Workshop 2013 of Quantitative Finance

1. Quantitative Finance: stochastic volatility market models Closed Solution for Heston PDE by Geometrical Transformations XIV WorkShop of Quantitative Finance Mario Dell’Era Pisa University June 24, 2014 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

2. 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 νS 2 ∂2 f ∂S2 + ρναS ∂2 f ∂S∂ν + 1 2 να 2 ∂2 f ∂ν2 + κ(Θ − ν) ∂f ∂ν + rS ∂f ∂S − rf = 0 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

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 νS 2 ∂2 f ∂S2 + ρναS ∂2 f ∂S∂ν + 1 2 να 2 ∂2 f ∂ν2 + κ(Θ − ν) ∂f ∂ν + rS ∂f ∂S − rf = 0 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

4. 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 Closed Solution for Heston PDE by Geometrical Transformations

12. Quantitative Finance: stochastic volatility market models Coordinate Transformations technique We have elaborated a new methodology based on changing of variables which is independent of payoffs and does not need to use the inverse Fourier transform algorithm or numerical methods as Finite Difference and Monte Carlo simulations. In particular, we will compute the price of Vanilla Options, in order to validate numerically the Geometrical Transformations technique. Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

13. Quantitative Finance: stochastic volatility market models 1st Transformations: 8 >< >: x = ln S, x ∈ (−∞, +∞) ˜ν = ν/α, ˜ν ∈ [0, +∞) f(t, S, ν) = f1(t, x, ˜ν)e−r(T−t) (1) thus one has: ∂f1 ∂t + 1 2 ˜ν ∂2 f1 ∂x2 + 2ρ ∂2 f1 ∂x∂˜ν + ∂2 f1 ∂˜ν2 ! + „ r − 1 2 α˜ν « ∂f1 ∂x + κ α (θ − α˜ν) ∂f1 ∂˜ν = 0 f1(T, x, ˜ν) = Φ1(x) ρ ∈ (−1, +1), α ∈ R + x ∈ (−∞, +∞) ˜ν ∈ [0, +∞) t ∈ [0, T] Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

14. Quantitative Finance: stochastic volatility market models 1st Transformations: 8 >< >: x = ln S, x ∈ (−∞, +∞) ˜ν = ν/α, ˜ν ∈ [0, +∞) f(t, S, ν) = f1(t, x, ˜ν)e−r(T−t) (1) thus one has: ∂f1 ∂t + 1 2 ˜ν ∂2 f1 ∂x2 + 2ρ ∂2 f1 ∂x∂˜ν + ∂2 f1 ∂˜ν2 ! + „ r − 1 2 α˜ν « ∂f1 ∂x + κ α (θ − α˜ν) ∂f1 ∂˜ν = 0 f1(T, x, ˜ν) = Φ1(x) ρ ∈ (−1, +1), α ∈ R + x ∈ (−∞, +∞) ˜ν ∈ [0, +∞) t ∈ [0, T] Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

15. Quantitative Finance: stochastic volatility market models 2nd Transformations: 8 >< >: ξ = x − ρ˜ν ξ ∈ (−∞, +∞) η = −˜ν p 1 − ρ2 η ∈ (−∞, 0] f1(t, x, ˜ν) = f2(t, ξ, η) (2) Again we have: ∂f2 ∂t − αη 2 p 1 − ρ2 (1 − ρ 2 ) ∂2 f2 ∂ξ2 + ∂2 f2 ∂η2 ! + αη 2 p 1 − ρ2 „ 1 − 2κρ α « ∂f2 ∂ξ − αη 2 p 1 − ρ2 „ 2κ α p 1 − ρ2 « ∂f2 ∂η + „ r − κρθ α « ∂f2 ∂ξ − θκ α p 1 − ρ2 ∂f2 ∂η = 0 f2(T, ξ, η) = Φ2(ξ, η), ρ ∈ (−1, +1), α ∈ R + . ξ ∈ (−∞, +∞), η ∈ (−∞, 0], t ∈ [0, T]. Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

