2 The Financial Model: Smile Tale 2 Revisited
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442
CHAPTER 14 JUMP–DIFFUSION PROCESSES
strategy in greater detail. The arbitrageur observes a smile curve which is roughly ﬂat
for values of the strike above the at-the-money level, and which rises monotonically for
strikes below the at-the-money level. For instance, the ‘market’ smile the trader observes
could be of the form shown in Figure 12.10. If she believes that she inhabits a Blackand-Scholes world, it is natural for her to enter the following transactions (see Table 14.1
and Figure 14.1):
Table 14.1 Two sub-portfolios, before (left box) and after
(right box) the jump. For each of the two boxes, the ﬁrst
column refers to the sub-portfolio with the out-of-the-money
option, and the second to the sub-portfolio with the at-themoney option. The stock price before the jump is at 100 and
the residual maturity 1 year; the two strikes are at 100 at 80.
The amount of cash before the jump makes each sub-portfolio
worth zero. A ‘Ratio’ amount of portfolio 2 makes the Total
Portfolio vega and gamma neutral. When the jump occurs, the
stock price moves from 100 to 80.
Strike
80.0000
100.0000
80.0000
100.0000
6.3725
21.1859
6.3725
21.1859
Price
−0.4602
−0.8451
−0.1120
−0.4602
Delta
0.0248
0.0149
0.0248
0.0149
Gamma
31.7562
19.0534
19.0534
39.6953
Vega
−43.1862 −88.7954 −12.3906 −53.9828
Cash
0.0000
−0.0000
2.9456
4.0169
Port
Ratio
1.6667
0.4800
−0.0000
−1.0175
TotalPort
0.0002
Portfolio value
0
97.5
98
98.5
99
99.5
100
100.5
101
−0.0002
−0.0004
−0.0006
−0.0008
Series1
−0.001
Stock price
Figure 14.1 The P&L of the total portfolio described in Table 14.1 as a function of the post-jump
value of the stock. Note how the total portfolio is gamma neutral at the origin.
14.2 THE FINANCIAL MODEL: SMILE TALE 2 REVISITED
443
• she sells an out-of-the-money put;
• she constructs a zero-cost, delta-neutral replicating portfolio to cancel the delta exposure of the out-of-the-money put;
• she neutralizes the vega exposure of the out-of-the-money put by buying an amount
of at-the-money put with the same vega;
• she eliminates the residual delta exposure introduced by the last transaction by
dealing in an appropriate (delta) amount of stock and by borrowing or lending cash.
Note that, for the moment, the delta and vega transactions carried out by the arbitrageur
are those suggested by the standard Black formula used with the at-the-money volatility.
This is consistent with an arbitrageur who does not ‘believe in jumps’ and considers the
higher implied volatilities purely a result of the desire for insurance of the fund players.
Also, we have, rather arbitrarily, assumed that the arbitrageur is trying to obtain the
‘biggest bang for the buck’, i.e. that she has chosen to hedge herself by buying the option
which has, at the same time, the maximum difference in implied volatility and is closest
in strike to the chosen out-of-the-money put. Given the assumed shape of the equity smile
curve she observes, the option chosen as a ‘buy’ is therefore the at-the-money option.
This assumption is not necessary, and the choice of option strikes will be discussed later
in the chapter.
If the trader has correctly guessed the volatility of the purely time-dependent diffusive component of the process, the overall hedged portfolio will make exactly no gains
and no losses until the ﬁrst jump. (I have assumed continuous, frictionless trading. See
Section 14.11 and Figure 14.33 in particular for a more realistic treatment.) When the
ﬁrst jump occurs the total portfolio (options plus hedges) will no longer have an overall
zero value, and, as shown above, over a downward jump the overall portfolio will always
make a loss. This is because the portfolio made up of the out-of-the-money put, which
the arbitrageur has shorted, and of the accompanying delta amount of stock will increase
in absolute value by more than the associated at-the-money put and its delta hedges.
The effect of a particular downward jump on a portfolio that has been kept balanced as
explained above at all times before the jump event is shown in Table 14.1. The portfolio
P&L over jumps of different amplitude (upward and downward) is shown in Figure 14.1.
