Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability
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208 Chapter 5
Numerical data are the following:
(0.05,0.90,0.05),
0.6 0.3 0.1 1.05 1.03 1.02
0.02 0.96 0.02 , 1.05 1.03 1.02 ,
0.6 0.35 0.05 1.06 1.04 1.03
1.07 1.10 1.20 0.5 0.7 0.8
1.07 1.10 1.20 , 0.6 0.7 0.8
1.07 1.09 1.15 0.65 0.7 0.8
=
⎡⎤⎡⎤
⎢⎥⎢⎥
==
⎢⎥⎢⎥
⎢⎥⎢⎥
⎣⎦⎣⎦
⎡⎤⎡⎤
⎢⎥⎢⎥
==
⎢⎥⎢
⎢⎥⎢
⎣⎦⎣⎦
a
Pr
UD.
⎥
⎥
(10.35)
For both examples, we will consider a European call option with
0
100 95.SandK==
Results are given in Table 10.1.
S 100
K 95
Example 10.1
transition a1 a2 a3 p(ij) r(ij) u(ij) d(ij) q(ij)
Cij(100,1) Ci(100,1) C(100,1)
0 to 0 0.95 0.05 0.98 1.03 1.3 0.7 0.55 18.68932
18.57744
0 to 1 0.02 1.05 1.1 0.5 0.9167
13.09524
1 to 0 0.6 1.05 1.06 0.4 0.9848
10.31746 13.1484
1 to 1 0.4 1.03 1.2 0.6 0.7167
17.39482
18.27158
Example 10.2
0.05 0.9 0.05
bad to bad 0.6 1.05 1.07 0.5 0.9649
11.02757 11.56895
bad to normal 0.3 1.03 1.1 0.7 0.825
12.01456
bad to good 0.1 1.02 1.2 0.8 0.55
13.48039
normal to bad 0.02 1.05 1.07 0.6 0.9574
10.94225 12.02243
normal to normal 0.96 1.03 1.07 0.7 0.8919
12.01456
normal to good 0.02 1.02 1.07 0.8 0.8148
13.48039
good to bad 0.6 1.02 1.2 0.65 0.6727
11.05121 7.67948
good to normal 0.35 1.02 1.2 0.7 0.64
11.7357
good to good 0.05 1.03 1.15 0.8 0.6571
12.76006
11.82274
Table 10.1: European call option examples
Black & Scholes extensions 209
10.2 The Multi-Period Discrete Markov Chain Model
Let us now consider a multi-period model over the time interval
[
]
0, n , n being an
integer larger than 1 representing the maturity time of the option, always under
the assumption of absence of arbitrage as in section 1.
To obtain useful results, we will still follow the fundamental presentation of the
CRR model (Cox, Rubinstein (1985)) but adapted for our Markov environment in
such a way that tractable results may be found.
1)
result with knowledge of
0
,...,
n
J
J
Let us begin with a discrete time model with n periods and suppose that given
00
,..., , (0)
n
J
JS S
=
with
0
,,
n
J
iJ j
=
=
the up and down parameters, the non-
risky interest rate and the probabilities of an up movement for each period are the
same for all periods and given respectively by , , ,
ij ij ij ij
udrq.
Then, the asset value
S(n) at time n is given by:
01 1
0
()
nn
jj j j
Sn V V S
−
=
⋅⋅ (10.36)
where the conditional distributions of the random variables
V are defined as:
1
with probability ,
,.
with probability 1- ,
nn
ij ij
JJ
ij ij
uq
VijI
dq
−
⎧
⎪
=∈
⎨
⎪
⎩
(10.37)
Moreover, we suppose that, for each
n, the random variables
01 1
,...,
nn
J
JJJ
VV
−
are
conditionally independent given
0
,..., .
n
J
J
Now if the random variable
n
M
represents the total number of up movements on
[]
0, n , the asset value at time n is given by:
0
() ( ) ( )
nn
MnM
ij ij
Sn u d S
−
= (10.38)
and consequently:
0
()
ln ln ( )ln .
