Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability

188 Chapter 5

Figure 4.1: a typical trajectory

4.4 Pricing The Call With The Black-Scholes-Samuelson

Model

4.4.1 The Hedging Portfolio

The problem consists in pricing the value of a European call of maturity T and

exercise price K at every time t belonging to [0,T] as a function of t or the

maturity at time t,

Tt

τ

=−

, and of the value of the asset at time t, S=S(t)

knowing that the non-risky instantaneous interest rate is r, so that if i is the non-

risky annual rate, we have:

1

r

ei

=

+ . (4.26)

We will use the notation C(S,t) or, more frequently, ( , )CS

τ

.

As in the CRR model, we introduce a portfolio P containing, at every time t of a

call and a proportion

α

, which may be negative, shares of the underlying asset.

The stochastic differential of P(t) is given by:

() ( ,) ()dP t dC S t dS t

α

=

+

(4.27)

or, from relation (4.16):

() ( ,) () () ().dPt dCSt Stdt StdBt

α

μασ

=+ +

(4.28)

Using the Itô formula, in a correct form as proved by Bartels (1995) of the first

initial form given by Black and Scholes (1973), we get:

()

2

22

1

)(,) (,) (,) ()

2

() ( ,) ().

CCC

dPt St S St St S St dt

StS

C

St St S dBt

S

∂∂∂

μσαμ

∂∂∂

∂

ασ σ

∂

⎡⎤

=++ +

⎢⎥

⎣⎦

⎡⎤

++

⎢⎥

⎣⎦

(4.29)

Now, using the principle of AOA, this variation must be identical to that of the

same amount invested at the non-risky interest, that is:

Black & Scholes extensions 189

[

]

.),()( dtStSCrdttrP

α

+

=

(4.30)

So, we get the following relation:

() ()rP t dt dP t

=

(4.31)

[

]

2

22

(,)

1

(,) (,) (,) ()

2

() ( ,) ().

rCSt Sdt

CCC

St S St St S St dt

StS

C

St St S dBt

S

α

∂∂∂

μσαμ

∂∂∂

∂

ασ σ

∂

+=

⎡⎤

++ +

⎢⎥

⎣⎦

⎡⎤

++

⎢⎥

⎣⎦

(4.32)

By identification, we get:

[]

2

22

(,)

1

(,) (,) (,) () 0,

2

() ( ,) 0.

rCSt Sdt

CCC

St S St St S St dt

StS

C

St St S

S

α

∂∂∂

μσαμ

∂∂∂

∂

ασ σ

∂

+−

⎡⎤

++ + =

⎢⎥

⎣⎦

⎡⎤

+=

⎢⎥

⎣⎦

(4.33)

From the last equality, we get:

(,).

C

St

S

∂

α

∂

=− (4.34)

Substituting this value in the first equality of (4.33), we get after simplification:

2

22

1

(,) (,) (,) (,) 0,

2

CCC

rCSt StS St St S

StS

∂∂∂

σ

∂∂∂

⎡⎤

−−+ =

⎢⎥

⎣⎦

(4.35)

or finally

2

22

1

(,) (,) (,) (,) 0,

2

CC C

rCSt r StS St St S

St S

∂∂ ∂

σ

∂∂∂

−+ + + = (4.36)

a linear partial derivative equation of order 2 for the unknown function C(S,t)

with as initial condition

[

)

{}

0, 0, ,

(,)

max 0, ,

tT

CSt

SKtT

⎧

∈

⎪

=

⎨

−

=

⎪

⎩

(4.37)

Using results from the heat equation in physics, for which an explicit solution is

given in terms of a so-called Green function, known in this case, Black and

Scholes (1973) got the following explicit form for the call value:

()

12

2

1

21

(,) ( ) ( ),

1

log ( )( ) ,

2

,

().

rT t

CSt S d Ke d

S

drTt

K

Tt

dd Tt

SSt

σ

−−

=Φ − Φ

⎡

⎤

=++−

⎢

⎥

−

⎣

⎦

=− −

=

(4.38)

190 Chapter 5

Remark 4.2 Using relation (4.31), we get relation (3.15) for t=0 or

T

τ

=

.

The interpretation is of course already given in section 3.

4.4.2 The Risk Neutral Measure And The Martingale Property

As for the CRR model, it is possible to construct another probability measure Q

on

()

,,( )

t

Ωℑ ℑ , called the risk neutral measure, such that the value of the call

given by formula (4.38) is simply the expectation value of the present value of

the “gain” at maturity time T.

