Applications of fractional diffusion in option pricing

Jan Korbel

Slides available at: slides.com/jankorbel

Personal web: jankorbel.eu

VCMF 2025

Option pricing

  • Black-Scholes model 
    • Based on ordinary Brownian motion
    • ​In financial crises or in complex markets, the model cannot catch realistic market dynamics
      • large drops
      • sudden shocks
      • memory effects
  • Finite moment log-stable model (Carr and Wu 2003)
    • based on Lévy-stable fractional diffusion
    • enables large drops
  • We generalize the models by using the space-time fractional diffusion equation

Fractional diffusion

  • Black-Scholes model is based on diffusion equation

$$ \left( \partial_t - \mu \partial_{xx}^2 \right) g(x,t) = 0$$

  • We extend it to the case of fractional diffusion

$$ \left({}^*_0 \mathcal{D}^\gamma_t - \mu \ {}^\theta \mathcal{D}_x^{\alpha}\right) g(x,t) = 0$$

 

  • Caputo derivative: \( {}^*_{t_0} \mathcal{D}^\gamma_t f(t) = \frac{1}{\Gamma(\lceil \gamma \rceil - \gamma)} \int_{t_0}^t \mathrm{d} \tau \frac{f^{\lceil \gamma \rceil}(\tau)}{(t-\tau)^{\gamma + 1 - \lceil \gamma \rceil}}\)
  • Riesz-Feller derivative: \(\mathcal{F}[{}^{\theta} \mathcal{D}^\alpha_x f(x)](k) = -|k|^\alpha e^{i \, \mathrm{sign}(k) \theta \pi/2} \mathcal{F}[f(x)](k) \)
  • \(\gamma=1\) - Carr-Wu model
  • \(\gamma=1,\alpha=2\) - Black-Scholes model

[1] Physica A 449 (2016) 200-214

Solution of the Fractional diffusion

Solution can be found by taking the Fourier-Laplace transform

$$g_{\alpha,\theta,\gamma}(k,s) = \frac{s^{\gamma-1}} {s^{\gamma} -{|k|}^\alpha e^{i \, \mathrm{sign}(k) \theta \pi/2}  }$$

 

It can be transformed back in terms of Mellin-Barnes transform

$$g_{\alpha,\theta,\gamma}(x,t) = \frac{1}{2 \pi i} \frac{1}{\alpha x} \int_{c-i \infty}^{c+i \infty} \frac{\Gamma(\frac{y}{\alpha}) \Gamma(1-\frac{y}{\alpha})\Gamma(1-y)}{\Gamma(1-\frac{\gamma}{\alpha} y)\Gamma(\frac{\alpha-\theta}{2 \alpha} y) \Gamma(1- \frac{\alpha-\theta}{2 \alpha} y)} \left(\frac{x}{-\mu t}\right)^y \mathrm{d} y$$

Solution of fractional diffusion

  • \(\gamma=1, \alpha=2\) - ordinary Gaussian diffusion

 

 

  • \(\gamma=1, \alpha<2\) - Lévy-stable diffusion

 

 

  • \(\gamma \neq 1, \alpha=2\) - diffusion with memory

 

 

  • \(\gamma \neq 1, \alpha<2\) - space-time fractional diffusion

[6] Mathematics 7 (9) (2019) 796

Space-time fractional option pricing

Price of European call option: $$C(S,K,\tau) = \int_{-\infty}^\infty \max\{S e^{(r+\mu) \tau + x}-K,0\} g_{\alpha,\theta,\gamma}(x,\tau) \mathrm{d} x$$

Interpretation of parameters:

  • \(\theta = \max\{-\alpha, \alpha-2\}\) 
    • maximally asymmetric distribution
    • power-law probability of drops (negative Lévy tail)
    •  Gaussian probability of rises (positive exponential tail
  • \(\alpha < 2\) - risk redistribution to large drops
  • \(\gamma \) - risk redistribution in time
    • \(\gamma < 1\) shorter contracts are more risky
    • \(\gamma > 1\) longer contracts are more risky

