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


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.



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: 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)







(2007)









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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 = 1/12Perturbative method Fourier method for κ = 0

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 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




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



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)

),


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.




PricingError = F((

2φ∂2

∂δ∂τ+ φ2 ∂

2

∂τ2

)f (t ,S, ν)

),


2√

1−ρ2.





PricingError = F((

2φ∂2

∂δ∂τ+ φ2 ∂

2

∂τ2

)f (t ,S, ν)

),


2√

1−ρ2.





PricingError = F((

2φ∂2

∂δ∂τ+ φ2 ∂

2

∂τ2

)f (t ,S, ν)

),


2√

1−ρ2.





PricingError = F((

2φ∂2

∂δ∂τ+ φ2 ∂

2

∂τ2

)f (t ,S, ν)

),


2√

1−ρ2.




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.





2√

1−ρ2

)of Heston’s

PDE.






2√

1−ρ2

)of Heston’s

PDE.






2√

1−ρ2

)of Heston’s

PDE.



Economy & Finance

Workshop 2012 of Quantitative Finance