workshop 2011 of quantitative finance
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Quantitative Finance: stochastic volatility market models
Supervisors:Roberto Reno, Claudio Pacati
Geometrical Approximation and Perturbativemethod for PDEs in Finance
PhD Program in Mathematics for Economic Decisions
Mario Dell’Era
Leonardo Fibonacci School
November 28, 2011
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Stochastic Volatility Market Models
dSt = rStdt + a2(σt ,St )dW (1)t
dσt = b1(σt )dt + b2(σt )dW (2)t
dBt = rBtdt
f (T ,ST ) = φ(ST )
under a risk-neutral martingale measure Q.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Heston Model
dSt = rStdt +√νtStdW (1)
t S ∈ [0,+∞)
dνt = K (Θ− νt )dt + α√νtdW (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
+12να2 ∂
2f∂ν2 +κ(Θ−ν)
∂f∂ν
+rS∂f∂S−rf = 0
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Heston Model
dSt = rStdt +√νtStdW (1)
t S ∈ [0,+∞)
dνt = K (Θ− νt )dt + α√νtdW (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
+12να2 ∂
2f∂ν2 +κ(Θ−ν)
∂f∂ν
+rS∂f∂S−rf = 0
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical methods(1) Fourier Transform: Heston, S.L., (1993)
(2) Finite Difference: Kluge, T., (2002)
(3) Monte Carlo: Jourdain, B., (2005)
Approximation method(1) Analytic and Geometric Methods for Heat Kernel: Avramidi, I.,
(2007)
(2) Implied Volatility: Forde, M., Jacquier, A. (2009)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Geometrical Approximation method: Heston model
The proposed technique is based on a stochastic approximation ofthe Cauchy condition.
We use Φ (ST eεT ) where εT = ρ(ν − νT )/α,
instead of the standard pay-off function Φ(ST ).
εT is a stochastic quantity and ν is the expected value of νT varianceprocess. Define stochastic error:
eεT = eρ{[(ν0−Θ)e−κ(T )+Θ]−νT}
α .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Geometrical Approximation method: Heston model
The proposed technique is based on a stochastic approximation ofthe Cauchy condition.
We use Φ (ST eεT ) where εT = ρ(ν − νT )/α,
instead of the standard pay-off function Φ(ST ).
εT is a stochastic quantity and ν is the expected value of νT varianceprocess. Define stochastic error:
eεT = eρ{[(ν0−Θ)e−κ(T )+Θ]−νT}
α .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Geometrical Approximation method: Heston model
The proposed technique is based on a stochastic approximation ofthe Cauchy condition.
We use Φ (ST eεT ) where εT = ρ(ν − νT )/α,
instead of the standard pay-off function Φ(ST ).
εT is a stochastic quantity and ν is the expected value of νT varianceprocess. Define stochastic error:
eεT = eρ{[(ν0−Θ)e−κ(T )+Θ]−νT}
α .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Its distribution is obtained via simulation for sensible parameter values:
ρ = −0.64, ν0 = 0.038, Θ = 0.04, κ = 1.15, α = 0.38, T = 1-year.
0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.0250
5
10
15
20
25Geometrical Approximation method and Heston model
Numb
er of ev
ents
Stochastic Error
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
we consider for a Call: (ST eεT − E)+, instead of (ST − E)+; and for aPut:(E − ST eεT )+ instead of (E − ST )+.
The Call option price is give by:
Cρ,α,Θ,κ(t ,St , νt ) = (Steεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
Pρ,α,Θ,κ(t ,St , νt ) = Eeδρ2 N(−dρ2 )− (Steεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
we consider for a Call: (ST eεT − E)+, instead of (ST − E)+; and for aPut:(E − ST eεT )+ instead of (E − ST )+.
The Call option price is give by:
Cρ,α,Θ,κ(t ,St , νt ) = (Steεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
Pρ,α,Θ,κ(t ,St , νt ) = Eeδρ2 N(−dρ2 )− (Steεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
we consider for a Call: (ST eεT − E)+, instead of (ST − E)+; and for aPut:(E − ST eεT )+ instead of (E − ST )+.
