implied volatility - saurabh.com singal saturday, august 14 ... rule of thumb for atm option price...

Implied VolatilitySingapore Management University

QF 301Saurabh Singal

Saturday, August 14, 2010

• What is it?

• How to calculate?

• Bisection Method and Newton-Raphson Method

• Smile

• Term structure

Implied Volatility


Bisection Method (Ms Nan’s MATLAB code)


!"#$%&'()*"$+%,)

)

)

-%)#")+./")

)

)

!"#$%&'()*"$+%,)

)

)

-%)#")+./")

)

)

!"#$%&'()*"$+%,)

)

)

-%)#")+./")

)

)

!"#$%&'()*"$+%,)

)

)

-%)#")+./")

)

)


Drop in Implied Vols

• Before an important political announcement, like vote on a no-confidence motion, budget announcement, FOMC meeting, trading grinds to a halt. “Actual volatility” is low. But implied vols are high. After the event, vols drop as if a ballon that is punctured.

• Calls expensive before earnings!

• Weekend effect: cautious of holding options over the weekend owing to theta. Therefore option prices drop and people adjust vols lower. of Monday


ATMOption=σ T2π

and 12π = 0.3989

Rule of Thumb for ATM Option Price


Volatility Smile for SPX options (SKEW on Bloomberg)


Term Structure of Volatility, SPX options (TRMS on Bloomberg)


• Fx Markets use sticky-Delta to convey smile informations (Implied Volatility is quoted in delta rather than strike, e.g., 25 delta strike). If FX spot moves, the implied volatilities vs Delta stay the same but they change vs strike.

• Implied volatility quotes are given in terms of a straddle, risk reversal and butterfly (which is [strangle-straddle]/2)

Volatility Quotation in FX Derivatives


Vanna-Volga Method

In case one is given a value of σBF25(1vol) from the market, and wants to use it to build an impliedsmile curve, one may proceed the following way:

pseudo-algorithm 2

1. Select an initial guess for σBF25(2vol)

2. Construct an implied smile curve using σBF25(2vol) and market value of σRR25

3. Compute the value of σBF25(1vol) (for instance following guidelines of pseudo-algorithm 1)

4. Compare σBF25(1vol) you obtained in 3 to the market-given one.

5. If the difference between the two values exceeds some tolerance, adapt the value σBF25(2vol) andgo back to 2.

To close this section on the broker’s Strangle issue, let us clarify another enigmatic concept of FXmarkets often used by practitioners, the so-called Vega-weighted Strangle quote. This is in fact anapproximation for the value of σSTG25(1vol). To show this, we start from equality (11). First we assumeK∗

25P = K25P and K∗25C = K25C . Next, we develop both sides in a first order Taylor expansion in σ

around σATM . After canceling repeating terms on the left and right-hand side, we are left with:

(σSTG25(1vol) − σATM) · (V(K25P ,σATM) + V(K25C ,σATM))≈ (σ25∆P − σATM) · V(K25P ,σATM) + (σ25∆C − σATM) · V(K25C ,σATM)

(12)

where V(K, σ) represents the Vega of the option, namely the sensitivity of the option price P withrespect to a change of the implied volatility: V = ∂P

∂σ . Solving this for σSTG25(1vol) yields:

σSTG25(1vol) ≈σ25∆P · V(K25P ,σATM) + σ25∆C · V(K25C ,σATM)

V(K25P ,σATM) + V(K25C ,σATM)(13)

which corresponds to the average (weighted by Vega) of the call and put implied volatilities.

4 The Vanna-Volga Method

The Vanna-Volga method consists in adjusting the Black-Scholes TV by the cost of a portfolio whichhedges three main risks associated to the volatility of the option, the Vega, the Vanna and the Volga.The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate: Vanna = ∂V

∂S .Similarly, the Volga is the sensitivity of the Vega with respect to a change of the implied volatility σ:Volga = ∂V

∂σ . The hedging portfolio will be composed of the following three strategies:

ATM =12Straddle(KATM)

RR = Call(Kc,σ(Kc))− Put(Kp,σ(Kp))

BF =12Strangle(Kc,Kp)−

12Straddle(KATM)

where KATM represents the ATM strike, Kc/p the 25-Delta call/put strikes obtained by solving theequations ∆call(Kc,σATM) = 1

4 and ∆put(Kp,σATM) = −14 and σ(Kc/p) the corresponding volatilities

evaluated from the smile surface.

