A lookback option is an exotic that allows investors to look back at the underlying prices occurring over life of option, and exercise right assets optimal point. This paper proposes a mean-reverting stock model investigate in uncertain environment. The call put options pricing formulas are derived, corresponding numerical algorithms designed compute these two optio...

There is a compelling need to accurately and efficiently compute option values. Existing literature shows that models based on constant stock volatilities have been widely used in option valuation. However, stock volatilities change constantly in real life situations. The introduction of the Auto Regressive Conditional Heteroskedasticity (ARCH) model and subsequently, the Generalized Auto Regre...

In the case of early exercise of an American-style call option, we consider the issue of the existence of a “treshhold value,” namely a boundary which, once exceeded, early exercise is optimal for all values of the underlying asset which exceed that value. We discuss optimal exercise thresholds for call options for two-period models, under alternate contexts: geometric Brownian motion vs. mean-...

We consider series solutions for the location of the optimal exercise boundary of an American option close to expiry. By using Monte Carlo methods, we compute the expected value of an option if the holder uses the approximate location given by such a series as his exercise strategy, and compare this value to the actual value of the option. This gives an alternative method to evaluate approximat...

This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that the stock price is log-normally distributed and that the universe is riskneutral. Then, using Ito’s Lemma, we will justify the use of the risk-neutral rate in these initial calculations. Finally, we will prove put-call parity in order to price European...

a r t i c l e i n f o This paper examines the relationship between option trading activity and stock market volatility. Although the option market is uniquely suited for trading on volatility information, there is little analysis on how trading activity in this market is linked to stock price volatility. The bulk of the discussion tends to focus on whether trading activity in the stock market i...

We use Malliavin calculus and the Clark-Ocone formula to derive the hedging strategy of an arithmetic Asian Call option in general terms. Furthermore we derive an expression for the density of the integral over time of a geometric Brownian motion, which allows us to express hedging strategy and price of the Asian option as an analytic, that is closed form, expression. Numerical computations whi...

We see that the price of an european call option in a stochastic volatility framework can be decomposed in the sum of four terms, which identify the main features of the market that affect to option prices: the expected future volatility, the correlation between the volatility and the noise driving the stock prices, the market price of volatility risk and the difference of the expected future v...

The duality principle in option pricing aims at simplifying valuation problems that depend on several variables by associating them to the corresponding dual option pricing problem. Here we analyze the duality principle for options that depend on several assets. The asset price processes are driven by general semimartingales, and the dual measures are constructed via an Esscher transformation. ...

The double barrier option is characterized by pay-off with strike K, maturity T, upper Su and lower Sd barrier levels and the corresponding rebates φu and φd which can be time dependent. We divide last four quantities by strike K and introduce new variables x = ln(S/K), xu = ln(Su/K), xd = ln(Sd/K). The value of European double barrier call option U(t, x) satisfies the extended Black-Scholes eq...