چکیده:
This paper focuses on two main issues that are based on two important concepts: exponential Levy process and minimal entropy martingale measure. First, we intend to obtain risk measurement such as value-at-risk (VaR) and conditional value-at-risk (CvaR) using Monte-Carlo methodunder minimal entropy martingale measure (MEMM) for exponential Levy process. This Martingale measure is used for the exponential type of the processes such as exponential Levy process. Also, it can be said MEMM is a kind of important sampling method where the probability measure with minimal relative entropy replaces the main probability. Then we are going to obtain VaR and CVaR by Monte-Carlo simulation. For this purpose, we have to calculate option price, implied volatility and returns under MEMM and then obtain risk measurement by proposed algorithm. Finally, this model is simulated for exponential variance gamma process. Next, we intend to develop two theorems for implied volatility under minimal entropy martingale measure by examining the conditions. These theorems consider the asymptotic implied volatility for the case that time to maturity tends to zero and infinity.
خلاصه ماشینی:
First, we intend to obtain risk measurement such as value-at-risk (VaR) and conditional value-at-risk (CvaR) using Monte-Carlo method under minimal entropy martingale measure (MEMM) for exponential Levy process.
In [19] has been concentrated on calculating the value to the minimal relative entropy between the main probability with respect to all of the equivalent martingale measure for the Levy model by Monte-Carlo and quasi Monte-Carlo using low-discrepancy sequence.
Finally, the methods for consider implied volatility and theorems presented short maturity asymptotic for at the money option and flattening of the smile far from maturity for Levy process under MEMM Preliminaries The primary concepts are stated in the following.
Discussion and Results The topics discussed in this section are considered in two parts, first we are going to obtain the value of VaR and CVaR under minimal entropy martingale measure for exponential Levy process by Monte- Carlo simulation.
Computing VaR and CVaR under MEMM for Levy Process Here, we are going to obtain the value of VaR and CVaR under minimal entropy martingale measure for exponential Levy process by Monte-Carlo simulation.
These two purpose are considering and computing the implied volatility and risk measurement and those two important concepts are Levy process and minimal entropy martingale measure.
These two purpose are considering and computing the implied volatility and risk measurement and those two important concepts are Levy process and minimal entropy martingale measure.
Also, we obtained VaR and CvaR using Monte-Carlo method for exponential variance gamma process by calculate the value of option price and returns.