8

4.1 The general framework

In this section we present the Vanna-Volga methodology.The simplest formulation [17] suggests that the Vanna-Volga price XVV of an exotic instrument

X is given by

XVV = XBS +Vanna(X)Vanna(RR)

wRR

RRcost +Volga(X)Volga(BF)

wBF

BFcost (14)

where by XBS we denoted the Black-Scholes price of the exotic and the Greeks are calculated withATM volatility. Also, for any instrument I we define its ‘smile cost’ as the difference between its pricecomputed with/without including the smile effect: Icost = Imkt − IBS, and in particular

RRcost = [Call(Kc,σ(Kc))− Put(Kp,σ(Kp))]− [Call(Kc,σATM)− Put(Kp,σATM)]

BFcost =12

[Call(Kc,σ(Kc)) + Put(Kp,σ(Kp))]

−12

[Call(Kc,σATM) + Put(Kp,σATM)] (15)

The rationale behind (14) is that one can extract the smile cost of an exotic option by measuring thesmile cost of a portfolio designed to hedge its Vanna and Volga risks. The reason why one chooses thestrategies BF and RR to do this is because they are liquid FX instruments and they carry respectivelymainly Volga and Vanna risks. The weighting factors wRR and wBF in (14) represent respectively theamount of RR needed to replicate the option’s Vanna, and the amount of BF needed to replicate theoption’s Volga. The above approach ignores the small (but non-zero) fraction of Volga carried by theRR and the small fraction of Vanna carried by the BF. It further neglects the cost of hedging the Vegarisk. This has led to a more general formulation of the Vanna-Volga method in which one considersthat within the BS assumptions the exotic option’s Vega, Vanna and Volga can be replicated by theweighted sum of three instruments:

Xi = wATM ATMi + wRR RRi + wBF BFi i=vega, vanna, volga (16)

where the weightings are obtained by solving the following system:

x = Aw (17)

with

A =

ATMvega RRvega BFvega

ATMvanna RRvanna BFvanna

ATMvolga RRvolga BFvolga

w =

wATM

wRR

wBF

x =

Xvega

Xvanna

Xvolga

(18)

Given this replication, the Vanna-Volga method adjusts the BS price of an exotic option by the smilecost of the above weighted sum (note that the ATM smile cost is zero by construction):

XVV = XBS + wRRRRmkt − RRBS

+ wBF

BFmkt − BFBS

= XBS + xT (AT )−1I = XBS + Xvega Ωvega + Xvanna Ωvanna + Xvolga Ωvolga

(19)

9

4.1 The general framework

In this section we present the Vanna-Volga methodology.The simplest formulation [17] suggests that the Vanna-Volga price XVV of an exotic instrument

X is given by

XVV = XBS +Vanna(X)Vanna(RR)

wRR

RRcost +Volga(X)Volga(BF)

wBF

BFcost (14)

where by XBS we denoted the Black-Scholes price of the exotic and the Greeks are calculated withATM volatility. Also, for any instrument I we define its ‘smile cost’ as the difference between its pricecomputed with/without including the smile effect: Icost = Imkt − IBS, and in particular

RRcost = [Call(Kc,σ(Kc))− Put(Kp,σ(Kp))]− [Call(Kc,σATM)− Put(Kp,σATM)]

BFcost =12

[Call(Kc,σ(Kc)) + Put(Kp,σ(Kp))]

−12

[Call(Kc,σATM) + Put(Kp,σATM)] (15)

The rationale behind (14) is that one can extract the smile cost of an exotic option by measuring thesmile cost of a portfolio designed to hedge its Vanna and Volga risks. The reason why one chooses thestrategies BF and RR to do this is because they are liquid FX instruments and they carry respectivelymainly Volga and Vanna risks. The weighting factors wRR and wBF in (14) represent respectively theamount of RR needed to replicate the option’s Vanna, and the amount of BF needed to replicate theoption’s Volga. The above approach ignores the small (but non-zero) fraction of Volga carried by theRR and the small fraction of Vanna carried by the BF. It further neglects the cost of hedging the Vegarisk. This has led to a more general formulation of the Vanna-Volga method in which one considersthat within the BS assumptions the exotic option’s Vega, Vanna and Volga can be replicated by theweighted sum of three instruments:

