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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 dS t = σ t S t dW t . ❚❤❡ t✐♠❡ ✵ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ˆ σ ❢♦r ♠❛t✉r✐t② T ❛♥❞ str✐❦❡ K s❛t✐s✜❡s √ � T � σ 2 t S 2 t ∂ 2 S C BS ( S t , ˆ � σ T − t, K ) dt 0 E σ 2 = ˆ . ✭✶✮ √ � T � � S 2 t ∂ 2 S C BS ( S t , ˆ σ T − t, K ) dt 0 E ❲❡ r❡❝❛❧❧ t❤❛t S C BS ( S, σ √ τ, K ) = σ √ τ n ( d + ( S, σ √ τ, K )) , S S 2 ∂ 2 d + ( S, σ √ τ, K ) = ln( S/K ) 2 σ √ τ. + 1 σ √ τ

In terms of S only: local volatility ❲❡ ❤❛✈❡ √ � T � � σ 2 loc ( t, S t ) S 2 t ∂ 2 S C BS ( S t , ˆ σ T − t, K ) dt 0 E σ 2 = ˆ √ � T � � S 2 t ∂ 2 S C BS ( S t , ˆ σ T − t, K ) dt 0 E ✇❤❡r❡ σ 2 � σ 2 � loc ( t, S t ) = E t | S t .

In terms of σ only: local gamma dollar ❲❡ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ✈♦❧❛t✐❧✐t② ♣❛t❤ σ 0 T = σ ( σ t , 0 ≤ t ≤ T ) ✿ ✐❢ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ σ ( W t , 0 ≤ t ≤ T ) , √ � � S 2 t ∂ 2 S C BS ( S t , ˆ = S 2 0 ∂ 2 S C BS ( S 0 , δ t , K ) σ T − t, K ) | σ 0 T ✭✷✮ E ✇❤❡r❡ � t σ 2 ( T − t ) δ 2 σ 2 t = s ds + ˆ 0 s♦ t❤❛t � T 0 E [ σ 2 t S 2 0 ∂ 2 S C BS ( S 0 , δ t , K )] dt σ 2 = ˆ ✭✸✮ � T 0 E [ S 2 0 ∂ 2 S C BS ( S 0 , δ t , K )] dt

Constant vol ❲❤❡♥ t❤❡ ✈♦❧❛t✐❧✐t② ♣❛t❤ ✐s ❝♦♥st❛♥t❧② ˆ σ ✱ ✭✷✮ r❡❛❞s √ √ � � S 2 t ∂ 2 S C BS ( S t , ˆ = S 2 0 ∂ 2 S C BS ( S 0 , ˆ σ T − t, K ) | σ 0 T ≡ ˆ σ σ T, K ) . E ❚❤✐s ✐s ♥♦r♠❛❧ ❜❡❝❛✉s❡ ✐♥ t❤❡ ❇❙ ♠♦❞❡❧ ✇✐t❤ ✈♦❧ σ, ˆ √ S 2 t ∂ 2 S C BS ( S t , ˆ σ T − t, K ) ✐s ❛ ♠❛rt✐♥❣❛❧❡✳

Deterministic vol ❲❤❡♥ σ t ✐s ❞❡t❡r♠✐♥✐st✐❝✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ✐♠♣❧✐❡❞ ✈♦❧ ˆ σ s❛t✐s✜❡s � T σ 2 T = σ 2 ˆ t dt 0 ❜✉t ❤♦✇ ❝❛♥ ②♦✉ r❡tr✐❡✈❡ t❤✐s ❡①♣r❡ss✐♦♥ ❢r♦♠ ✭✶✮ ♦r ✭✸✮❄ ❲❤❡♥ σ t ✐s ♥♦t ❝♦♥st❛♥t✱ t❤❡ ✇❡✐❣❤t w t = E [ S 2 0 ∂ 2 S C BS ( S 0 , δ t , K )] = S 2 0 ∂ 2 S C BS ( S 0 , δ t , K ) ✐s ♥♦t ❝♦♥st❛♥t✦

