Workshop 2011 of Quantitative Finance

1. Quantitative Finance: stochastic volatility market models Supervisors: Roberto Ren´o, Claudio Pacati Geometrical Approximation and Perturbative method 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

2. Quantitative Finance: stochastic volatility market models Stochastic Volatility Market Models dSt = rSt dt + a2(σt , St )d ˜W (1) t dσt = b1(σt )dt + b2(σt )d ˜W (2) t dBt = rBt dt f(T, ST ) = φ(ST ) under a risk-neutral martingale measure Q. Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

3. Quantitative Finance: stochastic volatility market models Heston Model dSt = rSt dt + √ νt St d ˜W (1) t S ∈ [0, +∞) dνt = K(Θ − νt )dt + α √ νt d ˜W (2) t ν ∈ (0, +∞) under a risk-neutral martingale measure Q. From Itˆo’s lemma we have the following PDE: ∂f ∂t + 1 2 νS2 ∂2 f ∂S2 +ρναS ∂2 f ∂S∂ν + 1 2 να2 ∂2 f ∂ν2 +κ(Θ−ν) ∂f ∂ν +rS ∂f ∂S −rf = 0 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

4. Quantitative Finance: stochastic volatility market models Heston Model dSt = rSt dt + √ νt St d ˜W (1) t S ∈ [0, +∞) dνt = K(Θ − νt )dt + α √ νt d ˜W (2) t ν ∈ (0, +∞) under a risk-neutral martingale measure Q. From Itˆo’s lemma we have the following PDE: ∂f ∂t + 1 2 νS2 ∂2 f ∂S2 +ρναS ∂2 f ∂S∂ν + 1 2 να2 ∂2 f ∂ν2 +κ(Θ−ν) ∂f ∂ν +rS ∂f ∂S −rf = 0 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

5. 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

12. Quantitative Finance: stochastic volatility market models Geometrical Approximation method: Heston model The proposed technique is based on a stochastic approximation of the 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 variance process. Deﬁne stochastic error: eεT = e ρ{[(ν0−Θ)e−κ(T)+Θ]−νT } α . Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

15. 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 1.025 0 5 10 15 20 25 Geometrical Approximation method and Heston model Numberofevents Stochastic Error Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

16. Quantitative Finance: stochastic volatility market models Vanilla Options we consider for a Call: (ST eεT − E) + , instead of (ST − E) + ; and for a Put:(E − ST eεT ) + instead of (E − ST ) + . The Call option price is give by: Cρ,α,Θ,κ(t, St , νt ) = (St eεt ) eδρ 1 N(˜dρ 1 ) − Eeδρ 2 N(˜dρ 2 ); and for a Put: Pρ,α,Θ,κ(t, St , νt ) = Eeδρ 2 N(−˜dρ 2 ) − (St eεt ) eδρ 1 N(−˜dρ 1 ). Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

19. Quantitative Finance: stochastic volatility market models Numerical Experiments r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.64, St = E 1 ± 10% √ ΘT T = 6/12 G.A. Fourier F.D.M. M.C ATM 5.3034 5.5707 5.4806 5.4925 INM 6.1828 6.4481 6.4158 6.4250 OTM 4.5057 4.7549 4.7347 4.7502 T = 9/12 G.A. Fourier F.D.M. M.C ATM 6.8930 7.0500 6.9430 6.9628 INM 7.9923 8.1346 8.0769 8.928 OTM 5.8918 6.0392 6.0156 6.381 T = 1 G.A. Fourier F.D.M. M.C ATM 8.3329 8.3816 8.2619 8.2887 INM 9.6192 9.6351 9.5843 9.6030 OTM 7.1577 7.2112 7.1562 7.1357 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

22. Quantitative Finance: stochastic volatility market models Numerical Experiments r = 3%, Θ = 0.04, κ = 1.15, α = 0.39, ρ = −0.28, St = E 1 ± 10% √ ΘT T = 6/12 G.A. Fourier F.D.M. M.C. ATM 5.9827 5.9957 5.9972 5.9975 INM 6.7329 6.8154 6.7964 6.7954 OTM 5.2918 5.2646 5.2597 5.2618 T = 9/12 G.A. Fourier F.D.M. M.C. ATM 7.5188 7.4963 7.5040 7.4966 INM 8.5418 8.5108 8.4832 8.4736 OTM 6.6719 6.5941 6.5994 6.5948 T = 1 G.A. Fourier F.D.M. M.C. ATM 8.9847 8.8488 8.8614 8.8258 INM 9.9273 10.0035 10.0177 9.9790 OTM 7.8896 7.7832 7.7936 7.7617 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

25. 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 12 2 3 4 5 6 7 8 9 10 11 Approximation method in the Heston with drift zero Maturity date EuropeanCalloptionprice ans(Approximation method) ans(Fourier transform) Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