Using a change of probability measure for going from P to Q, with the famous

Girsanov theorem (see for example Gikhman and Skorokhod,vol.III

(1975),p.250), it can be shown that the new measure Q, which moreover is

unique, can be defined by replacing in the stochastic differential equation (4.16)

the trend

μ

by r.

Doing so, the explicit form of S(t) given by relation (4.26) becomes:

2

'( )

0

()

rt

B

t

St Se e

σ

⎛⎞

−

⎜⎟

⎝⎠

= (4.39)

where the process

'

B

still an adapted standard Brownian motion and the value of

C can be computed as the present value of the expectation of the final “gain” of

the call at time T:

{

}

(

)

()

(,) sup () ,0 .

rT t

Q

CSt e E ST K

−−

=− (4.40)

The risk neutral measure gives another important property for the process of

present values of the asset values on [0,T]:

[

]

{

}

.,0),( TttSe

rt

∈

−

(4.41)

Indeed, under Q, this process is a martingale, so that (see Chapter 1, section 8)

for all s and t such that :

s

t<

()

() ( ) ( ).

rt

s

E

eSt st Ss

−

ℑ<= (4.42)

This means that at every time s, the best statistical estimation of S(t) is given by

the observed value at time s, a result consistent with the assumption of an

efficient market.

From relation (4.42), we get in particular:

(

)

0

() .

rt

E

eSt S

−

=

(4.43)

So, on average, the present value of the asset at any time t equals its value at time

0.

In conclusion, we see that the knowledge of the risk neutral measure avoids the

resolution of the partial derivative equation and replaces it by the computation of

an expectation, which is in general easier, as it only uses the marginal

distribution of S(T).

Black & Scholes extensions 191

But we must add that, for more complicated derivative products, it may be more

interesting, from the numerical point of view, to solve this partial derivative

equation with the finite difference method, and particularly in the case of

American options.

4.4.3 The Call-Put Parity Relation

From section 1, we know that the value of a put at maturity time T and exercise

price K is given by:

()

{

}

(), max0, ().PST K K ST=− (4.44)

As for the call, we have:

(

)

{

}

(), max0,() ,CST K ST K=− (4.45)

and so, we get:

()

(

)

(), (), () .CST K PST K ST K

−

=− (4.46)

And so, for the expectations:

(

)

(

)

(

)

(

)

(

)

(), (), () .

E

CST K EPST K EST K

−

=− (4.47)

Using the principle of mathematical expectation for pricing the call and the put,

we get:

(

)

00

(,0) (,0) () .

rT rT

eCS ePS EST K

−

=−

(4.48)

We call this relation the general call-put parity relation as it gives the value of

the put knowing the value of the call and vice versa

Now, under the assumption of an efficient market, we can use property (4.43) to

get

KeSSPeSCe

rTrTrT

−=−

000

)0,()0,(

(4.49)

and so the put value is given by:

000

( ,0) ( ,0) .

rT

PS CS S e K

−

=−+

(4.50)

Remark 4.3 We can interpret this relation as follows: assume a portfolio having

at time 0 a share of value S

0

, a put on the same asset with maturity T and an

exercise price K and a sold call with the same maturity and exercise price; the

value of the portfolio at time T is always K, whatever the value of S(T) is.

From the call-put parity relation, we easily get the value of a put having the same

maturity time T and exercise price K as for the call:

()

(,) (,) ,

rT t

PSt CSt S e K

−−

=−+

(4.51)

and using the Black and Scholes result, we obtain:

192 Chapter 5

).(

,

,))(

2

(log

1

),()(),(

12

2

1

12

)(

tSS

tTdd

tTr

K

S

tT

d

dSdKetSP

tTr

=

−−=

⎥

⎦

⎤

⎢

⎣

⎡

−++

−

=

−Φ−−Φ=

−−

σ

(4.52)

5 EXERCISE ON OPTION PRICING

Exercise 5.1 Let us consider a portfolio with

Δ

shares of unit price 1000 Euro

and an amount B invested at the non-risky interest rate of 4% per period.

1°) What is the price C of a European call having 1050 Euro as exercise price, of

maturity 2 periods if per period, the share increases by a quarter of its value with

probability 0.75 and decreases by a third of its value with probability 0.25?

What are the intermediate values of the call?