[1] Physica A 449 (2016) 200-214; [3] Fractal Fract. 2 (1) (2018) 15

Differential 2-form representation

We can rewrite the option price via Mellin-Barnes representation: $$C(S,K,\tau) = \frac{K e^{-r \tau}}{\alpha} \int_{c_1 - i \infty}^{c_1 + i \infty} \int_{c_2 - i \infty}^{c_2 + i \infty} (-1)^{t_2} \frac{\Gamma(t_2)\Gamma(1-t_2)\Gamma(-1-t_1+t_2)}{\Gamma(1-\frac{\gamma}{\alpha}t_1)}$$

$$\times\left(- \log\frac{S}{K}-(r+\mu) \tau\right)^{1+t_1-t_2}(-\mu\tau)^{-t_1/\alpha} \frac{\mathrm{d}t_1}{2\pi i}\wedge \frac{\mathrm{d} t_2}{2 \pi i}$$

This can be compactly represented as an integral over a complex differential 2-form:

$$C(S,K,\tau) = \frac{K e^{-r \tau}}{\alpha} \int_{\vec{c} + i \mathbb{R}^2} \omega$$

Its characteristic vector is \(\Delta = (-1+\frac{\gamma}{\alpha},1)\)

As a result, the integral can be expressed as a sum of residues, which can be simply written as a double sum

Residues of the differential 2-form

Slide title

By using residue summation in \(\mathbb{C}^2\) it is possible to express the price in terms of rapidly-convergent double series ( \(\mathcal{L} = \log \frac{S}{K} + r \tau\) ) 

$$C(S,K,\tau) = \frac{K e^{-r \tau}}{\alpha}  \sum_{n=0}^\infty \sum_{m=1}^\infty \frac{1}{n! \Gamma\left(1 + \frac{m-n}{\alpha}\right)}(\mathcal{L}+\mu \tau)^{n}(-\mu \tau)^{\frac{m-n}{2}}$$

[4] Fract. Calc. Appl. Anal. 21 (4) (2018) 981-1004

Applications to exotic options and the Greeks

  • Analogous results can be applied to  exotic types of options
    • Cash-or-nothing \(C(S,K,0) = \Theta(S-K)\)
    • Asset-or-nothing \(C(S,K,0) = S \times \Theta(S-K)\)
  • Additionally, similar formulas can be derived for risk sensitivities (the Greeks), including
    • Delta \(\Delta = \frac{\partial C}{\partial S}\)
    • Rho \(\rho = \frac{\partial C}{\partial r}\)
    • Gamma \(\Gamma = \frac{\partial^2 C}{\partial S^2}\)
  • Specific results can be found in
    • [5] Risks 7 (2) (2019) 36; 10.3390/risks7020036
    • [6] Mathematics 7 (9) (2019) 796; 10.3390/math7090796

Subordinator representation

\(g_{\alpha,\theta,\gamma}(x,t)\) can be represented as a subordinated process

$$g_{\alpha,\theta,\gamma}(x,t) = \int_0^\infty \mathrm{d} l K_\gamma(t,l) L_{\alpha}^\theta(l,x)$$

  • \(L_{\alpha}^\theta(l,x)\) - Lévy-stable distribution with scaling parameter \(l\) 
  • \(K_\gamma(t,l\) - subordinator (smearing kernel)
    • \(K_\gamma(t,l) =  \frac{t}{l^{1+1/\gamma} \gamma}  L_\gamma^{\gamma}\left(\frac{t}{l^{1/\gamma}}\right)\)
  • We compare with other subordinated models
    • Variance gamma \(K_\lambda(t,l) =  \lambda  e^{\lambda  (-t/l)}\)
    • Negative inverse-gamma \(K_{\alpha,\beta}(t,l) = \frac{e^{-\frac{\beta }{t/l}} \left(\frac{\beta }{t/l}\right)^{\alpha }}{t/l \Gamma (\alpha )}\)