16. Quantitative Finance: stochastic volatility market models 2nd Transformations: 8 >< >: ξ = x − ρ˜ν ξ ∈ (−∞, +∞) η = −˜ν p 1 − ρ2 η ∈ (−∞, 0] f1(t, x, ˜ν) = f2(t, ξ, η) (2) Again we have: ∂f2 ∂t − αη 2 p 1 − ρ2 (1 − ρ 2 ) ∂2 f2 ∂ξ2 + ∂2 f2 ∂η2 ! + αη 2 p 1 − ρ2 „ 1 − 2κρ α « ∂f2 ∂ξ − αη 2 p 1 − ρ2 „ 2κ α p 1 − ρ2 « ∂f2 ∂η + „ r − κρθ α « ∂f2 ∂ξ − θκ α p 1 − ρ2 ∂f2 ∂η = 0 f2(T, ξ, η) = Φ2(ξ, η), ρ ∈ (−1, +1), α ∈ R + . ξ ∈ (−∞, +∞), η ∈ (−∞, 0], t ∈ [0, T]. Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

17. Quantitative Finance: stochastic volatility market models 3rd Transformations: 8 >>>< >>>: γ = ξ + ` r − κρθ α ´ (T − t) γ ∈ (−∞, +∞) φ = −η + κθ α p 1 − ρ2(T − t) φ ∈ [0, +∞) τ = 1 2 R T t νsds τ ∈ [0, +∞) f2(t, ξ, η) = f3(τ, γ, φ) which give us the following PDE: ∂f3 ∂τ = (1 − ρ 2 ) ∂2 f3 ∂γ2 + ∂2 f3 ∂φ2 ! − „ 1 − 2κρ α « ∂f3 ∂γ − „ 2κ α p 1 − ρ2 « ∂f3 ∂φ = 0 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

18. Quantitative Finance: stochastic volatility market models 3rd Transformations: 8 >>>< >>>: γ = ξ + ` r − κρθ α ´ (T − t) γ ∈ (−∞, +∞) φ = −η + κθ α p 1 − ρ2(T − t) φ ∈ [0, +∞) τ = 1 2 R T t νsds τ ∈ [0, +∞) f2(t, ξ, η) = f3(τ, γ, φ) which give us the following PDE: ∂f3 ∂τ = (1 − ρ 2 ) ∂2 f3 ∂γ2 + ∂2 f3 ∂φ2 ! − „ 1 − 2κρ α « ∂f3 ∂γ − „ 2κ α p 1 − ρ2 « ∂f3 ∂φ = 0 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

19. Quantitative Finance: stochastic volatility market models and imposing: f3(τ, γ, φ) = eaτ+bγ+cφ f4(τ, γ, φ), where 8 >>< >>: a = −(1 − ρ2 )(b2 + c2 ); b = (1− 2κρ α ) 2(1−ρ2) ; c = κ α √ 1−ρ2 ; ﬁnally one has: ∂f4 ∂τ = (1 − ρ 2 ) ∂2 f4 ∂γ2 + ∂2 f4 ∂φ2 ! f4(0, γ, φ) = Φ4(γ, φ) τ ∈ [0, +∞) φ ∈ [0, +∞) γ ∈ (−∞, +∞), Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

20. Quantitative Finance: stochastic volatility market models and imposing: f3(τ, γ, φ) = eaτ+bγ+cφ f4(τ, γ, φ), where 8 >>< >>: a = −(1 − ρ2 )(b2 + c2 ); b = (1− 2κρ α ) 2(1−ρ2) ; c = κ α √ 1−ρ2 ; ﬁnally one has: ∂f4 ∂τ = (1 − ρ 2 ) ∂2 f4 ∂γ2 + ∂2 f4 ∂φ2 ! f4(0, γ, φ) = Φ4(γ, φ) τ ∈ [0, +∞) φ ∈ [0, +∞) γ ∈ (−∞, +∞), Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