26. 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 12 2 3 4 5 6 7 8 9 10 Maturity date EuropeanCalloptionprice 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

27. 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 12 2 3 4 5 6 7 8 9 Approximation method in the Heston with drift zero Maturity date EuropeanCalloptionprice ans(Approximation method) ans(Fourier transform) Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

28. Quantitative Finance: stochastic volatility market models SABR Model In the SABR model one supposes that under a martingale measure Q the forward price follows the SDEs: dFt = σt Fβ t dW (1) t β ∈ (0, 1] dσt = ασt dW (2) t α ∈ R dBt = rBt dt. For Itˆo’s lemma, in the case β = 1, we have: ∂f ∂t + 1 2 (σ)2 F2 ∂2 f ∂F2 + 2ρFα ∂2 f ∂F∂σ + α2 ∂2 f ∂σ2 − rf = 0 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

31. Quantitative Finance: stochastic volatility market models Implied Volatility method: Hagan (2002) Hagan et al. (2002) derive and study the approximate formulas for the implied Black and Bachelier volatilities in the SABR model, which can be represented as follows: ˆσ(E, T) = σ0 (F0/E)(1−β)/2 “ 1 + (1−β)2 24 ln2 (F0/E) + (1−β)4 1920 ln4 (F0/E) + ..... ” × z χ(z) ( 1 + " (1 − β)2 σ2 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 the time 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 ( p 1 − 2ρz + z2 + z − ρ 1 − ρ ) . Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

34. 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 variance process. Deﬁne stochastic error: eεT = e ρ 2 6 4σ0e „ α2 2 T « −σT 3 7 5 α . Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

37. 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 1.25 0 5 10 15 20 25 Geometrical Approximation method and SABR model Stochastic Error Numberofevents Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

38. Quantitative Finance: stochastic volatility market models Vanilla Options We consider for a Call: (FT eεT − E) + , instead of (FT − E) + ; and for a Put:(E − FT eεT ) + instead of (E − FT ) + . The Call option price is give by: C(t, Ft , σt ) = (Ft eεt ) eδρ 1 N(˜dρ 1 ) − Eeδρ 2 N(˜dρ 2 ); and for a Put: P(t, Ft , σt ) = Eeδρ 2 N(−˜dρ 1 ) − (Ft eεt ) eδρ 1 N(−˜dρ 1 ). Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

42. Quantitative Finance: stochastic volatility market models Numerical Experiments r = 3%, σ0 = 20%, α = 0.29, ρ = −0.71, Ft = E 1 ± 10% σ2 0T T = 1/12 G.A. Hagan ATM 2.3426 2.2956 INM 3.0008 2.9492 OTM 1.7655 1.6605 T = 3/12 G.A. Hagan ATM 3.9097 3.9495 INM 5.0110 5.1039 OTM 2.9481 2.8821 T = 6/12 G.A. Hagan ATM 5.3064 5.5295 INM 6.8070 7.1942 OTM 4.0023 4.0742 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

48. Quantitative Finance: stochastic volatility market models In the money: Ft = E “ 1 + 10% q σ2 0 T ” , r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1 1 3 6 9 12 2 3 4 5 6 7 8 9 10 Maturity date EuropeanCalloptionprice 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

49. 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 12 2 3 4 5 6 7 8 Maturity date EuropeanCalloptionprice 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

50. Quantitative Finance: stochastic volatility market models Out the money: Ft = E “ 1 − 10% q σ2 0 T ” , r = 3%, σ0 = 20%, α = 0.29, ρ = −0.1 1 3 6 9 12 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Geometrical Approximation method and SABR model Maturity date EuropeanCalloptionprie ans(Geometrical Approximation method) ans(Hagan method) Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

51. Quantitative Finance: stochastic volatility market models Perturbative Method: Heston model with zero drift In this case we have discussed a particular choice of the volatility price of risk in the Heston model, namely such that the drift term of the risk-neutral stochastic volatility process is zero: dSt = rSt dt + √ νt St d ˜W (1) t , dνt = α √ νt d ˜W (2) t , α ∈ R+ d ˜W (1) t d ˜W (2) t = ρdt, ρ ∈ (−1, +1) dBt = rBt dt. 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

52. Quantitative Finance: stochastic volatility market models Perturbative Method: Heston model with zero drift In this case we have discussed a particular choice of the volatility price of risk in the Heston model, namely such that the drift term of the risk-neutral stochastic volatility process is zero: dSt = rSt dt + √ νt St d ˜W (1) t , dνt = α √ νt d ˜W (2) t , α ∈ R+ d ˜W (1) t d ˜W (2) t = ρdt, ρ ∈ (−1, +1) dBt = rBt dt. 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

