de...