The Call option price is give by:
Cρ,α,Θ,κ(t ,St , νt ) = (Steεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
Pρ,α,Θ,κ(t ,St , νt ) = Eeδρ2 N(−dρ2 )− (Steεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.64,St = E
(1± 10%
√ΘT)
T = 6/12G.A. Fourier F.D.M. M.C
ATM 5.3034 5.5707 5.4806 5.4925INM 6.1828 6.4481 6.4158 6.4250OTM 4.5057 4.7549 4.7347 4.7502
T = 9/12G.A. Fourier F.D.M. M.C
ATM 6.8930 7.0500 6.9430 6.9628INM 7.9923 8.1346 8.0769 8.928OTM 5.8918 6.0392 6.0156 6.381
T = 1G.A. Fourier F.D.M. M.C
ATM 8.3329 8.3816 8.2619 8.2887INM 9.6192 9.6351 9.5843 9.6030OTM 7.1577 7.2112 7.1562 7.1357
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.64,St = E
(1± 10%
√ΘT)
T = 6/12G.A. Fourier F.D.M. M.C
ATM 5.3034 5.5707 5.4806 5.4925INM 6.1828 6.4481 6.4158 6.4250OTM 4.5057 4.7549 4.7347 4.7502
T = 9/12G.A. Fourier F.D.M. M.C
ATM 6.8930 7.0500 6.9430 6.9628INM 7.9923 8.1346 8.0769 8.928OTM 5.8918 6.0392 6.0156 6.381
T = 1G.A. Fourier F.D.M. M.C
ATM 8.3329 8.3816 8.2619 8.2887INM 9.6192 9.6351 9.5843 9.6030OTM 7.1577 7.2112 7.1562 7.1357
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.64,St = E
(1± 10%
√ΘT)
T = 6/12G.A. Fourier F.D.M. M.C
ATM 5.3034 5.5707 5.4806 5.4925INM 6.1828 6.4481 6.4158 6.4250OTM 4.5057 4.7549 4.7347 4.7502
T = 9/12G.A. Fourier F.D.M. M.C
ATM 6.8930 7.0500 6.9430 6.9628INM 7.9923 8.1346 8.0769 8.928OTM 5.8918 6.0392 6.0156 6.381
T = 1G.A. Fourier F.D.M. M.C
ATM 8.3329 8.3816 8.2619 8.2887INM 9.6192 9.6351 9.5843 9.6030OTM 7.1577 7.2112 7.1562 7.1357
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.28,St = E
(1± 10%
√ΘT)
T = 6/12G.A. Fourier F.D.M. M.C.
ATM 5.9827 5.9957 5.9972 5.9975INM 6.7329 6.8154 6.7964 6.7954OTM 5.2918 5.2646 5.2597 5.2618
T = 9/12G.A. Fourier F.D.M. M.C.
ATM 7.5188 7.4963 7.5040 7.4966INM 8.5418 8.5108 8.4832 8.4736OTM 6.6719 6.5941 6.5994 6.5948
T = 1G.A. Fourier F.D.M. M.C.
ATM 8.9847 8.8488 8.8614 8.8258INM 9.9273 10.0035 10.0177 9.9790OTM 7.8896 7.7832 7.7936 7.7617
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.28,St = E
(1± 10%
√ΘT)
T = 6/12G.A. Fourier F.D.M. M.C.
ATM 5.9827 5.9957 5.9972 5.9975INM 6.7329 6.8154 6.7964 6.7954OTM 5.2918 5.2646 5.2597 5.2618
T = 9/12G.A. Fourier F.D.M. M.C.
ATM 7.5188 7.4963 7.5040 7.4966INM 8.5418 8.5108 8.4832 8.4736OTM 6.6719 6.5941 6.5994 6.5948
T = 1G.A. Fourier F.D.M. M.C.
ATM 8.9847 8.8488 8.8614 8.8258INM 9.9273 10.0035 10.0177 9.9790OTM 7.8896 7.7832 7.7936 7.7617
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.28,St = E
(1± 10%
√ΘT)
T = 6/12G.A. Fourier F.D.M. M.C.
ATM 5.9827 5.9957 5.9972 5.9975INM 6.7329 6.8154 6.7964 6.7954OTM 5.2918 5.2646 5.2597 5.2618
T = 9/12G.A. Fourier F.D.M. M.C.
ATM 7.5188 7.4963 7.5040 7.4966INM 8.5418 8.5108 8.4832 8.4736OTM 6.6719 6.5941 6.5994 6.5948
T = 1G.A. Fourier F.D.M. M.C.