21. Quantitative Finance: stochastic volatility market models The solution is known in the literature (Andrei D. Polyanin, Handbook of Linear Partial Differential Equations, 2002, p. 188), and it can be written as integral, whose kernel G(0, γ , φ |τ, γ, δ) is a bivariate gaussian function: G(0, γ , φ |τ, γ, φ) = 1 4πτ(1 − ρ2) 2 4e − (γ −γ)2+(φ −φ)2 4τ(1−ρ2) − e − (γ −γ)2+(φ +φ)2 4τ(1−ρ2) 3 5 , therefore f4(τ, γ, φ) = Z +∞ 0 dφ Z +∞ −∞ dγ f4(0, γ , φ )G(0, γ , φ |τ, γ, φ) + (1 − ρ 2 ) Z τ 0 du Z +∞ −∞ dγ f4(u, γ , 0) » ∂G(0, γ , δ |τ − u, γ, δ) ∂φ – φ =0 . Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

22. Quantitative Finance: stochastic volatility market models The solution is known in the literature (Andrei D. Polyanin, Handbook of Linear Partial Differential Equations, 2002, p. 188), and it can be written as integral, whose kernel G(0, γ , φ |τ, γ, δ) is a bivariate gaussian function: G(0, γ , φ |τ, γ, φ) = 1 4πτ(1 − ρ2) 2 4e − (γ −γ)2+(φ −φ)2 4τ(1−ρ2) − e − (γ −γ)2+(φ +φ)2 4τ(1−ρ2) 3 5 , therefore f4(τ, γ, φ) = Z +∞ 0 dφ Z +∞ −∞ dγ f4(0, γ , φ )G(0, γ , φ |τ, γ, φ) + (1 − ρ 2 ) Z τ 0 du Z +∞ −∞ dγ f4(u, γ , 0) » ∂G(0, γ , δ |τ − u, γ, δ) ∂φ – φ =0 . Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

23. Quantitative Finance: stochastic volatility market models Using the natural variables we may rewrite the solution as follows: f(t, S, ν) = e −r(T−t)+aτ+bγ+cδ Z +∞ 0 dφ Z +∞ −∞ dγ f4(0, γ , φ )G(0, γ , φ |τ, γ, φ) + (1 − ρ 2 )e −r(T−t)+aτ+bγ+cφ Z τ 0 du Z +∞ −∞ dγ f4(u, γ , 0) × » ∂G(0, γ , φ |τ − u, γ, φ) ∂φ – φ =0 . Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

24. Quantitative Finance: stochastic volatility market models Vanilla Option Pricing In order to test above option pricing formula, we are going to consider as option a Vanilla Call with strike price K and maturity T. In the new variable the payoff (ST − K)+ is equal to e−bγ−cφ (eγ+ρφ/ √ 1−ρ2 − K)+ . Substituting this latter in the above equation we have: f(t, S, ν) = e −r(T−t)+aτ+bγ+cφ × Z +∞ 0 dφ Z +∞ −∞ dγ e −bγ −cφ (e γ +ρφ / √ 1−ρ2 − K) + G(0, γ , φ |τ, γ, φ) +(1−ρ 2 )e −r(T−t)+aτ+bγ+cφ Z τ 0 du Z +∞ −∞ dγ f4(u, γ , 0) » ∂G(0, γ , φ |τ − u, γ, φ) ∂φ – φ =0 , Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

25. Quantitative Finance: stochastic volatility market models Vanilla Option Pricing In order to test above option pricing formula, we are going to consider as option a Vanilla Call with strike price K and maturity T. In the new variable the payoff (ST − K)+ is equal to e−bγ−cφ (eγ+ρφ/ √ 1−ρ2 − K)+ . Substituting this latter in the above equation we have: f(t, S, ν) = e −r(T−t)+aτ+bγ+cφ × Z +∞ 0 dφ Z +∞ −∞ dγ e −bγ −cφ (e γ +ρφ / √ 1−ρ2 − K) + G(0, γ , φ |τ, γ, φ) +(1−ρ 2 )e −r(T−t)+aτ+bγ+cφ Z τ 0 du Z +∞ −∞ dγ f4(u, γ , 0) » ∂G(0, γ , φ |τ − u, γ, φ) ∂φ – φ =0 , Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

