Mario Dell'Era - Academia.edu (original) (raw)

Papers by Mario Dell'Era

Social Science Research Network, Jul 12, 2012

Social Science Research Network, 2019

Social Science Research Network, 2019

RePEc: Research Papers in Economics, May 5, 2010

We propose to discuss a new technique to derive an good approximated solution for the price of a ... more We propose to discuss a new technique to derive an good approximated solution for the price of a European Vanilla options, in a market model with stochastic volatility. In particular, the models that we have considered are the Heston and SABR(for β = 1). These models allow arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation.

RePEc: Research Papers in Economics, Mar 25, 2008

We propose to discuss the efficiency of the spectral method for computing the value of Double Bar... more We propose to discuss the efficiency of the spectral method for computing the value of Double Barrier Options. Using this method, one may write the option price as a Fourier series, with suitable coefficients. We propose a simple approach for its computing. One consider the general case, in which the volatility is time dependent, but it is immediate extend our methodology also in the case of constant volatility. The advantage to write the arbitrage price of the Double Barrier Options as Fourier series, is matter of computation complexity. The methods used to evaluate options of this kind have a high value of computation complexity, furthermore, them have not the capacity to manage it, while using our method, one can define, through an easy analytical report, the computation complexity of the problem, and also one can choice its accuracy.

Social Science Research Network, 2012

One presents and discusses an alternative solution writeable in closed form of the Heston's PDE, ... more One presents and discusses an alternative solution writeable in closed form of the Heston's PDE, for which the solution is known in literature, up to an inverse Fourier transform. Since the algorithm to compute the inverse Fourier Transform is not able to be applied easily for every payoff, one has elaborated a new methodology based on changing of variables which is independent of payoffs and does not need to use the inverse Fourier transform algorithm or numerical methods as Finite Difference and Monte Carlo simulation. In particular, one will compute the price of Vanilla Options for small maturities in order to validate numerically the Geometrical Transformations technique. The principal achievement is to use an analytical formula to compute the prices of derivatives, in order to manage, balance any portfolio through the Greeks, that by the proposed solution one is able to compute analytically. The above mentioned numerical techniques are computationally expensive in the stochastic volatility market models and for this reason is usually employed the Black Scholes model, that is unsuitable, as it has been widely proven in literature to compute the sensitivities of a portfolio and the price of derivatives. The present article wants to introduce a new approach to solve PDEs complicated, such as, those coming out from the stochastic volatility market models, with the achievement to reduce the computational cost and thus the time machine; besides, the proposed solution is easy to be generalized by adding jump processes as well. The present research work is rather technical and one does wide use of functional analysis. For the conceptual simplicity of the technique, one confides which many applications and studies will follow, extending the applications of the Geometrical Transformations technique to other derivative contracts of different styles and asset classes.

RePEc: Research Papers in Economics, Mar 10, 2010

We propose to discuss a new technique to derive an good approximated solution for the price of a ... more We propose to discuss a new technique to derive an good approximated solution for the price of a European call and put options, in a market model with stochastic volatility. In particular, the model that we have considered is the Heston's model. This allows arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation. 1

RePEc: Research Papers in Economics, Mar 25, 2008

We discuss the efficiency of the spectral method for computing the value of the European Call Opt... more We discuss the efficiency of the spectral method for computing the value of the European Call Options, which is based upon the Fourier series expansion. We propose a simple approach for computing accurate estimates. We consider the general case, in which the volatility is time dependent, but it is immediate extend our methodology at the case of constant volatility. The advantage to write the arbitrage price of the European Call Options as Fourier series, is matter of computation complexity. Infact, the methods used to evaluate options of this kind have a high value of computation complexity, furthermore, them have not the capacity to manage it. We can define, by an easy analytical relation, the computation complexity of the problem in the framework of general theory of the "Function Analysis", called The Spectral Theory.

