2 Derivation of the formulae for calls, puts and simple digitals
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the Black–Scholes formulae and the ‘greeks’ Chapter 7
independent of the underlying. This is a big step forward, made possible by the lognormality
of the underlying asset. We are nearly there.
The last step is simple, but the motivation is not so obvious. Write
x = ξ + r − 12 σ 2 τ ,
and U = W (x, τ ). This is just a ‘translation’ of the coordinate system. It’s a bit like using the
forward price of the asset instead of the spot price as a variable. After this change of variables
the Black–Scholes becomes the simpler
∂ 2W
∂W
= 12 σ 2 2 .
∂τ
∂x
(7.2)
To summarize,
V (S, t) = e−r(T −t) U (S, t) = e−rτ U (S, T − τ ) = e−rτ U (eξ , T − τ )
= e−rτ U e
1
x− r− 2 σ 2 τ
,T
−τ
= e−rτ W (x, τ ).
To those of you who already know the Black–Scholes formulae for calls and puts the variable
x will ring a bell:
x = ξ + r − 12 σ 2 τ = log S + r − 12 σ 2 (T − t).
Having turned the original Black–Scholes equation into something much simpler, let’s take
a break for a moment while I explain where we are headed.
I’m going to derive an expression for the value of any option whose payoff is a known
function of the asset price at expiry. This includes calls, puts and digitals. This expression will
be in the form of an integral. For special cases, I’ll show how to rewrite this integral in terms
of the cumulative distribution function for the Normal distribution. This is particularly useful
since the function can be found on spreadsheets, calculators and in the backs of books. But
there are two steps before I can write down this integral.
The ﬁrst step is to ﬁnd a special solution of (7.2), called the fundamental solution. This
solution has useful properties. The second step is to use the linearity of the equation and the
useful properties of the special solution to ﬁnd the general solution of the equation. Here we go.
I’m going to look for a special solution of (7.2) of the following form
W (x, τ ) = τ α f
(x − x )
,
τβ
(7.3)
where x is an arbitrary constant. And I’ll call this special solution Wf (x, τ ; x ). Note that the
unknown function depends on only one variable (x − x )/τ β . As well as ﬁnding the function f
we must ﬁnd the constant parameters α and β. We can expect that if this approach works, the
equation for f will be an ordinary differential equation since the function only has one variable.
This reduction of dimension is an example of a similarity reduction, discussed in Chapter 6.
Substituting expression (7.3) into (7.2) we get
τ α−1 αf − βη
df
dη
= 12 σ 2 τ α−2β
d 2f
,
dη2
(7.4)
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Part One mathematical and ﬁnancial foundations
where
x−x
.
τβ
Examining the dependence of the two terms in (7.4) on both τ and η we see that we can only
have a solution if
η=
α − 1 = α − 2β
i.e. β = 12 .
I want to ensure that my ‘special solution’ has the property that its integral over all ξ is
independent of τ , for reasons that will become apparent. To ensure this, I require
∞
−∞
τ α f ((x − x )/τ β ) dx
to be constant. I can write this as
∞
−∞
τ α+β f (η) dη
and so I need
α = −β = − 12 .
The function f now satisﬁes
−f − η
df
d 2f
= σ2 2 .
dη
dη
This can be written
σ2
d(ηf )
d 2f
= 0,
+
2
dη
dη
which can be integrated once to give
σ2
df
+ ηf = a,
dη
where a is a constant. For my special solution I’m going to choose a = 0. This equation can
be integrated again to give
f (η) = be−η
2 /(2σ 2 )
.
I will choose the constant b such that the integral of f from minus inﬁnity to plus inﬁnity is
one:
1
2
2
e−η /(2σ ) .
f (η) = √
2πσ
This is the special solution I have been seeking:1
Wf (x, τ ) = √
1
e−((x−x )
2 /2σ 2 τ )
.
2πτ σ
Now I will explain why it is useful in our quest for the Black–Scholes formulae.
1
It is just the probability density function for a Normal random variable with mean zero and standard deviation σ .
the Black–Scholes formulae and the ‘greeks’ Chapter 7
5
4.5
W
4
3.5
3
2.5
2
1.5
1
0.5
−2
x
0
−1
0
1
2
3
4
x'
Figure 7.1 The fundamental solution.
In Figure 7.1 is plotted Wf as a function of x for several values of τ . Observe how the
function rises in the middle but decays at the sides. As τ → 0 this becomes more pronounced.
The ‘middle’ is the point x = x. At this point the function grows unboundedly, and away from
this point the function decays to zero as τ → 0. Although the function is increasingly conﬁned
to a narrower and narrower region its area remains ﬁxed at one. These properties of decay
away from one point, unbounded growth at that point and constant area, result in a Dirac delta
function δ(x − x) as τ → 0. The delta function has one important property, namely
δ(x − x)g(x ) dx = g(x)
where the integration is from any point below x to any point above x. Thus the delta function
‘picks out’ the value of g at the point where the delta function is singular i.e. at x = x. In the
limit as τ → 0 the function Wf becomes a delta function at x = x . This means that
1
√
τ →0 σ 2πτ
lim
∞
−∞
e
2
− (x −x)
2
2σ τ
g(x ) dx = g(x).
