Quantitative Finance: stochastic volatility market models

Closed Solution for Heston PDE byGeometrical Transformations

XIV WorkShop of Quantitative Finance

Mario Dell’Era

Pisa University

June 24, 2014

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Heston Model

dSt = rSt dt +√νt St dW (1)

t S ∈ [0,+∞)

dνt = K (Θ− νt )dt + α√νt dW (2)

t ν ∈ (0,+∞)

under a risk-neutral martingale measure Q.From Ito’s lemma we have the following PDE:

∂f∂t

+12νS2 ∂

2f∂S2

+ ρναS∂2f∂S∂ν

+12να

2 ∂2f

∂ν2+ κ(Θ− ν)

∂f∂ν

+ rS∂f∂S− rf = 0

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Heston Model

dSt = rSt dt +√νt St dW (1)

t S ∈ [0,+∞)

dνt = K (Θ− νt )dt + α√νt dW (2)

t ν ∈ (0,+∞)

under a risk-neutral martingale measure Q.From Ito’s lemma we have the following PDE:

∂f∂t

+12νS2 ∂

2f∂S2

+ ρναS∂2f∂S∂ν

+12να

2 ∂2f

∂ν2+ κ(Θ− ν)

∂f∂ν

+ rS∂f∂S− rf = 0

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical methods(1) Fourier Transform: S.L. Heston (1993)

(2) Finite Difference: T. Kluge (2002)

(3) Monte Carlo: B. Jourdain (2005)

Approximation method(1) Analytic and Geometric Methods for Heat Kernel: I. Avramidi

(2007)

(2) Implied Volatility: M. Forde, A. Jacquier (2009)

(3) Geometrical Approximation method: M. Dell’Era (2010)

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Coordinate Transformations techniqueWe have elaborated a new methodology based on changing of variableswhich is independent of payoffs and does not need to use the inverse Fouriertransform algorithm or numerical methods as Finite Difference and MonteCarlo simulations. In particular, we will compute the price of Vanilla Options,in order to validate numerically the Geometrical Transformations technique.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

1stTransformations:

8><>:x = ln S, x ∈ (−∞,+∞)

ν = ν/α, ν ∈ [0,+∞)

f (t ,S, ν) = f1(t , x , ν)e−r(T−t)

(1)

thus one has:

∂f1∂t

+12ν

∂2f1∂x2

+ 2ρ∂2f1∂x∂ν

+∂2f1∂ν2

!+

„r −

12αν

«∂f1∂x

+κ

α(θ − αν)

∂f1∂ν

= 0

f1(T , x, ν) = Φ1(x) ρ ∈ (−1,+1), α ∈ R+

x ∈ (−∞,+∞) ν ∈ [0,+∞) t ∈ [0, T ]

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

1stTransformations:

8><>:x = ln S, x ∈ (−∞,+∞)

ν = ν/α, ν ∈ [0,+∞)

f (t ,S, ν) = f1(t , x , ν)e−r(T−t)

(1)

thus one has:

∂f1∂t

+12ν

∂2f1∂x2

+ 2ρ∂2f1∂x∂ν

+∂2f1∂ν2

!+

„r −

12αν

«∂f1∂x

+κ

α(θ − αν)

∂f1∂ν

= 0

f1(T , x, ν) = Φ1(x) ρ ∈ (−1,+1), α ∈ R+

x ∈ (−∞,+∞) ν ∈ [0,+∞) t ∈ [0, T ]

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

2nd Transformations:

8><>:ξ = x − ρν ξ ∈ (−∞,+∞)

η = −νp

1− ρ2 η ∈ (−∞, 0]

f1(t , x , ν) = f2(t , ξ, η)

(2)

Again we have:

∂f2∂t−

αη

2p

1− ρ2(1− ρ2)

∂2f2∂ξ2

+∂2f2∂η2

!+

αη

2p

1− ρ2

„1−

2κρα

«∂f2∂ξ

−αη

2p

1− ρ2

„2κα

p1− ρ2

«∂f2∂η

+

„r −

κρθ

α

«∂f2∂ξ−θκ

α

p1− ρ2

∂f2∂η

= 0

f2(T , ξ, η) = Φ2(ξ, η), ρ ∈ (−1,+1), α ∈ R+.