nij n ij
Sn
M
unM d
S
=+− (10.39)
Given
00 0
,..., , (0)
nn
J
jJjS S===, the conditional distribution of
n
M
is a
binomial distribution with parameters ( , )
ij
nq . It follows that:
00 0
0
()
ln ,..., , (0) ( ln (1 )ln )
n n ij ij ij ij
Sn
E
Jj JjS S nqu q d
S
⎛⎞
==== +−
⎜⎟
⎝⎠
.(10.40)
Concerning the conditional variance, we get:
2
00 0
0
()
var ln ,..., , (0) (1 ) ln .
ij
n n ij ij
ij
u
Sn
Jj JjS S nq q
Sd
⎡
⎤
⎛⎞
⎛⎞
⎢
⎥
====−
⎜⎟
⎜⎟
⎜⎟
⎢
⎥
⎝⎠
⎝⎠
⎣
⎦
(10.41)
210 Chapter 5
Choosing now for the up probability on the n periods, the risk neutral probability
given by relation (10.10):
11
1
11
ij ij
ij
ij ij
rd
q
ud
−
=
−
, (10.42)
it is now clear that, under our assumptions, for each n, given
00
,..., , (0)
n
J
JS S=
with
0
,,
n
J
iJ j== we have a CRR model, so that their results recalled in the
beginning of this chapter concerning the European call are valid. Consequently,
we get the value of the European call with exercise price and maturity n as the
present value of the expectation of the “gain” at time n under the risk neutral
measure, that is:
{}
0001
0
0
( , ) ( , , ,..., )
1
(1 ) max .
ij n
n
knkknk
ij ij ij ij
n
k
ij
CSn CSnJ iJ J j
n
qq udSK
k
ν
−−
=
===
⎛⎞
=− −
⎜⎟
⎝⎠
∑
(10.43)
After some computation, we can obtain the following expression (see Cox and
Rubinstein (1985)):
0
00 1
(;,') (;,), if ,
( , , ,..., )
0 if ,
ij ij ij ij ij
n
ij
n
ij
K
SBa nq Ba nq a n
CS nJ iJ J j
an
ν
⎧
−
<
⎪
===
⎨
⎪
>
⎩
(10.44)
where ( ; , )Bxm
α
is the value of the complementary binomial distribution
function complementary with parameters ,m
α
at point
x
and
0
ln( / )
1,
ln( / )
'.
n
ij
ij
ij ij
ij
ij ij
ij
KdS
a
ud
u
qq
r
⎡
⎤
=+
⎢
⎥
⎢
⎥
⎣
⎦
=
(10.45)
The result (10.44) can be seen as the discrete time extension of the Black and
Scholes formula given the environment:
00
,..., , (0)
n
J
iJ jS S
=
==. (10.46)
2) result with knowledge of
0
J
i
=
If we only know the initial state of the environment
0
J
i
=
, it is clear that the
value of the call is given by
()
00
1
(,) (,)
m
n
iijij
j
CSn p C Sn
=
=
∑
(10.47)
where, of course:
()
.
nn
ij
p
⎡⎤
=
⎣⎦
P (10.48)
3) result with knowledge of
n
J
j
=
Black & Scholes extensions 211
Proceeding as in the preceding section, the use of Bayes formula gives the
following result now on n periods instead of one:
()
(
)
()
0
0
()
()
0
,
n
n
n
n
iij
m
n
kkj
k
PJ iJ j
PJ iJ j
PJ j
ap
ap
=
=
=
===
=
=
∑
(10.49)
and so the value of the call given
n
J
=j, represented by
0
(,)
j
CSn, is given by:
()
00
()
1
0
(,) (,).
n
m
iij
j
ij
m
n
i
kkj
k
ap
CSn CSn
ap
=
=
=
∑
∑
(10.50)
4) result with no environment knowledge
Finally if we have no knowledge of the initial environment state but just its
probability distribution given by (10.1), the value of the call denoted
0
(,)CS nis
given by
00
1
(,) (,)
m
ii
i
CS n aC S n
=
=
∑
(10.51)
or by
()
00
11
(,) (,).
mm
nj
kkj
jk
CS n ap C S n
==
=
∑∑
(10.52)
10.3 The Multi-Period Discrete Markov Chain Limit Model
To construct our continuous time model on the time interval [0,t], t being the
maturity time of the considered option, let us begin to consider a multi-period
discrete Markov chain model with n periods where each period has length h so
that we have equidistant observations at time 0,h,2h,...,nh with /nth=
⎢
⎥
⎣
⎦
.