26. Quantitative Finance: stochastic volatility market models Considering the particular case, for τ which goes to zero (i.e T → 0), the solution reduces itself to: f(t, St , νt ) = St » N “ −ψ1(0), −a1,1 p 1 − ρ2 ” − e −2 “ ρ− κ α ”“ νt α + κ α θ(T−t) ” N “ −ψ2(0), −a1,2 p 1 − ρ2 ”– −Ke −r(T−t) » N “ − ˜ψ1(0), −a2,1 p 1 − ρ2 ” − e 2 κ α “ νt α + κ α θ(T−t) ” N “ − ˜ψ2(0), −a2,2 p 1 − ρ2 ”– , Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

27. Quantitative Finance: stochastic volatility market models ψ1(0) = − h νt α + κ α θ(T − t) + (ρ − κ α ) R T t νsds i qR T t νsds , ψ2(0) = h νt α + κ α θ(T − t) − (ρ − κ α ) R T t νsds i qR T t νsds , ˜ψ1(0) = − h νt α + κ α θ(T − t) − κ α R T t νsds i qR T t νsds , ˜ψ2(0) = h νt α + κ α θ(T − t) + κ α R T t νsds i qR T t νsds , Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

28. Quantitative Finance: stochastic volatility market models a1,1 = h ln(K/St ) − r(T − t) − 1 2 R T t νsds i q (1 − ρ2) R T t νsds , a1,2 = h ln(K/St ) + 2 ρ α νt − (r − 2 κθρ α )(T − t) − 1 2 R T t νsds i q (1 − ρ2) R T t νsds , a2,1 = h ln(K/St ) − r(T − t) + 1 2 R T t νsds i q (1 − ρ2) R T t νsds , a2,2 = h ln(K/St ) + 2 ρ α νt − (r − 2 κθρ α )(T − t) + 1 2 R T t νsds i q (1 − ρ2) R T t νsds . Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

29. Quantitative Finance: stochastic volatility market models Numerical Validation The approximation τ → 0 will be here interpreted as option pricing for few days. From 1 day up to 10 days are suitable maturities to prove our validation hypothesis, at varying of volatility. Parameter values are those in Bakshi, Cao and Chen (1997) namely κ = 1.15, Θ = 0.04, α = 0.39 and ρ = −0.64. We have chosen r = 10% K = 100, and three different maturities T. In what follows we use the expected value of the variance process EP[νs] instead of νs in the term R T t νsds. In the tables hereafter one can see the results of numerical experiments: Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

30. Quantitative Finance: stochastic volatility market models Numerical Validation The approximation τ → 0 will be here interpreted as option pricing for few days. From 1 day up to 10 days are suitable maturities to prove our validation hypothesis, at varying of volatility. Parameter values are those in Bakshi, Cao and Chen (1997) namely κ = 1.15, Θ = 0.04, α = 0.39 and ρ = −0.64. We have chosen r = 10% K = 100, and three different maturities T. In what follows we use the expected value of the variance process EP[νs] instead of νs in the term R T t νsds. In the tables hereafter one can see the results of numerical experiments: Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

31. Quantitative Finance: stochastic volatility market models Table: At the money, S0 = 100, K = 100, with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 1 day. Volatility Fourier method Dell’Era method 30% 0.6434 0.6442 40% 0.8543 0.8541 50% 1.0643 1.0641 60% 1.2743 1.2742 70% 1.4843 1.4845 80% 1.6943 1.6949 90% 1.9042 1.9055 100% 2.1142 2.1162 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

32. Quantitative Finance: stochastic volatility market models Table: At the money, S0 = 100, K = 100, with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 5 days. Volatility Fourier method Dell’Era method 30% 1.4763 1.4748 40% 1.9430 1.9407 50% 2.4101 2.4081 60% 2.8772 2.8769 70% 3.3444 3.3472 80% 3.8115 3.8190 90% 4.2785 4.2927 100% 4.7454 4.7683 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

33. Quantitative Finance: stochastic volatility market models Table: At the money, S0 = 100, K = 100, with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 10 days. Volatility Fourier method Dell’Era method 30% 2.1234 2.1191 40% 2.7787 2.7722 50% 3.4348 3.4294 60% 4.0912 4.0905 70% 4.7477 4.7557 80% 5.4040 5.4254 90% 6.0601 6.1002 100% 6.7158 6.7806 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