ATM 8.9847 8.8488 8.8614 8.8258INM 9.9273 10.0035 10.0177 9.9790OTM 7.8896 7.7832 7.7936 7.7617
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
In the money: St = E“
1 + 10%√
ΘT”
, r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.1
1 3 6 9 122
3
4
5
6
7
8
9
10
11Approximation method in the Heston with drift zero
Maturity date
Europ
ean C
all op
tion pr
ice
ans(Approximation method)ans(Fourier transform)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
At the money: St = E , r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.1
1 3 6 9 122
3
4
5
6
7
8
9
10
Maturity date
Europ
ean C
all op
tion pr
ice
Approximation method in the Heston with drift zero
ans(Approximation method)ans(Fourier transform)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Out the money: St = E“
1− 10%√
ΘT”
, r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.1
1 3 6 9 122
3
4
5
6
7
8
9Approximation method in the Heston with drift zero
Maturity date
Europ
ean C
all op
tion pr
ice
ans(Approximation method)ans(Fourier transform)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
SABR Model
In the SABR model one supposes that under a martingale measure Qthe forward price follows the SDEs:
dFt = σtFβt dW (1)
t β ∈ (0,1]
dσt = ασtdW (2)t α ∈ R
dBt = rBtdt .
For Ito’s lemma, in the case β = 1, we have:
∂f∂t
+12
(σ)2(
F 2 ∂2f
∂F 2 + 2ρFα∂2f∂F∂σ
+ α2 ∂2f
∂σ2
)− rf = 0
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
SABR Model
In the SABR model one supposes that under a martingale measure Qthe forward price follows the SDEs:
dFt = σtFβt dW (1)
t β ∈ (0,1]
dσt = ασtdW (2)t α ∈ R
dBt = rBtdt .
For Ito’s lemma, in the case β = 1, we have:
∂f∂t
+12
(σ)2(
F 2 ∂2f
∂F 2 + 2ρFα∂2f∂F∂σ
+ α2 ∂2f
∂σ2
)− rf = 0
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
SABR Model
In the SABR model one supposes that under a martingale measure Qthe forward price follows the SDEs:
dFt = σtFβt dW (1)
t β ∈ (0,1]
dσt = ασtdW (2)t α ∈ R
dBt = rBtdt .
For Ito’s lemma, in the case β = 1, we have:
∂f∂t
+12
(σ)2(
F 2 ∂2f
∂F 2 + 2ρFα∂2f∂F∂σ
+ α2 ∂2f
∂σ2
)− rf = 0
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Implied Volatility method: Hagan (2002)
Hagan et al. (2002) derive and study the approximate formulas for theimplied Black and Bachelier volatilities in the SABR model, which canbe represented as follows:
σ(E, T ) =σ0
(F0/E)(1−β)/2“
1 + (1−β)224 ln2(F0/E) + (1−β)4
1920 ln4(F0/E) + .....”×
zχ(z)
(1 +
"(1− β)2σ2
0
24(F0E)(1−β)+
ρβσ0α
4(F0E)(1−β)/2+
(2− 3ρ2)α2
24
#T + .......
),
where E is the strike price, F0 is the underlying asset value at thetime t = 0 and σ0 is the value of the volatility at time t = 0,
z =α
σ0(F0/E)(1−β)/2 ln(F0/E), χ(z) = ln
(p1− 2ρz + z2 + z − ρ
1− ρ
).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Implied Volatility method: Hagan (2002)
Hagan et al. (2002) derive and study the approximate formulas for theimplied Black and Bachelier volatilities in the SABR model, which canbe represented as follows:
σ(E, T ) =σ0
(F0/E)(1−β)/2“
1 + (1−β)224 ln2(F0/E) + (1−β)4
1920 ln4(F0/E) + .....”×
zχ(z)
(1 +
"(1− β)2σ2
0
24(F0E)(1−β)+
ρβσ0α
4(F0E)(1−β)/2+
(2− 3ρ2)α2
24
#T + .......
),
where E is the strike price, F0 is the underlying asset value at thetime t = 0 and σ0 is the value of the volatility at time t = 0,
z =α
σ0(F0/E)(1−β)/2 ln(F0/E), χ(z) = ln
(p1− 2ρz + z2 + z − ρ
1− ρ
).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Implied Volatility method: Hagan (2002)
Hagan et al. (2002) derive and study the approximate formulas for theimplied Black and Bachelier volatilities in the SABR model, which canbe represented as follows:
σ(E, T ) =σ0
(F0/E)(1−β)/2“
1 + (1−β)224 ln2(F0/E) + (1−β)4
1920 ln4(F0/E) + .....”×
zχ(z)
(1 +
"(1− β)2σ2
0
24(F0E)(1−β)+
ρβσ0α
4(F0E)(1−β)/2+
(2− 3ρ2)α2
24
#T + .......
