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2 Derivation of the formulae for calls, puts and simple digitals

# 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

= 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 .

This can be written

σ2

d(ηf )

d 2f

= 0,

+

2

which can be integrated once to give

σ2

df

+ ηf = a,

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).

113

114

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) = √

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

117

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.

119

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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