Nowadays, options are common financial derivatives. For this reason, by increase of applications for these financial derivatives, the problem of options pricing is one of the most important economic issues. With the development of stochastic models, the need for randomly computational methods caused the generation of a new field called financial engineering. In the financial engineering the pre...

The most important application of the Itô calculus, derived from the Itô lemma, in financial mathematics is the pricing of options. The most famous result in this area is the Black-Scholes formulae for pricing European vanilla call and put options. As a consequence of the formulae, both in theoretical and practical applications, Robert Merton and Myron Scholes were awarded the Nobel Prize for E...

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 the paper by Melnikov and Petrachenko ‘On option pricing in binomial market with transaction costs,’ Finance Stoch. 9 (2005), 141–149, a procedure is put forward for pricing and replicating an arbitrary European contingent claim in the binomial model with bid-ask spreads. We present a counter-example to show that the option pricing formula stated in that paper can in fact lead to arbitrage. ...

Nonlinear Black–Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor’s preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option pr...

this paper is a translation of a chapter of the hook written by jonathan e. ingersoll jr. the farsi translation will he of great help to iranian students studying option pricing models.

We consider a financial model with permanent price impact. Continuous time trading dynamics are derived as the limit of discrete rebalancing policies. We then study the problem of super-hedging a European option. Our main result is the derivation of a quasi-linear pricing equation. It holds in the sense of viscosity solutions. When it admits a smooth solution, it provides a perfect hedging stra...

The Black-Scholes formula [6] provides with an elegant and simple method to price financial derivatives under the assumption that the stock price is log-normally distributed. However, the actual distribution of most assets is rarely log-normal, and theoretical prices of options with different strikes generated by the Black-Scholes formula differ from observed market prices. One way to reconcile...

Recent events in financial markets have highlighted the frequency with which large scale financial losses occur. Extreme value theory focusses on the statistical properties of such extreme events. In this paper, extreme value distributions are applied in the context of option pricing. Analytic solutions to the standard European option problem based on the assumption that returns are Gumbel dist...