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On validity of a singular perturbation expansion of European options - - PowerPoint PPT Presentation
On validity of a singular perturbation expansion of European options - - PowerPoint PPT Presentation
On validity of a singular perturbation expansion of European options and implied volatility Masaaki Fukasawa CSFI, Osaka Univ. 1 [Stochastic volatility model: Notation] Suppose that S t : an asset price, Z t = log( S t ) : log price, r : risk
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In a stochastic volatility model, stylized facts such as
- the (Black-Scholes) implied volatility skew/smile,
- time varying volatility,
- leverage effect,
- mean-reverting volatility
are explained. It is a natural extension of the Black-Scholes model, including the Heston model, SABR model. Unfortunately, no simple analytic formula for the option prices is available in general.
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[Singular perturbation expansion] Fouque, Papanicolaou and Sircar (2000): consider
dZt = (r − ϕ(Xt)2/2) + ϕ(Xt)dWt, dXt = −aXt − b ǫ dt + Λ(Xt) √ǫ dt + c √ǫdW′
t
where ǫ is small. They obtained, by a formal calculation, that
P(t, z, x) = P0(t, z) + √ǫP1(t, z) + O(ǫ),
where P0 is the Black-Scholes price.
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The validity of the singular perturbation expansion is proved by
- Fouque, Papanicolaou, Sircar and Solna (2003)
- Conlon and Sullivan (2005)
- higher order expansion for call option
- Khasminskii and Yin (2005)
- ergodic diffusion on compact set and smooth payoff
- Fukasawa (2008)
- general ergodic diffusion and payoff
- Edgeworth expansion for ergodic diffusions
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[Extended fast mean-reverting model] More generally, consider
dZt = (r − ϕ(Xt)2/2)dt + ϕ(Xt)dWt, dXt = b(Xt) + Λǫ(Xt) ǫ dt + c(Xt) √ǫ dW′
t ,
where Λǫ → 0 (in a weak sense) as ǫ → 0. Intuition: put ˆ
Xt = Xǫt and ˆ W′ = ǫ−1/2W′
ǫt. Let Λǫ = 0 for brevity. Then
dZt = (r − ϕ( ˆ Xt/ǫ)2/2)dt + ϕ( ˆ Xt/ǫ)dWt, d ˆ Xt = b( ˆ Xt)dt + c( ˆ Xt)d ˆ W′
t .
Notice that the law of ˆ
X does not depend on ǫ.
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Intuition (continued): putting
ˆ Xt = Xǫt, ˆ W′ = ǫ−1/2W′
ǫt,
ˆ W = ǫ−1/2Wǫt,
we have
Zt = Z0 + ǫ ∫ t/ǫ (r − ϕ( ˆ Xs)2/2)ds + √ǫ ∫ t/ǫ ϕ( ˆ Xs)d ˆ Ws,
where
d ˆ Xt = b( ˆ Xt)dt + c( ˆ Xt)d ˆ W′
t .
Note that the law of ( ˆ
X, ˆ W, ˆ W′) does not depend on ǫ, so that Zt = ǫ ∫ t/ǫ ϕ( ˆ Xs)2ds → tπ[ϕ2] as ǫ → 0
for the ergodic distribution π of ˆ
- X. Consequently, by the martingale CLT,
Zt ⇒ Z0 + (r − π[ϕ2]/2)t + √ π[ϕ2]Wt.
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[Edgeworth expansion for ergodic diffusions: Fukasawa(2008, PTRF)] For a diffusion X with scale function S :
S ′(x) = s(x), s(x) = exp { −2 ∫ x
x0
b(y) c(y)2dy } , M = ∫ dx c(x)2s(x) < ∞
we have
IT[ϕ] = 1 T ∫ T ϕ(Xt)dt → π[ϕ], √ T(IT[ϕ] − π[ϕ]) ⇒ N(0, σ2),
where π(dx)/dx = 1/Mc(x)2s(x). As a refinement of this CLT, we can show
E[H( √ T(A(IT[ϕ]) − A(π[ϕ])))] = ∫ H(w)N(dw) + T−1/2 ∫ H(w)p(w)N(dw) + O(T−1)
under a moment condition and a smoothness condition.
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[First result: Edgeworth approach] Recall that
dZt = (r − ϕ(Xt)2/2)dt + ϕ(Xt)dWt, dXt = ǫ−1bǫ(Xt)dt + ǫ−1/2c(Xt)dW′
t ,
where bǫ = b + Λǫ.
- the state space of X is supposed to be R.
- assume that supǫ≥0 |bǫ|, c, 1/c, ϕ are locally bounded.
- the option price for payoff h is given as
P(t, z, x) = E[e−r(T−t)h(S T)|Zt = z, Xt = x].
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[Assumptions] Lǫ; the generator of ˆ
X, ˆ Xt = Xǫt ψ(x) = ∫ x dyϕ(y)/c(y),
- Lǫψ → L0ψ on an interval I and L0ψ is continuous on I,
- L0ψ is not constant on I,
- there exist κ± > 0 such that
γ± > 2κ±, sup
x≥0
1 + ϕ(x)2 eκ+xc(x)2 < ∞, sup
x≤0
1 + ϕ(x)2 e−κ−xc(x)2 < ∞,
where
γ+ = − lim sup
x→∞,ǫ→0
bǫ(x) c(x)2, γ− = lim inf
x→−∞,ǫ→0
bǫ(x) c(x)2.
