implied volatility at long maturities
TRANSCRIPT
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Implied volatility at long maturities
Michael Tehranchi
Statistical LaboratoryUniversity of Cambridge
Vienna, 10 February 2009
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
The Black–Scholes formula
Black–Scholes: The price of a European call option with strike Kand maturity T
Ct(K ,T ) = StΦ(d+)− Ke−r(T−t)Φ(d−)
I St underlying stock price (no dividends)
I r risk-free yield
I d± =log(St/K
)σ√
T − t+( r
σ± σ
2
)√T − t,
I Φ(x) =
∫ x
−∞
e−t2/2
√2π
dt
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
The Black–Scholes formula
I σ is the volatility of the underlying stock
I Unlike other parameters, not directly observed
σ2 =1
TVar(log ST )
I Liquid options priced by market already
I Options often quoted in terms of implied volatility
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
The Black–Scholes formula
The assumptions
I No arbitrage (existence of martingale measure)
I Calls of all maturities and strikes liquidly traded
I Zero interest rate
Let (St)t≥0 be a non-negative local martingale.
Price of European call option with strike K and maturity T
Ct(K ,T ) = St − E[ST ∧ K |Ft ]
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
The Black–Scholes formula
Warning:
E[(ST − K )+|Ft ] = E[ST − ST ∧ K |Ft ]
≤ St − E[ST ∧ K |Ft ]
= Ct(T ,K )
with equality if and only if S is a true martingale.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
The Black–Scholes formula
Black–Scholes call price function
BS(k, v) = Φ
(− k√
v+
√v
2
)− ekΦ
(− k√
v−√
v
2
)
DefinitionThe random variable Σt(k, τ) is defined on {St > 0} by
E[St+τ
St∧ ek
∣∣Ft
]= 1− BS
(k, τ Σt(k, τ)
2)
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
The Black–Scholes formula
Assumption: P(St > 0) > 0 all t ≥ 0, but St → 0 almost surely.Equivalently:
I There exists a k ∈ R such that τΣ(k, τ)2 ↑ ∞I τΣ(k, τ)2 ↑ ∞ for all k ∈ R.
I C (K ,T ) ↑ S0 for all K > 0.
I There exists a K > 0 such that C (K ,T ) ↑ S0
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
With no loss, let S0 = 1.
Theorem
τΣ(k, τ)2 = −8 log E(Sτ ∧ ek)| − 4 log[− log E(Sτ ∧ ek)]+4k − 4 log π + ε(k, τ)
where
sup−M≤k≤M
|ε(k, τ)|+ sup−M≤k1<k2≤M
|ε(k2, τ)− ε(k1, τ)|k2 − k1
→ 0
for all M > 0.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Corollary
supk∈[−M,M]
∣∣∣∣ τΣ(k, τ)2
−8 log E(Sτ ∧ 1)− 1
∣∣∣∣→ 0
as τ ↑ ∞ for all M > 0.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Towards Ross’s conjecture
Long implied volatility can never fall
Theorem (Rogers–T 2008)
For any k1, k2 ∈ R we have
lim supτ↑∞
Σt(k1, τ)− Σs(k2, τ) ≥ 0
for t ≥ s ≥ 0. There exist examples for which the inequality isstrict.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Towards Ross’s conjecture
Theorem (Dybvig–Ingersoll–Ross 1996)
Let ft(τ) be the instantaneous forward interest rate with long rate
lim supτ↑∞
ft(τ) = `t .
Then`s ≤ `t
for 0 ≤ s ≤ t.
See Hubalek–Klein–Teichmann (2002) for a nice proof.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Towards Ross’s conjecture
Theorem (Rogers–T. 2008)
SupposeΣt(k, τ) = Σ0(k, τ) + ξt
for some process (ξt)t≥0.
I Then ξt ≥ 0.
I If log E(S1/2s ) + log E(S
1/2t ) ≤ log E(S
1/2s+t), then ξt = 0.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Towards Ross’s conjecture
Theorem (Balland 2002)
If ξt = 0 for all t ≥ 0 then log S has independent, stationaryincrements.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Corollary
Let Q be the measure locally equivalent to P with densitydQtdPt
= St . Then
∂
∂kτΣ(k, τ)2 = 4
(Q(Sτ < ek)− ekP(Sτ ≥ ek)
Q(Sτ < ek) + ekP(Sτ ≥ ek)
)+ ε′(k, τ)
if the distribution of Sτ continuous at ek . In particular,
lim supτ↑∞
supk∈[−M,M]
∣∣∣∣ ∂∂kτΣ(k, τ)2
∣∣∣∣ ≤ 4
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
For comparison:
I∂
∂kΣ(k, τ)2 <
4
τfor all k ≥ 0
I∂
∂kΣ(k, τ)2 > −4
τfor all k ≤ 0
I∂
∂kΣ(k, τ)2 <
2
τfor all k ≥ k+(τ), for some k+
I∂
∂kΣ(k, τ)2 ≥ −2
τfor all k ≤ k−(τ) for some k−
Gatheral (1999), Carr–Wu (2003), Lee (2004), Benhaim–Friz(2008), Rogers–T. (2008).
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Motivation:S ∧ 1 ≤ Sp
for all 0 ≤ p ≤ 1 and S ≥ 0, implies
lim infτ↑∞
τΣ(k, τ)2
−8 inf0≤p≤1 log E(Spτ )≥ 1.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Letψt(p) = log E(Sp
t 1{St>0}).
Properties of ψt
I ψt(0) = log P(St > 0) ≤ 0, ψt(1) = log E(St) ≤ 0
I finite-valued on (0, 1) ⇒ real-analytic
I convex
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Intuition: For many models
1
tψt(p) → ψ(p)
Let p∗ be the minimizer of ψ. Three cases
I 0 < p∗ < 1
I p∗ = 0 or p∗ = 1
I p∗ < 0 or p∗ > 1
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Assumption: There exists a 0 < p∗ < 1 and a positive increasingfunction C with C (τ) ↑ ∞ such that
ψτ
(p∗ + i
θ
C (τ)
)− ψτ (p
∗) → −θ2/2
as τ ↑ ∞ for all real θ
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Theorem
supk∈[−M,M]
∣∣∣∣ τΣ(k, τ)2
−8ψτ (p∗)− 1
∣∣∣∣→ 0
Proof: Cramer’s large deviation principle.
See Lewis (2000), Jacquier (2007)
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
φτ (θ) =1√2π
exp
[ψτ
(p∗ + i
θ
C (τ)
)− ψτ (p
∗)
].
TheoremIf ∫ ∞
−∞
|φτ (θ)|1 + θ2/C (τ)2
dθ → 1
then
τΣ(k, τ)2 = −8ψτ (p∗)+4k(2p∗−1)+8 log
(C (τ)p∗(1− p∗)√
−ψτ (p∗)/2
)+δ(k, τ)
where supk∈[−M,M] |δ(k, τ)| → 0 as τ ↑ ∞ for each M > 0.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Assumption: There exists a p∗ ∈ {0, 1} and a positive increasingfunction C with C (τ) ↑ ∞ such that
ψτ
(p∗ + i
θ
C (τ)
)− ψτ (p
∗) → −θ2/2
as τ ↑ ∞ for all real θ
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
TheoremIf p∗ = 1 then
τΣ(k, τ)2 = −8ψτ (1)− 4 log[−ψτ (1)] + 4k − 4 log(π/4) + δ(k, τ),
and if p∗ = 0, then
τΣ(k, τ)2 = −8ψτ (0)− 4 log[−ψτ (0)]− 4k − 4 log(π/4) + δ(k, τ)
where supk∈[−M,M] |δ(k, τ)| → 0 for all M > 0.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
Assumption: There exists a p∗ such that either
1. p∗ > 1 and ψτ (p∗)− ψτ (1) → −∞, or
2. p∗ < 0 and ψτ (p∗)− ψτ (0) → −∞.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Regular caseBorderline casesIrregular cases
TheoremIf p∗ > 1 then
τΣ(τ, k)2 = −8ψτ (1)− 4 log[−ψτ (1)] + 4k − 4 log π + δ(k, τ),
and if p∗ < 0 then
τΣ(τ, k)2 = −8ψτ (0)− 4 log[−ψτ (0)]− 4k − 4 log π + δ(k, τ),
where supk∈[−M,M] |δ(k, τ)| → 0 for all M > 0.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Example: Independent, stationary increments
1
tψt(p) = ψ1(p)
If
infq 6=0
<ψ1(p∗ + iq)− ψ1(p
∗)
q2 ∧ 1< 0,
where < denotes the real part of a complex number, then the fullasymptotic formula holds.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Example: Binomial model. Suppose Sτ+1 = ξτ+1Sτ whereP(ξτ = eb) = 1
eb+1= 1− P(ξτ = e−b) so
ψ1(p) = log
(cosh[b(p − 1/2)]
cosh(b/2)
).
Integrability fails:
τΣ(k, τ)2 = 8τ log cosh(b/2) + 4 log
(b2F (k, τ)2
8 log cosh(c/2)
)+ δ(k, τ)
where
F (k) =∑n∈Z
(−1)nτ cos(knπ/b)
1 + 4n2π2/b26≡ 1.
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Example: Affine models. Lewis (2000), Jacquier (2007),Keller-Ressel (2008)
Michael Tehranchi Implied volatility at long maturities
IntroductionKey approximation
An analogue of DIRThe smile flattens
Cumulant generating functionExamples
Example: CEV model.
dSt = S2t dWt .
Note
E(St) = 2Φ
(1√t
)− 1,
Irregular case with p∗ > 1
τΣ(k, τ)2 = 4 log τ − 4 log log τ + 4k + δ(k, τ).
Michael Tehranchi Implied volatility at long maturities