Computing Greeks using multilevel path simulation

We investigate the extension of the multilevel Monte Carlo method [2, 3] to the calculation of Greeks. The pathwise sensitivity analysis [5] differentiates the path evolution and effectively reduces the smoothness of the payoff. This leads to new challenges: the use of naive algorithms is often impossible because of the inapplicability of pathwise sensitivities to discontinuous payoffs.\ud \ud These challenges can be addressed in three different ways: payoff smoothing using conditional expectations of the payoff before maturity [5]; an approximation of the above technique using path splitting for the final timestep [1]; the use of a hybrid combination of pathwise sensitivity and the Likelihood Ratio Method [4]. We discuss the strengths and weaknesses of these alternatives in different multilevel Monte Carlo settings

Investigation into Vibrato Monte Carlo for the Computation of\ud Greeks of Discontinuous Payoffs

Monte Carlo simulation is a popular method in computational finance. Its basic theory is relatively simple, it is also quite easy to implement and allows nevertheless an efficient pricing of financial options, even in high-dimensional problems (basket options, interest rates products...).\ud \ud The pricing of options is just one use of Monte Carlo in finance. More important than the prices themselves are their sensitivities to input parameters (underlying asset value, interest rates, market volatility...). Indeed we need those sensitivities (also known as "Greeks") to hedge against market risk.\ud \ud In this paper, we will first recall classical approaches to the computation of Greeks through Monte Carlo simulation: finite differences, Likelihood Ratio method (LRM) and Pathwise Sensitivities (PwS). Each of those approaches has particular limitations in the case of options with discontinuous payoffs. We will expound those limitations and introduce a new hybrid method proposed by Prof. Mike Giles, the Vibrato Monte Carlo, which combines both Pathwise Sensitivity and Likelihood Ratio methods to get around their shortcomings.\ud \ud We will discuss the possible use of Vibrato Monte Carlo ideas for options with discontinuous payoffs. My personal contribution is an improvement to the standard Vibrato Monte Carlo yielding both computational savings and an improved accuracy. I will call it Allargando Vibrato Monte Carlo (AVMC). I then also extend the Vibrato Monte Carlo technique to discretely sampled path dependent options (digital option with discretely sampled barrier, lookback option with discretely sampled maximum)

Investigation into Vibrato Monte Carlo for the Computation of Greeks of Discontinuous Payoffs

Monte Carlo simulation is a popular method in computational finance. Its basic theory is relatively simple, it is also quite easy to implement and allows nevertheless an efficient pricing of financial options, even in high-dimensional problems (basket options, interest rates products...). The pricing of options is just one use of Monte Carlo in finance. More important than the prices themselves are their sensitivities to input parameters (underlying asset value, interest rates, market volatility...). Indeed we need those sensitivities (also known as "Greeks") to hedge against market risk. In this paper, we will first recall classical approaches to the computation of Greeks through Monte Carlo simulation: finite differences, Likelihood Ratio method (LRM) and Pathwise Sensitivities (PwS). Each of those approaches has particular limitations in the case of options with discontinuous payoffs. We will expound those limitations and introduce a new hybrid method proposed by Prof. Mike Giles, the Vibrato Monte Carlo, which combines both Pathwise Sensitivity and Likelihood Ratio methods to get around their shortcomings. We will discuss the possible use of Vibrato Monte Carlo ideas for options with discontinuous payoffs. My personal contribution is an improvement to the standard Vibrato Monte Carlo yielding both computational savings and an improved accuracy. I will call it Allargando Vibrato Monte Carlo (AVMC). I then also extend the Vibrato Monte Carlo technique to discretely sampled path dependent options (digital option with discretely sampled barrier, lookback option with discretely sampled maximum)

The computation of Greeks with multilevel Monte Carlo

We study the use of the multilevel Monte Carlo technique in the context of the calculation of Greeks. The pathwise sensitivity analysis differentiates the path evolution and reduces the payoff's smoothness. This leads to new challenges: the inapplicability of pathwise sensitivities to non-Lipschitz payoffs often makes the use of naive algorithms impossible. These challenges can be addressed in three different ways: payoff smoothing using conditional expectations of the payoff before maturity; approximating the previous technique with path splitting for the final timestep; using of a hybrid combination of pathwise sensitivity and the Likelihood Ratio Method. We investigate the strengths and weaknesses of these alternatives in different multilevel Monte Carlo settings.