),
where E is the strike price, F0 is the underlying asset value at thetime t = 0 and σ0 is the value of the volatility at time t = 0,
z =α
σ0(F0/E)(1−β)/2 ln(F0/E), χ(z) = ln
(p1− 2ρz + z2 + z − ρ
1− ρ
).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Geometrical Approximation method: SABR model β = 1
Also in this case, as done for the Heston model,
we use Φ (FT eεT ) where εT = ρ(σ − σT )/α, instead of the standard
pay-off function Φ(FT ).
εT is a stochastic quantity and σ is the expected value of σT varianceprocess. Define stochastic error:
eεT = e
ρ
264σ0e
„α22 T
«−σT
375α .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Geometrical Approximation method: SABR model β = 1
Also in this case, as done for the Heston model,
we use Φ (FT eεT ) where εT = ρ(σ − σT )/α, instead of the standard
pay-off function Φ(FT ).
εT is a stochastic quantity and σ is the expected value of σT varianceprocess. Define stochastic error:
eεT = e
ρ
264σ0e
„α22 T
«−σT
375α .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Geometrical Approximation method: SABR model β = 1
Also in this case, as done for the Heston model,
we use Φ (FT eεT ) where εT = ρ(σ − σT )/α, instead of the standard
pay-off function Φ(FT ).
εT is a stochastic quantity and σ is the expected value of σT varianceprocess. Define stochastic error:
eεT = e
ρ
264σ0e
„α22 T
«−σT
375α .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Its distribution is obtained via simulation for sensible parameter values:
ρ = −0.71, σ0 = 20% α = 0.29, β = 1, T = 1-year.
0.9 0.95 1 1.05 1.1 1.15 1.2 1.250
5
10
15
20
25Geometrical Approximation method and SABR model
Stochastic Error
Numb
er of ev
ents
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
We consider for a Call: (FT eεT − E)+, instead of (FT − E)+; and for aPut:(E − FT eεT )+ instead of (E − FT )+.
The Call option price is give by:
C(t ,Ft , σt ) = (Fteεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
P(t ,Ft , σt ) = Eeδρ2 N(−dρ1 )− (Fteεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
We consider for a Call: (FT eεT − E)+, instead of (FT − E)+; and for aPut:(E − FT eεT )+ instead of (E − FT )+.
The Call option price is give by:
C(t ,Ft , σt ) = (Fteεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
P(t ,Ft , σt ) = Eeδρ2 N(−dρ1 )− (Fteεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
We consider for a Call: (FT eεT − E)+, instead of (FT − E)+; and for aPut:(E − FT eεT )+ instead of (E − FT )+.
The Call option price is give by:
C(t ,Ft , σt ) = (Fteεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
P(t ,Ft , σt ) = Eeδρ2 N(−dρ1 )− (Fteεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Vanilla Options
We consider for a Call: (FT eεT − E)+, instead of (FT − E)+; and for aPut:(E − FT eεT )+ instead of (E − FT )+.
The Call option price is give by:
C(t ,Ft , σt ) = (Fteεt ) eδρ1 N(dρ1 )− Eeδ
ρ2 N(dρ2 );
and for a Put:
P(t ,Ft , σt ) = Eeδρ2 N(−dρ1 )− (Fteεt ) eδ
ρ1 N(−dρ1 ).
