Explicit_derivation_of_black_scholes_delta.pdf

Find an Explicit Solution for Delta in Black-Scholes Ophir Gottlieb 11/7/2007

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Introduction

We have seen through the creation of a replicating portfolio that the delta required to hedge an European call option is simply ∂C ∂S . Now we will explicitly compute delta by differentiating the closed form Black-Scholes Formula once with respect to the underlying stock. We recall the Black-Scholes formula for an European call option today (t=0 ) expiring at time t = T with constant interest rate (r ), constant volatility (σ) and strike price K as: C = S · Φ(d1 ) − e−rT · K · Φ(d2 )

S ) + (r + ln( K √ d1 = σ T

(1)

σ2 2 )T

√ d2 = d1 − σ T

where Φ(.) is the standard normal cumulative distribution. We will also write φ(.) as the standard normal probability density.

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Finding

∂C ∂S

First we note that by using the chain rule we find: ∂d1 ∂d2 1 1 = = · √ ∂S ∂S S σ T

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We can then differentiate equation (1) with respect to S to find: ∂C 1 1 = Φ(d1 ) + · √ [S · φ(d1 ) − e−rT · K · φ(d2 )] ∂S S σ T

(2)

We now make the claim that [S · φ(d1 ) − e−rT · K · φ(d2 )] = 0 and are thus are left with the result that ∆ = ∂C ∂S = Φ(d1 ).

Proof: Starting with a simple substitution for d2 and then moving through the algebra: √ e−rT · K · φ(d2 ) = e−rT · K · φ(d1 − σ T )

−(d1 −σ 1 2 = e−rT · K · √ · e 2π

√ 2 T)

√

−rT

=e

−d2 −(−2d1 σ T +σ 2 T ) 1 1 2 ·K · √ ·e 2 ·e 2π

= e−rT · K · φ(d1) · ed1 σ

= K · φ(d1) · e−rT −

σ2 T 2

√

T

· e−σ

2

S +ln( K )+rT + σ 2T

= K · φ(d1) ·

S K

e−rT · K · φ(d2 ) = S · φ(d1 )

2

2T

Therefore, we have from equation (2) above that: ∂C 1 1 = Φ(d1 ) + · √ [S · φ(d1 ) − e−rT · K · φ(d2 )] ∂S S σ T

= Φ(d1 ) +

1 1 · √ [S · φ(d1 ) − S · φ(d1 )] S σ T

∂C = Φ(d1 ) ∂S And we have thus verified the well known property of Black-Scholes; namely that ∆ = ∂C ∂S = Φ(d1 ). This in turn yields a nice interpretation of the first term in the Black-Scholes formula in equation (1). That is S · Φ(d1 ) is the value of the long position in the stock required to replicate the European call option. Note that ∆ is a function, not a constant.

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