Let me stress again that, even if we had assumed that in the real world the likelihood
of occurrence of upward jumps is the same as the probability of downward jumps, the
presence of jumps, of whichever sign, will introduce a ﬁnite variance to the terminal value
of the portfolio. If the arbitrageur is risk averse, and perceives risk in terms of portfolio
variance, she will therefore demand some compensation for this form of risk even if the
expected jump amplitude ratio were one (or, up to a point, even if it were positive). If, in
addition, jumps are more likely to be downwards, the arbitrageur will demand additional
compensation. The total ‘expected’ jump amplitude will therefore be made up of an
‘actuarial’ component, plus another part as compensation for the uncertainty introduced
by the existence of jumps of any sign.
It is important to point out that in the model we have outlined the true process of
the underlying equity index is mixed diffusive–jump in nature. The jumps, in particular,
are of random amplitude, and, as discussed below, the market is incomplete with respect
to their occurrence. In this setting a risk-neutral valuation will therefore fail to provide
a unique option price, and the result will depend to some extent on the arbitrageurs’
444
CHAPTER 14 JUMP–DIFFUSION PROCESSES
appetite for risk. This feature obviously is computationally unpleasant, but, in a way, it
is one of the central features employed in this chapter in order to explain and account for
the stylized facts that characterize equity smiles. Because of its importance, the topic of
market completeness in the presence of jumps is discussed in the next section.
14.3
Hedging and Replicability in the Presence of Jumps:
First Considerations
Several pricing models, starting from Merton’s (1990), have superimposed a discontinuous
(jump) component to a diffusion. The treatments that can be found in the literature differ
in the degree of hedging allowed, and in the nature of the jump process (log-normally- or
otherwise-distributed jumps, a ﬁnite or an inﬁnite number of possible jump amplitudes,
etc.). It is therefore important to state clearly the nature of the jumps and the trading
universe of ‘fundamental securities’ assumed to exist at the trader’s disposal for hedging
purposes. More precisely, if the process of the underlying is a jump–diffusion with a
continuous jump amplitude three situations can arise.
1. In the ﬁrst case, the hedging of an option with anything but the underlying stock (or
index) is disallowed. The market is incomplete, and for a general path realization
the trader cannot expect to replicate the ﬁnal payoff of a plain-vanilla option even
if she could trade without friction in continuous time.
2. The second pricing framework allows hedging of an option (e.g. out-of-the-money
puts) with one or more other options (e.g. the at-the-money calls/puts). In other
words, the second approach recognizes that some degree of hedging is possible, and
is likely to be entered into by the market participants, but that also in this case the
resulting portfolio will not exactly replicate the terminal payoff of the plain-vanilla
option. The market is still incomplete, but, with a judicious choice of the hedging
instrument, the variance of returns from the overall portfolio (option to hedge plus
imperfectly hedging options) can be signiﬁcantly reduced.
3. In the third case it is assumed that the trader can include an inﬁnite number of options
in her trading strategy. In this case a perfect hedge against the inﬁnite number of
possible realizations of the jump amplitude would seem to be possible, and the
market would appear completable. See, however, the caveat in Section 14.3.1.
If the jump–diffusion process is such that only a ﬁnite number of possible jump
amplitudes are possible, then, depending on the degree of hedging possible, the following
situations can arise.
1. If the hedging of an option with anything but the underlying stock (or index) is
disallowed, then the market is still incomplete. Therefore, once again, for a general
path realization the trader cannot expect to replicate the ﬁnal payoff of a plain-vanilla
option.
2. If as many hedging options as possible jump amplitudes are allowed, then it is
in theory possible to complete the market, and to set up an exactly replicating
portfolio. Since the number of possible amplitudes has been assumed to be ﬁnite,
14.3 HEDGING AND REPLICABILITY IN THE PRESENCE OF JUMPS
445
the completion of the market only requires a ﬁnite number of plain-vanilla options.
Also in this case, however, see the reservations I raise in Section 14.3.1.
The second statement is certainly correct and would appear to be quite powerful.
This is because, as I show in Section 14.7, a continuous-amplitude jump process can
often be closely approximated by a similar process with a ﬁnite (and sometimes rather
small) number of possible jump amplitude ratios. It would therefore seem that one could
efﬁciently ‘approximate’ a continuous-amplitude jump process with a process with only a
ﬁnite number of possible jump amplitude ratios. One could then invoke the completeness
of the latter setting to impute ‘almost’ unique pricing by no arbitrage also in the more
general case. Unfortunately, this approach is less useful than it might appear. I show in
the next section, in fact, that, in order to determine the holdings of the hedging options
required to create a riskless portfolio one must know not only their prices today, but also
their prices in all possible future states of the world.
