Proof that the Black-Scholes-Merton (BSM) option pricing equation

Proof that the Black-Scholes-Merton (BSM) option pricing equation
satisfies the Black-Scholes-Merton differential equation
The Task at Hand: Confirm that the value of a European call option given by the BlackScholes-Merton pricing equation: f = SN (d1 ) − Ke−r(T −t) N (d2 ) satisfies the Black-Scholes∂f
∂f
1
∂2f
Merton differential equation (14.16), which is
+ rS
+ σ 2 S 2 2 = rf.
∂t
∂S 2
∂S
Solution: So long as the underlying asset’s price movements conform to the geometric Brownian motion equation, then one can invoke delta hedging arguments involving the underlying
asset and the derivative in order to obtain a risk neutral valuation relationship between these
two securities. This risk neutral valuation relationship is given by the Black-Scholes-Merton
differential equation. As I show in my “Derivation and Comparative Statics of the Black-Scholes
Call and Put Option Pricing Equations” teaching note, the call option’s “theta” corresponds
∂f
to the rate at which the value of the call option decays with the passing of time; i.e.,
=
∂t
.5σ
∂f
−rKe−r(T −t) N (d2 )−Sn(d1 ) √
< 0. Furthermore, the call option’s delta
= N (d1 ) > 0,
∂S
T −t
∂N (d1 )
∂N (d1 ) ∂d1
∂d1
n(d1 )
∂2f
√
=
=
= n(d1 )
=
> 0.
and it’s gamma is
2
∂S
∂S
∂d1 ∂S
∂S
Sσ T − t
Substituting these expressions for theta, delta, and gamma into the BSM PDE, we complete our
proof; i.e., the Black-Scholes-Merton (BSM) pricing model for a European call option satisfies
the BSM non-stochastic partial differential equation (PDE):
∂f
1
∂2f
∂f
+ rS
+ σ2S 2 2
∂t
∂S 2
∂S
.5σ
n(d1 )
−r(T
−t)
−r(T
−t)
r[SN (d1 ) − Ke
N (d2 )] = −rKe
N (d2 ) − Sn(d1 ) √
+ rSN (d1 ) + 0.5σ 2 S 2 √
T −t
Sσ T − t
.5σ
.5σ
−r(T
−t)
−r(T
−t)
r[SN (d1 ) − SN (d1 ) + Ke
N (d2 ) − Ke
N (d2 )] = −Sn(d1 ) √ + Sn(d1 ) √ = 0.
t
t
rf =