Download - Workshop 2012 of Quantitative Finance
Quantitative Finance: stochastic volatility market models
Vanilla Option Pricing in Stochastic Volatilitymarket models
XIII WorkShop of Quantitative Finance
Mario Dell’Era
Scuola Superiore Sant’Anna
January 21, 2013
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
Stochastic Volatility Market Models
dSt = rStdt + a(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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 = 1/12Perturbative method Fourier method for κ = 0
ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410
T = 3/12Perturbative method Fourier method for κ = 0
ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499
T = 6/12Perturbative method Fourier method for κ = 0
ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
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 = 1/12Perturbative method Fourier method for κ = 0
ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410
T = 3/12Perturbative method Fourier method for κ = 0
ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499
T = 6/12Perturbative method Fourier method for κ = 0
ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
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 = 1/12Perturbative method Fourier method for κ = 0
ATM 2.4305 2.4261INM 2.7337 2.7341OTM 2.1503 2.1410
T = 3/12Perturbative method Fourier method for κ = 0
ATM 4.3755 4.3524INM 4.9037 4.8942OTM 3.8871 3.8499
T = 6/12Perturbative method Fourier method for κ = 0
ATM 6.3790 6.3765INM 7.1214 7.1322OTM 5.6925 5.6358
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 Vanilla Option Pricing in Stochastic Volatility market models
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 for κ = 0
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 Vanilla Option Pricing in Stochastic Volatility market models
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 for κ = 0
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 Vanilla Option Pricing in Stochastic Volatility market models
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 for κ = 0
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 Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
Pricing ErrorIn order to estimate the error of Perturbative method, we use anempirical idea. We can evaluate the magnitude of neglected terms,and we put in relation the magnitude of φ with the pricing error thatwe have obtained numerically:
PricingError = F((
2φ∂2
∂δ∂τ+ φ2 ∂
2
∂τ2
)f (t ,S, ν)
),
where φ = α(T−t)
2√
1−ρ2.
Following this approach we are able to conclude that for values ofφ ∼ 10−3, the price error is around 1% for maturity lesser than 1-year.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
Pricing ErrorIn order to estimate the error of Perturbative method, we use anempirical idea. We can evaluate the magnitude of neglected terms,and we put in relation the magnitude of φ with the pricing error thatwe have obtained numerically:
PricingError = F((
2φ∂2
∂δ∂τ+ φ2 ∂
2
∂τ2
)f (t ,S, ν)
),
where φ = α(T−t)
2√
1−ρ2.
Following this approach we are able to conclude that for values ofφ ∼ 10−3, the price error is around 1% for maturity lesser than 1-year.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
Pricing ErrorIn order to estimate the error of Perturbative method, we use anempirical idea. We can evaluate the magnitude of neglected terms,and we put in relation the magnitude of φ with the pricing error thatwe have obtained numerically:
PricingError = F((
2φ∂2
∂δ∂τ+ φ2 ∂
2
∂τ2
)f (t ,S, ν)
),
where φ = α(T−t)
2√
1−ρ2.
Following this approach we are able to conclude that for values ofφ ∼ 10−3, the price error is around 1% for maturity lesser than 1-year.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
Pricing ErrorIn order to estimate the error of Perturbative method, we use anempirical idea. We can evaluate the magnitude of neglected terms,and we put in relation the magnitude of φ with the pricing error thatwe have obtained numerically:
PricingError = F((
2φ∂2
∂δ∂τ+ φ2 ∂
2
∂τ2
)f (t ,S, ν)
),
where φ = α(T−t)
2√
1−ρ2.
Following this approach we are able to conclude that for values ofφ ∼ 10−3, the price error is around 1% for maturity lesser than 1-year.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
Pricing ErrorIn order to estimate the error of Perturbative method, we use anempirical idea. We can evaluate the magnitude of neglected terms,and we put in relation the magnitude of φ with the pricing error thatwe have obtained numerically:
PricingError = F((
2φ∂2
∂δ∂τ+ φ2 ∂
2
∂τ2
)f (t ,S, ν)
),
where φ = α(T−t)
2√
1−ρ2.
Following this approach we are able to conclude that for values ofφ ∼ 10−3, the price error is around 1% for maturity lesser than 1-year.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
ConclusionsPerturbative method intends to be an alternative method for pricingoptions in stochastic volatility market models. We offer an analytical
solution by perturbative expansion in φ,(φ = α(T−t)
2√
1−ρ2
)of Heston’s
PDE.
The proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we have not the problems whichplague the numerical methods. Besides this technique is a generalapproach and it can be used for pricing several Derivatives.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
ConclusionsPerturbative method intends to be an alternative method for pricingoptions in stochastic volatility market models. We offer an analytical
solution by perturbative expansion in φ,(φ = α(T−t)
2√
1−ρ2
)of Heston’s
PDE.
The proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we have not the problems whichplague the numerical methods. Besides this technique is a generalapproach and it can be used for pricing several Derivatives.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
ConclusionsPerturbative method intends to be an alternative method for pricingoptions in stochastic volatility market models. We offer an analytical
solution by perturbative expansion in φ,(φ = α(T−t)
2√
1−ρ2
)of Heston’s
PDE.
The proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we have not the problems whichplague the numerical methods. Besides this technique is a generalapproach and it can be used for pricing several Derivatives.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models
Quantitative Finance: stochastic volatility market models
ConclusionsPerturbative method intends to be an alternative method for pricingoptions in stochastic volatility market models. We offer an analytical
solution by perturbative expansion in φ,(φ = α(T−t)
2√
1−ρ2
)of Heston’s
PDE.
The proposed method has the advantage to compute a solution andthe greeks in closed form, therefore, we have not the problems whichplague the numerical methods. Besides this technique is a generalapproach and it can be used for pricing several Derivatives.
Mario Dell’Era Vanilla Option Pricing in Stochastic Volatility market models