14.3.1 What Is Really Required To Complete the Market?
I show analytically in Section 14.5 to what extent it is possible to hedge a contingent claim
when the process for the underlying is a mixed jump–diffusion. It is useful, however, to
gain an intuitive understanding of the nature of the problem by looking at a very simple
discrete-time setting. This case study will also be of help in understanding the origin and
limitations of statements such as:
If only a ﬁnite number of jump amplitudes are possible, then the market can be
completed by introducing among the set of hedging instruments as many plain-vanilla
calls as possible jump amplitudes.
Completion of the market, in this context, means that, in the absence of frictions,
an arbitrary payoff could be exactly replicated via a self-ﬁnancing trading strategy that
only involves the underlying, a bond and as many plain-vanilla options as possible jump
amplitudes. If this is the case, we know from Chapter 2 that we can arrive at a unique,
preference-independent price by invoking no arbitrage.
Let us consider a one-period problem.2 I extend the replication construction presented
in Chapter 2 to incorporate the possibility of jumps. For simplicity we will assume that
only one jump amplitude is possible, and therefore the underlying can move either to an
‘up’ or a ‘down’ diffusive state, or to a ‘jump’ state.3 We know the values of the stock
today and in all the three possible states that can be reached tomorrow, but we do not
claim to know the probabilities of reaching them. Similarly, we know the values today and
in all these possible future states of the world of a bond, B, and of the (call) option, O1 , of
strike K1 that we want to use for hedging purposes. To ensure unambiguous (i.e. modelindependent) knowledge of the value of the option O1 in all the possible future states,
we impose that its expiry should take place in one period’s time. In this case the value
2I
wish to thank Mark Joshi for pointing out to me this very clear example.
do we know that the up and down states are associated with the diffusive part, and the third state
with the jump component? The only way to tell is to examine how the value of the stock
√ changes in the various
t, while the magnitude
states as we change the length of the time-step, t. The diffusive steps will scale as
of the jump will remain unchanged as the time-step is reduced. See Section 2.4.1 in Chapter 2.
3 How
446
CHAPTER 14 JUMP–DIFFUSION PROCESSES
after the time-step of the hedging option will simply be equal to its payoff:
Payoff(O1 ) = max[Sj − K1 , 0],
j = up, down, jump
(14.1)
We also know the values that the option, O2 , that we want to price will attain in all the
possible future states of the world. This is because we assume that it also is an option
expiring in one period’s time. Therefore it is, say, a call with a different strike, K2 :
Payoff(O2 ) = max[Sj − K2 , 0],
j = up, down, jump
(14.2)
See Figure 14.2.
We want to construct a portfolio made up of the stock, the bond and the hedging option
with weights α, β and γ , respectively, such that, in all states of the world, it will have
the same value as the option to be hedged. As we discussed in Chapter 2, the problem of
ﬁnding the ‘fair’ price of the option O2 today is therefore solved by determining weights
α, β and γ such that
αSup + βBup + γ O1,up = O2,up
(14.3)
αSdown + βBdown + γ O1,down = O2,down
(14.4)
αSjump + βBjump + γ O1,jump = O2,jump
(14.5)
This gives rise to a 3 × 3 system of linear equations. Subject to some obvious constraints
on the known terms, this system admits a unique solution {α, β, γ }. Absence of arbitrage
Sup
S (0)
O 1, up = max[S up − K1, 0]
B up
B (0)
O1(0)
S down
B down
O 1, down = max[S down − K1, 0]
S jump
B jump
O 1, jump = max[S jump − K1, 0]
O 2, up = max[S up − K2, 0]
O2(0)
O 2, down = max[S down − K2, 0]
O 2, jump = max[S jump − K2, 0]
Figure 14.2 The values of the stock, the bond, the hedging option and the option to be hedged
today and after one time-step.
