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The study analyses some problems arising in stochastic volatility models by using Ito’s lemma and its applications to boundary Cauchy problem by giving the solution of vanilla option pricing models satisfying the partial differential equation obtained by assuming stochastic volatility in replication problems and risk neutral probability.

In finance Wiener process and geometric Brown process are largely used. The name came from George Brown in the 1827 noted that the volatility of a small particle suspended in a liquid increased with the time. Wiener gave a mathematical formal assumption on the phenomena from the term of Wiener process. The main property of the Wiener process is that it is a forward process such that we may integrate it although it is a function of infinite variation; the main idea is that the process converges to the discrete process because the limit tends toward the discrete process when it is shared in sub intervals. From this we may approximate the Wiener process in the instant as

The geometric Brown process is used in finance to indicate a formal assumption for the dynamic of the prices that does not permit to assume negative value, formally we have:

where

We may assume the following for the Wiener process:

This means that a Wiener process is a forward process, the uncertainty is to the end of the process in T + dt. From this we may obtain explanation of Ito’s lemma by using Taylor series, if we take a function of S as F(S) we may write Ito’s lemma in the following way:

We may note that:

From this we obtain as dt tends to zero:

By substituting dS we obtain Ito’s lemma:

We may see now as to obtain the expected value of a normal distribution as such we have the following:

As such we have the following:

This may be rewritten by:

From this we may obtain explanation for Ito’s lemma, if we take a function of S as

where:

As result:

Because:

Now we may analyze the following parabolic problem:

Subject to the following constraint:

The solution it is easy to solve, because if we take Ito’s lemma and we take the expectation we obtain that the solution to the parabolic problem is given by:

It is interesting to introduce the concept of stochastic volatility, as such we may write Ito’s lemma in the following form:

where

where:

The PDE that an option must satisfy by assuming stochastic volatility is given by the following:

where:

The solution it is easy to solve, because if we take Ito’s lemma and we take the expectation we obtain that the solution to the parabolic problem is given by the following by using the integrant factor

The final pay off of a Call and Put option is given by the following:

The prices of the options are given by the expectation of the final pay off discounted for the Call options, instead, for the Put options we assume that to replicate the value of options we have to have available the amount of money K that will produce a risk free earnings:

So from 33 we may note that for the Put options will be discounted only the process

As result we obtain the following pricing formula for the options by using the respective numeraire:

where:

where:

We may compare the model with [

As results we have the following figures for Put options (Tables 4-6).

We may note from the numerical results that for rational value of parameters [

Spot price (S) | 1 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (T ? t) | 2 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.03 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Heston call price | 0.0896 |

Bivariate call price | 0.0970 |

Black scholes call price | 0.0887 |

Spot price (S) | 1.5 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (T ? t) | 2 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.03 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Heston call price | 0.5582 |

Bivariate call price | 0.5885 |

Black scholes call price | 0.5583 |

Spot price (S) | 2 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (T ? t) | 2 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.03 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Heston call price | 1.0582 |

Bivariate call price | 1.0986 |

Black scholes call price | 1.0582 |

Spot price (S) | 1 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (T ? t) | 2 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.03 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Heston put price | 0.0313 |

Bivariate put price | 0.0347 |

Black scholes put price | 0.0305 |

Spot price (S) | 0.75 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (T ? t) | 2 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.03 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Heston put price | 0.1918 |

Bivariate put price | 0.2345 |

Black scholes put price | 0.1945 |

Spot price (S) | 0.5 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (T ? t) | 2 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.03 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Heston put price | 0.4463 |

Bivariate put price | 0.4899 |

Black scholes put price | 0.4418 |

Spot price (S) | 1 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (days) | 730 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.022 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Number of simulations | 1.000 |

Bivariate MC call price | 0.0964 |

Spot price (S) | 1.5 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (days) | 730 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.022 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Number of simulations | 1.000 |

Bivariate MC call price | 0.5815 |

Spot price (S) | 2 |
---|---|

Strike price (K) | 1 |

Risk free rate (r) | 0.03 |

Time to maturity (days) | 730 |

Rho (ρ) | −0.5 |

Kappa (κ) | 0.2 |

Theta (θ) | 0.022 |

Lambda (λ) | 2 |

Volatility of variance (σ) | 0.1 |

Current variance (v) | 0.01 |

Number of simulations | 1.000 |

Bivariate MC call price | 1.0943 |

Indeed, with the simulations the normal distribution is more skewed, so we used a calibrated parameters drift, to compare the two approaches [

The study permits to obtain closed form solution for option pricing with stochastic volatility by assuming normal distribution obtained by the properties of the bivariate standardized normal distribution that is the solution of the Cauchy problem with stochastic volatility due to the properties of Ito’s lemma.

RossanoGiandomenico, (2015) Option Pricing with Stochastic Volatility. Journal of Applied Mathematics and Physics,03,1645-1653. doi: 10.4236/jamp.2015.312189

To run the simulation we used the following VBA code.