We also suppose that in the approximated discrete time model, the environment
process is now a homogeneous ergodic Markov chain defined by relations (10.1)
and (10.2) and that (see Cox and Rubinstein (1985),p. 200) or relations (3.8),
section 3.1 of this chapter), for each n, given
00
,..., , (0)
n
J
JS S= with
0
,,
n
J
iJ j== we select, in each subinterval
[
]
,( 1)kh k h+ , the following up and
down parameters:
212 Chapter 5
11
1
,,
11
,
22
ij ij
kk kk
kk
tt
nn
jj jj
ij
jj
ij
uede
t
q
n
σσ
μ
σ
++
+
−
==
=+
(10.53)
depending thus on the two
mm
×
non-negative matrices:
,
ij ij
μ
σ
⎡
⎤⎡ ⎤
⎣
⎦⎣ ⎦
. (10.54)
From relations (10.40) and (10.41), it follows that, for all n:
00 0
0
()
ln ,..., , (0) ,
nn ij
Sn
E
Jj JjS S t
S
μ
⎛⎞
====
⎜⎟
⎝⎠
(10.55)
2
00 0
0
()
var ln ,..., , (0) .
nn ij
Sn
J
jJjS S t
S
σ
⎛⎞
====
⎜⎟
⎝⎠
(10.56)
As our conditioning implies that we can follow the reasoning of Cox and
Rubinstein (1985), we know that, for n →+∞:
2
0
()
ln ( , ),
ij ij
St
N
tt
S
μσ
≺ (10.57)
where j
0
=i as the initial environment state observed at t=0 and j the environment
state at time t.
Concerning the non-risky interest rates, we also suppose that, for all i and j, there
exists 1
ij
ν
> such that the new return rate for all the periods
(
)
,( 1) )kh k h+ ,
denoted
ˆ
ij
r , for n →+∞, satisfies the following condition:
ˆ
(1 ) (1 )
nt
ij ij
rr+→+. (10.58)
Now let
0
(,)
ij
CSnrepresent the value at time 0 of a European call option with
maturity n and exercise price K.
Using the proof of the Black and Scholes formula given by Cox and Rubinstein
((1985), pp. 205-208) but here with our parameters depending on all of the
environment states i and j, we get under the conditions (10.53) and (10.58), for
fixed t:
00
(,) (,)
ij ij
CSn CSt→ (10.59)
where:
001 ,2
0
1
,1
,2 ,1
(,) ( ) ( ),
ln
1
,
2
.
tt
t
t
t
ij ij ij ij
ij
ij ij
ij
ij ij ij
CSt S d Kr d
S
Kr
dt
t
dd t
σ
σ
σ
−
−
=Φ − Φ
=+
=−
(10.60)
Black & Scholes extensions 213
This result gives the value of the call at time 0 with i as initial environment state
and j as environment state observed at time t, represented from now on by
.
t
J
If we want to use the classical notation in the Black and Scholes (1973)
framework, we can define the instantaneous interest rate intensity
ij
ρ
such that:
ij
ij
re
ρ
= (10.61)
so that the preceding formula (10.60) becomes now:
001 ,2
2
,1
,2 ,1
(,) ( ) ( ),
1
ln ,
2
.
ij
ttt
t
t
t
ij ij ij
ij
ij ij
ij
ij ij ij
CSt S d Ke d
S
dt
K
t
dd t
ρ
σ
ρ
σ
σ
−
=Φ − Φ
⎛⎞
⎛⎞
⎜⎟
=+−
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=−
(10.62)
10.4 The Extension Of The Black-Scholes Pricing Formula
With Markov Environnment: The Janssen-Manca Formula
The last result (10.62) gives a first extension of the Black and Scholes formula in
continuous time from the knowledge of the initial and final environment states,
respectively
0
,
t
J
J where
t
J
represents, as said above, the state of the
environment at time t.