53. Quantitative Finance: stochastic volatility market models From Itˇo’s lemma we have: ∂f ∂t + 1 2 ν S2 ∂2 f ∂S2 + 2ραS ∂2 f ∂S∂ν + α2 ∂2f ∂ν2 + rS ∂f ∂S − rf = 0 After three coordinate transformations we have: ∂f3 ∂τ − (1 − ρ2 ) ∂2 f3 ∂γ2 + ∂2 f3 ∂δ2 + 2φ ∂2 f3 ∂δ∂τ + φ2 ∂2 f2 ∂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

56. Quantitative Finance: stochastic volatility market models This allowed us to illustrate a methodology for solving the pricing PDE in an approximate way, in which we have imposed to be worthless some terms of the PDE, recovering a pricing formula which in this particular case, turn out to be simple, for Vanilla Options and Barrier Options: for European Call: C(t, S, ν) = e ν(T−t) 4(1−ρ2) S » N “ d1, a0,1 p 1 − ρ2 ” − e “ −2 ρ α ν ” N “ d2, a0,2 p 1 − ρ2 ”– − e ν(T−t) 4(1−ρ2) Ee −r(T−t) h N “ ˜d1, ˜a0,1 p 1 − ρ2 ” − N “ ˜d2, ˜a0,2 p 1 − ρ2 ”i ; for Down-and-out Call: C out L (t, S, ν) = e −(bρr(T−t)) » e cρν(T−t) N(h1) − e − ρν α(1−ρ2) N(h2) – × 8 >< >: S ∗ 2 6 4N(d1) − „ L S « 1−2ρ2 1−ρ2 N(d2) 3 7 5 − e ν(T−t) 2(1−ρ2) E ∗ " N(˜d1) − „ S L « 1 1−ρ2 N(˜d2) # 9 >= >; . Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

59. 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 = 1/12 Approximation method Fourier ATM 2.4305 2.4261 INM 2.7337 2.7341 OTM 2.1503 2.1410 T = 3/12 Approximation method Fourier ATM 4.3755 4.3524 INM 4.9037 4.8942 OTM 3.8871 3.8499 T = 6/12 Approximation method Fourier ATM 6.3790 6.3765 INM 7.1214 7.1322 OTM 5.6925 5.6358 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

62. 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 = 1/12 down-and-out Call Vanilla Call ATM 1.77384 2.4305 INM 2.0727 2.7337 OTM 1.5048 2.1503 T = 3/12 down-and-out Call Vanilla Call ATM 3.0715 4.3755 INM 3.5822 4.9037 OTM 2.6123 3.8871 T = 6/12 down-knock-out Call Vanilla Call ATM 4.3145 6.3790 INM 5.0229 7.1214 OTM 3.6785 5.6925 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

65. 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 4.3196 ATM 30% 6.4678 6.4593 40% 8.2098 8.4480 20% 5.1092 4.9654 INM 30% 7.6807 7.6785 40% 9.9626 9.9847 20% 3.6172 3.4234 OTM 30% 5.7154 5.7209 40% 6.5834 6.5061 Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

68. Quantitative Finance: stochastic volatility market models Theoretical Error The theoretical error in Perturbative method can be evaluated by computing 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

71. Quantitative Finance: stochastic volatility market models Conclusions The G.A. and Perturbative method intend to be two alternative methods for pricing options in stochastic volatility market models. In the ﬁrst case our idea is to approximate the exact solution obtained using a different Cauchy’s condition, rather than searching a numerical solution to the PDE with the exact Cauchy’s condition, and in the second case we offer an analytical solution by perturbative expansion of PDE. Advantage The proposed method has the advantage to compute a solution and the greeks in closed form, therefore, we do not have the problems which plague the numerical methods. Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

74. Quantitative Finance: stochastic volatility market models Publications: International Review (1) Dell’Era, M. (2010): “Geometrical Approximation method and Stochastic Volatility market models”, International review of applied Financial issues and Economics, Volume 2, Issue 3, IRAFIE ISSN: 9210-1737. (2) Dell’Era, M. (2011): “Vanilla Option pricing in Stochastic Volatility market models”, International review of applied Financial issues of Economics, IRAFIE ISSN: 9210-1737, in press. (3) Dell’Era, M. (2011): “Perturbative method: Barrier Option Pricing in Stochastic Volatility market models”, submitted to International review of Finance, June 1, 2011. Mario Dell’Era Geometrical Approximation and Perturbative method for PDEs in Finance

75. Quantitative Finance: stochastic volatility market models Publications: National Review (1) Valutazione di Derivati in un Modello a Volatilit´a Stocastica, AIAF journal, 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

Workshop 2011 of Quantitative Finance

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