Generalised Fractional-Black-Scholes Equation: pricing and hedging Álvaro Cartea Birkbeck College, University of London April 2004
Outline Lévy processes Fractional calculus Fractional-Black-Scholes 1
Definition 1 Lévy process. Let X(t) be a random variable dependent on time t. Then the stochastic process X(t), for 0 < t < and X(0) = 0, is a Lévy process iff it has independent and stationary increments. Theorem 1 Lévy-Khintchine representation. Let X(t) be a Lévy process. Then the natural logarithm of the characteristic function can be written as ln E[e iθx(t) ] = aitθ 1 2 σ2 tθ 2 + t (e iθx 1 iθxi x <1 ) W (dx), where a R, σ 0, I is the indicator function and the Lévy measure W must satisfy R/0 min{1, x2 }W (dx) <. (1) 2
Which Lévy process? Why? Brownian motion; Bachelier. α-stable or Lévy Stable; Mandelbrot. Jump Diffusion; Merton. GIG and Generalised Hyperbolic Distribution; Barndorff-Nielsen. Variance Gamma; Madan et al. 3
CGMY, Carr et al. KoBol, Tempered Stable; Koponen. FMLS; Carr and Wu. others. When specifying a particular Lévy process we are basically asking how do we want to specify the behaviour of the jumps, in other words how is the Lévy density w(x) (ie W (dx) = w(x)dx) chosen. For example
Size and sign of jumps Frequency of jumps Existence of moments Simplicity
The CGMY process A simple answer is then to consider a Lévy density of the form { C x w CGMY (x) = 1 Y e G x for x < 0, Cx 1 Y e Mx for x > 0, and the log of the characteristic function is given by Ψ CGMY (θ) = tcγ(y ){(M iθ) Y M Y + (G + iθ) Y G Y }. Here C > 0, G 0, M 0 and Y < 2. 4
The Damped-Lévy process { Cq x 1 α e λ x for x < 0, w DL (x) = Cpx 1 α e λx for x > 0, and the natural logarithm of the characteristic equation is given by Ψ DL (θ) = tκ α { p(λ iθ) α + q(λ + iθ) α λ α iθαλ α 1 (q p) }, for 1 < α 2 and p + q = 1. 5
The Lévy-Stable process Is a pure jump process with Lévy density { Cq x 1 α for x < 0, w LS (x) = Cpx 1 α for x > 0, Hence the log of the characteristic function is Ψ(θ) = { κ α θ α {1 iβ sign(θ) tan(απ/2)} for α 1, κ θ { 1 + 2iβ π sign(θ) ln θ } for α = 1, here C > 0 is a scale constant, p 0 and q 0, with p + q = 1 and β = p q is the skewness parameter. 6
Fractional Integrals For an n-fold integral there is the well known formula x x a x f(x)dx = 1 a a (n 1)! x a (x t)n 1 f(t)dt. Note that since (n 1)! = Γ(n) the expression above may have a meaning for non-integer values of n. Definition 2 The Riemann-Liouville Fractional Integral. The fractional integral of order α > 0 of a function f(x) is given by and D α a + f(x) = 1 Γ(α) D α b f(x) = 1 Γ(α) x a (x ξ)α 1 f(ξ)dξ, b x (ξ x)α 1 f(ξ)dξ. 7
Definition 3 The Riemann-Liouville Fractional Derivative. D α 1 a + f(x) = Γ(n α) dx n and Db α f(x) = ( 1)n d n Γ(n α) dx n d n x a (x ξ)n α 1 f(ξ)dξ, b x (ξ x)n α 1 f(ξ)dξ. The Fourier Transform View Note that if we let a = and b = we have F{D+ α f(x)} = ( iξ)α ˆf(ξ) and F{D α f(x)} = (iξ)α ˆf(ξ). 8
The Lévy-Stable Fractional-Black-Scholes. Under the physical measure the price process follows a geometric LS process d(ln S) = µdt + σdl LS, where L S α (dt 1/α, β, 0) with 1 < α < 2, 1 β 1 and σ > 0. And under the risk-neutral measure (McCulloch) it follows d(ln S) = (r βσ α sec(απ/2))dt + dl LS + dl DL where dl LS and dl DL are independent. rv = where V (x, t) + (r βσ α V (x, t) sec(απ/2)) κ α 2 t x sec(απ/2)dα + V (x, t) +κ α 1 sec(απ/2) ( V (x, t) e x D α e x V (x, t) ), κ α 2 = 1 β 2 σα and κ α 1 = 1 + β 2 σα. 9