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, σ0 = 20%, α = 0.29, ρ = −0.71, Ft = E
(1± 10%
√σ2
0T)
T = 1/12G.A. Hagan
ATM 2.3426 2.2956INM 3.0008 2.9492OTM 1.7655 1.6605
T = 3/12G.A. Hagan
ATM 3.9097 3.9495INM 5.0110 5.1039OTM 2.9481 2.8821
T = 6/12G.A. Hagan
ATM 5.3064 5.5295INM 6.8070 7.1942OTM 4.0023 4.0742
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, σ0 = 20%, α = 0.29, ρ = −0.71, Ft = E
(1± 10%
√σ2
0T)
T = 1/12G.A. Hagan
ATM 2.3426 2.2956INM 3.0008 2.9492OTM 1.7655 1.6605
T = 3/12G.A. Hagan
ATM 3.9097 3.9495INM 5.0110 5.1039OTM 2.9481 2.8821
T = 6/12G.A. Hagan
ATM 5.3064 5.5295INM 6.8070 7.1942OTM 4.0023 4.0742
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, σ0 = 20%, α = 0.29, ρ = −0.71, Ft = E
(1± 10%
√σ2
0T)
T = 1/12G.A. Hagan
ATM 2.3426 2.2956INM 3.0008 2.9492OTM 1.7655 1.6605
T = 3/12G.A. Hagan
ATM 3.9097 3.9495INM 5.0110 5.1039OTM 2.9481 2.8821
T = 6/12G.A. Hagan
ATM 5.3064 5.5295INM 6.8070 7.1942OTM 4.0023 4.0742
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, σ0 = 20%, α = 0.29, ρ = −0.1, Ft = E
(1± 10%
√σ2
0T)
T = 1/12G.A. Hagan
ATM 2.2855 2.2983INM 2.9702 2.9389OTM 1.7152 1.6764
T = 3/12G.A. Hagan
ATM 3.9241 3.9654INM 5.0839 5.0795OTM 2.9615 2.9351
T = 6/12G.A. Hagan
ATM 5.4885 5.5684INM 7.0892 7.1575OTM 4.1643 4.1901
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, σ0 = 20%, α = 0.29, ρ = −0.1, Ft = E
(1± 10%
√σ2
0T)
T = 1/12G.A. Hagan
ATM 2.2855 2.2983INM 2.9702 2.9389OTM 1.7152 1.6764
T = 3/12G.A. Hagan
ATM 3.9241 3.9654INM 5.0839 5.0795OTM 2.9615 2.9351
T = 6/12G.A. Hagan
ATM 5.4885 5.5684INM 7.0892 7.1575OTM 4.1643 4.1901
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experimentsr = 3%, σ0 = 20%, α = 0.29, ρ = −0.1, Ft = E
(1± 10%
√σ2
0T)
T = 1/12G.A. Hagan
ATM 2.2855 2.2983INM 2.9702 2.9389OTM 1.7152 1.6764
T = 3/12G.A. Hagan
ATM 3.9241 3.9654INM 5.0839 5.0795OTM 2.9615 2.9351
T = 6/12G.A. Hagan
ATM 5.4885 5.5684INM 7.0892 7.1575OTM 4.1643 4.1901
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
In the money: Ft = E“
1 + 10%qσ2
0T”
, r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1
1 3 6 9 122
3
4
5
6
7
8
9
10
Maturity date
Europ
ean C
all op
tion pr
ice
Geometrical Approximation method and SABR model
ans(Geometrical Approximation method )ans(Hagan method)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
At the money: Ft = E , r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1
1 3 6 9 122
3
4
5
6
7
8
Maturity date
Europ
ean C
all op
tion pr
ice
Geometrical Approximation method and SABR model
ans(Geometrical Approximation method)ans(Hagan method)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Out the money: Ft = E“
1− 10%qσ2
0T”
, r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1
1 3 6 9 121.5
2
2.5
3
3.5
4
4.5
5
5.5
6Geometrical Approximation method and SABR model
Maturity date
Europ
ean C
all op
tion pr
ie
ans(Geometrical Approximation method)ans(Hagan method)
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Perturbative Method: Heston model with zero drift
In this case we have discussed a particular choice of the volatilityprice of risk in the Heston model, namely such that the drift term ofthe risk-neutral stochastic volatility process is zero:
dSt = rStdt +√νtStdW (1)
t ,
dνt = α√νtdW (2)
t , α ∈ R+
dW (1)t dW (2)
t = ρdt , ρ ∈ (−1,+1)
dBt = rBtdt .
f (T ,S, ν) = Φ(ST )
under a risk-neutral martingale measure Q.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Perturbative Method: Heston model with zero drift
In this case we have discussed a particular choice of the volatilityprice of risk in the Heston model, namely such that the drift term ofthe risk-neutral stochastic volatility process is zero:
dSt = rStdt +√νtStdW (1)
t ,
dνt = α√νtdW (2)
t , α ∈ R+
dW (1)t dW (2)
t = ρdt , ρ ∈ (−1,+1)
dBt = rBtdt .
f (T ,S, ν) = Φ(ST )
under a risk-neutral martingale measure Q.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
From Ito’s lemma we have:
∂f∂t
+12ν
(S2 ∂
2f∂S2 + 2ραS
∂2f∂S∂ν
+ α2 ∂2f
∂ν2
)+ rS
∂f∂S− rf = 0
After three coordinate transformations we have:
∂f3∂τ− (1− ρ2)
(∂2f3∂γ2 +
∂2f3∂δ2 + 2φ
∂2f3∂δ∂τ
+ φ2 ∂2f2∂τ2
)+ r
∂f3∂γ
= 0
where φ = α(T−t)
2√
1−ρ2.
