Workshop 2011 of Quantitative Finance

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


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.



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



Heston Model


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



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)







(2007)








(2007)








(2007)








(2007)








(2007)








(2007)




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 .









α .









α .



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

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25Geometrical Approximation method and Heston model

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Stochastic Error



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 ).



Vanilla Options




ρ2 N(dρ2 );

and for a Put:


ρ1 N(−dρ1 ).



Vanilla Options




ρ2 N(dρ2 );

and for a Put:


ρ1 N(−dρ1 ).



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


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




(1± 10%

√ΘT)


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


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


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




(1± 10%

√ΘT)


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


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


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




(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


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




(1± 10%

√ΘT)


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


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


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




(1± 10%

√ΘT)


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


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


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



In the money: St = E“

1 + 10%√

ΘT”

, r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.1

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11Approximation method in the Heston with drift zero

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ans(Approximation method)ans(Fourier transform)



At the money: St = E , r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.1

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Approximation method in the Heston with drift zero




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

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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



SABR Model



t β ∈ (0,1]


dBt = rBtdt .


∂f∂t

+12

(σ)2(

F 2 ∂2f

∂F 2 + 2ρFα∂2f∂F∂σ

+ α2 ∂2f

∂σ2

)− rf = 0



SABR Model



t β ∈ (0,1]


dBt = rBtdt .


∂f∂t

+12

(σ)2(

F 2 ∂2f

∂F 2 + 2ρFα∂2f∂F∂σ

+ α2 ∂2f

∂σ2

)− rf = 0



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

).





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 + .......

),


z =α

σ0(F0/E)(1−β)/2 ln(F0/E), χ(z) = ln

(p1− 2ρz + z2 + z − ρ

1− ρ

).





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 + .......

),


z =α

σ0(F0/E)(1−β)/2 ln(F0/E), χ(z) = ln

(p1− 2ρz + z2 + z − ρ

1− ρ

).



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 .








eεT = e

ρ

264σ0e

„α22 T

«−σT

375α .








eεT = e

ρ

264σ0e

„α22 T

«−σT

375α .



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

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25Geometrical Approximation method and SABR model

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Vanilla Options

We consider for a Call: (FT eεT − E)+, instead of (FT − E)+; and for aPut:(E − FT eεT )+ instead of (E − FT )+.


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 ).



Vanilla Options




ρ2 N(dρ2 );

and for a Put:


ρ1 N(−dρ1 ).



Vanilla Options




ρ2 N(dρ2 );

and for a Put:


ρ1 N(−dρ1 ).



Vanilla Options




ρ2 N(dρ2 );

and for a Put:


ρ1 N(−dρ1 ).



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




(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




(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




(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




(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




(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



In the money: Ft = E“

1 + 10%qσ2

0T”

, r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1

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Geometrical Approximation method and SABR model

ans(Geometrical Approximation method )ans(Hagan method)



At the money: Ft = E , r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1

1 3 6 9 122

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Geometrical Approximation method and SABR model

ans(Geometrical Approximation method)ans(Hagan method)



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

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ans(Geometrical Approximation method)ans(Hagan method)



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:


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 )




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:


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 )




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.




∂f∂t

+12ν

(S2 ∂

2f∂S2 + 2ραS

∂2f∂S∂ν

+ α2 ∂2f

∂ν2

)+ rS

∂f∂S− rf = 0


∂f3∂τ− (1− ρ2)

(∂2f3∂γ2 +

∂2f3∂δ2 + 2φ

∂2f3∂δ∂τ

+ φ2 ∂2f2∂τ2

)+ r

∂f3∂γ

= 0


2√

1−ρ2.


√1− ρ2)−1 ∼ 10−1; thus φ ∼ 10−3, φ2 ∼ 10−6.





∂f∂t

+12ν

(S2 ∂

2f∂S2 + 2ραS

∂2f∂S∂ν

+ α2 ∂2f

∂ν2

)+ rS

∂f∂S− rf = 0


∂f3∂τ− (1− ρ2)

(∂2f3∂γ2 +

∂2f3∂δ2 + 2φ

∂2f3∂δ∂τ

+ φ2 ∂2f2∂τ2

)+ r

∂f3∂γ

= 0


2√

1−ρ2.


√1− ρ2)−1 ∼ 10−1; thus φ ∼ 10−3, φ2 ∼ 10−6.




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>=>; .




for European Call:


4(1−ρ2) S»

N“

d1, a0,1

p1− ρ2

”− e

“−2 ρ

αν”

N“

d2, a0,2

p1− ρ2

”–

− eν(T−t)


d1, a0,1

p1− ρ2

”− N

“d2, a0,2

p1− ρ2

”i;




− 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>=>; .




for European Call:


4(1−ρ2) S»

N“

d1, a0,1

p1− ρ2

”− e

“−2 ρ

αν”

N“

d2, a0,2

p1− ρ2

”–

− eν(T−t)


d1, a0,1

p1− ρ2

”− N

“d2, a0,2

p1− ρ2

”i;




− 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>=>; .



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


ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499


ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358




r = 3%, ν0 = 0.04, α = 0.1, ρ = −0.64, E = 100,St = E

(1± 10%

√ΘT)


ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410


ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499


ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358




r = 3%, ν0 = 0.04, α = 0.1, ρ = −0.64, E = 100,St = E

(1± 10%

√ΘT)


ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410


ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499


ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358



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


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




L = 70, E = 100, St = E(

1± 10%√

ΘT)


ATM 1.77384 2.4305INM 2.0727 2.7337OTM 1.5048 2.1503


ATM 3.0715 4.3755INM 3.5822 4.9037OTM 2.6123 3.8871


ATM 4.3145 6.3790INM 5.0229 7.1214OTM 3.6785 5.6925




L = 70, E = 100, St = E(

1± 10%√

ΘT)


ATM 1.77384 2.4305INM 2.0727 2.7337OTM 1.5048 2.1503


ATM 3.0715 4.3755INM 3.5822 4.9037OTM 2.6123 3.8871


ATM 4.3145 6.3790INM 5.0229 7.1214OTM 3.6785 5.6925




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




L = 80, E = 100, St = E(

1± 10%√

ΘT)


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




L = 80, E = 100, St = E(

1± 10%√

ΘT)


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



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),


2√

1−ρ2, for which the error is around 1% for maturity

lesser than 1-year.




Err =(

2φ ∂2

∂δ∂τ + φ2 ∂2

∂τ2

)f (t ,S, ν),


2√


lesser than 1-year.




Err =(

2φ ∂2

∂δ∂τ + φ2 ∂2

∂τ2

)f (t ,S, ν),


2√


lesser than 1-year.



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.









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.



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.


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