ξ ∈ (−∞,+∞), η ∈ (−∞, 0], t ∈ [0, T ].

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

2nd Transformations:

8><>:ξ = x − ρν ξ ∈ (−∞,+∞)

η = −νp

1− ρ2 η ∈ (−∞, 0]

f1(t , x , ν) = f2(t , ξ, η)

(2)

Again we have:

∂f2∂t−

αη

2p

1− ρ2(1− ρ2)

∂2f2∂ξ2

+∂2f2∂η2

!+

αη

2p

1− ρ2

„1−

2κρα

«∂f2∂ξ

−αη

2p

1− ρ2

„2κα

p1− ρ2

«∂f2∂η

+

„r −

κρθ

α

«∂f2∂ξ−θκ

α

p1− ρ2

∂f2∂η

= 0

f2(T , ξ, η) = Φ2(ξ, η), ρ ∈ (−1,+1), α ∈ R+.

ξ ∈ (−∞,+∞), η ∈ (−∞, 0], t ∈ [0, T ].

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

3rd Transformations:

8>>><>>>:γ = ξ +

`r − κρθ

α

´(T − t) γ ∈ (−∞,+∞)

φ = −η + κθα

p1− ρ2(T − t) φ ∈ [0,+∞)

τ = 12

R Tt νsds τ ∈ [0,+∞)

f2(t , ξ, η) = f3(τ, γ, φ)

which give us the following PDE:

∂f3∂τ

= (1− ρ2)

∂2f3∂γ2

+∂2f3∂φ2

!−„

1−2κρα

«∂f3∂γ−„

2κα

p1− ρ2

«∂f3∂φ

= 0

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

3rd Transformations:

8>>><>>>:γ = ξ +

`r − κρθ

α

´(T − t) γ ∈ (−∞,+∞)

φ = −η + κθα

p1− ρ2(T − t) φ ∈ [0,+∞)

τ = 12

R Tt νsds τ ∈ [0,+∞)

f2(t , ξ, η) = f3(τ, γ, φ)

which give us the following PDE:

∂f3∂τ

= (1− ρ2)

∂2f3∂γ2

+∂2f3∂φ2

!−„

1−2κρα

«∂f3∂γ−„

2κα

p1− ρ2

«∂f3∂φ

= 0

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

and imposing:

f3(τ, γ, φ) = eaτ+bγ+cφf4(τ, γ, φ),

where8>><>>:a = −(1− ρ2)(b2 + c2);

b =(1− 2κρ

α )2(1−ρ2)

;

c = κ

α√

1−ρ2;

finally one has:

∂f4∂τ

= (1− ρ2)

∂2f4∂γ2

+∂2f4∂φ2

!f4(0, γ, φ) = Φ4(γ, φ)

τ ∈ [0,+∞) φ ∈ [0,+∞) γ ∈ (−∞,+∞),

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

and imposing:

f3(τ, γ, φ) = eaτ+bγ+cφf4(τ, γ, φ),

where8>><>>:a = −(1− ρ2)(b2 + c2);

b =(1− 2κρ

α )2(1−ρ2)

;

c = κ

α√

1−ρ2;

finally one has:

∂f4∂τ

= (1− ρ2)

∂2f4∂γ2

+∂2f4∂φ2

!f4(0, γ, φ) = Φ4(γ, φ)

τ ∈ [0,+∞) φ ∈ [0,+∞) γ ∈ (−∞,+∞),

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

The solution is known in the literature (Andrei D. Polyanin, Handbook ofLinear Partial Differential Equations, 2002, p. 188), and it can be written asintegral, whose kernel G(0, γ′, φ′|τ, γ, δ) is a bivariate gaussian function:

G(0, γ′, φ′|τ, γ, φ) =1

4πτ(1− ρ2)

24e− (γ′−γ)2+(φ′−φ)2

4τ(1−ρ2) − e− (γ′−γ)2+(φ′+φ)2

4τ(1−ρ2)

35 ,therefore

f4(τ, γ, φ) =

Z +∞

0dφ′

Z +∞

−∞dγ′f4(0, γ′, φ′)G(0, γ′, φ′|τ, γ, φ)

+ (1− ρ2)