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[Theorem] If h is bounded, then for every (t, z, x),
P(t, z, x) = P0(t, z) + √ǫP1(t, z) + O(ǫ),
where P0 is the Black-Scholes price with volatility σ =
√ πǫ[ϕ2]; P0(t, z) = e−rτ ∫ h(exp(z + (r − σ2/2)τ + σ √τw))N(dw), τ = T − t, and P1(t, z) = e−rτ ∫ h(exp(z + (r − σ2/2)τ + σ √τw)) × δ σ2 1 − w2 + w3 − 3w σ √τ N(dw), δ = −ρ ∫ ∞
−∞
∫ x
−∞
(ϕ(y)2 − σ2)πǫ(dy)ϕ(x) c(x)dx.
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[Corollary] The put option price admits
P(t, z, x) = P0(t, z) − δ √ǫ σ2 ezφ(d1)d2 + O(ǫ),
where P0 is the Black-Scholes put price, and (d1, d2) is the usual one with σ. The Black-Scholes implied volatility IV admits IV = σ −
δ √ǫd2 σ2 √ T − t + O(ǫ).
Note that IV = c1
log(K/S t) T − t + c2 + O(ǫ)
for strike K, where c1 and c2 are constants of O( √ǫ) and O(1) respectively.
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[Example: the Heston model]
dZt = (r − Vt/2)dt + √ VtdWt, dVt = −aVt − b ǫ dt + c √Vt √ǫ dW′
t
with W, W′t = ρt. We have σ2 = b/a and δ = ρbc/2a2, so that IV =
√ b a − ρc √ǫd2 2a √ T − t + O(ǫ).
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[Alternative conditions for a diffusion being not exponentially mixing]
- Lǫψ → L0ψ on an interval I and L0ψ is continuous on I,
- L0ψ is not constant on I,
- there exist p ≥ 0 such that
2γ + 1 > 4p, lim sup
|x|→∞
1 + ϕ(x)2 |x|p−2c(x)2 < ∞,
where
γ = − lim sup
x→∞,ǫ→0
xbǫ(x) c(x)2 .
The same conclusion as before is obtained under these conditions.
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[What is new?]
- a probabilistic approach.
- the Heston model (or, a model with irregular coefficients) is incorporated.
- digital options (or, irregular payoff functions) are incorporated.
- volatility process X with slow decay of the autocorrelation function is in-
- corporated. For example,
dXt = −α Xt 1 + X2
t
+ βdWt.
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[Second result: martingale approach] Let us consider a more general model for Z = log(S ),
Z = Z0 + A + M + g · W + h · C,
where M is a local martingale, g and h are adapted cag processes, C is a time-changed compound Poisson process with intensity Λ, W is a std BM in- dependent (M, g, h, Λ),
At = rt − 1 2Mt − 1 2 ∫ t |gs|2ds − ∫ t ∫
R
(ehsz − 1)ν(dz)Λ(ds).
We consider the following perturbation:
M = Mn, g = gn, h = hn, Λ = Λn, ν = νn
with ZT/Σn → T for a sequence Σn as n → ∞ so that the martingale CLT is
- applied. Hence, the Black-Scholes model is recovered as n → ∞.
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In this framework, putting rn = ǫ2
n → 0,
- slowly varying volatility: (Sircar, Lee, ...)
dXn
t = (rnb1(Xn t ) + r2 nb2(Xn t ))dt + rnc(Xn t )dWt,
- small volatility of volatility: (Hull and White, Lewis, Sørensen et.al., ..)
dXn
t = b(Xn t )dt + rnc(Xn t )dWt,
- fast mean-reverting: (Fouque et.al., Khasminskii et.al., Fukasawa )
dXn
t = r−2 n b(Xn t )dt + r−1 n λ(Xn t )dt + r−1 n c(Xn t )dWt,
are included.
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[Theorem] Under several conditions, we have for any bounded function H,
E[H(ZT)] = ∫
R
H(rT − Σn/2 + √ Σnz)φn(z)dz + o(rn)
with a sequence rn → 0 as n → ∞,
φn(z) = φ(z) { 1 + rn 2 { β(z2 − 1) + ( γ + 1 3α ) (z3 − 3z) − √ Σn {( β + √Σn 3 α ) z + γ(z2 − 1) }}} .
Here α, β, γ are functions of T. [Proof] Apply Yoshida’s formula for martingale expansion.
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[Corollary] The put option price e−rTE[(K − S T)+|S 0 = S ] is expanded as
P0(σn) + 1 2rnσn √ TS φ(d1) { β − γd2 − 1 3α(d2 − σn √ T) } + o(rn),
where P0(σn) is the Black-Scholes put price with volatility σn and σ2
nT = Σn
The Black-Scholes implied volatility IV is expanded as
σn { 1 + rn 2 { β − γd2 − 1 3α(d2 − σn √ T) }} + o(rn).
Note that IV = c1 log(K/S 0) + c2 + o(rn) for strike K, where c1 and c2 are functions of T, of O(rn) and O(1) respectively.
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[What is new?]
- a probabilistic approach.
- various perturbations are treated in a unified way.
- non-Markovian model is incorporated.
- jump is also incorporated.
- multi-dimensional volatility process is incorporated.