Xi = wATM ATMi + wRR RRi + wBF BFi i=vega, vanna, volga (16)

where the weightings are obtained by solving the following system:

x = Aw (17)

with

A =

ATMvega RRvega BFvega

ATMvanna RRvanna BFvanna

ATMvolga RRvolga BFvolga

w =

wATM

wRR

wBF

x =

Xvega

Xvanna

Xvolga

(18)

Given this replication, the Vanna-Volga method adjusts the BS price of an exotic option by the smilecost of the above weighted sum (note that the ATM smile cost is zero by construction):

XVV = XBS + wRRRRmkt − RRBS

+ wBF

BFmkt − BFBS

= XBS + xT (AT )−1I = XBS + Xvega Ωvega + Xvanna Ωvanna + Xvolga Ωvolga

(19)

9


VIX

• The VIX or Volatility Index is a model free estimate of expected 30 day volatility of S&P500 options.

• Often moves opposite to equity markets and is sharply mean-reverting.

• Details on CBOE’s website (http://www.cboe.com/micro/vix/vixwhite.pdf)

• The main thing from this paper is how the vix is calculated, shown in next slide.


http://www.cboe.com/micro/vix/vixwhite.pdf




!

!"#$%&'()'*+,-'.%/01('2-,*(0%!"#$%&'()*+*,--.*!(&/0'"*1"0%2*3#)&"45*67/(04'89*:4/"%#"%0)82;*<==*%&'()5*%858%>82;!

"!

TTTTHE HE HE HE VIXVIXVIXVIX CCCCALCULATION ALCULATION ALCULATION ALCULATION SSSSTEPTEPTEPTEP----BYBYBYBY----SSSSTEPTEPTEPTEP!%#$%&'! ()*+,+-.! -/&0! 1-! $0+!#23!455.! 16+! &17&/71$+*!/-()8! $0+! 96(&+-! %:! $0+(6! &%;9%)+)$!-$%&'-<!!=1&0!()*+,!+;97%>-!6/7+-!$01$!8%?+6)!$0+!-+7+&$(%)!%:!&%;9%)+)$!-+&/6($(+-!1)*!1!:%6;/71!$%!&17&/71$+!()*+,!?17/+-<!!!!@AB! (-! 1!?%71$(7($>! ()*+,!&%;96(-+*!%:!!"#$!%&' 61$0+6! $01)!-$%&'-.!C($0! $0+!96(&+!%:!+1&0!%9$(%)!6+:7+&$()8!$0+!;16'+$D-!+,9+&$1$(%)!%:!:/$/6+!?%71$(7($><!!E('+!&%)?+)$(%)17!()*+,+-.!@AB! +;97%>-! 6/7+-! :%6! -+7+&$()8! &%;9%)+)$! %9$(%)-! 1)*! 1! :%6;/71! $%! &17&/71$+! ()*+,!?17/+-<!!!!F0+!8+)+617(G+*!:%6;/71!/-+*!()!$0+!@AB!&17&/71$(%)3!(-H!!

!!!!4%5I

5I

JJ

KLI

"#

$%&

'((

)* ()

*(+,

((

* $-*

$ $

$ %%%%%%%%%%%%%%%%%%678%

WWWWHEREHEREHEREHERE…………

%!' (-! ! J55

./0 !+!@AB!M!!!N!J55!

F! ! ! F(;+!$%!+,9(61$(%)!

O! ! ! O%6C16*!()*+,!7+?+7!*+6(?+*!:6%;!()*+,!%9$(%)!96(&+-!!

P5! O(6-$!-$6('+!Q+7%C!$0+!:%6C16*!()*+,!7+?+7.!O!!!P$! #$6('+!96(&+!%:!$#1'%/$R%:R$0+R;%)+>'%9$(%)S!1!&177!(:!P$TP5!1)*!1!9/$!

(:!P$U!P5S!Q%$0!9/$!1)*!&177!(:!P$MP5<!!)P(! A)$+6?17! Q+$C++)! -$6('+! 96(&+-! V! 017:! $0+! *(::+6+)&+! Q+$C++)! $0+!

-$6('+!%)!+($0+6!-(*+!%:!P(H!!!!!

!!!!!!!!!!!!!!!!!!!!!!)P(!M!I

JJ (, ( $$ ((!

!L2!#,H!!)P!:%6!$0+!7%C+-$!-$6('+!(-!-(;97>!$0+!*(::+6+)&+!Q+$C++)!$0+!7%C+-$! -$6('+! 1)*! $0+! )+,$! 0(80+6! -$6('+<! ! E('+C(-+.! )P! :%6! $0+!0(80+-$! -$6('+! (-! $0+! *(::+6+)&+! Q+$C++)! $0+! 0(80+-$! -$6('+! 1)*! $0+!)+,$!7%C+6!-$6('+<K!!

W! ! ! W(-'R:6++!()$+6+-$!61$+!$%!+,9(61$(%)!

XLP(K! F0+!;(*9%()$!%:!$0+!Q(*R1-'!-96+1*!:%6!+1&0!%9$(%)!C($0!-$6('+!P(<!!!

3!!37+1-+!-++!Y3!4,'#15%'6!7',8,4'95%#,:'#!';%!9'5<!7#'8!=5#$=$#6'&95"&Z!Q>!P6+-(;(6![+;+$+6:(.!=;1)/+7![+6;1).!\(&01+7!P1;17!1)*!]%-+90!^%/.!_%7*;1)!#1&0-!X/1)$($1$(?+!#$61$+8(+-!W+-+16&0!`%$+-.!\16&0!Jaaa<

!

!"#$%&'()'*+,-'.%/01('2-,*(0%!"#$%&'()*+*,--.*!(&/0'"*1"0%2*3#)&"45*67/(04'89*:4/"%#"%0)82;*<==*%&'()5*%858%>82;!

"!


!!!!4%5I

5I

JJ

KLI

"#

$%&

'((

)* ()

*(+,

((

* $-*

$ $

$ %%%%%%%%%%%%%%%%%%678%


%!' (-! ! J55

./0 !+!@AB!M!!!N!J55!

F! ! ! F(;+!$%!+,9(61$(%)!

O! ! ! O%6C16*!()*+,!7+?+7!*+6(?+*!:6%;!()*+,!%9$(%)!96(&+-!!

P5! O(6-$!-$6('+!Q+7%C!$0+!:%6C16*!()*+,!7+?+7.!O!!!P$! #$6('+!96(&+!%:!$#1'%/$R%:R$0+R;%)+>'%9$(%)S!1!&177!(:!P$TP5!1)*!1!9/$!


-$6('+!%)!+($0+6!-(*+!%:!P(H!!!!!

!!!!!!!!!!!!!!!!!!!!!!)P(!M!I

JJ (, ( $$ ((!

!L2!#,H!!)P!:%6!$0+!7%C+-$!-$6('+!(-!-(;97>!$0+!*(::+6+)&+!Q+$C++)!$0+!7%C+-$! -$6('+! 1)*! $0+! )+,$! 0(80+6! -$6('+<! ! E('+C(-+.! )P! :%6! $0+!0(80+-$! -$6('+! (-! $0+! *(::+6+)&+! Q+$C++)! $0+! 0(80+-$! -$6('+! 1)*! $0+!)+,$!7%C+6!-$6('+<K!!

W! ! ! W(-'R:6++!()$+6+-$!61$+!$%!+,9(61$(%)!

XLP(K! F0+!;(*9%()$!%:!$0+!Q(*R1-'!-96+1*!:%6!+1&0!%9$(%)!C($0!-$6('+!P(<!!!

3!!37+1-+!-++!Y3!4,'#15%'6!7',8,4'95%#,:'#!';%!9'5<!7#'8!=5#$=$#6'&95"&Z!Q>!P6+-(;(6![+;+$+6:(.!=;1)/+7![+6;1).!\(&01+7!P1;17!1)*!]%-+90!^%/.!_%7*;1)!#1&0-!X/1)$($1$(?+!#$61$+8(+-!W+-+16&0!`%$+-.!\16&0!Jaaa<

!

!"#$%&'()'*+,-'.%/01('2-,*(0%!"#$%&'()*+*,--.*!(&/0'"*1"0%2*3#)&"45*67/(04'89*:4/"%#"%0)82;*<==*%&'()5*%858%>82;!

"!


!!!!4%5I

5I

JJ

KLI

"#

$%&

'((

)* ()

*(+,

((

* $-*

$ $

$ %%%%%%%%%%%%%%%%%%678%


%!' (-! ! J55

./0 !+!@AB!M!!!N!J55!

F! ! ! F(;+!$%!+,9(61$(%)!

O! ! ! O%6C16*!()*+,!7+?+7!*+6(?+*!:6%;!()*+,!%9$(%)!96(&+-!!

P5! O(6-$!-$6('+!Q+7%C!$0+!:%6C16*!()*+,!7+?+7.!O!!!P$! #$6('+!96(&+!%:!$#1'%/$R%:R$0+R;%)+>'%9$(%)S!1!&177!(:!P$TP5!1)*!1!9/$!


-$6('+!%)!+($0+6!-(*+!%:!P(H!!!!!

!!!!!!!!!!!!!!!!!!!!!!)P(!M!I

JJ (, ( $$ ((!

!L2!#,H!!)P!:%6!$0+!7%C+-$!-$6('+!(-!-(;97>!$0+!*(::+6+)&+!Q+$C++)!$0+!7%C+-$! -$6('+! 1)*! $0+! )+,$! 0(80+6! -$6('+<! ! E('+C(-+.! )P! :%6! $0+!0(80+-$! -$6('+! (-! $0+! *(::+6+)&+! Q+$C++)! $0+! 0(80+-$! -$6('+! 1)*! $0+!)+,$!7%C+6!-$6('+<K!!

W! ! ! W(-'R:6++!()$+6+-$!61$+!$%!+,9(61$(%)!

XLP(K! F0+!;(*9%()$!%:!$0+!Q(*R1-'!-96+1*!:%6!+1&0!%9$(%)!C($0!-$6('+!P(<!!!

3!!37+1-+!-++!Y3!4,'#15%'6!7',8,4'95%#,:'#!';%!9'5<!7#'8!=5#$=$#6'&95"&Z!Q>!P6+-(;(6![+;+$+6:(.!=;1)/+7![+6;1).!\(&01+7!P1;17!1)*!]%-+90!^%/.!_%7*;1)!#1&0-!X/1)$($1$(?+!#$61$+8(+-!W+-+16&0!`%$+-.!\16&0!Jaaa<

!

!"#$%&'()'*+,-'.%/01('2-,*(0%!"#$%&'()*+*,--.*!(&/0'"*1"0%2*3#)&"45*67/(04'89*:4/"%#"%0)82;*<==*%&'()5*%858%>82;!

"!


!!!!4%5I

5I

JJ

KLI

"#

$%&

'((

)* ()

*(+,

((

* $-*

$ $

$ %%%%%%%%%%%%%%%%%%678%


%!' (-! ! J55

./0 !+!@AB!M!!!N!J55!

F! ! ! F(;+!$%!+,9(61$(%)!

O! ! ! O%6C16*!()*+,!7+?+7!*+6(?+*!:6%;!()*+,!%9$(%)!96(&+-!!

P5! O(6-$!-$6('+!Q+7%C!$0+!:%6C16*!()*+,!7+?+7.!O!!!P$! #$6('+!96(&+!%:!$#1'%/$R%:R$0+R;%)+>'%9$(%)S!1!&177!(:!P$TP5!1)*!1!9/$!


-$6('+!%)!+($0+6!-(*+!%:!P(H!!!!!

!!!!!!!!!!!!!!!!!!!!!!)P(!M!I

JJ (, ( $$ ((!

!L2!#,H!!)P!:%6!$0+!7%C+-$!-$6('+!(-!-(;97>!$0+!*(::+6+)&+!Q+$C++)!$0+!7%C+-$! -$6('+! 1)*! $0+! )+,$! 0(80+6! -$6('+<! ! E('+C(-+.! )P! :%6! $0+!0(80+-$! -$6('+! (-! $0+! *(::+6+)&+! Q+$C++)! $0+! 0(80+-$! -$6('+! 1)*! $0+!)+,$!7%C+6!-$6('+<K!!

W! ! ! W(-'R:6++!()$+6+-$!61$+!$%!+,9(61$(%)!

XLP(K! F0+!;(*9%()$!%:!$0+!Q(*R1-'!-96+1*!:%6!+1&0!%9$(%)!C($0!-$6('+!P(<!!!

3!!37+1-+!-++!Y3!4,'#15%'6!7',8,4'95%#,:'#!';%!9'5<!7#'8!=5#$=$#6'&95"&Z!Q>!P6+-(;(6![+;+$+6:(.!=;1)/+7![+6;1).!\(&01+7!P1;17!1)*!]%-+90!^%/.!_%7*;1)!#1&0-!X/1)$($1$(?+!#$61$+8(+-!W+-+16&0!`%$+-.!\16&0!Jaaa<


VIX Index


Other VIX-like Indices


VIX like indices on OIL, Gold,Corn...


implied volatility - saurabh.com singal saturday, august 14 ... rule of thumb for atm option price...

Documents