The most popular market model in continuous time is the Black-Scholes model. It assumes for the u... more The most popular market model in continuous time is the Black-Scholes model. It assumes for the underlying process, a geometric Brownian motion with constant volatility, that is dS_t = rS_t dt σS_t dW_t, dB_t = rB_t dt, where r is the constant risk-free rate, St is the stock and σ is the constant volatility of the stock.Under these assumptions, closed form solutions for the values of European call and put options, are derived by use of the PDE method. We want to discuss by present work, the PDE approach in most complicated cases of market models. Our objective is to use three different techniques that are respectively Spectral Methods, Geometrical Approximation and Perturbative method (these last introduced by us), in order to compute the price of the derivatives for the following kinds of contracts: Vanilla Options, Barrier Options and Double Barrier Options.We have structured the work in five chapters: Chapters 1 and 2, respectively show the theoretical foundations of the parabolic PDE, and the Black Scholes market model. In chapter 3, we are going to consider Double Barrier Options, in Black Scholes model, that belongs to the kind of exotic options, in which case we have a deterministic volatility function σ(t).For example we consider the value of a Knock-out, down-and-out Call option, that is given by the solution of the Black-Scholes equation with appropriate boundary conditions, but we are able to discuss also the cases in which we have Knock-in options, or we have a Put option and do not a Call. To grant the existence and uniqueness of the solution, it is necessary to define the boundary condition and the initial condition. Also we require that when the value of the underlying asset hits the two barriers, lower (L) and upper (H), the option is cancelled in our case, but it could be activated for knock-in options . The best method to solve the above problem, is the using of the Spectral Theory, which allows to write the price of the Knock-out or Knock-in options, as series expansion. We are going to compare the spectral method with others, studying also the computational complexity.In chapter 4, we propose a new technique, that we have called the Geometrical Approximation method, applying this one to the stochastic volatility market models as Heston and SABR. The assumption of constant volatility isn’t reasonable in a real market, since we require different values for the volatility parameter for different strikes and different expiries to match market prices. The volatility parameter that is required in the Black-Scholes formula to reproduce market prices is called the implied volatility. To obtain market prices of options maturing at a certain date, volatility needs to be a function of the strike. This function is the so called volatility skew or smile.Furthermore for a fixed strike we also need different volatility parameters to match the market prices of options maturing on different dates written on the same underlying, hence volatility is a function of both the strike and the expiry date of the derivative security. This bivariate function is called the volatility surface. There are two prominent ways of working around this problem, namely, local volatility models and stochastic volatility models. For local volatility models the assumption of constant volatility made in Black and Scholes (1973) is relaxed. The underlying risk-neutral stochastic process becomes dS_t = r(t) S_t dt σ(t, S_t) S_t dW_t, where r(t) is the instantaneous forward rate of maturity t implied by the yield curve and the function σ(St, t) is chosen (calibrated) such that the model is consistent with market data, see Dupire (1994), Derman and Kani (1994) and (Wilmott, 2000). It is claimed in Hagan et al. (2002) that local volatility models predict that the smile shifts to higher prices (resp. lower prices) when the price of the underlying decreases (resp. increases). This is in contrast with the market behaviour where the smile shifts to higher prices (resp. lower prices) when the price of the underlying increases (resp. decreases). Another way of working around the inconsistency introduced by constant volatility is by introducing a stochastic process for the volatility itself; such models are called stochastic volatility models. The major advances in stochastic volatility models are Hull and White (1987), Heston (1993) and Hagan et al. (2002). Such models have a general form and varying its parameters we can obtain them: for δ = 1, j = 1, α = 0, a2(S) = Sβ, β ∈ (0, 1] and b1 = 0, we get the SABR model, by Hagan; for δ = 1, j = 2, α = 0, a2(S) = S and b1 = k(θ − σjt ), we get Heston model, by Heston; for δ = 1, α = 0 and b1 = 0 we get Black-Scholes model with constant volatility, by Black Scholes Merton; where the tradable security S_t and its volatility σ_t are correlated. Using the above indicated general market model, from Itˆo’s lemma, it is possible to derive, under mild additional assumptions, the partial differential…

RePEc: Research Papers in Economics, Oct 4, 2010

We want to discuss the option pricing on stochastic volatility market models, in which we are goi... more We want to discuss the option pricing on stochastic volatility market models, in which we are going to consider a generic function β(ν t) for the drift of volatility process. It is our intention choose any equivalent martingale measure, so that the drift of volatility process, respect at the new measure, is zero. This technique is possible when the Girsanov theorem is satisfied, since the stochastic volatility models are uncomplete markets, thus one has to choice an arbitrary risk price of volatility. In all this cases we are able to compute the price of Vanilla options in a closed form. To name a few, we can think to the popular Heston's model, in which the solution is known in literature, unless of an inverse Fourier transform.