2°) What is the composition of the hedging portfolio at time 0?

3°) If the maturity has for value 2 weeks and the period is the day, give an

estimation of the volatility and the trend of the considered asset.

Solution:

1°)

512.5, 0,

315.38, 0,

194.08.

uu ud dd

ud

CCC

CC

C

===

==

=

2°)

u

d

where:

C

= 54.07%( part of the asset),

()

uC

346.57 (loan at the non-risky rate from the bank).

()

d

u

CSB

C

Su d

dC

BF

ud

=Δ +

−

Δ=

−

==−

−

3°)

We know that:

Black & Scholes extensions 193

14 14

5

1000 1000 ,

4

2

1000 1000 ,

3

:

14 ,

1 ,

:

55

14 14 ln ,

44

25

14 14 ln .

34

, :

1

0.2231436 0.0079694,

28

360 0.0079694 2.868994,

1

0.2231

214

tt

nn

tt

nn

year

e

or

tdays

nday

so

e

Finally we get

μσ

μ

σ

+

−

+

−

×= ×

=

=⇒+=

=⇒−=

==

=× =

=

436 0.0298188,

360.0.0298188 0.565772.

year

σ

=

==

6 THE GREEK PARAMETERS

6.1 Introduction

The technical management of the trader of options, particularly by the brokers,

uses the so-called Greek parameters to measure the impacts of small variations

of parameters involved in formulas (4.38) and (4.52) for the pricing of options:

,,,,SrK

σ

τ

.

(i) The delta coefficient

This is an indicator concerning the influence of small variations S

Δ

of the asset

price defined as follows:

(,)(,)(),

(,).

CS St CSt S

C

St

S

∂

+

Δ≈ +ΔΔ

Δ=

(6.1)

194 Chapter 5

This parameter is often used to cancel the variations of the asset value in the

hedging portfolio.

(

ii) The gamma coefficient

It is defined as:

2

(,)

C

St

S

γ

∂

=

∂

(6.2)

and so it may be seen as the delta of the delta.

It gives a measure of the acceleration of the variation of the call and a refinement

of the measure of the variation of the call using the Taylor formula of order 2:

2

1

(,)(,) .

2

CS St CSt t t

γ

+Δ ≈ +ΔΔ + Δ

(6.3)

(iii) The theta coefficient

It gives the dependence of C with respect to the maturity ( )Tt

τ

=

− , and so also

from the time t:

.

CC

t

∂∂

θ

∂

∂τ

⎛⎞

=− =

⎜⎟

⎝⎠

(6.4)

It follows the first order approximation:

(, ) (,) .CSt t CSt t

θ

+

Δ≈ −Δ

(6.5)

For the maturity variations

τ

=

T

−

t

, we get:

(, ) (,) .CS CS

τ

ττθτ

+

Δ≈ +Δ

(6.6)

(iv) The elasticity coefficient

Recall the economic definition of this coefficient which gives here:

(,) (,)

(,)

CS

eSt St

SCSt

∂

=× (6.7)

and so:

(,)(,)

()(,).

(,)

CCS StCSt S

eSt

CCSt S

Δ+Δ− Δ

=≈

(6.8)

(v) The vega coefficient

It is the indicator concerning the measure of small variations of the volatility

σ

and so:

(,)

C

St

∂

υ

∂

σ

=

. (6.9)

Thus, we have approximately for small variations

,

σ

Δ

(,)(,).CS St CSt

υ

ο

+

Δ≈ +Δ

(6.10)

Black & Scholes extensions 195

(vi) The rhô coefficient

It concerns the non-risky instantaneous rate r and so:

(,).

C

St

r

ρ

∂

=

∂

(6.11)

6.2 Values Of The Greek Parameters

The following table gives the values of the Greek parameters first for the call and

then for the put.

