Stratégies Options - Time And Volatility Equivalence

I - A first equation

It's been shown that for small asset(S) movements, if V(S) is the value of an option for S as an asset level,

V(S1) - V(S0) = ∂V(S0)/∂S . ( S1 - S0 )

If,
Δ(S0) = ∂V(S0)/∂S is the delta for S = S0
dS = ( S1 - S0 ) is the asset variation

V(S1) - V(S0) = Δ(S0) . dS

One is able to be much more accurate incorporating the gamma

V(S1) - V(S0) = ∂V(S0)/∂S . ( S1 - S0 ) + 0.5 . ∂²V(S0)/∂S² . ( S1 - S0 )²

then if we note the gamma Γ(S0) calculated at S = S0

V(S1) - V(S0) = Δ(S0) . dS + 0.5 . Γ(S0) . (dS)²

If dV is the option value variation

dV = V(S1) - V(S0) = Δ(S0) . dS + 0.5 . Γ(S0) . (dS)²
dV = Δ . dS + 0.5 . Γ . (dS)²

That works out for a very short time period.
If we take into account the time

dV = Δ . dS + 0.5 . Γ . (dS)² + Θ . dt

where Θ is time decay during the period dt

II - A riskless portfolio Π

If we build a portfolio made of one option V and short Δ underlying,

That leads to : Π = V - Δ . S

For small time period dt and asset variations dS, this portfolio moves by dΠ and,

dΠ = dV - Δ . dS

But we know that dV = Δ . dS + 0.5 . Γ . (dS)² + Θ . dt

Hence,
dΠ = Δ . dS + 0.5 . Γ . (dS)² + Θ . dt - Δ . dS
dΠ = 0.5 . Γ . (dS)² + Θ . dt

We know that,
dS = S . σ . √dt
(dS)² = ( S . σ . √dt)² = S² . σ² . dt

Thus,
dΠ = 0.5 . Γ . (dS)² + Θ . dt = 0.5 . Γ . S² . σ² . dt + Θ . dt
dΠ = ( 0.5 . Γ . S² . σ² + Θ ) . dt

This porfolio is perfectly hedged. There is no risk. If there is no risk, it's a riskfree investment and it must have the same yield as for a cash account. If an interest rate r is paid for a cash account it must be paid for this portfolio too!

That leads to,
dΠ = r . Π . dt

and
dΠ = ( 0.5 . Γ . S² . σ² + Θ ) . dt

Thus
( 0.5 . Γ . S² . σ² + Θ ) = r . Π
( 0.5 . Γ . S² . σ² + Θ ) = r . ( V - Δ . S )
( 0.5 . Γ . S² . σ² + Θ ) = .r . V - r . Δ . S

If r<>0
V = Δ . S + ( 1/r ) . ( 0.5 . Γ . S² . σ² + Θ )

If r = 0
0.5 . Γ . S² . σ² = - Θ

This is the starting point of the Black & Scholes model.

For a delta hedged portfolio that is worth $1 and that is built with a long option and a short delta underlying position:

( 0.5 . Γ . S² . σ² + Θ ) = r

Next : Gamma Г And Vega υ Relationship
Previous : At The Money Forward Relationships 4

Relationships Between Option Sensitivities - INDEX Relationships Between Option Sensitivities - CHAPTER I
Relationships Between Option Sensitivities - CHAPTER II
Relationships Between Option Sensitivities - CHAPTER III

Strategies Options