RePEc: Research Papers in Economics, Jun 20, 2009

The attention to environmental conditions of the planet drives many scientists to study and to an... more The attention to environmental conditions of the planet drives many scientists to study and to analyze the externalities of the economic activities and their relapses on nature. The issue is quite complex because of the non-linear interactions between human and natural phenomena. Our intention is to study the particular case of tourist activities. Starting from the specification of the concept of sustainable development, using a simple model we characterize the conditions for which there exists an optimal equilibrium between nature and tourism. Then, trough several simulations we study which policies are able to guarantee the better synergies between economy and environmental quality.

The Thesis investigates the benefits of Spectral Methods, which are found to be an appealing nume... more 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 sensibl...

Virology, 1973

With high resolution SLSpolyacrylamide gel electrophoresis (Maizel, 1971) and a new method to ext... more With high resolution SLSpolyacrylamide gel electrophoresis (Maizel, 1971) and a new method to extract the basic proteins, it w&9 ascertained that adenovirus type 2 contains at least 10 distinct polypcptides (II, III, IIla, IV-X) and possibly more. Five proteins (V, VI, VII, VIII, and S) were purified by selective extraction in urea at high ionic strength, and low pI1 followed by preparative polyacrylamide electrophoresis toward the cathode at ~113.4. .4ntisera, produced against proteins V, VI, and VII were used to reveal t.hat these proteins were antigenically distinct and unrelated to hexons, pentons, and fibers. The location of the polypeptides was investigated by two methods of virion degradation (Prage el al., 1970). Polypeptides V and VI1 were associated with the l>NAcont.aining core. Polypcptide VI appeared to be associat,ed with all hexons of the virion. llexons from the triangular facets of the capsid cbontained in addition to polypeptides 11 and VI, polypept,ide IX, and possibly alvo polypeptide VIII. Hexons purified from infected cells contained only polypeptide II. Polypeptide IIla was preferentially released together with the peripentonal hexons. INTROI~UCTIOS peptides have been identified and t.heir Adenoviruses are well suited for st,udics on localization in the capsid has been studied by virion proteins since the outer capsid pro-different methods to disintegrate the sdenoteins are soluble under native condiions. viriOn. SDS-polyacrvlamide gel clectrophoresis re-,4 hypothetical model for the topography vealed a minimum of nine polypeptidcs of the polypcptides of adenovirions was (II-X) (Maize1 et al., 1968a; Muizel, 1971). constructed on t.hc basis of t.hesc results. Fiveof t.hese (II, III, IV, V, VII) areintcpral MATEItIAI,d AXI) RIKTlCOl)S parts of hexons, pentons, fibers, or cores

We want to discuss the option pricing on stochastic volatility market models, in which we are goi... more We want to discuss the option pricing on stochastic volatility market models, in which we are going to consider a generic function β(ν t) for the drift of volatility process. It is our intention choose any equivalent martingale measure, so that the drift of volatility process, respect at the new measure, is zero. This technique is possible when the Girsanov theorem is satisfied, since the stochastic volatility models are uncomplete markets, thus one has to choice an arbitrary risk price of volatility. In all this cases we are able to compute the price of Vanilla options in a closed form. To name a few, we can think to the popular Heston's model, in which the solution is known in literature, unless of an inverse Fourier transform.