Since α ∼ 10−1 , for maturity date lesser than 1-year the term(T − t) ∼ 10−1 and (2
√1− ρ2)−1 ∼ 10−1; thus φ ∼ 10−3, φ2 ∼ 10−6.
Thus it is reasonable to approximate φ ' 0.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
From Ito’s lemma we have:
∂f∂t
+12ν
(S2 ∂
2f∂S2 + 2ραS
∂2f∂S∂ν
+ α2 ∂2f
∂ν2
)+ rS
∂f∂S− rf = 0
After three coordinate transformations we have:
∂f3∂τ− (1− ρ2)
(∂2f3∂γ2 +
∂2f3∂δ2 + 2φ
∂2f3∂δ∂τ
+ φ2 ∂2f2∂τ2
)+ r
∂f3∂γ
= 0
where φ = α(T−t)
2√
1−ρ2.
Since α ∼ 10−1 , for maturity date lesser than 1-year the term(T − t) ∼ 10−1 and (2
√1− ρ2)−1 ∼ 10−1; thus φ ∼ 10−3, φ2 ∼ 10−6.
Thus it is reasonable to approximate φ ' 0.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
From Ito’s lemma we have:
∂f∂t
+12ν
(S2 ∂
2f∂S2 + 2ραS
∂2f∂S∂ν
+ α2 ∂2f
∂ν2
)+ rS
∂f∂S− rf = 0
After three coordinate transformations we have:
∂f3∂τ− (1− ρ2)
(∂2f3∂γ2 +
∂2f3∂δ2 + 2φ
∂2f3∂δ∂τ
+ φ2 ∂2f2∂τ2
)+ r
∂f3∂γ
= 0
where φ = α(T−t)
2√
1−ρ2.
Since α ∼ 10−1 , for maturity date lesser than 1-year the term(T − t) ∼ 10−1 and (2
√1− ρ2)−1 ∼ 10−1; thus φ ∼ 10−3, φ2 ∼ 10−6.
Thus it is reasonable to approximate φ ' 0.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
This allowed us to illustrate a methodology for solving the pricing PDEin an approximate way, in which we have imposed to be worthlesssome terms of the PDE, recovering a pricing formula which in thisparticular case, turn out to be simple, for Vanilla Options and BarrierOptions:
for European Call:
C(t,S, ν) = eν(T−t)
4(1−ρ2) S»
N“
d1, a0,1
p1− ρ2
”− e
“−2 ρ
αν”
N“
d2, a0,2
p1− ρ2
”–
− eν(T−t)
4(1−ρ2) Ee−r(T−t)hN“
d1, a0,1
p1− ρ2
”− N
“d2, a0,2
p1− ρ2
”i;
for Down-and-out Call:
CoutL (t,S, ν) = e−(bρ r(T−t))
»ecρν(T−t)N(h1)− e
− ρν
α(1−ρ2) N(h2)
–×8><>:S ∗
264N(d1)−„
LS
« 1−2ρ2
1−ρ2N(d2)
375− eν(T−t)
2(1−ρ2) E ∗"
N(d1)−„
SL
« 11−ρ2
N(d2)
#9>=>; .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
This allowed us to illustrate a methodology for solving the pricing PDEin an approximate way, in which we have imposed to be worthlesssome terms of the PDE, recovering a pricing formula which in thisparticular case, turn out to be simple, for Vanilla Options and BarrierOptions:
for European Call:
C(t,S, ν) = eν(T−t)
4(1−ρ2) S»
N“
d1, a0,1
p1− ρ2
”− e
“−2 ρ
αν”
N“
d2, a0,2
p1− ρ2
”–
− eν(T−t)
4(1−ρ2) Ee−r(T−t)hN“
d1, a0,1
p1− ρ2
”− N
“d2, a0,2
p1− ρ2
”i;
for Down-and-out Call:
CoutL (t,S, ν) = e−(bρ r(T−t))
»ecρν(T−t)N(h1)− e
− ρν
α(1−ρ2) N(h2)
–×8><>:S ∗
264N(d1)−„
LS
« 1−2ρ2
1−ρ2N(d2)
375− eν(T−t)
2(1−ρ2) E ∗"
N(d1)−„
SL
« 11−ρ2
N(d2)
#9>=>; .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
This allowed us to illustrate a methodology for solving the pricing PDEin an approximate way, in which we have imposed to be worthlesssome terms of the PDE, recovering a pricing formula which in thisparticular case, turn out to be simple, for Vanilla Options and BarrierOptions:
for European Call:
C(t,S, ν) = eν(T−t)
4(1−ρ2) S»
N“
d1, a0,1
p1− ρ2
”− e
“−2 ρ
αν”
N“
d2, a0,2
p1− ρ2
”–
− eν(T−t)
4(1−ρ2) Ee−r(T−t)hN“
d1, a0,1
p1− ρ2
”− N
“d2, a0,2
p1− ρ2
”i;
for Down-and-out Call:
CoutL (t,S, ν) = e−(bρ r(T−t))
»ecρν(T−t)N(h1)− e
− ρν
α(1−ρ2) N(h2)
–×8><>:S ∗
264N(d1)−„
LS
« 1−2ρ2
1−ρ2N(d2)
375− eν(T−t)
2(1−ρ2) E ∗"
N(d1)−„
SL
« 11−ρ2
N(d2)
#9>=>; .
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a European Call option
r = 3%, ν0 = 0.04, α = 0.1, ρ = −0.64, E = 100,St = E
(1± 10%
√ΘT)
T = 1/12Approximation method Fourier
ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410
T = 3/12Approximation method Fourier
ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499
T = 6/12Approximation method Fourier
ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a European Call option
r = 3%, ν0 = 0.04, α = 0.1, ρ = −0.64, E = 100,St = E
(1± 10%
√ΘT)
T = 1/12Approximation method Fourier
ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410
T = 3/12Approximation method Fourier
ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499
T = 6/12Approximation method Fourier
ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a European Call option
r = 3%, ν0 = 0.04, α = 0.1, ρ = −0.64, E = 100,St = E
(1± 10%
√ΘT)
T = 1/12Approximation method Fourier
ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410
T = 3/12Approximation method Fourier
ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499
T = 6/12Approximation method Fourier
ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a Down-and-out Call option
L = 70, E = 100, St = E(
1± 10%√
ΘT)
T = 1/12down-and-out Call Vanilla Call
ATM 1.77384 2.4305INM 2.0727 2.7337OTM 1.5048 2.1503
T = 3/12down-and-out Call Vanilla Call
ATM 3.0715 4.3755INM 3.5822 4.9037OTM 2.6123 3.8871
T = 6/12down-knock-out Call Vanilla Call
ATM 4.3145 6.3790INM 5.0229 7.1214OTM 3.6785 5.6925
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a Down-and-out Call option
L = 70, E = 100, St = E(
1± 10%√
ΘT)
T = 1/12down-and-out Call Vanilla Call
ATM 1.77384 2.4305INM 2.0727 2.7337OTM 1.5048 2.1503
T = 3/12down-and-out Call Vanilla Call
ATM 3.0715 4.3755INM 3.5822 4.9037OTM 2.6123 3.8871
T = 6/12down-knock-out Call Vanilla Call
ATM 4.3145 6.3790INM 5.0229 7.1214OTM 3.6785 5.6925
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a Down-and-out Call option
L = 70, E = 100, St = E(
1± 10%√
ΘT)
T = 1/12down-and-out Call Vanilla Call
ATM 1.77384 2.4305INM 2.0727 2.7337OTM 1.5048 2.1503
T = 3/12down-and-out Call Vanilla Call
ATM 3.0715 4.3755INM 3.5822 4.9037OTM 2.6123 3.8871
T = 6/12down-knock-out Call Vanilla Call
ATM 4.3145 6.3790INM 5.0229 7.1214OTM 3.6785 5.6925
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a Down-and-out Call option
L = 80, E = 100, St = E(
1± 10%√
ΘT)
(T = 6/12)Volatility Perturbative method Fourier method
20% 4.3361 4.3196ATM 30% 6.4678 6.4593
40% 8.2098 8.448020% 5.1092 4.9654
INM 30% 7.6807 7.678540% 9.9626 9.984720% 3.6172 3.4234
OTM 30% 5.7154 5.720940% 6.5834 6.5061
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a Down-and-out Call option
L = 80, E = 100, St = E(
1± 10%√
ΘT)
(T = 6/12)Volatility Perturbative method Fourier method
20% 4.3361 4.3196ATM 30% 6.4678 6.4593
40% 8.2098 8.448020% 5.1092 4.9654
INM 30% 7.6807 7.678540% 9.9626 9.984720% 3.6172 3.4234
OTM 30% 5.7154 5.720940% 6.5834 6.5061
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Numerical Experiments: for a Down-and-out Call option
L = 80, E = 100, St = E(
1± 10%√
ΘT)
(T = 6/12)Volatility Perturbative method Fourier method
20% 4.3361 4.3196ATM 30% 6.4678 6.4593
40% 8.2098 8.448020% 5.1092 4.9654
INM 30% 7.6807 7.678540% 9.9626 9.984720% 3.6172 3.4234
OTM 30% 5.7154 5.720940% 6.5834 6.5061
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Theoretical ErrorThe theoretical error in Perturbative method can be evaluated bycomputing the terms that we have before neglected
Err =(
2φ ∂2
∂δ∂τ + φ2 ∂2
∂τ2
)f (t ,S, ν),
where φ = α(T−t)
2√
1−ρ2, for which the error is around 1% for maturity
lesser than 1-year.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Theoretical ErrorThe theoretical error in Perturbative method can be evaluated bycomputing the terms that we have before neglected