Now, always with the assumption that the Markov chain with matrix
P is
ergodic, we can extend results (10.43), (10.47) (10.50) and (10.52) valid for our
discrete multi-period model to our continuous time model thus giving the
following main result.
Proposition 10.1 (Janssen and Manca (1999))
Under the assumption that the Markov chain of matrix
P of the environment
process is ergodic and that given the initial environment state
iI∈ and the
environment state at time t is j I
∈
, the non-risky rate is given by
ij
ρ
and the
annual volatility by
ij
σ
, then we have the following results concerning the
European call price at time 0 with exercise price K and maturity t:
(1) with knowledge of state
0
,
t
J
iJ j
=
= , the call value is given by result (10.62),
(2) with knowledge of state
0
J
i
=
, the call value represented by
0
(,)
i
CSt is
given by:
00
1
(,) (,),
m
ijij
j
CSt C St
π
=
=
∑
(10.63)
(3) with knowledge of state
t
J
j
=
, the call value represented by
0
(,)
j
CSt is
given by:
214 Chapter 5
00
1
(,) (,),
m
j
iij
i
CSt aCSt
=
=
∑
(10.64)
(4) without any environment knowledge, the call value represented by
0
(,)CS t is
given by:
00
1
(,) (,)
m
ii
i
CS t aC S t
=
=
∑
(10.65)
or
00
1
(,) (,)
m
j
j
j
CS t C S t
π
=
=
∑
. (10.66)
Proof
Result (1) is proved above.
Result (2) follows from relation (10.47) letting n go to
∞
+
and then using result
(1) and the assumption of ergodicity on the environment matrix chain
P .
Result (3) can easily be deduced from result (2) and relation (10.50).
Finally, result (4) follows immediately from relations (10.51) or (10.52) and
results (2) and (3).
Example
The Example 10.1 is now treated in Table 10.2 with the following annual
volatility matrix
0.20 0.30
0.25 0.35
⎡
⎤
=
⎢
⎥
⎣
⎦
σ
Using for the matrix
P being given by relation (10.33), using relation 2(9.77), the
asymptotic distribution is given by:
[]
[
]
12
, 0.977742,0.032258
ππ
==π
Example 10.1
K 80 K 95
S 100 S 100
0 to 0 0 to 0
1-t Cij(100,t) 1-t Cij(100,t)
0.25 20.45 0.25 7.23
0.5 21.10 0.5 8.98
0.75 21.84 0.75 10.42
1 22.61 1 11.67
1 to 0 0.25 20.57 0.25 8.08
Black & Scholes extensions 215
0.5 21.52 0.5 10.24
0.75 22.56 0.75 11.98
1 23.59 1 13.49
0 to 1 0.25 20.79 0.25 8.97
0.5 22.13 0.5 11.54
0.75 23.48 0.75 13.57
1 24.76 1 15.33
1 to 1 0.25 21.12 0.25 9.88
0.5 22.87 0.5 12.85
0.75 24.53 0.75 15.18
1 26.07 1 17.18
? To 1 0.25
20.8065
0.25
9.0155
0.5
22.167
0.5
11.6055
0.75
23.5325
0.75
13.6505
1
24.8255
1
15.4225
? To 0 0.25
20.456
0.25
7.2725
0.5
21.121
0.5
9.043
0.75
21.876
0.75
10.498
1
22.659
1
11.761
? To ? 0.25
20.46731
0.25
7.328726
0.5
21.15474
0.5
9.125661
0.75
21.92944
0.75
10.59969
1
22.72889
1
11.87911
Table 10.2
In conclusion, the Janssen-Manca approach gives for the first time a new family
of Black and Scholes formulae taking into account the economic and social
environment showing that:
- a “good” extension of the classical Cox-Rubinstein model is possible,
-the model also extends the Black and Scholes model
-numerical results are possible.
Moreover, as the JM formulas are linear combinations of the classical BS results,
the Greek parameters can also be computed and will be linear combinations of
the Greek parameters given in section 6 and similarly for hedging coefficients.
We also add that, in our point of view, one of the main potential applications of
our new model concerns the possibility to get a new way of acting with the Black
216 Chapter 5
and Scholes formula with information related to the economic, financial and even
political environment provided it can be modelled by an ergodic homogeneous
Markov chain.
