Jan Korbel
Workshop Fractional Differential Equations, Applications and Complex Networks, Lorentz Center, Leiden
slides available at: slides.com/jankorbel
References:
[1] Physica A 449 (2016) 200-214; 10.1016/j.physa.2015.12.125
[2] Fract. Calc. Appl. Anal. 19 (6) (2016) 1414-1433; 10.1515/fca-2016-0073
[3] Fractal Fract. 2 (1) (2018) 15; 10.3390/fractalfract2010015
[4] Fract. Calc. Appl. Anal. 21 (4) (2018) 981-1004; 10.1515/fca-2018-0054
[5] Risks 7 (2) (2019) 36; 10.3390/risks7020036
[6] Mathematics 7 (9) (2019) 796; 10.3390/math7090796
[7] Fract. Calc. Appl. Anal. 23 (4) (2020) 996-1012; 10.1515/fca-2020-0052
[8] Risks 8 (4) (2020) 124; 10.3390/risks8040124
[9] Mathematics 9(24), 3198;10.3390/math9243198
Work [2] has been first discussed with Yuri Luchko here
more than 7 years ago
Review paper [6]
Recent paper [7] with Živorad and Johan
The STFD equation is defined as
(0∗Dtγ−μ θDxα)g(x,t)=0
Caputo derivative: t0∗Dtγf(t)=Γ(⌈γ⌉−γ)1∫t0tdτ(t−τ)γ+1−⌈γ⌉f⌈γ⌉(τ)
Riesz-Feller derivative: F[θDxαf(x)](k)=−∣k∣αeisign(k)θπ/2F[f(x)](k)
Solution can be defined in terms of Mellin-Barnes transform
gα,θ,γ(x,t)=2πi1αx1∫c−i∞c+i∞Γ(1−αγy)Γ(2αα−θy)Γ(1−2αα−θy)Γ(αy)Γ(1−αy)Γ(1−y)(−μtx)ydy
[1] Physica A 449 (2016) 200-214
[6] Mathematics 7 (9) (2019) 796
Price of European call option: C(S,K,τ)=∫−∞∞max{Se(r+μ)τ+x−K,0}gα,θ,γ(x,τ)dx
Interpretation of parameters:
[1] Physica A 449 (2016) 200-214; [3] Fractal Fract. 2 (1) (2018) 15
We can rewrite the option price via Mellin-Barnes representation: C(S,K,τ)=αKe−rτ∫c1−i∞c1+i∞∫c2−i∞c2+i∞(−1)t2Γ(1−αγt1)Γ(t2)Γ(1−t2)Γ(−1−t1+t2)
×(−logKS−(r+μ)τ)1+t1−t2(−μτ)−t1/α2πidt1∧2πidt2
This can be compactly represented as an integral over a complex differential 2-form:
C(S,K,τ)=αKe−rτ∫c+iR2ω
Its characteristic vector is Δ=(−1+αγ,1)
As a result, the integral can be expressed as a sum of residues, which can be simply written as a double sum
By using residue summation in C2 it is possible to express the price in terms of rapidly-convergent double series ( L=logKS+rτ )
C(S,K,τ)=αKe−rτ n=0∑∞m=1∑∞n!Γ(1+αm−n)1(L+μτ)n(−μτ)2m−n
[4] Fract. Calc. Appl. Anal. 21 (4) (2018) 981-1004
gα,θ,γ(x,t) can be represented as a subordinated process
gα,θ,γ(x,t)=∫0∞dlKγ(t,l)Lαθ(l,x)
[1] Physica A 449 (2016) 200-214; [8] Risks 8 (4) (2020) 124
[3] FCAA 19 (6) (2016) 1414-1433
[3] FCAA 23 (4) (2020) 996-1012
u(x,t)=c+u+(x,t)ξx≥0(x)+c−u−(x,t)ξx<0(x)
u+(x,t)=n=0∑∞ Φ−2n(x)(−κ)n eρ,μn+1,ωγn(t)
u−(x,t)=n=0∑∞(2n+1)!κn+1∣x∣2n+1eρ,−μ(n+1)+1,ω−γ(n+1)(t)
The fundamental solution of the FDE:
Thus, the option price can be expressed as
C(S,K,τ)=c+C+(S,K,τ)+c−C−(S,K,τ)
C+(S,K,τ) = max{Se(r+q)τ−K,0}+ ϕK(Se(r+q)τ)n=1∑∞(−κ)neρ,μn+1,ωγn(τ)
C−(S,K,τ)=n=0∑∞(2n+1)!κn+11eρ,−μ(n+1)+1,ω−γ(n+1)(τ) ×(e(r+q)τγ(2(n+1),log(Se(r+q)τ/K))2(n+1)K (log(Se(r+q)τ/K))2(n+1))
where ϕK(x)=x if x>K and ϕK(x)=0 if x<K, and γ is the incomplete gamma function.
Finally, we obtain that