Quantitative Finance: stochastic volatility market models Closed Solution for Heston PDE by Geometrical 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
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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
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
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
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
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
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
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
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
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
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