Err =(
2φ ∂2
∂δ∂τ + φ2 ∂2
∂τ2
)f (t ,S, ν),
where φ = α(T−t)
2√
1−ρ2, for which the error is around 1% for maturity
lesser than 1-year.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Theoretical ErrorThe theoretical error in Perturbative method can be evaluated bycomputing the terms that we have before neglected
Err =(
2φ ∂2
∂δ∂τ + φ2 ∂2
∂τ2
)f (t ,S, ν),
where φ = α(T−t)
2√
1−ρ2, for which the error is around 1% for maturity
lesser than 1-year.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
ConclusionsThe G.A. and Perturbative method intend to be two alternativemethods for pricing options in stochastic volatility market models. Inthe first case our idea is to approximate the exact solution obtainedusing a different Cauchy’s condition, rather than searching anumerical solution to the PDE with the exact Cauchy’s condition, andin the second case we offer an analytical solution by perturbativeexpansion of PDE.AdvantageThe proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we do not have the problemswhich plague the numerical methods.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
ConclusionsThe G.A. and Perturbative method intend to be two alternativemethods for pricing options in stochastic volatility market models. Inthe first case our idea is to approximate the exact solution obtainedusing a different Cauchy’s condition, rather than searching anumerical solution to the PDE with the exact Cauchy’s condition, andin the second case we offer an analytical solution by perturbativeexpansion of PDE.AdvantageThe proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we do not have the problemswhich plague the numerical methods.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
ConclusionsThe G.A. and Perturbative method intend to be two alternativemethods for pricing options in stochastic volatility market models. Inthe first case our idea is to approximate the exact solution obtainedusing a different Cauchy’s condition, rather than searching anumerical solution to the PDE with the exact Cauchy’s condition, andin the second case we offer an analytical solution by perturbativeexpansion of PDE.AdvantageThe proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we do not have the problemswhich plague the numerical methods.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Publications: International Review
(1) Dell’Era, M. (2010): “Geometrical Approximation method andStochastic Volatility market models”, International review of appliedFinancial issues and Economics, Volume 2, Issue 3, IRAFIE ISSN:9210-1737.
(2) Dell’Era, M. (2011): “Vanilla Option pricing in Stochastic Volatilitymarket models”, International review of applied Financial issues ofEconomics, IRAFIE ISSN: 9210-1737, in press.
(3) Dell’Era, M. (2011): “Perturbative method: Barrier Option Pricing inStochastic Volatility market models”, submitted to Internationalreview of Finance, June 1, 2011.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
Quantitative Finance: stochastic volatility market models
Publications: National Review
(1) Valutazione di Derivati in un Modello a Volatilita Stocastica, AIAFjournal, ISSN: 1128-3475 published, volume 3, March 2010.
(2) Modello di Mercato SABR/LIBOR, AIAF journal, ISSN:1128-3475,published January 2011.
Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance
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