We propose to discuss a new technique to derive an good approximated solution for the price of a ... more We propose to discuss a new technique to derive an good approximated solution for the price of a European call and put options, in a market model with stochastic volatility. In particular, the model that we have considered is the Heston's model. This allows arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation. 1

Social Science Research Network, Jul 12, 2012

Social Science Research Network, 2019

Social Science Research Network, 2019

RePEc: Research Papers in Economics, May 5, 2010

We propose to discuss a new technique to derive an good approximated solution for the price of a ... more We propose to discuss a new technique to derive an good approximated solution for the price of a European Vanilla options, in a market model with stochastic volatility. In particular, the models that we have considered are the Heston and SABR(for β = 1). These models allow arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation.

RePEc: Research Papers in Economics, Mar 25, 2008

We propose to discuss the efficiency of the spectral method for computing the value of Double Bar... more We propose to discuss the efficiency of the spectral method for computing the value of Double Barrier Options. Using this method, one may write the option price as a Fourier series, with suitable coefficients. We propose a simple approach for its computing. One consider the general case, in which the volatility is time dependent, but it is immediate extend our methodology also in the case of constant volatility. The advantage to write the arbitrage price of the Double Barrier Options as Fourier series, is matter of computation complexity. The methods used to evaluate options of this kind have a high value of computation complexity, furthermore, them have not the capacity to manage it, while using our method, one can define, through an easy analytical report, the computation complexity of the problem, and also one can choice its accuracy.

Social Science Research Network, 2012

One presents and discusses an alternative solution writeable in closed form of the Heston's PDE, ... more One presents and discusses an alternative solution writeable in closed form of the Heston's PDE, for which the solution is known in literature, up to an inverse Fourier transform. Since the algorithm to compute the inverse Fourier Transform is not able to be applied easily for every payoff, one has elaborated a new methodology based on changing of variables which is independent of payoffs and does not need to use the inverse Fourier transform algorithm or numerical methods as Finite Difference and Monte Carlo simulation. In particular, one will compute the price of Vanilla Options for small maturities in order to validate numerically the Geometrical Transformations technique. The principal achievement is to use an analytical formula to compute the prices of derivatives, in order to manage, balance any portfolio through the Greeks, that by the proposed solution one is able to compute analytically. The above mentioned numerical techniques are computationally expensive in the stochastic volatility market models and for this reason is usually employed the Black Scholes model, that is unsuitable, as it has been widely proven in literature to compute the sensitivities of a portfolio and the price of derivatives. The present article wants to introduce a new approach to solve PDEs complicated, such as, those coming out from the stochastic volatility market models, with the achievement to reduce the computational cost and thus the time machine; besides, the proposed solution is easy to be generalized by adding jump processes as well. The present research work is rather technical and one does wide use of functional analysis. For the conceptual simplicity of the technique, one confides which many applications and studies will follow, extending the applications of the Geometrical Transformations technique to other derivative contracts of different styles and asset classes.

RePEc: Research Papers in Economics, Mar 10, 2010

We propose to discuss a new technique to derive an good approximated solution for the price of a ... more We propose to discuss a new technique to derive an good approximated solution for the price of a European call and put options, in a market model with stochastic volatility. In particular, the model that we have considered is the Heston's model. This allows arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation. 1

RePEc: Research Papers in Economics, Mar 25, 2008

We discuss the efficiency of the spectral method for computing the value of the European Call Opt... more We discuss the efficiency of the spectral method for computing the value of the European Call Options, which is based upon the Fourier series expansion. We propose a simple approach for computing accurate estimates. We consider the general case, in which the volatility is time dependent, but it is immediate extend our methodology at the case of constant volatility. The advantage to write the arbitrage price of the European Call Options as Fourier series, is matter of computation complexity. Infact, the methods used to evaluate options of this kind have a high value of computation complexity, furthermore, them have not the capacity to manage it. We can define, by an easy analytical relation, the computation complexity of the problem in the framework of general theory of the "Function Analysis", called The Spectral Theory.