Z τ

0duZ +∞

−∞dγ′f4(u, γ′, 0)

»∂G(0, γ′, δ′|τ − u, γ, δ)

∂φ′

–φ′=0

.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

The solution is known in the literature (Andrei D. Polyanin, Handbook ofLinear Partial Differential Equations, 2002, p. 188), and it can be written asintegral, whose kernel G(0, γ′, φ′|τ, γ, δ) is a bivariate gaussian function:

G(0, γ′, φ′|τ, γ, φ) =1

4πτ(1− ρ2)

24e− (γ′−γ)2+(φ′−φ)2

4τ(1−ρ2) − e− (γ′−γ)2+(φ′+φ)2

4τ(1−ρ2)

35 ,therefore

f4(τ, γ, φ) =

Z +∞

0dφ′

Z +∞

−∞dγ′f4(0, γ′, φ′)G(0, γ′, φ′|τ, γ, φ)

+ (1− ρ2)

Z τ

0duZ +∞

−∞dγ′f4(u, γ′, 0)

»∂G(0, γ′, δ′|τ − u, γ, δ)

∂φ′

–φ′=0

.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Using the natural variables we may rewrite the solution as follows:

f (t,S, ν) = e−r(T−t)+aτ+bγ+cδZ +∞

0dφ′

Z +∞

−∞dγ′f4(0, γ′, φ′)G(0, γ′, φ′|τ, γ, φ)

+ (1− ρ2)e−r(T−t)+aτ+bγ+cφZ τ

0duZ +∞

−∞dγ′f4(u, γ′, 0)

×»∂G(0, γ′, φ′|τ − u, γ, φ)

∂φ′

–φ′=0

.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Vanilla Option PricingIn order to test above option pricing formula, we are going to consider asoption a Vanilla Call with strike price K and maturity T. In the new variable the

payoff (ST − K )+ is equal to e−bγ−cφ(eγ+ρφ/√

1−ρ2 − K )+. Substituting thislatter in the above equation we have:

f (t,S, ν) = e−r(T−t)+aτ+bγ+cφ

×Z +∞

0dφ′

Z +∞

−∞dγ′e−bγ′−cφ′ (eγ

′+ρφ′/√

1−ρ2− K )+G(0, γ′, φ′|τ, γ, φ)

+(1−ρ2)e−r(T−t)+aτ+bγ+cφZ τ

0duZ +∞

−∞dγ′f4(u, γ′, 0)

»∂G(0, γ′, φ′|τ − u, γ, φ)

∂φ′

–φ′=0

,

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Vanilla Option PricingIn order to test above option pricing formula, we are going to consider asoption a Vanilla Call with strike price K and maturity T. In the new variable the

payoff (ST − K )+ is equal to e−bγ−cφ(eγ+ρφ/√

1−ρ2 − K )+. Substituting thislatter in the above equation we have:

f (t,S, ν) = e−r(T−t)+aτ+bγ+cφ

×Z +∞

0dφ′

Z +∞

−∞dγ′e−bγ′−cφ′ (eγ

′+ρφ′/√

1−ρ2− K )+G(0, γ′, φ′|τ, γ, φ)

+(1−ρ2)e−r(T−t)+aτ+bγ+cφZ τ

0duZ +∞

−∞dγ′f4(u, γ′, 0)

»∂G(0, γ′, φ′|τ − u, γ, φ)

∂φ′

–φ′=0

,

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Considering the particular case, for τ which goes to zero (i.e T → 0), thesolution reduces itself to:

f (t,St , νt )

= St

»N“−ψ1(0),−a1,1

p1− ρ2

”− e−2

“ρ− κ

α

”“ νtα

+ καθ(T−t)

”N“−ψ2(0),−a1,2

p1− ρ2

”–

−Ke−r(T−t)»

N“−ψ1(0),−a2,1

p1− ρ2

”− e2 κ

α

“ νtα

+ καθ(T−t)

