Implied volatility at long maturities

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




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






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






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 ]






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.






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)






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





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.





Corollary

supk∈[−M,M]

∣∣∣∣ τΣ(k, τ)2

−8 log E(Sτ ∧ 1)− 1

∣∣∣∣→ 0

as τ ↑ ∞ for all M > 0.





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.






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.






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.






Theorem (Balland 2002)

If ξt = 0 for all t ≥ 0 then log S has independent, stationaryincrements.





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





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).





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.






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






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






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






Theorem

supk∈[−M,M]

∣∣∣∣ τΣ(k, τ)2

−8ψτ (p∗)− 1

∣∣∣∣→ 0

Proof: Cramer’s large deviation principle.

See Lewis (2000), Jacquier (2007)






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.






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






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.






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) → −∞.






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.





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.





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.





Example: Affine models. Lewis (2000), Jacquier (2007),Keller-Ressel (2008)





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).


Implied volatility at long maturities

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