This property of the special solution, together with the linearity of the Black–Scholes equation,
are all that are needed to ﬁnd some explicit solutions.
Now is the time to consider the payoff. Let’s call it
Payoff(S).
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Part One mathematical and ﬁnancial foundations
This is the value of the option at time t = T . It is the ﬁnal condition for the function V ,
satisfying the Black–Scholes equation:
V (S, T ) = Payoff(S).
With our new variables, this ﬁnal condition is
W (x, 0) = Payoff(ex ).
(7.5)
I claim that the solution of this for τ > 0 is
∞
W (x, τ ) =
−∞
Wf (x, τ ; x )Payoff(ex ) dx .
(7.6)
To show this, I just have to demonstrate that the expression satisﬁes the equation (7.2) and
the ﬁnal condition (7.5). Both of these are straightforward. The integration with respect to x
is similar to a summation, and since each individual component satisﬁes the equation so does
the sum/integral. Alternatively, differentiate (7.6) under the integral sign to see that it satisﬁes
the partial differential equation. That it satisﬁes the condition (7.5) follows from the special
properties of the fundamental solution Wf .
Retracing our steps to write our solution in terms of the original variables, we get
V (S, t) =
∞
e−r(T −t)
√
σ 2π(T − t)
e
2
1
− log(S/S )+ r− 2 σ 2 (T −t)
2σ 2 (T −t)
Payoff(S
)
0
dS
,
S
(7.7)
where I have written x = log S .
This is the exact solution for the option value in terms of the arbitrary payoff function. In
the next sections I will manipulate this expression for special payoff functions.
7.2.1 Formula for a Call
The call option has the payoff function
Payoff(S) = max(S − E, 0).
Expression (7.7) can then be written as
e−r(T −t)
√
σ 2π(T − t)
∞
e
2
1
− log(S/S )+ r− 2 σ 2 (T −t) /2σ 2 (T −t)
(S
− E)
E
dS
.
S
Return to the variable x = log S , to write this as
e−r(T −t)
√
σ 2π (T − t)
=
∞
e
2
1
− −x +log S+ r− 2 σ 2 (T −t) /2σ 2 (T −t) x
(e
− E) dx
log E
e−r(T −t)
√
σ 2π(T − t)
∞
e
2
1
− −x +log S+ r− 2 σ 2 (T −t) /2σ 2 (T −t) x
e
dx
log E
e−r(T −t)
−E √
σ 2π (T − t)
∞
e
log E
2
1
− −x +log S+ r− 2 σ 2 (T −t) /2σ 2 (T −t)
dx
.
the Black–Scholes formulae and the ‘greeks’ Chapter 7
Both integrals in this expression can be written in the form
∞
1
e− 2 x dx
2
d
for some d (the second is just about in this form already, and the ﬁrst just needs a completion
of the square).
Apart from a couple of minor differences, this integral is just like the cumulative distribution
function for the standardized Normal distribution2 deﬁned by
x
1
N (x) = √
2π
1 2
−∞
e− 2 φ dφ.
This function, plotted in Figure 7.2, is the probability that a Normally distributed variable is
less than x.
Thus the option price can be written as two separate terms involving the cumulative distribution function for a Normal distribution:
Call option value = SN(d1 ) − Ee−r(T −t) N (d2 )
where
d1 =
log(S/E) + (r + 12 σ 2 )(T − t)
√
σ T −t
CDF
1
0.8
0.6
0.4
0.2
f
−2.8 −2.4
−2
−1.6 −1.2 −0.8 −0.4
0
0
0.4
0.8
1.2
1.6
2
2.4
2.8
Figure 7.2 The cumulative distribution function for a standardized Normal random variable, N(x).
2
i.e. having zero mean and unit standard deviation.
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Part One mathematical and ﬁnancial foundations
and
d2 =
log(S/E) + (r − 12 σ 2 )(T − t)
.
√
σ T −t
When there is continuous dividend yield on the underlying, or it is a currency, then
Call option value
Se
d1 =
−D(T −t)
N (d1 ) − Ee−r(T −t) N (d2 )
log(S/E) + (r − D + 12 σ 2 )(T − t)
√
σ T −t
log(S/E) + (r − D − 12 σ 2 )(T − t)
√
σ T −t
√
= d1 − σ T − t
d2 =
The option value is shown in Figure 7.3 as a function of the underlying asset at a
ﬁxed time to expiry. In Figure 7.4 the value of the at-the-money option is shown as
a function of time, and expiry is t = 1. In Figure 7.5 is the call value as a function
of both the underlying and time.