This model also gives the possibility of taking into account anticipations made
by the investors in such a way as to incorporate them in their own option pricing
and can also be used for models with financial crashes as well as to construct
scenarios and particularly in the case of stress in a VaR type approach.
11 THE EXTENSION OF THE BLACK-SCHOLES
PRICING FORMULA WITH A MARKOV
ENVIRONMENT: THE SEMI-MARKOVIAN JANSSEN-
MANCA-VOLPE FORMULA
11.1 Introduction
In this section, we present another semi-Markov extension of the Black and
Scholes formula to the so-called Janssen-Manca-Volpe model to eliminate one of
the restrictions of the Black and Scholes model that is the assumption of constant
volatility upon time.
If there have been a lot of attempts to slacken this condition, as for example in
the model of Hull and White (1985) where the concept of stochastic volatility is
introduced, nevertheless, to our knowledge, in practice, no generalised model
really supplants the classical Black and Scholes model.
Whilst comparing with the Markovian Janssen-Manca model of the preceding
section, we develop another type of model. More precisely we present new semi-
Markov models for the evolution of the volatility of the underlying asset.
In fact, the SM model presented here supposes a type of SM evolution for the
volatility of an initial Black-Scholes model presented for the first time in an oral
communication in the ETH Zurich (1995) by J. Janssen and in a different
approach by E. Çinlar in an oral communication at the First Euro-Japanese
meeting on Insurance, Finance and Reliability, held in Brussels in 1998, and
leading to a generalization of the classical Black and Scholes formula for the
pricing of European calls with easy numerical applications.
11.2 The Janssen-Manca-Çinlar Model
Hereby, we present our initial model of 1995 close to the oral presentation of
Çinlar but he gives the formula for the pricing of a call option using the Markov
renewal theory.
Black & Scholes extensions 217
11.2.1 The JMC (Janssen-Manca-Çinlar) Semi-Markov Model (1995, 1998)
Let us consider a two-dimensional positive (J-X) process of kernel Q with as
state space:
{
}
1,..., .Im= (11.1)
This means that on the probability space
(
)
Ω
,,
ℑ
P , we define the three-
dimensional process
(
)
(
)
,( , ) , 0
nnn
JX n
σ
≥ (11.2)
with:
,( , ) ,
nnn
JIX
σ
+
+
∈∈× (11.3)
such that:
()
()
1
, , ,( , ) , 0.1..... 1
(, ), ..
n
nn n kkk
Jj
PX x J j J X k n
Qxps
σσ σ
σ
−
≤≤= = −
=
(11.4)
We know that the , ,
ij
Qij I∈ can be written in the following form:
(, ) (, )
ij ij ij
Qx pFx
σ
σ
=
(11.5)
where:
(
)
1
,1,
ij n k n
p
PJ jJ k n J i
−
== ≤−=, (11.6)
()
1
( , ) , ( ,( , )), 1, .
ij n n k k k n
F
xPXx JX knJi
σσσσ
−
=≤≤ ≤−= (11.7)
We also introduce the following r.v.:
{}
1
()
,0,
() sup : , 0,
() , 0.
nn
n
Nt
TX Xn
Nt nT t t
Zt J t
=++ ≥
=
≤≥
=≥
(11.8)
As usual, the transition probability for the process
(
)
(), 0ZZtT
=
≥ is designed
by:
(
)
() () ()
ij
tPZt jZti
φ
=
==
(11.9)
and the stochastic processes ( ( ), ),( ( ), )Nt t Zt t
+
+
∈∈are respectively the
Markov renewal counting and the semi-Markov processes.
To give the financial interpretation of our model, let us define on the probability
space
()
,,PΩℑ , the following filtration ( , )
t
t
+
ℑ= ℑ ∈ ,
(( ,( , )), ( )).
tnnn
J
XnNt
σ
σ
ℑ= ≤ (11.10)
Given
ℑ
t
, let us consider the random time interval
() ()1
,
Nt Nt
TT
+
⎡
⎤
⎣
⎦
on which we
define the new stochastic process
((), )St t
+
∈ , representing the value of the
considered financial asset, as the solution of the stochastic differential equation: