mardi 22 janvier 2013

Delta Symmetry

La réponse à cette question est très pratique et comme on dit, "A Tout Seigneur Tout Honneur".

J'ai repris l'article de Global Hedge qui s'intitutle "On Delta Symmetry, Gamma-Vega Parity : Ultimate Symmetry" écrit par Morgane TRAMASAYGUES .
Voici l'article, en anglais dans le texte :


After « Call Put Parity », « Call Put Symmetry », « SuperSymmetry »....now..... « Ultimate Symmetry ».
Here we are !


Ultimate Symmetry is a particular relationship for options with the same maturity that is really interesting from a trader’s point of view. Despite being derived in a world where  implied volatility remains the same for every strike on a given expiry, it helps traders to find options that would naturally match their greeks in the real world.




I - A « Flat Vol’ » World

Assume that all strikes for a maturity T have the same implied volatility – a « Flat Vol’ World » ( It doesn’t mean that every maturity needs to have the same implied volatility, but every strike for each maturity has the same one).

If one set for that maturity T the Delta Neutral Straddle Strike as

KDelta Neutral Straddle = S . exp(( r - q + 0.5.σ²).T)

Where
S underlying spot
K strike price
r continuously compounded interest rate
q continuously compounded dividend rate
T expiry (in year(s))
σ implied volatility

(Which is the strike where Call and Put have deltas that exhibit the same absolute value, it’s also the strike where vega is maximum and then volga is nil (∂Vega /∂σ = 0 )).


Then,

For every Call struck at KCall with the same maturity T it exists a Put struck at KPut with maturity T that matches simultaneously the absolute values of delta, gamma and vega, and,


KPut = (KDelta Neutral Straddle )² / KCall


For that Strike We have,

Delta Symmetry :      Δ Put = - Δ Call
The delta of a Call struck at K has the opposite value as the delta of a Put struck at (KDelta Neutral Straddle)²/K .
ΔC( S, K, r, q, σ, T ) = - ΔP( S, ( KDelta Neutral Straddle)²/K, r, q, σ, T )


Gamma Parity :  Γ Put = Γ Call
The gamma of a Call struck at K has the same value as the gamma of a Put struck at (KDelta Neutral Straddle)²/K .
Γ C( S, K, r, q, σ, T ) = Γ P( S, ( KDelta Neutral Straddle)²/K, r, q, σ, T )


Vega Parity : νega Put = vega Call
The vega of a Call struck at K has the same value as the vega of a Put struck at (KDelta Neutral Straddle)²/K .
Vega C( S, K, r, q, σ, T ) = Vega P( S, ( KDelta Neutral Straddle)²/K, r, q, σ, T )






In a Black-Scholes world, if you’ve found for a given asset which is worth XXX USD,  KDelta Neutral Straddle = 110 for a given maturity T,
→ You already know how to hedge the delta of a long position of 500 Calls struck at 179 with the same maturity by buying 500 Puts struck at KPut = (110)² / 179 = 67.59
The 67.59 Put and de 179 Call with the same maturity will have opposite deltas, the same gamma and the same vega.





Risk Reversal.
As far as you know that KDelta Neutral Straddle = 110 , you know how to build a Risk Reversal that is gamma neutral and vega neutral using for example a long 120 Call with the same maturity T, by selling a Put struck at KPut= (110)² / 120 = 100.83 with maturity T.

Vega Weighted Butterfly
If KDelta Neutral Straddle = 110 , you know how to build a Vega Weighted Butterfly using the 120 call with the same maturity T, and the Put struck at  (110²)/120 = 100.83 with maturity T by selling 1 Straddle struck at KDelta Neutral Straddle = 110 and buying (Vega Call Delta Neutral Straddle / Vega Call 120 ) . Strangle (120 Call + 100.83 Put )







II - A Real World

In a flat vol’world, that would match (-)delta, gamma and vega. You won’t even have to care about ΔCall  ΓCall and νega Call values, you already know that they will naturally match. There would be no need to price them.

Our world is not a flat BS Implied Volatility World, that’s a fact, but the relationship remains valid as an approximation. It provides a simple way to approximate the real solution. It may even provide tools to price for a given delta how far from the BS strike the market strike is, as a pricing tool for the skew. It provides also a different way to replicate knock out / knock in options using a risk reversal.









Derivations



Δ is the delta for the strike K
Γ is the gamma for the strike K
S underlying spot
K strike price
r continuously compounded interest rate
q continuously compounded dividend rate
T maturity (in year(s))
σ implied volatility

d1= ( ln(S/K) + (r-q + 0.5. σ²)T ) / σ√T




Delta Symmetry :      Δ Put = - Δ Call

For  KPut = (KDelta Neutral Straddle )² / KCall

We have

d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(S/(K²DeltaNeutralStraddle / KCall) + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(S. KCall /(K²DeltaNeutralStraddle) + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(S. KCall /( S . exp(( r-q + 0.5.σ²).T)²) + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(KCall /(S.(exp(( r-q + 0.5.σ²).T)²) + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(KCall /(S) -ln(exp(( r-q + 0.5.σ²).T)²) + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(KCall /(S) -2ln(exp(( r-q + 0.5.σ²).T)) + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(KCall /(S) -2( r-q + 0.5.σ²).T + (r-q + 0.5. σ²)T) / σ√T
d1= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = (ln(KCall /(S) -( r-q + 0.5.σ²).T) / σ√T

d1Put= (ln(S/KPut) + (r-q + 0.5. σ²)T) / σ√T = - (ln(S / KCall) +( r-q + 0.5.σ²).T / σ√T) = -d1Call
d1Put= -d1Call            hence   -d1Put=  d1Call

N(-d1Put) =N (d1Call )
Exp(- qT).N(-d1Put) = Exp(- qT).N (d1Call )

Then,  -Δ Put =  Δ Call







Gamma Parity :  Γ Put = Γ Call

If d1Put= -d1Call

Then,

Since n(x) = 1/(√2π)exp(-x²) is an even function
n(-d1Put) = n (d1Call )
Exp(- qT). n(-d1Put) / ( S σ√T ) = Exp(- qT). n (d1Call )  / ( S σ√T )

And  Γ Put = Γ Call




Vega Parity : νega Put = vega Call

Since vega = Γ . σ . S² . T

 if Γ Put = Γ Call

 Then, (σ . S² . T). Γ Put = (σ . S² . T). Γ Call

 And ,  νega Put   =  vega Call


Références et pdfOn Delta Symmetry, Gamma-Vega Parity : Ultimate Symmetry

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