Two cases: classical Black-Scholes and the fractional FMLS Case α = 2, Black-Scholes rv (x, t) = V (x, t) t + (r σ 2 V (x, t) ) x + σ 2 2 V (x, t) x 2. Case α > 1 and β = 1, FMLS rv (x, t) = V (x, t) + (r + σ α V (x, t) sec(απ/2)) t x σ α sec(απ/2)d+ α V (x, t). 10
Proposition 1 CGMY Fractional-Black-Scholes equation. Let the risk-neutral log-stock price dynamics follow a CGMY process d(ln S) = (r w)dt + dl CGMY. (2) The value of a European-style option with final payoff Π(x, T ) satisfies the following fractional differential equation rv (x, t) = where σ = CΓ( Y ). V (x, t) V (x, t) + (r w) t x +σ(m Y + G Y )V (x, t) +σe Mx D Y ( e Mx V (x, t) ) +σe Gx D Y + ( e Gx V (x, t) ), 11
Proof 1 V (x, t) = e r(t t) E t [Π(x T, T )]. 2 V (x, t) = e r(t t) 2π E t [ +iν +iν e ix T ξ ˆΠ(ξ)dξ ]. 3 ˆV (ξ, t) = e r(t t) e iξµ(t t) (T e t)ψ( ξ)ˆπ(ξ). 4 ˆV (ξ, t) = (r + iξµ Ψ( ξ))ˆv (ξ, t) t with boundary condition ˆV (ξ, T ) = ˆΠ(ξ, T ). 12
Dynamic Hedging: Delta hedging, Delta-Gamma hedging, Variance minimisation. The Taylor Expansion View dv = V t V dt + S ds + 1 2 V 2 S 2 ds2 +.... Portfolio P (S, t) = V 1 (S, t) S bv 2 (S, t) = V 1(S, t) S V 2(S, t) b, S b = 2 V 1 (S, t)/ S 2 2 V 2 (S, t)/ S 2. 13
30 FMLS Daily Deltra hedge with α=1.5, S=K=100, T=1month 25 20 Frequency 15 10 5 0 90 80 70 60 50 40 30 20 10 0 10 Proit and Loss ( ) 14
9 FMLS Min Variance with α=1.5, S=K=100, T=1month 8 7 6 Frequency 5 4 3 2 1 0 40 35 30 25 20 15 10 5 0 5 Profit and Loss 15
120 FMLS Delta and Gamma Hedging with α=1.5, S=K=100, T=1month 100 Frequency 80 60 Min= 35.33 Max=10.6 std=0.70 mean= 0.0006 40 20 0 40 30 20 10 0 10 20 Profit and Loss ( ) 16
FMLS Black-Scholes: the Taylor expansion view dv (x, t) = (Samko et al 1993). V (x, t) V (x, t) dt + dx t x + 1 Γ(2 α) Dα + V (x, t)(dx)α +. Therefore it seems natural, in the FMLS case, to delta and fractional-gamma hedge the portfolio P (x, t) = V 1 (x, t) e x bv 2 (x, t), hence and a = V 1(x, t) x 1 e x V 2(x, t) 1 x e xb b = ex D α + V 1(x, t) V 1 (x, t)/ xd α + ex e x D α + V 2(x, t) V 2 (x, t)/ xd α + ex. 17
120 FMLS Delta and Fractional Hedging with α=1.5, S=K=100, T=1month 100 Frequency 80 60 Min= 1.2 Max=8.4 std=0.17 mean=0.001 40 20 0 2 0 2 4 6 8 10 Profit and Loss ( ) 18
In General might want to do... V (x, t) V (x, t) rv (x, t) = + µ + GV (x, t), (3) t x where G is an operator containing the fractional derivatives. Therefore we require and P (x, t) = V 1 (s, t; T 1 ) ae x bv 2 (s, t; T 2 ) (4) a = V 1(x, t) x 1 e x V 2(x, t) x 1 e xb b = ex GV 1 (x, t) V 1 (x, t)/ xge x e x GV 2 (x, t) V 2 (x, t)/ xge x so the portfolio is both Delta and Fractional-Gamma-neutral, ie P (x, t) x = 0 and GP (x, t) = 0. 19
LS Delta Hedging, α=1.7, S=K=100, T=1month 20 15 Min= 408 Max=5.2764 Mean= 0.055 Frequency 10 5 0 60 40 20 0 20 40 Pofit and Loss ( ) 20
LS Delta and Gamma Hedging, α=1.7, S=K=100, T=1month 70 60 Min= 56 Max=2.89 Mean=0.0024 50 Frequency 40 30 20 10 0 8 6 4 2 0 2 4 6 8 Profit and Loss ( ) 21
Levy Stable Delta and Fractional Hedging, α=1.7, S=K=100, T=1month 60 50 Min= 20.64 Max=2.29 Mean= 0.0001 40 Frequency 30 20 10 0 1.5 1 0.5 0 0.5 1 1.5 Profit and Loss 22
CONCLUSIONS and FURTHER WORK For Lévy processes with Lévy densities that have a polynomial singularity at the origin and exponential decay at the tails we can recast the pricing equation in terms of Fractional derivatives. The non-local property of the fractional operators can be useful when dynamically hedging options. Using well established numerical schemes for Fractional operators it might be possible to price American options. Moreover, for these processes we can derive Fractional Fokker-Planck equations that may also be used in the pricing of American options. 23