[1] Physica A 449 (2016) 200-214; [8] Risks 8 (4) (2020) 124

Slide title

Space-time fractional option pricing with derivatives of varying order

  • One of the important aspects of financial markets is switching between different regimes - boom vs crisis
  • Long-term scaling properties remain stable for each stock.
  • This requires a time-dependent description by fractional diffusion of varying order.
  • For each time period, we estimate the parameters

[2] FCAA 19 (6) (2016) 1414-1433 

  • Let us define intervals \(T_i = (t_i ; t_{i+1})\)
  • Dynamics described by a space-time fractional diffusion in each interval $$ \left({}^*_{t_i} \mathcal{D}^{\gamma_i}_t - \mu \ {}^\theta \mathcal{D}_x^{\Omega \gamma_i}\right) g(x,t) = 0$$ with initial conditions: \(g_i(x,t_i) := g_{i-1}(x,t_i)\), \(g_0(x,0) := f(x)\)
  • For \(\gamma_i > 1\) we add \(\frac{\partial g_i(x,t)}{\partial t} |_{t=t_i} = 0\)  
  • The overall solution is given as a convolution $$g(x,t) = f(x) \star g_0(x,t_1-t_0) \star \dots \star g_i(x,t-t_i) \qquad \mathrm{for} \ t \in T_i$$

Space-time fractional option pricing with derivatives of varying order

  • The stable parameter is defined as \(\alpha_i = \Omega \gamma_i\) so that \(\Omega = \frac{\alpha_i}{\gamma_i}\) remains the same among all intervals \(T_i\) and characterizes the scaliung of \(g(x,t) \mathrm{d}{x} = \frac{1}{t^{\Omega}} g\left(\frac{x}{t^{\Omega}}\right) \mathrm{d}{x}\)
  • Omega can be estimated e.g., from diffusion entropy analysis \(S(t) = - \int g(x,t) \ln g(x,t) = S(1) + \Omega \ln t\)
  • This can be used, e.g., to model diffusion in a temporally abnormal period (crisis)
  • We distinguish two intervals
    • short-term behavior affected by immediate dynamics
    • long-term behavior characterized by scaling properties

Space-time fractional option pricing with derivatives of varying order

  • This regime switch can be described by \(g(x, t)\)  as an overlap between space-time fractional
    diffusion (\(t \leq \tau\)  ) and Lévy flight (\tau \leq t).
  • Considering \(\Omega\) as a system-characterized scaling exponent we obtain  $$g(x,t) = \left\{ \begin{array}{ll} g^{\theta}_{\Omega \gamma,\gamma}(x,t) & t \leq \tau \\ \left[ g^{\theta}_{\Omega \gamma,\gamma}(\tau) \star L_\Omega^\theta(t-\tau)  \right](x) & t > \tau \end{array}\right.$$ 

Space-time fractional option pricing with derivatives of varying order

Space-time fractional option pricing with derivatives of varying order

Extensions to more sophisticated fractional derivatives

[7] FCAA 23 (4) (2020) 996-1012 

  • We also generalize the time-fractional derivative to a more generalized operator - Hilfer-Prabhakar derivative.
  • We define the following functions:
  • Generalized Mittag-Leffler function $$E^{\gamma}_{\rho,\mu}(t) := \sum_{k=0}^{\infty} \frac{\Gamma(\gamma+k)}{\Gamma(\gamma)\Gamma(\rho k + \mu)}t^k$$ 
  • Rescaled GML function $$e^\gamma_{\rho,\mu,\omega}(t) := t^{\mu-1} E^{\gamma}_{\rho,\mu}(\omega t^\rho) $$
  • Prabhakar integral $$(I^{\gamma}_{\rho,\mu,\omega,0+} f)(t) := \int_0^t (t-y)^{\mu-1} E^{\gamma}_{\rho,\mu}[\omega(t-y)^\rho]f(y) \mathrm{d} y$$