A framework for implied vol proxys ❲❡ ❝❛♥ ✇r✐t❡ � T � + σ 2 loc ( t, y ) ϕ ( t, y, ˆ σ ) dydt σ 2 = 0 R ∗ ˆ � T � + ϕ ( t, y, ˆ σ ) dydt 0 R ∗ ✇✐t❤ √ T − t, K ) 2 � � y − d + ( y, ˆ σ ϕ ( t, y, ˆ σ ) = exp f S t ( y ) . ✭✹✮ � 2 σ ˆ 2 π ( T − t ) ❆ ♣r♦①② ❢♦r ϕ ❧❡❛❞s t♦ ❛ ♣r♦①② ❢♦r σ. ˆ ϕ r❡❞✉❝❡s t♦ √ � � T,K ) 2 − d + ( S 0 , ˆ σ S 0 2 πT exp t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ S 0 ✇❤❡♥ t ❣♦❡s √ 2 ˆ σ t♦ ③❡r♦✱ ❛♥❞ t♦ K 2 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❡❞✿ � T 1 E [ σ 2 loc ( t, S t ) | S T = K ] dt. T 0 ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ ❛♣♣r♦①✐♠❛t✐♥❣ ϕ ( t, y, ˆ σ ) ❜② ϕ 2 ( t, y ) ✱ t❤❡ ❝♦♥❞✐✲ t✐♦♥❛❧ ❞❡♥s✐t② ♦❢ S t ❛t ♣♦✐♥t y ❣✐✈❡♥ S T = K ✳ ■♥❞❡❡❞✱ � T � � T + σ 2 loc ( t, y ) ϕ 2 ( t, y ) dy dt = 1 0 R ∗ E [ σ 2 loc ( t, S t ) | S T = K ] dt. � T T � + ϕ 2 ( t, y ) dy dt 0 0 R ∗ ◆♦t❡ t❤❛t ϕ 2 ( t, y ) r❡❞✉❝❡s t♦ ♦♥❡ t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ S 0 ✇❤❡♥ t ❣♦❡s t♦ ③❡r♦✱ ❛♥❞ t♦ ♦♥❡ t✐♠❡s t❤❡ ❉✐r❛❝ ♠❛ss ✐♥ K ✇❤❡♥ t ❣♦❡s t♦ T ✳

❍♦✇ t♦ ❝♦♠♣✉t❡ E [ σ 2 loc ( t, S t ) | S T = K ] ? � ¯ S t | ¯ ✇❤❡r❡ d ¯ σ ( t ) ¯ � • ❙t❡♣ ✶✿ L ( S t | S T = K ) ≃ L S T = K S t = ¯ S t dW t ✱ � ¯ S t | ¯ � ❢♦r s♦♠❡ ❞❡t❡r♠✐♥✐st✐❝ ¯ σ ( t ) ❀ L S T = K ✐s ❦♥♦✇♥ ✕ t❤❡ ❧❛✇ ✈❡rs✐♦♥✿ ¯ σ ( t ) = ¯ σ ∞ ( t ) ✇❤❡r❡ ¯ σ ∞ ( t ) = lim n →∞ ¯ σ n ( t ) ✇✐t❤ ¯ σ 0 ( t ) = σ ( t, S 0 ) ✭♦r ❛♥② ♦t❤❡r r❡❛s♦♥❛❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐✲ σ n +1 ( t ) = E [ σ loc ( t, ¯ � ¯ S n,T = K ] ✇❤❡r❡ d ¯ � t✐♦♥✮ ❛♥❞ ¯ S n,t ) S n,t = σ n ( t ) ¯ ¯ S n,t dW t ❀ ✕ t❤❡ ♠❡❛♥ ♣❛t❤ ✈❡rs✐♦♥✿ ✐❞❡♠ ❡①❝❡♣t t❤❛t σ n +1 ( t ) ¯ = σ loc ( t, E [ ¯ � ¯ S n,T = K ]) ❀ E [ ¯ � ¯ � � S n,t S n,t S n,T = K ] ❤❛s ❛ ❝❧♦s❡❞ ❢♦r✲ ♠✉❧❛✳ • ❙t❡♣ ✷✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ E [ σ 2 loc ( t, S t ) | S T = K ] loc ( t, ¯ S t ) | ¯ ✕ t❤❡ ❧❛✇ ✈❡rs✐♦♥✿ E [ σ 2 loc ( t, S t ) | S T = K ] ≃ E [ σ 2 S T = K ]; E [ σ 2 ✕ t❤❡ ♠❡❛♥ ♣❛t❤ ✈❡rs✐♦♥✿ loc ( t, S t ) | S T = K ] ≃ � ¯ S t | ¯ σ 2 � loc ( t, E S T = K ) .

❈♦♥❝❧✉s✐♦♥s✿ • ❚❤❡ ✏♠❡❛♥ ♣❛t❤ ❛t ❛❧❧ st❡♣s✑ ♠❡t❤♦❞ ✐s t♦t❛❧❧② ♦✛✳ ■♥❞❡❡❞✱ ✇r✐t✲ � ¯ S t | ¯ ✐♥❣ E [ σ 2 loc ( t, S t ) | S T = K ] ≃ σ 2 � loc ( t, E S T = 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❡ ¯ σ n +1 ( t ) ✐s ❤✐❣❤❡r t❤❛♥ ✐♥ t❤❡ ✏♠❡❛♥ ♣❛t❤ ❛t st❡♣ ✶ ♦♥❧②✑ ♠❡t❤♦❞✱ ❞✉❡ t♦ t❤❡ ❝♦♥✈❡①✐t② ♦❢ t❤❡ ❧♦❝❛❧ ✈♦❧✱ s♦ t❤❛t ❣✐✈❡♥ S T = K ✱ S t ❤❛s ❣r❡❛t❡r ✈❛r✐❛♥❝❡✱ ❛♥❞ � T 0 E [ σ 2 ( t, S t ) | S T = 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 ϕ 1 ( t, y, ¯ σ, σ Γ ) ✱ ❞❡✜♥❡❞ ❜② √ � � loc ( t, ¯ S t ) ¯ S C BS ( ¯ σ 2 S 2 t ∂ 2 S t , σ Γ ( t ) T − t, K ) E � σ 2 = loc ( t, y ) ϕ 1 ( t, y, ¯ σ, σ Γ ) dy R ∗ + ✇❤❡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❡ σ Γ ( t ) ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❡q✉❛❧ t♦ ˆ σ ✳

◆♦t❡ t❤❛t √ T − t, K ) 2 � � 1 − d + ( y, σ Γ ( t ) ϕ 1 ( t, y, ¯ σ, σ Γ ) = exp � 2 σ Γ ( t ) 2 π ( T − t ) √ t, S 0 ) 2 1 � − d + ( y, ˇ σ t � √ 2 πt exp 2 σ t ˇ 1 = � 2 πσ Γ ( t )ˇ σ t t ( T − t ) √ √ T − t, K ) 2 + d + ( y, ˇ t, S 0 ) 2 � � − d + ( y, σ Γ ( t ) σ t exp 2

s♦ t❤❛t S 0 e x ϕ 1 ( t, S 0 e x , ¯ σ, σ Γ )    2   S 0  − 1 1 = exp  λ t  x − x K + µ t     σ 2 � 2 Γ ( t )( T − t )  2 πσ Γ ( t )ˇ σ t t ( T − t ) 1 + σ 2 ˇ t t ✇❤❡r❡ � t t = 1 σ 2 σ ( u ) 2 du, ˇ ¯ t 0 x K = ln K , S 0 1 1 λ t = t t + Γ ( t )( T − t ) , σ 2 σ 2 ˇ � 2 � � σ 2 t t + σ 2 µ t = d + S 0 , ˇ Γ ( t )( T − t ) , K .