## Thesis etd-12282011-120101 |

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Thesis type

Tesi di dottorato di ricerca

Author

DELL'ERA, MARIO

email address

m.dellera@sssup.it

URN

etd-12282011-120101

Thesis title

From Spectral Methods to the Geometrical Approximation for PDEs in Finance

Academic discipline

SECS-S/06

Course of study

MATEMATICA PER LE DECISIONI ECONOMICHE

Supervisors

**tutor**Prof. Renò, Roberto

**correlatore**Prof. Pacati, Claudio

Keywords

- market
- models
- Option
- PDE
- Pricing
- Stocastic
- volatility

Graduation session start date

11/11/2011

Availability

Full

Summary

The Thesis investigates the benefits of Spectral Methods, which are found to be an appealing numerical technique when the solution in closed form doesn't exist, but unfortunately it cannot be used in every case. A remarkable case in which it is possible to use the Spectral Methods is for pricing the Double Barrier Options as we have seen in Chapter 3.

The main achievement of this Thesis is the introduction of two methods, that we have called Geometrical Approximation and Perturbative Method respectively, by which is possible to evaluate the fair option prices in the Heston and SABR market model. Both proposed methods can be generalised to other market models and for pricing other derivatives contracts, although, in order to show the above methodologies, we have chosen to pricing Options of only two kinds: Vanilla Options and knock-out Barrier Options.

On the first, we have that the G. A. method intends to be an alternative method, which can be particularly convenient for sensible values of the model parameters, which allows computation of closed-form expressions of approximated option prices.

The option price is approximated since we can get closed-form solutions for the PDE at the cost of modifying the Cauchy's condition, rather than looking for a numerical solution to the PDE with the exact Cauchy's condition. The proposed method has the advantage to compute a solution in closed form, therefore, we do not have the problems which plague the numerical methods.

For example, one can consider the inverse Fourier transform method, in which we have to compute an integral between zero and infinity. In this case in fact, there is always some problem in order to define (in practice) the correct domain of integration; or equivalently, considering also the finite difference method, in which we have to define a suitable grid, in other words we have some problems about the choice of the grid's meshes.

In the present work we have used the Geometrical Approximation method in the Heston model and in the SABR model, comparing the Vanilla Option price obtained with these computed by inverse Fourier transform, Monte-Carlo simulation and Finite Difference method or again the Implied Volatility method.

The Geometrical Approximation method is more reliable for low values of the correlation between price and variance shocks. In this case, our numerical experiments in a specific but sensible case show that the difference with the Fourier method is of the order of 1%. Markets in which the price/volatility correlation is low, and thus the G. A. method seems more promising, are the Electricity Markets.

Besides it is possible, through the G. A. method, get the Vanilla Option price by a strategy, whose price at time zero is equal to the sum of the option price with modified payoff and a bond price, so that, this one is equal to the difference between |S_{0}e^{\varepsilon_{0}}-E| capitalised at rate r. This strategy gives us, for every correlation value, a price higher than the Heston price around some percent, but in this way the writer of the Options is fully hedged. This strategy can be very useful for Banks and Institutions that write derivatives contracts.

On the second, the Perturbative Method, we have elaborated another approximating approach, illustrated in Chapter 5, in which we have discussed a particular choice of the volatility price of risk in the Heston model, namely such that the drift term of the risk-neutral stochastic volatility process is zero. This allowed us to illustrate an alternative methodology for solving the pricing PDE in an approximate way, in which we have neglected some terms of the PDE, recovering a pricing formula which in this particular case, turn out to be simple, for Vanilla Options and Barrier Options. The approximating formulas give an accurate price close to that obtained by Fourier transform for the Vanilla options, and Down-knock-out Call options. The Perturbative method can be used for pricing several derivatives contracts and we are sure that manifold applications will follow.

The main achievement of this Thesis is the introduction of two methods, that we have called Geometrical Approximation and Perturbative Method respectively, by which is possible to evaluate the fair option prices in the Heston and SABR market model. Both proposed methods can be generalised to other market models and for pricing other derivatives contracts, although, in order to show the above methodologies, we have chosen to pricing Options of only two kinds: Vanilla Options and knock-out Barrier Options.

On the first, we have that the G. A. method intends to be an alternative method, which can be particularly convenient for sensible values of the model parameters, which allows computation of closed-form expressions of approximated option prices.

The option price is approximated since we can get closed-form solutions for the PDE at the cost of modifying the Cauchy's condition, rather than looking for a numerical solution to the PDE with the exact Cauchy's condition. The proposed method has the advantage to compute a solution in closed form, therefore, we do not have the problems which plague the numerical methods.

For example, one can consider the inverse Fourier transform method, in which we have to compute an integral between zero and infinity. In this case in fact, there is always some problem in order to define (in practice) the correct domain of integration; or equivalently, considering also the finite difference method, in which we have to define a suitable grid, in other words we have some problems about the choice of the grid's meshes.

In the present work we have used the Geometrical Approximation method in the Heston model and in the SABR model, comparing the Vanilla Option price obtained with these computed by inverse Fourier transform, Monte-Carlo simulation and Finite Difference method or again the Implied Volatility method.

The Geometrical Approximation method is more reliable for low values of the correlation between price and variance shocks. In this case, our numerical experiments in a specific but sensible case show that the difference with the Fourier method is of the order of 1%. Markets in which the price/volatility correlation is low, and thus the G. A. method seems more promising, are the Electricity Markets.

Besides it is possible, through the G. A. method, get the Vanilla Option price by a strategy, whose price at time zero is equal to the sum of the option price with modified payoff and a bond price, so that, this one is equal to the difference between |S_{0}e^{\varepsilon_{0}}-E| capitalised at rate r. This strategy gives us, for every correlation value, a price higher than the Heston price around some percent, but in this way the writer of the Options is fully hedged. This strategy can be very useful for Banks and Institutions that write derivatives contracts.

On the second, the Perturbative Method, we have elaborated another approximating approach, illustrated in Chapter 5, in which we have discussed a particular choice of the volatility price of risk in the Heston model, namely such that the drift term of the risk-neutral stochastic volatility process is zero. This allowed us to illustrate an alternative methodology for solving the pricing PDE in an approximate way, in which we have neglected some terms of the PDE, recovering a pricing formula which in this particular case, turn out to be simple, for Vanilla Options and Barrier Options. The approximating formulas give an accurate price close to that obtained by Fourier transform for the Vanilla options, and Down-knock-out Call options. The Perturbative method can be used for pricing several derivatives contracts and we are sure that manifold applications will follow.

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