”N“−ψ2(0),−a2,2

p1− ρ2

”–,

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

ψ1(0) = −

hνtα + κ

α θ(T − t) + (ρ− κα )R T

t νsds

iqR T

t νsds,

ψ2(0) =

hνtα + κ

α θ(T − t)− (ρ− κα )R T

t νsds

iqR T

t νsds,

ψ1(0) = −

hνtα + κ

α θ(T − t)− κα

R Tt νsds

iqR T

t νsds,

ψ2(0) =

hνtα + κ

α θ(T − t) + κα

R Tt νsds

iqR T

t νsds,

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

a1,1 =

hln(K/St )− r(T − t)− 1

2

R Tt νsds

iq

(1− ρ2)R T

t νsds,

a1,2 =

hln(K/St ) + 2 ρανt − (r − 2κθρα )(T − t)− 1

2

R Tt νsds

iq

(1− ρ2)R T

t νsds,

a2,1 =

hln(K/St )− r(T − t) + 1

2

R Tt νsds

iq

(1− ρ2)R T

t νsds,

a2,2 =

hln(K/St ) + 2 ρανt − (r − 2κθρα )(T − t) + 1

2

R Tt νsds

iq

(1− ρ2)R T

t νsds.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical ValidationThe approximation τ → 0 will be here interpreted as option pricing for fewdays. From 1 day up to 10 days are suitable maturities to prove our validationhypothesis, at varying of volatility. Parameter values are those in Bakshi, Caoand Chen (1997) namely κ = 1.15, Θ = 0.04, α = 0.39 and ρ = −0.64. Wehave chosen r = 10% K = 100, and three different maturities T . In whatfollows we use the expected value of the variance process EP[νs] instead ofνs in the term

R Tt νsds. In the tables hereafter one can see the results of

numerical experiments:

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Numerical ValidationThe approximation τ → 0 will be here interpreted as option pricing for fewdays. From 1 day up to 10 days are suitable maturities to prove our validationhypothesis, at varying of volatility. Parameter values are those in Bakshi, Caoand Chen (1997) namely κ = 1.15, Θ = 0.04, α = 0.39 and ρ = −0.64. Wehave chosen r = 10% K = 100, and three different maturities T . In whatfollows we use the expected value of the variance process EP[νs] instead ofνs in the term

R Tt νsds. In the tables hereafter one can see the results of

numerical experiments:

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: At the money, S0 = 100,K = 100, with parameter values: κ = 1.5,θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 1 day.

Volatility Fourier method Dell’Era method30% 0.6434 0.644240% 0.8543 0.854150% 1.0643 1.064160% 1.2743 1.274270% 1.4843 1.484580% 1.6943 1.694990% 1.9042 1.9055100% 2.1142 2.1162

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: At the money, S0 = 100,K = 100, with parameter values: κ = 1.5,θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 5 days.

Volatility Fourier method Dell’Era method30% 1.4763 1.474840% 1.9430 1.940750% 2.4101 2.408160% 2.8772 2.876970% 3.3444 3.347280% 3.8115 3.819090% 4.2785 4.2927100% 4.7454 4.7683

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: At the money, S0 = 100,K = 100, with parameter values: κ = 1.5,θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 10 days.

Volatility Fourier method Dell’Era method30% 2.1234 2.119140% 2.7787 2.772250% 3.4348 3.429460% 4.0912 4.090570% 4.7477 4.755780% 5.4040 5.425490% 6.0601 6.1002100% 6.7158 6.7806

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: In the money, S0 = K“

1 + 10%pθ(T − t)

”, with parameter values:

κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 1 day.

Volatility Fourier method Dell’Era method30% 0.6991 0.699440% 0.9094 0.908950% 1.1191 1.118760% 1.3289 1.328770% 1.5377 1.538980% 1.7488 1.749490% 1.9588 1.9600100% 2.1688 2.1708

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: In the money, S0 = K“

1 + 10%pθ(T − t)

”, with parameter values:

κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 5 days.

Volatility Fourier method Dell’Era method30% 1.6049 1.601240% 2.0700 2.066150% 2.5362 2.533160% 3.0030 3.001970% 3.4700 3.472380% 3.9372 3.944590% 4.4044 4.4186100% 4.8715 4.8947

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: In the money, S0 = K“

1 + 10%pθ(T − t)

”, with parameter values:

κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 10 days.

