Monte-Carlo Methods in Finance


  • ▹ Asian Options :
    This project presents a method for calculating the price of an Asian option. A common approximation is used to value such options (it has the advantage of being a closed-form formula). We examine the validity of this approximation by comparing it with a Monte Carlo valuation within the framework of the Black–Scholes model. We also investigate the effect of discretising observations of the mean.

  • ▹ Barrier Options :
    This project deals with the pricing of an option on an asset that can be exercised on a predefined date only if the price of the asset has not fallen below a predefined level (barrier). We work within the framework of the Black–Scholes model. We first consider the case without a barrier, which yields a simple analytical solution that we then compare with the Monte Carlo calculation. We then examine the price of the barrier option and the influence of discretising observations of barrier crossings.

  • ▹ Basket Options :
    This project sets out to calculate the price of an option on a basket of two assets. There is no closed-form formula for valuing such options, but an approximation is commonly used (it has the advantage of being a closed-form formula). We examine the validity of this approximation by comparing it with a valuation using the Monte Carlo method within the framework of the Black–Scholes model.

  • ▹ Bermudean Options :
    This project deals with the pricing of an option on an asset that can be exercised on any date chosen from a predefined set of dates. This raises the question of the optimal exercise date. We first examine the case of two possible exercise dates, which has the advantage of being straightforward within the framework of the Black–Scholes model, since the price of the European option can be expressed in closed-form. We then consider the more complex case of three exercise dates, where we employ a Longstaff-Schwarz regression technique, which is very frequently used in practice, to approximate the optimal exercise choice.

  • ▹ BestOf Options :
    This project deals with the pricing of an option on the maximum value of a basket of assets. This option is traditionally known as a Best-of option. More specifically, we first consider the Best-of forward contract, and then we examine the case of a Best-of put option. As this option depends on the joint distribution of the assets, we focus in particular on this aspect. We use the Monte Carlo valuation method within the framework of the Black–Scholes model.

  • ▹ Greeks :
    Sensitivity analysis plays a crucial role in finance as it forms the basis of portfolio hedging. This project presents the calculation of three key sensitivities using two different methods based on the Monte Carlo technique within the framework of the Black–Scholes model. We focus firstly on the theoretical calculation in the specific case of a vanilla option, which we then compare with the Monte Carlo calculation. We then examine the more general case of a basket option, for which there is no closed-form formula.

  • ▹ Importance Sampling :
    This project presents a study of a generic variance reduction method applicable to numerical estimates of expectations using Monte Carlo techniques. We first examine the theoretical aspects of the method, then apply it to the pricing of a vanilla option, where we will demonstrate its effectiveness, particularly when the option is out-of-the-money. Finally, we apply it to the pricing of two more exotic options for which no closed-form formula exists.

  • ▹ Spread Options :
    This project presents a method for calculating the price of a spread option (the difference between two prices) using the Monte Carlo method within the framework of the Black–Scholes model. We begin by examining the theoretical calculation in the specific case of a zero-strike option, which we then compare with the Monte Carlo calculation. We then study the general case of a non-zero strike, for which there is no closed-form formula.

  • ▹ Var Swaps :
    This project presents a method for calculating the price of a variance swap. This is a product that has become commonplace in the markets. The principle behind the payoff is to replicate the realised variance of an asset over a given period. As this is not directly observable, the contractual payoff is based on an approximation. However, practitioners generally disregard this approximation. This is examined within the framework of a stochastic volatility model in the first part. Furthermore, the realised variance can be quasi-replicated statically using a continuum of vanilla options. This replication is examined in the second part.

  • ▹ Valeurs numériques