 

 

Extensions to more sophisticated fractional derivatives

  • Prabhakar derivative $$(D^{\gamma}_{\rho,\mu,\omega,0+} f)(x) := \frac{d^m}{d x^m}(I^{-\gamma}_{\rho,m-\mu,\omega,0+}f)(x)$$
  • Generalized Hilfer-Prabhakar derivative $$(D^{\gamma,\mu,\nu}_{\rho,\omega,0+} f)(x):= (I^{-\gamma \nu}_{\rho,\nu(n-\mu),\omega,0+} \frac{\mathrm{d}^n}{\mathrm{d} x^n} I^{-(\gamma)(1-\nu)}_{\rho,(1-\nu)(n-\mu),\omega,0+} f)(x)$$
  • The log-price evolution is then driven by the fractional diffusion equation $$D^{\gamma}_{\rho,\mu,\omega,0+} g(x,t) = \kappa \frac{\mathrm{d}^2}{\mathrm{d} x^2} g(x,t)$$
  • The initial conditions are: \(g(x,0) = f(x)\), \(\frac{\partial g(x,t)}{\partial t} |_{t=0} = h(x)\)
  • We can then calculate the value of the European call option if the underlying stock is driven by the aforementioned FDE

Extensions to more sophisticated fractional derivatives

$$g(x,t) =c^+ g^+(x,t) \xi_{x\geq 0}(x)+c^- g^-(x,t) \xi_{x<0}(x)$$
$$g^+(x,t)=\sum_{n=0}^\infty  \Phi_{-2n}(x) (-\kappa)^n  e^{\gamma n}_{\rho,\mu n+1,\omega}(t)$$
$$g^-(x,t)=\sum_{n=0}^{\infty}\frac{|x|^{2n+1}}{(2n+1)!\kappa^{n+1}}{e}^{-\gamma(n+1)}_{\rho,-\mu(n +1)+1,\omega}(t)$$

The fundamental solution of the FDE:

Thus, the option price can be expressed as

$$C(S,K,\tau) = c^+ C^+(S,K,\tau) + c^- C^-(S,K,\tau)$$

Extensions to more sophisticated fractional derivatives

$$ C^+(S,K,\tau)  =  \max\{S e^{(r+q)\tau}-K,0\} +  \phi_K(S e^{(r+q)\tau}) \sum_{n=1}^\infty (-\kappa)^n e^{\gamma n}_{\rho,\mu n+1,\omega}(\tau)$$

$$C^-(S,K,\tau) = \sum_{n=0}^{\infty}\frac{1}{(2n+1)!\kappa^{n+1}} {e}^{-\gamma(n+1)}_{\rho,-\mu(n+1)+1,\omega}(\tau)$$ $$ \times \left(e^{(r+q) \tau} \gamma(2(n+1),\log(Se^{(r+q)\tau}/K)) \frac{K \ (\log(S e^{(r+q)\tau}/K))^{2(n+1)}}{2(n+1)} \right)$$
where \(\phi_K(x) = x\) if \(x > K\) and \(\phi_K(x) =0\) if \(x< K\), and \(\gamma\) is the incomplete gamma function. One can similarly express the negative part.

The positive part of the call price is

csh.ac.at

This work has been done in collaboration with:

Hagen Kleinert (FU Berlin) [1]

Jean-Philippe Aguilar (Covea Finance) [3,4,5,6,8,9]

Cyril Coste (MAIF) [4]

Yuri Luchko (TU of Applied Science Berlin) [2,6]

Živorad Tomovski (University of Ostrava) [7]

Johann L. A. Dubbeldam (University of Delft) [7]

Justin Lars Kirkby (Georgia Institute of Technology) [8]

Nicolas Pesci (Covea Finance) [9]

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