1

-r

2

-r

21

2

I.For the calls:

C

1)delta(= )= ( ) 0

S

'( )

2)gamma(= ) 0

S

C

3)véga(= )= '( ) 0

C

4)rhô(= )= e ( ) 0

r

CS

5)théta(= )=rKe ( )+ '( ) 0

2

6) ( ) 0

II.For the puts:

P

1)delta(=

r

d

S

Sd

Kd

dd

C

ed

K

τ

∂

στ

∂

τ

∂σ

∂

τ

∂

∂σ

∂τ

τ

∂

−

Φ>

Φ

Δ

=>

Φ>

ΦΦ>

=− Φ <

[]

11

1

22

-r

12

)=( ( ) 1) ( )( 1) 0

S

'( )

2)gamma(= ) ( ) 0

S

P

3)véga(= )= '( )( ) 0

P

4)rhô(= )=- K ( ) K ( ) 1 ( K ) 0

r

P

5)théta(= )= '( ) Ke 1 ( ) (

2

C

rr r

C

dd

d

gamma

S

S d véga

ed ed rhôe

S

dr d rKe

ττ τ

τ

∂

στ

∂

τ

∂σ

∂

ττ τ

∂

∂σ

θ

∂τ

τ

−− −

−

Φ−=−Φ−=Δ−<

Φ

Δ

== >

Φ= >

Φ− = Φ − = − <

Φ− −Φ =−

22

)

6) ( ( ) 1) ( )( ) 0

r

rr r

C

PP

ed ed e

KK

τ

ττ τ

∂∂

−− −

=−Φ+=Φ−= + >

196 Chapter 5

These values give interesting results concerning the influence of the considered

parameters of the call and put values.

For example, we deduce that the call and put values are increasing functions of

the volatility, and the call increases as

S increases but the put decreases as S

increases.

6.3 Exercises

Exercise 6.1

Let us consider an asset of value 1700 Euro and having as weekly variance

0.000433.

(i) What is the value of a call of exercise price 1750 Euro with maturity 30 weeks

under a non-risky rate of 6%?

(ii) Under the anticipation of a rise of 100 Euro of the underlying asset and of a

rise of 0.000018 of the weekly variance, what will be the consequences of the

call and put values?

Solutions:

(i) The values of the parameters necessary to compute the call value using the

Black and Scholes formula are:

22

.. .

0.00043 52 0.00043 0.2236, 0.47286,

30 . 0.576923 , 1750, 1700,

6% ln(1 ) 0.05827.

week year year

weeks year K S

iri

σσ σ

τ

=⇒=×= =

== ==

=⇒=+=

It follows that:

2

11

1

21 2

1

ln ( ) 0.09760272,

2

( ) 0.5388762,

0.01637096, ( ) 0.4934692,

( , ) ( ) 81.07 .

r

S

drd

K

d

dd d

CS S d Ke Euro

τ

σ

τ

στ

τ

−

⎡⎤

=++⇒=

⎢⎥

⎣⎦

Φ=

⇒=− =− Φ =

=Φ − =

Using call-put parity relation; we get for the put value

73.07 .

r

PKe CS P Euro

τ

−

=+−⇒=

(ii)

Rise of the underlying asset:

We know that:

Black & Scholes extensions 197

1

(,)(,)(,),

(,) ( ),

so :

(1700 100, ) 81.07 100 0.5388762 135.95 .

C

CS S CS S S

S

C

Sd

S

CEuro

∂

ττ τ

∂

τ

∂

τ

+Δ = + Δ

=Φ

+=+× =

For the put, we obtain:

(,) ()()27.1.

r

PSS Ke CSSSS Euro

τ

−

+Δ = + +Δ − +Δ =

(iii) Rise of the volatility:

The value of the new weekly variance is now given by:

0.000433 0.00018 0.000613+=

and so the new yearly variance and volatility are given by

,1785385.0031876.0 =

and consequently, the variation of the yearly volatility is given by:

.284852.01500533.01785385.0

=

−=Δ

σ

As the increase in volatility comes after that of the asset value, we have

1

(, ,)(,,) ,

with:

().

C

CS S CS S

C

d

∂

σ

στ στ σ

∂σ

∂

τ

∂σ

+Δ +Δ = +Δ + Δ

′

=Φ

But:

,39704658.0

2

1

)(

2

1

2

1

==Φ

′

−

d

ed

π

and so:

542.84.

C

∂

σ

=

Finally, we get:

.41.150),,(),,( F

C

SSCSSC =Δ+Δ+=Δ+Δ+

σ

∂

σ

∂

τστσσ

For the variation for the put, we use the call-put parity relation and so:

( , ,) ( , ,) ( ) 42.56 .

r

PSS CSS Ke SS F

τ

σστ σστ

−

+Δ +Δ = +Δ +Δ + − +Δ =

Exercise 6.2

For the following data, compute the values of the call and the put and the Greek

parameters

100, 98, 30 days, 0,01664, 8%.

week

SK i

τ

σ

=== = =

Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability

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