34. Quantitative Finance: stochastic volatility market models Table: In the money, S0 = K “ 1 + 10% p θ(T − t) ” , with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 1 day. Volatility Fourier method Dell’Era method 30% 0.6991 0.6994 40% 0.9094 0.9089 50% 1.1191 1.1187 60% 1.3289 1.3287 70% 1.5377 1.5389 80% 1.7488 1.7494 90% 1.9588 1.9600 100% 2.1688 2.1708 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

35. Quantitative Finance: stochastic volatility market models Table: In the money, S0 = K “ 1 + 10% p θ(T − t) ” , with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 5 days. Volatility Fourier method Dell’Era method 30% 1.6049 1.6012 40% 2.0700 2.0661 50% 2.5362 2.5331 60% 3.0030 3.0019 70% 3.4700 3.4723 80% 3.9372 3.9445 90% 4.4044 4.4186 100% 4.8715 4.8947 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

36. Quantitative Finance: stochastic volatility market models Table: In the money, S0 = K “ 1 + 10% p θ(T − t) ” , with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 10 days. Volatility Fourier method Dell’Era method 30% 2.3098 2.3012 40% 2.9621 2.9527 50% 3.6168 3.6095 60% 4.2727 4.2708 70% 4.9291 4.9366 80% 5.5856 5.6072 90% 6.2421 6.2831 100% 6.8984 6.9647 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

37. Quantitative Finance: stochastic volatility market models Table: Out the money, S0 = K “ 1 − 10% p θ(T − t) ” , with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 1 day. Volatility Fourier method Dell’Era method 30% 0.5905 0.5918 40% 0.8013 0.8014 50% 1.0111 1.0112 60% 1.2210 1.2212 70% 1.4309 1.4313 80% 1.6407 1.6415 90% 1.8506 1.8519 100% 2.0605 2.0625 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

38. Quantitative Finance: stochastic volatility market models Table: Out the money, S0 = K “ 1 − 10% p θ(T − t) ” , with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 5 days. Volatility Fourier method Dell’Era method 30% 1.3539 1.3546 40% 1.8208 1.8201 50% 2.2878 2.2869 60% 2.7546 2.7551 70% 3.2214 3.2247 80% 3.6882 3.6959 90% 4.1547 4.1689 100% 4.6212 4.6438 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

39. Quantitative Finance: stochastic volatility market models Table: Out the money, S0 = K “ 1 − 10% p θ(T − t) ” , with parameter values: κ = 1.5, θ = 0.04, α = 0.39, ρ = −0.64, r = 0.10 and Maturity 10 days. Volatility Fourier method Dell’Era method 30% 1.9459 1.9459 40% 2.6019 2.5985 50% 3.2581 3.2548 60% 3.9142 3.9148 70% 4.5701 4.5787 80% 5.2257 5.2471 90% 5.8810 5.9204 100% 6.5359 6.5992 Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

40. Quantitative Finance: stochastic volatility market models Conclusions The proposed method is straightforward from theoretical viewpoint and seems to be promising from that numerical. We reduce the Heston’s PDE in a simpler, using , in a right order, suitable changing of variables, whose Jacobian has not singularity points, unless for ρ = ±1. This evidence gives us the safety that the variables chosen are well deﬁned. Besides, the idea to use the expected value of the variance process EP[νs], instead of νt , provides us, in concrete, a closed solution very easy to compute; and so, we are also able to know what is the error using the geometric transformation technique; which is equal to the variance of the variance process νt : Err = EP[(νt − EP[νt ])2 ]. While, using Fourier technique we are not able to know the numeric error directly, but we need to compare Fourier prices with Monte Carlo prices, for which one can manage the variance. We want to remark that the shown technique is independent to the payoff and therefore, the pricing activities have the same algorithmic complexity for every derivatives, unlike using Fourier Transform method, for which the complexity is tied to the payoff. Mario Dell’Era Closed Solution for Heston PDE by Geometrical Transformations

Workshop 2013 of Quantitative Finance

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