Volatility Fourier method Dell’Era method30% 2.3098 2.301240% 2.9621 2.952750% 3.6168 3.609560% 4.2727 4.270870% 4.9291 4.936680% 5.5856 5.607290% 6.2421 6.2831100% 6.8984 6.9647

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: Out the money, S0 = K“

1− 10%pθ(T − t)

”, with parameter values:

κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 1 day.

Volatility Fourier method Dell’Era method30% 0.5905 0.591840% 0.8013 0.801450% 1.0111 1.011260% 1.2210 1.221270% 1.4309 1.431380% 1.6407 1.641590% 1.8506 1.8519100% 2.0605 2.0625

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: Out the money, S0 = K“

1− 10%pθ(T − t)

”, with parameter values:

κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 5 days.

Volatility Fourier method Dell’Era method30% 1.3539 1.354640% 1.8208 1.820150% 2.2878 2.286960% 2.7546 2.755170% 3.2214 3.224780% 3.6882 3.695990% 4.1547 4.1689100% 4.6212 4.6438

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

Table: Out the money, S0 = K“

1− 10%pθ(T − t)

”, with parameter values:

κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 10 days.

Volatility Fourier method Dell’Era method30% 1.9459 1.945940% 2.6019 2.598550% 3.2581 3.254860% 3.9142 3.914870% 4.5701 4.578780% 5.2257 5.247190% 5.8810 5.9204100% 6.5359 6.5992

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

ConclusionsThe proposed method is straightforward from theoretical viewpoint andseems to be promising from that numerical. We reduce the Heston’s PDE ina simpler, using , in a right order, suitable changing of variables, whoseJacobian has not singularity points, unless for ρ = ±1. This evidence givesus the safety that the variables chosen are well defined.Besides, the idea to use the expected value of the variance process EP[νs],instead of νt , provides us, in concrete, a closed solution very easy tocompute; and so, we are also able to know what is the error using thegeometric transformation technique; which is equal to the variance of thevariance process νt : Err = EP[(νt − EP[νt ])

2]. While, using Fourier techniquewe are not able to know the numeric error directly, but we need to compareFourier prices with Monte Carlo prices, for which one can manage thevariance.We want to remark that the shown technique is independent to the payoff andtherefore, the pricing activities have the same algorithmic complexity forevery derivatives, unlike using Fourier Transform method, for which thecomplexity is tied to the payoff.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

ConclusionsThe proposed method is straightforward from theoretical viewpoint andseems to be promising from that numerical. We reduce the Heston’s PDE ina simpler, using , in a right order, suitable changing of variables, whoseJacobian has not singularity points, unless for ρ = ±1. This evidence givesus the safety that the variables chosen are well defined.Besides, the idea to use the expected value of the variance process EP[νs],instead of νt , provides us, in concrete, a closed solution very easy tocompute; and so, we are also able to know what is the error using thegeometric transformation technique; which is equal to the variance of thevariance process νt : Err = EP[(νt − EP[νt ])

2]. While, using Fourier techniquewe are not able to know the numeric error directly, but we need to compareFourier prices with Monte Carlo prices, for which one can manage thevariance.We want to remark that the shown technique is independent to the payoff andtherefore, the pricing activities have the same algorithmic complexity forevery derivatives, unlike using Fourier Transform method, for which thecomplexity is tied to the payoff.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Quantitative Finance: stochastic volatility market models

ConclusionsThe proposed method is straightforward from theoretical viewpoint andseems to be promising from that numerical. We reduce the Heston’s PDE ina simpler, using , in a right order, suitable changing of variables, whoseJacobian has not singularity points, unless for ρ = ±1. This evidence givesus the safety that the variables chosen are well defined.Besides, the idea to use the expected value of the variance process EP[νs],instead of νt , provides us, in concrete, a closed solution very easy tocompute; and so, we are also able to know what is the error using thegeometric transformation technique; which is equal to the variance of thevariance process νt : Err = EP[(νt − EP[νt ])

2]. While, using Fourier techniquewe are not able to know the numeric error directly, but we need to compareFourier prices with Monte Carlo prices, for which one can manage thevariance.We want to remark that the shown technique is independent to the payoff andtherefore, the pricing activities have the same algorithmic complexity forevery derivatives, unlike using Fourier Transform method, for which thecomplexity is tied to the payoff.

Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations