5.1 Derivation
49
Equating the two expressions for d from (5.4) and (5.5) and dividing by dt, we
obtain the differential equation that is linked to the names of its main contributors:
F. Black, M. Scholes, and R. Merton
−
∂c
∂t
=
1
2
σ
2 S
2 ∂
2 c
∂ S 2 + r f S
∂c
∂ S
− r f c .
(5.6)
It links the temporal evolution of the option c to that of the underlying asset S. It
is interesting that the temporal evolution of the option c is only determined by the
volatility σ and by the risk-free rate r f , rather than the assumed growth rate ρ of the
stock S.
We point out that (5.6) was derived by deliberately constructing the portfolio ,
defined in (5.3), to be risk free. This way of assembling a hedged portfolio of one
option and = ∂c/∂ S shares S is usually referred to as -hedging. We find that
the temporal evolution of the option c, given by (5.6), is a direct consequence of the
definition of in (5.3). Let us proceed to determine the value of c at a given time
before maturity. To do so, we need to integrate the partial differential equation from
(5.6), and this is the topic of the next section.
5.2 The Solution
We now continue to solve this equation for a call option with the boundary condition
c = max(S − K , 0) at time T. Inspired by [2], we note that this partial differential
equation is very similar to the diffusion equation, except for the additional term
−r f t and that time runs in the “wrong” direction due to the minus sign in front of
the temporal derivative. Since we are interested in the evolution of the option price
towards maturity at time T we introduce the new variable τ = T − t.
c(S, t) = e
−r f τ g(S, τ ) ,
(5.7)
which transforms (5.6) into
∂g
∂τ
=
1
2
σ
2 S
2 ∂
2 g
∂ S 2 + r f S
∂g
∂ S
.
(5.8)
Moreover, the powers of S in front of the derivatives indicate that it is beneficial to
introduce z = ln S/K . Here we chose to normalize the stock value S to the strike
price K , because it is the only variable that has units of monetary value in the problem.
After some algebra we find
∂g(z, τ )
∂τ
=
r f −
1
2
σ
2
∂g(z, τ )
∂z
+
1
2
σ
2 ∂
2 g(z, τ )
∂z 2
.
(5.9)
49
Equating the two expressions for d from (5.4) and (5.5) and dividing by dt, we
obtain the differential equation that is linked to the names of its main contributors:
F. Black, M. Scholes, and R. Merton
−
∂c
∂t
=
1
2
σ
2 S
2 ∂
2 c
∂ S 2 + r f S
∂c
∂ S
− r f c .
(5.6)
It links the temporal evolution of the option c to that of the underlying asset S. It
is interesting that the temporal evolution of the option c is only determined by the
volatility σ and by the risk-free rate r f , rather than the assumed growth rate ρ of the
stock S.
We point out that (5.6) was derived by deliberately constructing the portfolio ,
defined in (5.3), to be risk free. This way of assembling a hedged portfolio of one
option and = ∂c/∂ S shares S is usually referred to as -hedging. We find that
the temporal evolution of the option c, given by (5.6), is a direct consequence of the
definition of in (5.3). Let us proceed to determine the value of c at a given time
before maturity. To do so, we need to integrate the partial differential equation from
(5.6), and this is the topic of the next section.
5.2 The Solution
We now continue to solve this equation for a call option with the boundary condition
c = max(S − K , 0) at time T. Inspired by [2], we note that this partial differential
equation is very similar to the diffusion equation, except for the additional term
−r f t and that time runs in the “wrong” direction due to the minus sign in front of
the temporal derivative. Since we are interested in the evolution of the option price
towards maturity at time T we introduce the new variable τ = T − t.
c(S, t) = e
−r f τ g(S, τ ) ,
(5.7)
which transforms (5.6) into
∂g
∂τ
=
1
2
σ
2 S
2 ∂
2 g
∂ S 2 + r f S
∂g
∂ S
.
(5.8)
Moreover, the powers of S in front of the derivatives indicate that it is beneficial to
introduce z = ln S/K . Here we chose to normalize the stock value S to the strike
price K , because it is the only variable that has units of monetary value in the problem.
After some algebra we find
∂g(z, τ )
∂τ
=
r f −
1
2
σ
2
∂g(z, τ )
∂z
+
1
2
σ
2 ∂
2 g(z, τ )
∂z 2
.
(5.9)