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In the modern financial market the derivative pricing considers the use of historical or implied volatility which is actually the forward expectation of uncertainty. The common way of derivative pricing is to use the volatility as constant value in the well known Black Sholes equation. The aim of the current work was to develop a model where the uncertainty of volatility propagates to the derivative pricing and hedging according to the Black-Sholes PDE considering the volatility as stochastic process rather as a constant. A stochastic finite element method using generalized polynomial chaos was used to develop an algorithm of uncertainty propagation solving finally a deterministic problem for derivative pricing. The output of the method leads to derivative price distribution and the results of Monte Carlo Method for the derivative’s distribution were used as the exact solution against those rose from the new algorithm.

Several works were presented in the past for uncertainty quantification of derivative pricing due to random volatility. In [_{min} and σ_{max} was presented. The bound of volatility is computed by historical high-low peak of stock or option-implied volatilities and works as confidence interval for future volatility values. The derivative asset which arises as the volatility paths varies in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In [

In the current work a robust algorithm based on the stochastic finite element method using the generalized polynomial chaos was developed and it is considered as a general method for derivative pricing where the volatility input is considered as a random variable.

Suppose

where the drift rate μ and the volatility σ > 0 are assumed to be constant, and W(t) is a standard Brownian motion. The aim is to price a derivative of the form

In essence the solution of the problem is a function of the form

To solve the problem, we switch to the log-price process

The infinitesimal generator for this process has constant coefficients:

Thus by setting

In order to solve the problem according to the finite element method in the current paper we consider a linear

element with nodes

suming a test v function belongs to the space:

Using a test function v and integrating by parts over the domain D the variational formulation of the Black- Scholes equation has the following form:

Using the matrix notation the equation takes the following form:

where:

Assuming that the volatility of stock

The author has presented a stochastic finite element procedure to solve boundary problems using polynomial chaos [

where the order Q and the formula ψ of Polynomial Chaos are given in Appendix.

According that and using the inner product of the equation on each polynomial of the

Making the replacement of matrices and assuming

This is equivalent with:

To simplify the form make the following:

According to the previous replacement we discretize the Equation (14) using the theta-scheme with constant time step Δt. The finite element mesh considered as uniform and the Equation (14) takes the following form:

Equivalent

The statistical moments of the outcome derivative price arise by the properties of the Polynomial of Chaos expansion:

The expected value:

and the variance:

For the numerical example of derivative pricing a plain vanilla European Put Option was chosen. The historical data of S&P 500 was used to compute the mean value and the volatility of the index volatility. Based on the mean and max value of volatility and its mean volatility of volatility the option price has been calculated with expiration time T = 90/360 and strike price K = 1200. The statistical value of historical data for volatility for a period from 2000-2014 are given in

To verify the model we compare the results with those raised by the Monte Carlo method which is treated as the exact solution. The computational implementation of the Monte Carlo Method leads to the random process

generation of

The expected value and the variance are given by:

Volatility of S&P 500 2000-2014 | |||
---|---|---|---|

Min | Mean | Max | Vol of Vol |

5.01% | 17.20% | 86.35% | 10.58% |

Two different cases were carried out (

In the

A new algorithm of volatility uncertainty propagation in the derivative pricing and hedging procedure was presented according to the Black-Sholes PDE. A stochastic finite element method using generalized polynomial chaos was used to develop an algorithm of uncertainty propagation solving finally a deterministic problem for the derivative pricing. The results of Monte Carlo Method for the derivative’s distribution were used as the exact solution against those rose from the new algorithm. The method leads to high accuracy and eliminates the large number of the Monte Carlo Method’s simulations. The model applied using the historical data of S&P 500 for

Mean of volatility | Volatility of volatility | |
---|---|---|

Case I | 18.7% | 10.6% |

Case II | 86.35% | 10.6% |

the period 2000-2014. The mean value and the volatility of volatility of index are used for the pricing of a plain vanilla European put option considering the volatility parameter as a stochastic process. The effect of index volatility of volatility on the derivative price distribution was analyzed and the results were presented by the two methods. High uncertainty of index volatility leads to a highly non linear increase of option price volatility as the results of analysis are shown.

StefanosDrakos, (2016) Uncertain Volatility Derivative Model Based on the Polynomial Chaos. Journal of Mathematical Finance,06,55-63. doi: 10.4236/jmf.2016.61007

In order to solve the problem 3 we have to create the new space

Using the dyadic product of the space

The space

Assuming that the

The tensor product of the M

And using (A4)

where

And

Xiu & Karniadakis [

where:

For a 3^{rd} order of one dimension of uncertainty the Hermite Polynomial Chaos is given by: