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❋r♦♠ ▲♦❝❛❧ t♦ ■♠♣❧✐❡❞ ❱♦❧❛t✐❧✐t✐❡s

❏✉❧✐❡♥ ●✉②♦♥ ❊q✉✐t② ❉❡r✐✈❛t✐✈❡s ◗✉❛♥t✐t❛t✐✈❡ ❘❡s❡❛r❝❤ ❙♦❝✐été ●é♥ér❛❧❡

Contents

✶✳ ▼♦t✐✈❛t✐♦♥s ✷✳ ❆ ❢r❛♠❡✇♦r❦ ❢♦r ✐♠♣❧✐❡❞ ✈♦❧ ♣r♦①②s ✸✳ ❚✇♦ ♣r♦①②s

• ❚❤❡ ✏♠♦st ❧✐❦❡❧② ♣❛t❤✑ ♣r♦①②
• ❆ ♥❡✇ ♦♥❡

✹✳ ❙♦♠❡ ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

A link between the volatility path and the im- plied volatility

▲❡t ✉s ❝♦♥s✐❞❡r ❛♥ ❛ss❡t S ✇✐t❤ ❞②♥❛♠✐❝s dSt = σtSt dWt. ❚❤❡ t✐♠❡ ✵ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ˆ σ ❢♦r ♠❛t✉r✐t② T ❛♥❞ str✐❦❡ K s❛t✐s✜❡s ˆ σ2 = T

0 E

• σ2

t S2 t ∂2 SCBS(St, ˆ

σ √ T − t, K)

• dt

T

0 E

• S2

t ∂2 SCBS(St, ˆ

σ √ T − t, K)

• dt

. ✭✶✮ ❲❡ r❡❝❛❧❧ t❤❛t S2∂2

SCBS(S, σ√τ, K) =

S σ√τ n(d+(S, σ√τ, K)), d+(S, σ√τ, K) = ln(S/K) σ√τ + 1 2σ√τ.

In terms of S only: local volatility

❲❡ ❤❛✈❡ ˆ σ2 = T

0 E

• σ2

loc (t, St) S2 t ∂2 SCBS(St, ˆ

σ √ T − t, K)

• dt

T

0 E

• S2

t ∂2 SCBS(St, ˆ

σ √ T − t, K)

• dt

✇❤❡r❡ σ2

loc (t, St) = E

• σ2

t |St

• .

In terms of σ only: local gamma dollar

❲❡ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ✈♦❧❛t✐❧✐t② ♣❛t❤ σ0T = σ (σt, 0 ≤ t ≤ T) ✿ ✐❢ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ σ (Wt, 0 ≤ t ≤ T) , E

• S2

t ∂2 SCBS(St, ˆ

σ √ T − t, K)|σ0T

• = S2

0∂2 SCBS(S0, δt, K)

✭✷✮ ✇❤❡r❡ δ2

t =

t σ2

sds + ˆ

σ2 (T − t) s♦ t❤❛t ˆ σ2 = T

0 E [σ2 t S2 0∂2 SCBS(S0, δt, K)] dt

T

0 E [S2 0∂2 SCBS(S0, δt, K)] dt

✭✸✮

Constant vol

❲❤❡♥ t❤❡ ✈♦❧❛t✐❧✐t② ♣❛t❤ ✐s ❝♦♥st❛♥t❧② ˆ σ✱ ✭✷✮ r❡❛❞s E

• S2

t ∂2 SCBS(St, ˆ

σ √ T − t, K)|σ0T ≡ ˆ σ

• = S2

0∂2 SCBS(S0, ˆ

σ √ T, K). ❚❤✐s ✐s ♥♦r♠❛❧ ❜❡❝❛✉s❡ ✐♥ t❤❡ ❇❙ ♠♦❞❡❧ ✇✐t❤ ✈♦❧ ˆ σ, S2

t ∂2 SCBS(St, ˆ

σ √ T − t, K) ✐s ❛ ♠❛rt✐♥❣❛❧❡✳

Deterministic vol

❲❤❡♥ σt ✐s ❞❡t❡r♠✐♥✐st✐❝✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ✐♠♣❧✐❡❞ ✈♦❧ ˆ σ s❛t✐s✜❡s ˆ σ2T = T σ2

t dt

❜✉t ❤♦✇ ❝❛♥ ②♦✉ r❡tr✐❡✈❡ t❤✐s ❡①♣r❡ss✐♦♥ ❢r♦♠ ✭✶✮ ♦r ✭✸✮❄ ❲❤❡♥ σt ✐s ♥♦t ❝♦♥st❛♥t✱ t❤❡ ✇❡✐❣❤t wt = E [S2

0∂2 SCBS(S0, δt, K)] =

S2

0∂2 SCBS(S0, δt, K) ✐s ♥♦t ❝♦♥st❛♥t✦

A framework for implied vol proxys

❲❡ ❝❛♥ ✇r✐t❡ ˆ σ2 = T

• R∗

+ σ2

loc(t, y)ϕ(t, y, ˆ

σ)dydt T

• R∗

+ ϕ(t, y, ˆ

σ)dydt ✇✐t❤ ϕ(t, y, ˆ σ) = y ˆ σ

• 2π(T − t)

exp

• −d+(y, ˆ

σ √ T − t, K)2 2

• fSt(y). ✭✹✮

❆ ♣r♦①② ❢♦r ϕ ❧❡❛❞s t♦ ❛ ♣r♦①② ❢♦r ˆ σ. ϕ r❡❞✉❝❡s t♦

S0 ˆ σ √ 2πT exp

• −d+(S0,ˆ

σ √ T,K)2 2

• t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ S0 ✇❤❡♥ t ❣♦❡s

t♦ ③❡r♦✱ ❛♥❞ t♦ K2 t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ K ✇❤❡♥ t ❣♦❡s t♦ T✳ Pr♦①②s ❢♦r ϕ s❤♦✉❧❞ s❤❛r❡ t❤✐s ♣r♦♣❡rt②✳

Adil Reghai’s proxy

❆❞✐❧ ❘❡❣❤❛✐ ❤❛s ♣r♦♣♦s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦①② ❢♦r t❤❡ ✐♠♣❧✐❡❞ ✈♦❧ sq✉❛r❡❞✿ 1 T T E[σ2

loc(t, St)|ST = K] dt.

❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ ❛♣♣r♦①✐♠❛t✐♥❣ ϕ(t, y, ˆ σ) ❜② ϕ2(t, y)✱ t❤❡ ❝♦♥❞✐✲ t✐♦♥❛❧ ❞❡♥s✐t② ♦❢ St ❛t ♣♦✐♥t y ❣✐✈❡♥ ST = K✳ ■♥❞❡❡❞✱ T

• R∗

+ σ2

loc(t, y)ϕ2(t, y)dy dt

T

• R∗

+ ϕ2(t, y)dy dt

= 1 T T E[σ2

loc(t, St)|ST = K] dt.

◆♦t❡ t❤❛t ϕ2(t, y) r❡❞✉❝❡s t♦ ♦♥❡ t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ S0 ✇❤❡♥ t ❣♦❡s t♦ ③❡r♦✱ ❛♥❞ t♦ ♦♥❡ t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ K ✇❤❡♥ t ❣♦❡s t♦ T✳

❍♦✇ t♦ ❝♦♠♣✉t❡ E[σ2

loc(t, St)|ST = K] ?

• ❙t❡♣ ✶✿ L (St|ST = K) ≃ L

¯ St| ¯ ST = K

• ✇❤❡r❡ d ¯

St = ¯ σ(t) ¯ StdWt✱ ❢♦r s♦♠❡ ❞❡t❡r♠✐♥✐st✐❝ ¯ σ(t)❀ L ¯ St| ¯ ST = K

• ✐s ❦♥♦✇♥

✕ t❤❡ ❧❛✇ ✈❡rs✐♦♥✿ ¯ σ(t) = ¯ σ∞(t) ✇❤❡r❡ ¯ σ∞(t) = limn→∞ ¯ σn(t) ✇✐t❤ ¯ σ0(t) = σ(t, S0) ✭♦r ❛♥② ♦t❤❡r r❡❛s♦♥❛❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐✲ t✐♦♥✮ ❛♥❞ ¯ σn+1(t) = E[σloc(t, ¯ Sn,t)

• ¯

Sn,T = K] ✇❤❡r❡ d ¯ Sn,t = ¯ σn(t) ¯ Sn,tdWt❀ ✕ t❤❡ ♠❡❛♥ ♣❛t❤ ✈❡rs✐♦♥✿ ✐❞❡♠ ❡①❝❡♣t t❤❛t ¯ σn+1(t) = σloc(t, E[ ¯ Sn,t

• ¯

Sn,T = K])❀ E[ ¯ Sn,t

• ¯

Sn,T = K] ❤❛s ❛ ❝❧♦s❡❞ ❢♦r✲ ♠✉❧❛✳

• ❙t❡♣ ✷✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ E[σ2

loc(t, St)|ST = K]

✕ t❤❡ ❧❛✇ ✈❡rs✐♦♥✿ E[σ2

loc(t, St)|ST = K] ≃ E[σ2 loc(t, ¯

St)| ¯ ST = K]; ✕ t❤❡ ♠❡❛♥ ♣❛t❤ ✈❡rs✐♦♥✿ E[σ2

loc(t, St)|ST

= K] ≃ σ2

loc(t, E

¯ St| ¯ ST = K

• ).

❈♦♥❝❧✉s✐♦♥s✿

• ❚❤❡ ✏♠❡❛♥ ♣❛t❤ ❛t ❛❧❧ st❡♣s✑ ♠❡t❤♦❞ ✐s t♦t❛❧❧② ♦✛✳ ■♥❞❡❡❞✱ ✇r✐t✲

✐♥❣ E[σ2

loc(t, St)|ST = K] ≃ σ2 loc(t, E

¯ St| ¯ ST = K

• ) ❜♦✐❧s ❞♦✇♥ t♦

✐❣♥♦r✐♥❣ t❤❡ ❝♦♥✈❡①✐t② ♦❢ t❤❡ ❧♦❝❛❧ ✈♦❧❛t✐❧✐t②✳

• ❚❤❡ ✏♥♦ ♠❡❛♥ ♣❛t❤✑ ♠❡t❤♦❞ ❣✐✈❡s ❜❡tt❡r r❡s✉❧ts✱ ❜✉t t❤❡ s♠✐❧❡ ✐s

t♦♦ ❤✐❣❤✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐♥ s✉❝❤ ❛ ❝❛s❡ ¯ σn+1(t) ✐s ❤✐❣❤❡r t❤❛♥ ✐♥ t❤❡ ✏♠❡❛♥ ♣❛t❤ ❛t st❡♣ ✶ ♦♥❧②✑ ♠❡t❤♦❞✱ ❞✉❡ t♦ t❤❡ ❝♦♥✈❡①✐t② ♦❢ t❤❡ ❧♦❝❛❧ ✈♦❧✱ s♦ t❤❛t ❣✐✈❡♥ ST = K✱ St ❤❛s ❣r❡❛t❡r ✈❛r✐❛♥❝❡✱ ❛♥❞ T

0 E[σ2(t, St)|ST = K] ✐s ❜✐❣❣❡r✳

• ❚❤❡ ❜❡st ✜t ✐s ♦❜t❛✐♥❡❞ ❜② t❤❡ ✏♠❡❛♥ ♣❛t❤ ❛t st❡♣ ✶ ♦♥❧②✑ ♠❡t❤♦❞✳

❲❡ s❡❧❡❝t ✐t✳ ◆❡✈❡rt❤❡❧❡ss✱ ✐t s❡❡♠s t♦ ❣✐✈❡ ❛ s❧✐❣❤t❧② ❤✐❣❤✲❜✐❛s❡❞ ❡st✐♠❛t♦r ♦❢ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧✳

A new proxy

▲❡t ✉s ❛♣♣r♦①✐♠❛t❡ ϕ(t, y, ˆ σ) ❜② ϕ1(t, y, ¯ σ, σΓ)✱ ❞❡✜♥❡❞ ❜② E

• σ2

loc(t, ¯

St) ¯ S2

t ∂2 SCBS( ¯

St, σΓ(t) √ T − t, K)

• =
• R∗

+

σ2

loc(t, y)ϕ1(t, y, ¯

σ, σΓ)dy ✇❤❡r❡ ¯ S ❢♦❧❧♦✇s ❛ ❇❙ ❞②♥❛♠✐❝s ✇✐t❤ s♦♠❡ ❞❡t❡r♠✐♥✐st✐❝ ✈♦❧ ¯ σ(t)✳ ❚❤✐s ✐s ♥♦t t❤❡ ♥✉♠❡r❛t♦r ♦❢ ✭✶✮ ❜❡❝❛✉s❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ S ❛♥❞ ¯ S ❞✐✛❡r ❛♥❞ ❜❡❝❛✉s❡ σΓ(t) ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❡q✉❛❧ t♦ ˆ σ✳

◆♦t❡ t❤❛t ϕ1(t, y, ¯ σ, σΓ) = 1 σΓ(t)

• 2π(T − t)

exp

• −d+(y, σΓ(t)

√ T − t, K)2 2

• 1

ˇ σt √ 2πt exp

• −d+(y, ˇ

σt √ t, S0)2 2

• =

1 2πσΓ(t)ˇ σt

• t(T − t)

exp

• −d+(y, σΓ(t)

√ T − t, K)2 + d+(y, ˇ σt √ t, S0)2 2

s♦ t❤❛t S0exϕ1(t, S0ex, ¯ σ, σΓ) = S0 2πσΓ(t)ˇ σt

• t(T − t)

exp   −1 2   λt  x − 1 1 +

σ2

Γ(t)(T−t)

ˇ σ2

t t

xK  

2

+ µt    ✇❤❡r❡ ˇ σ2

t = 1

t t ¯ σ(u)2du, xK = ln K S0 , λt = 1 ˇ σ2

t t +

1 σ2

Γ(t)(T − t),

µt = d+

• S0,
• ˇ

σ2

t t + σ2 Γ(t)(T − t), K

2 .

E

• σ2

loc(t, ¯

St)∂2

SCBS( ¯

St, σ2

Γ(t), T − t, K) ¯

S2

t

• = S0
• R

σ2

loc(t, S0ex)ϕ1(t, S0ex, ¯

σ, σΓ)exdx = S0

• R

σ2

loc(t, S0ex)

1 2πσΓ(t)ˇ σt

• t(T − t)

exp   −1 2   λt  x − 1 1 +

σ2

Γ(t)(T−t)

ˇ σ2

t t

xK  

2

+ µt       dx = S0

• R

σ2

loc

 t, S0e

1 1+ σ2 Γ(t)(T−t) ˇ σ2 t t

xK +

1

√

λt z

 1 2π

• ˇ

σ2

t t + σ2 Γ(t)(T − t)

exp

• −1

2

• z2 + µt
• dz

= S0 √ 2π

• ˇ

σ2

t t + σ2 Γ(t)(T − t)

exp   − d+

• S0,
• ˇ

σ2

t t + σ2 Γ(t)(T − t), K

2 2    ×

• R

σ2

loc

 t, S0 exp   1 1 +

σ2

Γ(t)(T−t)

ˇ σ2

t t

xK + σΓ(t)ˇ σt

• t(T − t)
• ˇ

σ2

t t + σ2 Γ(t)(T − t)

z     1 √ 2π exp

• −1

2z2

• dz

❚❤✐s ❣✐✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦①② ❢♦r t❤❡ sq✉❛r❡❞ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✿ T

0 E

• σ2

loc(t, ¯

St)∂2

SCBS( ¯

St, σΓ(t) √ T − t, K) ¯ S2

t

• dt

T

0 E

• ∂2

SCBS( ¯

St, σΓ(t) √ T − t, K) ¯ S2

t

• dt

= T

0 wt

• R σ2

loc

• t, S0 exp
• 1

1+

σ2 Γ(t)(T−t) ˇ σ2 t t

xK +

σΓ(t)ˇ σt√ t(T−t)

√

ˇ σ2

t t+σ2 Γ(t)(T−t)z

• n(z)dzdt

T

0 wtdt

✭✺✮ ✇❤❡r❡ tˇ σ2

t =

t

0 ¯

σ(u)2du ❛♥❞ wt = 1

• ˇ

σ2

t t + σ2 Γ(t)(T − t)

exp   − d+

• S0,
• ˇ

σ2

t t + σ2 Γ(t)(T − t), K

2 2   

Perturbation at first order around ¯ σ

❲❤❡♥ ¯ σ(t) = σΓ(t) = ¯ σ✱ t❤✐s r❡❞✉❝❡s t♦ 1 T T

• R

σ2

loc

• t, S0 exp
• t

T xK + ¯ σ √ T

• t

T

• 1 − t

T

• z
• n(z)dzdt.✭✻✮

❚❤✐s ✐s r❡❧❛t❡❞ t♦ t❤❡ ❢❛❝t t❤❛t ✐♥ s✉❝❤ ❛ ❝❛s❡ ¯ S2

t ∂2 SCBS( ¯

St, ¯ σ √ T − t, K) ✐s ❛ ♠❛rt✐♥❣❛❧❡ s♦ t❤❛t E

• ¯

S2

t ∂2 SCBS( ¯

St, ¯ σ √ T − t, K)

• = ¯

S2

0∂2 SCBS( ¯

S0, ¯ σ √ T, K) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t✳ ■t ✐s ❡q✉❛❧ t♦ ¯ S0e−µ/2

1 ¯ σ √ 2πT ✇❤❡r❡ ✐♥ s✉❝❤ ❛ ❝❛s❡

µ ≡ µt ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t✳

❍♦✇ t♦ ❝❤♦♦s❡ ¯ σ(t)?

• ∂2

SCBS(St, σΓ(t), T −t, K) t❡♥❞s t♦ ❛ ❉✐r❛❝ ♠❛ss ❛t ♣♦✐♥t ST = K

✇❤❡♥ t t❡♥❞s t♦ T✱ s♦ t❤❛t ❛t t❤❡s❡ t✐♠❡s✱ t❤❡ ♦♥❧② ♣❛t❤s t❤❛t ❝♦♥tr✐❜✉t❡ t♦ E [σ2

loc(t, St)∂2 SCBS(St, σΓ(t), T − t, K)S2 t ] ❛r❡ t❤♦s❡

✇✐t❤ ST = K✳ ❆t t❤❡s❡ t✐♠❡s✱ ✐t ✐s ✇✐s❡ t♦ ❝❤♦♦s❡ ❛ ¯ σ(t) ❝❧♦s❡ t♦ σloc(T, K)✳

• ❙②♠♠❡tr✐❝❛❧❧②✱ ✐t ✐s ♥❛t✉r❛❧ t♦ ❝❤♦♦s❡ ❛ ¯

σ(t) ❝❧♦s❡ t♦ σloc(0, S0) ✇❤❡♥ t ✐s ❝❧♦s❡ t♦ ✵✳ ❲❡ ♠❛② t❛❦❡

• ¯

σ(t) = σloc

• t, S0 exp
• t

T ln K S0

• ♦r ¯

σ(t) = ¯ σ∞(t) ✇❤❡r❡ ¯ σ∞(t) = limn→∞ ¯ σn(t) ✇✐t❤ ¯ σ0(t) = σ(t, S0) ✭♦r ❛♥② ♦t❤❡r r❡❛s♦♥❛❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✮ ❛♥❞ ¯ σn+1(t) = σloc(t, E[ ¯ Sn,t

• ¯

Sn,T = K]) ✇❤❡r❡ d ¯ Sn,t = ¯ σn(t) ¯ Sn,tdWt✳

Numerical experiment

❲❡ ♣✐❝❦❡❞ ❛ ❤②♣❡r❜♦❧✐❝ ❧♦❝❛❧ ✈♦❧❛t✐❧✐t② ❢✉♥❝t✐♦♥✳ ❲❡ ❝♦♠♣❛r❡ ✸ ♠❡t❤✲ ♦❞s✿ ✶✳ t❤❡ ❢♦r✇❛r❞ P❉❊ ♠❡t❤♦❞✱ ✷✳ t❤❡ ϕ1 ♣r♦①② ✇✐t❤ ❝♦♥st❛♥t ¯ σ(t) = σΓ(t) = ¯ σ = σloc(0, S0) ❛♥❞ ❛ ❍❡r♠✐t❡ q✉❛❞r❛t✉r❡✱ ✐✳❡✳ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❛t ✜rst ♦r❞❡r ❛r♦✉♥❞ σloc(0, S0)✱ ✸✳ t❤❡ ϕ2 ♣r♦①②✳ ❋♦r ✈❛r✐♦✉s ♠❛t✉r✐t✐❡s✱ ✇❡ ❣r❛♣❤ t❤❡ ✸ s♠✐❧❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✸ ♠❡t❤♦❞s✱ ♣❧✉s t❤❡ ❧♦❝❛❧ ✈♦❧❛t✐❧✐t② ❛t t❤✐s ❞❛t❡✳

Data 1: the Eurostoxx 50 smile ❋✐rst ✇❡ ♣✐❝❦ ♣❛r❛♠❡t❡rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ ❊✉r♦st♦①① ✺✵ s♠✐❧❡✳ ❚❤❡ t❤r❡❡ s♠✐❧❡s ❤❛✈❡ t❤❡ s❛♠❡ s❤❛♣❡✱ ❡①❝❡♣t ❢♦r t❤❡ ϕ1 ❛♣♣r♦①✐♠❛t✐♦♥ ❛t s❤♦rt ♠❛t✉r✐t✐❡s ❛♥❞ ❢❛r ❢r♦♠ t❤❡ ♠♦♥❡②✳ ❚❤❡ ϕ1 ❛♥❞ ϕ2 ♣r♦①②s s❡❡♠ t♦ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ s♠✐❧❡s ❢♦r ❧❛r❣❡ ♠❛t✉r✐t✐❡s✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡s❡ s♠✐❧❡s ❛♣♣❡❛r t♦ ❜❡ ❛ ❧✐tt❧❡ ❜✐t ❤✐❣❤❡r t❤❛♥ t❤❡ ❡①❛❝t ✏❢♦r✇❛r❞ P❉❊✑ s♠✐❧❡✳

Data 2: A U-shaped smile ❲❡ ♣✐❝❦ ♣❛r❛♠❡t❡rs s♦ t❤❛t t❤❡ ❧♦❝❛❧ ✈♦❧✱ ❤❡♥❝❡ t❤❡ s♠✐❧❡✱ ✐s ❯✲ s❤❛♣❡❞✳ ❍❡r❡ t❤❡ ❧♦❝❛❧ ✈♦❧❛t✐❧✐t② ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥ t✐♠❡✳ ❚❤❡r❡ ✐s ❛ ❣♦♦❞ ❛❣r❡❡♠❡♥t ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ s♠✐❧❡s✳ ❚❤❡ ϕ1 ❛♣♣r♦①✐♠❛t✐♦♥ ❣✐✈❡s ❜❡tt❡r r❡s✉❧ts t❤❛♥ t❤❡ ϕ2 ♦♥❡ ❢♦r ❧♦♥❣ ♠❛t✉r✐t✐❡s✳

Data 3: another U-shaped smile ❍❡r❡ t❤❡ ❧♦❝❛❧ ✈♦❧ ✐s st✐❧❧ ❯✲s❤❛♣❡❞✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✐♠❡✱ ❜✉t ✇✐t❤ ♠♦r❡ st❡❡♣ ✇✐♥❣s✱ ❛♥❞ ❛ ♠✐♥✐♠✉♠ ❛r♦✉♥❞ ✶✵✺✪✳ ❚❤❡r❡ ✐s ❛ ♣♦♦r ❛❣r❡❡♠❡♥t ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ s♠✐❧❡s✱ ✇✐t❤ t❤❡ ✏❢♦r✇❛r❞ P❉❊✑ s♠✐❧❡ ❜❡✐♥❣ s②st❡♠❛t✐❝❛❧❧② ❛♥❞ s✐❣♥✐✜❝❛♥t❧② ❧♦✇❡r t❤❛♥ t❤❡ t✇♦ ♦t❤❡rs✳

Data 4: a slowly decreasing smile ❍❡r❡ t❤❡ ❧♦❝❛❧ ✈♦❧❛t✐❧✐t② ✐s s❧♦✇❧② ❞❡❝r❡❛s✐♥❣✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✐♠❡✳ ❚❤❡ t❤r❡❡ s♠✐❧❡s ❛❣r❡❡ r❡♠❛r❦❛❜❧②✳ ❍❡r❡ ✇❡ ❝❤❡❝❦ t❤❛t✱ ❛s ❡①♣❡❝t❡❞✱ ϕ1 ✐s ❛ ❣♦♦❞ ♣r♦①② ❢♦r ϕ ✇❤❡♥ t❤❡ ❧♦❝❛❧ ✈♦❧❛t✐❧✐t② ❢✉♥❝t✐♦♥ ❤❛s ❧✐tt❧❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ st♦❝❦ ✈❛❧✉❡✳ ❲❤❡♥ ✇❡ ❝♦♠♣❛r❡ ✇✐t❤ ❞❛t❛ ✸✱ ✐♥ ✇❤✐❝❤ t❤❡ ❧♦❝❛❧ ✈♦❧ ✐s s❧♦✇❧② ✈❛r②✐♥❣ t♦♦✱ ✐t s❡❡♠s t❤❛t t❤❡ s♠❛❧❧❡r t❤❡ ❝♦♥✈❡①✐t② ♦❢ t❤❡ ❧♦❝❛❧ ✈♦❧ ✐s✱ t❤❡ ❜❡tt❡r t❤❡ r❡s✉❧ts ♦❢ t❤❡ ϕ1 ❛♥❞ ϕ2 ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡✳

Other numerical experiments

❲❡ ❤❛✈❡ t❡st❡❞ ♦t❤❡r ♠❡t❤♦❞s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧✳ ❚❤❡② ❝♦♥s✐st❡❞ ✐♥✿ ✶✳ ■t❡r❛t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ σΓ ∈ R → T

0 E

• σ2

loc(t, ¯

St) ¯ S2

t ∂2 SCBS( ¯

St, σΓ √ T − t, K)

• dt

T

0 E

¯ S2

t ∂2 SCBS( ¯

St, σΓ √ T − t, K)

• dt

s✐♥❝❡ t❤❡ ❡①❛❝t ✐♠♣❧✐❡❞ ✈♦❧ ✐s t❤❡ ✜①❡❞ ♣♦✐♥t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥ ✇✐t❤ S ✐♥st❡❛❞ ♦❢ ¯ S✳ ❋♦r t❤❡ ❯✲s❤❛♣❡❞ ❡①❛♠♣❧❡s✱ t❤✐s ❧❡❛❞s t♦ ❤✐❣❤❡r ✐♠♣❧✐❡❞ ✈♦❧s t❤❛♥ t❤❡ s✐♥❣❧❡✲✐t❡r❛t✐♦♥✱ ✐✳❡✳ t❤❡ ϕ1 ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❤❡r❡❛s t❤❡ ❧❛tt❡r ✐s ❛❧r❡❛❞② t♦♦ ❤✐❣❤✳ ✷✳ ❚r②✐♥❣ ❞✐✛❡r❡♥t ¯ σ(t)✬s ✐♥st❡❛❞ ♦❢ ❛ ❝♦♥st❛♥t ¯ σ ✐♥ t❤❡ ϕ1 ❛♣♣r♦①✲ ✐♠❛t✐♦♥✳ ❚❤✐s ❧❡❛❞s t♦ ✈❡r② ♠✐♥♦r ❝❤❛♥❣❡s ✐♥ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧s✳ ❍❡♥❝❡ ✇❡ ❞♦ ♥♦t ❣r❛♣❤ t❤❡♠✳