When the asset is ‘at-the-money forward,’ i.e. S = Ee−(r−D)(T −t) , then there is a
simple approximation for the call value (Brenner & Subrahmanyam, 1994):
√
Call ≈ 0.4 Se−D(T −t) σ T − t.
200
180
160
140
120
Value
116
100
80
60
40
20
S
0
0
50
100
150
200
250
300
Figure 7.3 The value of a call option as a function of the underlying asset price at a ﬁxed time to
expiry.
the Black–Scholes formulae and the ‘greeks’ Chapter 7
14
12
Value
10
8
6
4
2
t
0
0
0.2
0.4
0.6
0.8
1
1.2
Figure 7.4 The value of an at-the-money call option as a function of time.
70
60
50
40
Value
30
20
10
Asset
Figure 7.5 The value of a call option as a function of asset and time.
150
135
120
105
90
75
0.3
60
45
30
15
0
0.9
0.75
0.6
Time
0.45
0.15
0
0
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Part One mathematical and ﬁnancial foundations
7.2.2 Formula for a Put
The put option has payoff
Payoff(S) = max(E − S, 0).
The value of a put option can be found in the same way as above, or using put-call parity
Put option value = −SN(−d1 ) + Ee−r(T −t) N (−d2 ),
with the same d1 and d2 .
When there is continuous dividend yield on the underlying,
or it is a currency, then
Put option value
−Se
−D(T −t)
N (−d1 ) + Ee−r(T −t) N (−d2 )
The option value is shown in Figure 7.6 against the underlying asset and in Figure 7.7 against time. In Figure 7.8 is the option value as a function of
both the underlying asset and time.
When the asset is at-the-money forward the simple approximation for the put value (Brenner
& Subrahmanyam, 1994) is
√
Put ≈ 0.4 Se−D(T −t) σ T − t.
100
90
80
70
60
Value
118
50
40
30
20
10
S
0
0
50
100
150
200
250
300
Figure 7.6 The value of a put option as a function of the underlying asset at a ﬁxed time to expiry.
the Black–Scholes formulae and the ‘greeks’ Chapter 7
4
3.5
3
Value
2.5
2
1.5
1
0.5
t
0
0
0.2
0.4
0.6
0.8
1
1.2
Figure 7.7 The value of an at-the money put option as a function of time.
100
90
80
70
60
Value
50
40
30
20
10
0
0.2
0.4
0.6
0.8
1
150
135
120
90
105
75
45
Asset
60
15
30
0
0
Time
Figure 7.8 The value of a put option as a function of asset and time.
7.2.3
Formula for a Binary Call
The binary call has payoff
Payoff(S) = H(S − E),
where H is the Heaviside function taking the value one when its argument is positive and zero
otherwise.
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Part One mathematical and ﬁnancial foundations
Incorporating a dividend yield, we can write the option value as
e−r(T −t)
√
σ 2π(T − t)
∞
e
2
1
− x −log S− r−D− 2 σ 2 (T −t) /2σ 2 (T −t)
dx
.
log E
This term is just like the second term in the call option equation and so
Binary call option value
e−r(T −t) N (d2 )
The option value is shown in Figure 7.9.
7.2.4 Formula for a Binary Put
The binary put has a payoff of one if S < E at expiry. It has a value of
Binary put option value
e−r(T −t) (1 − N (d2 ))
1
0.9
0.8
0.7
0.6
Value
120
0.5
0.4
0.3
0.2
0.1
S
0
0
50
100
Figure 7.9 The value of a binary call option.
150
200
250
300
the Black–Scholes formulae and the ‘greeks’ Chapter 7
1
0.9
0.8
0.7
Value
0.6
0.5
0.4
0.3
0.2
0.1
S
0
0
50
100
150
200
250
300
Figure 7.10 The value of a binary put option.
since a binary call and a binary put must add up to the present value of $1 received
at time T . The option value is shown in Figure 7.10.
7.3
DELTA
The delta, , of an option or a portfolio of options is the sensitivity of the option
or portfolio to the underlying. It is the rate of change of value with respect to
the asset:
=
∂V
∂S
Here V can be the value of a single contract or of a whole
portfolio of contracts. The delta of a portfolio of options is just
the sum of the deltas of all the individual positions.
The theoretical device of delta hedging, introduced in Chapter 5, for eliminating risk is far
more than that, it is a very important practical technique.
Roughly speaking, the ﬁnancial world is divided up into speculators and hedgers. The speculators take a view on the direction of some quantity such as the asset price (or more abstract
quantities such as volatility) and implement a strategy to take advantage of their view. Such
people may not hedge at all.
Then there are the hedgers. There are two kinds of hedger: the ones who hold a position
already and want to eliminate some very speciﬁc risk (usually using options) and the ones
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