The most popular market model in continuous time is the Black-Scholes model. It assumes for the u... more The most popular market model in continuous time is the Black-Scholes model. It assumes for the underlying process, a geometric Brownian motion with constant volatility, that is dS_t = rS_t dt σS_t dW_t, dB_t = rB_t dt, where r is the constant risk-free rate, St is the stock and σ is the constant volatility of the stock.Under these assumptions, closed form solutions for the values of European call and put options, are derived by use of the PDE method. We want to discuss by present work, the PDE approach in most complicated cases of market models. Our objective is to use three different techniques that are respectively Spectral Methods, Geometrical Approximation and Perturbative method (these last introduced by us), in order to compute the price of the derivatives for the following kinds of contracts: Vanilla Options, Barrier Options and Double Barrier Options.We have structured the work in five chapters: Chapters 1 and 2, respectively show the theoretical foundations of the parabolic PDE, and the Black Scholes market model. In chapter 3, we are going to consider Double Barrier Options, in Black Scholes model, that belongs to the kind of exotic options, in which case we have a deterministic volatility function σ(t).For example we consider the value of a Knock-out, down-and-out Call option, that is given by the solution of the Black-Scholes equation with appropriate boundary conditions, but we are able to discuss also the cases in which we have Knock-in options, or we have a Put option and do not a Call. To grant the existence and uniqueness of the solution, it is necessary to define the boundary condition and the initial condition. Also we require that when the value of the underlying asset hits the two barriers, lower (L) and upper (H), the option is cancelled in our case, but it could be activated for knock-in options . The best method to solve the above problem, is the using of the Spectral Theory, which allows to write the price of the Knock-out or Knock-in options, as series expansion. We are going to compare the spectral method with others, studying also the computational complexity.In chapter 4, we propose a new technique, that we have called the Geometrical Approximation method, applying this one to the stochastic volatility market models as Heston and SABR. The assumption of constant volatility isn’t reasonable in a real market, since we require different values for the volatility parameter for different strikes and different expiries to match market prices. The volatility parameter that is required in the Black-Scholes formula to reproduce market prices is called the implied volatility. To obtain market prices of options maturing at a certain date, volatility needs to be a function of the strike. This function is the so called volatility skew or smile.Furthermore for a fixed strike we also need different volatility parameters to match the market prices of options maturing on different dates written on the same underlying, hence volatility is a function of both the strike and the expiry date of the derivative security. This bivariate function is called the volatility surface. There are two prominent ways of working around this problem, namely, local volatility models and stochastic volatility models. For local volatility models the assumption of constant volatility made in Black and Scholes (1973) is relaxed. The underlying risk-neutral stochastic process becomes dS_t = r(t) S_t dt σ(t, S_t) S_t dW_t, where r(t) is the instantaneous forward rate of maturity t implied by the yield curve and the function σ(St, t) is chosen (calibrated) such that the model is consistent with market data, see Dupire (1994), Derman and Kani (1994) and (Wilmott, 2000). It is claimed in Hagan et al. (2002) that local volatility models predict that the smile shifts to higher prices (resp. lower prices) when the price of the underlying decreases (resp. increases). This is in contrast with the market behaviour where the smile shifts to higher prices (resp. lower prices) when the price of the underlying increases (resp. decreases). Another way of working around the inconsistency introduced by constant volatility is by introducing a stochastic process for the volatility itself; such models are called stochastic volatility models. The major advances in stochastic volatility models are Hull and White (1987), Heston (1993) and Hagan et al. (2002). Such models have a general form and varying its parameters we can obtain them: for δ = 1, j = 1, α = 0, a2(S) = Sβ, β ∈ (0, 1] and b1 = 0, we get the SABR model, by Hagan; for δ = 1, j = 2, α = 0, a2(S) = S and b1 = k(θ − σjt ), we get Heston model, by Heston; for δ = 1, α = 0 and b1 = 0 we get Black-Scholes model with constant volatility, by Black Scholes Merton; where the tradable security S_t and its volatility σ_t are correlated. Using the above indicated general market model, from Itˆo’s lemma, it is possible to derive, under mild additional assumptions, the partial differential…

RePEc: Research Papers in Economics, Oct 4, 2010

We want to discuss the option pricing on stochastic volatility market models, in which we are goi... more We want to discuss the option pricing on stochastic volatility market models, in which we are going to consider a generic function β(ν t) for the drift of volatility process. It is our intention choose any equivalent martingale measure, so that the drift of volatility process, respect at the new measure, is zero. This technique is possible when the Girsanov theorem is satisfied, since the stochastic volatility models are uncomplete markets, thus one has to choice an arbitrary risk price of volatility. In all this cases we are able to compute the price of Vanilla options in a closed form. To name a few, we can think to the popular Heston's model, in which the solution is known in literature, unless of an inverse Fourier transform.

RePEc: Research Papers in Economics, Jun 20, 2009

The attention to environmental conditions of the planet drives many scientists to study and to an... more The attention to environmental conditions of the planet drives many scientists to study and to analyze the externalities of the economic activities and their relapses on nature. The issue is quite complex because of the non-linear interactions between human and natural phenomena. Our intention is to study the particular case of tourist activities. Starting from the specification of the concept of sustainable development, using a simple model we characterize the conditions for which there exists an optimal equilibrium between nature and tourism. Then, trough several simulations we study which policies are able to guarantee the better synergies between economy and environmental quality.

The Thesis investigates the benefits of Spectral Methods, which are found to be an appealing nume... more 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 sensibl...

Virology, 1973

With high resolution SLSpolyacrylamide gel electrophoresis (Maizel, 1971) and a new method to ext... more With high resolution SLSpolyacrylamide gel electrophoresis (Maizel, 1971) and a new method to extract the basic proteins, it w&9 ascertained that adenovirus type 2 contains at least 10 distinct polypcptides (II, III, IIla, IV-X) and possibly more. Five proteins (V, VI, VII, VIII, and S) were purified by selective extraction in urea at high ionic strength, and low pI1 followed by preparative polyacrylamide electrophoresis toward the cathode at ~113.4. .4ntisera, produced against proteins V, VI, and VII were used to reveal t.hat these proteins were antigenically distinct and unrelated to hexons, pentons, and fibers. The location of the polypeptides was investigated by two methods of virion degradation (Prage el al., 1970). Polypeptides V and VI1 were associated with the l>NAcont.aining core. Polypcptide VI appeared to be associat,ed with all hexons of the virion. llexons from the triangular facets of the capsid cbontained in addition to polypeptides 11 and VI, polypept,ide IX, and possibly alvo polypeptide VIII. Hexons purified from infected cells contained only polypeptide II. Polypeptide IIla was preferentially released together with the peripentonal hexons. INTROI~UCTIOS peptides have been identified and t.heir Adenoviruses are well suited for st,udics on localization in the capsid has been studied by virion proteins since the outer capsid pro-different methods to disintegrate the sdenoteins are soluble under native condiions. viriOn. SDS-polyacrvlamide gel clectrophoresis re-,4 hypothetical model for the topography vealed a minimum of nine polypeptidcs of the polypcptides of adenovirions was (II-X) (Maize1 et al., 1968a; Muizel, 1971). constructed on t.hc basis of t.hesc results. Fiveof t.hese (II, III, IV, V, VII) areintcpral MATEItIAI,d AXI) RIKTlCOl)S parts of hexons, pentons, fibers, or cores

We want to discuss the option pricing on stochastic volatility market models, in which we are goi... more We want to discuss the option pricing on stochastic volatility market models, in which we are going to consider a generic function β(ν t) for the drift of volatility process. It is our intention choose any equivalent martingale measure, so that the drift of volatility process, respect at the new measure, is zero. This technique is possible when the Girsanov theorem is satisfied, since the stochastic volatility models are uncomplete markets, thus one has to choice an arbitrary risk price of volatility. In all this cases we are able to compute the price of Vanilla options in a closed form. To name a few, we can think to the popular Heston's model, in which the solution is known in literature, unless of an inverse Fourier transform.

We propose to discuss a new technique to derive an good approximated solution for the price of a ... more We propose to discuss a new technique to derive an good approximated solution for the price of a European call and put options, in a market model with stochastic volatility. In particular, the model that we have considered is the Heston's model. This allows arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation. 1