14.3 HEDGING AND REPLICABILITY IN THE PRESENCE OF JUMPS
447
then requires that the fair price today of the option O2 , i.e. the quantity that we were
looking for, should be equal to
O2 (0) = αS(0) + βB(0) + γ O1 (0)
(14.6)
As long as we deal with a one-period option to be hedged with one-period options of
different strikes, it is easy to see how this reasoning can be extended to more complex
cases where two, three, . . . , n, possible jump amplitudes exist: we would simply have to
add as many hedging options to our portfolio. For n possible jump amplitudes we would
have to solve an (n + 2) × (n + 2) linear system to ﬁnd the weights as above.
Why is this setting interesting? Because I show below that the price of most options
priced assumed a ﬁnite number of jump amplitudes and quickly converges to the price that
would obtain with a continuum of possible jump amplitudes, provided that the moments of
the continuous and discrete jump distributions are matched. This would be very important.
It would mean that, if ‘only’ n jump amplitudes were possible, and we could hedge with
as many plain-vanilla options, we would always have to agree about the price of any other
option, and our risk preferences would play no role in determining the price. To the extent
that a relatively small number of ‘well-chosen’ jump amplitudes produce a stock price
distribution very similar to the one produced by the continuous case, one would conclude
that in pricing complex options risk preferences would likely play a very limited role
even in the more general case (i.e. even in the case of random jump amplitudes).
This statement, formally correct, needs a strong qualiﬁcation. The origin of the problem
can be seen as follows. Let us extend our analysis to a two-period trading horizon. For
graphical clarity Figure 14.3 shows only the upper branch of the two-period bushy tree,
but similar branches can also be imagined emanating from the ‘down’ and ‘jump’ states.
Let us try to repeat the same reasoning presented above for this upper branch. Once
again we will try to hedge a two-period option (O2 ) with the stock, the bond and another
option, O1 , that now however expires in two periods’ time. The system of equations for
this upper branch now reads:
αSup,up + βBup,up + γ O1,up,up = O2,up,up
(14.7)
αSup,down + βBup,down + γ O1,up,down = O2,up,down
(14.8)
αSup,jump + βBup,jump + γ O1,up,jump = O2,up,jump
(14.9)
In order to avoid arbitrage, one must impose that the value of this replicating portfolio
should equal the price of the option to price in the ‘up’ state at time 1.
There is a formal similarity with the one-step problem discussed above, but the value
of the replicating portfolio now contains O1,up . This is the price of the hedging option 1
in the future if state ‘up’ prevails. Unlike O1 (0), the quantity O1,up , which is the future
conditional price of option 1 in the ‘up’ state, is not known from today’s market prices.
The same reasoning clearly applies to the other nodes (not shown in the ﬁgure): O1,down
and O1,jump are also not known from today’s market prices. Therefore, in order to price
option O2 by replication, one should know not just the price of the hedging option today,
O1 (0), but also its value in all the future possible states of the world.
Why are we troubled by this requirement? Have we not availed ourselves of similar
information when we have assumed to know the possible values of the stock price in all
S jump
S down
S up, jump
S up, down
B (0)
B jump
B down
B up
B up, jump
B up, down
B up, up
O 2(0)
O1(0)
O 2, jump
O 2, down
O 2, up
O 1, jump
O 1, down
O 1, up
O 2, up, jump = max[Sup, jump − K2, 0]
O 2, up, down = max[Sup, down − K2, 0]
O 2, up, up = max[Sup, up − K2, 0]
O 1, up, jump = max[Sup, jump − K1, 0]
O 1, up, down = max[Sup, down − K1, 0]
O 1, up, up = max[Sup, up − K1, 0]
Figure 14.3 The values of the stock, the bond, the hedging option and the option to be hedged today and after one time-step for all the reachable
states, and after two time-steps for the states emanating from the upper node at time 1.
S (0)
Sup
Sup, up
14.4 ANALYTIC DESCRIPTION OF JUMP–DIFFUSIONS
449
the future states of the world? Not really: recall that we have not speciﬁed the probabilities
of these states being reached, and therefore by assigning future possible stock values we
are basically simply deﬁning the space spanned by the price movements. Furthermore, in
the simple setting presented in Chapter 2, specifying the state of the world is ultimately
fully equivalent to assigning the realization of the stock price.4 Now we are assigning
to the same state the simultaneous prices of two assets (the underlying stock and the
hedging option). Clearly, we will not be able to do so without creating possibilities of
arbitrage unless we assume some quite detailed knowledge about the joint evolutions of the
stock and of the hedging option. But this is just the information typically provided as the
output of a calibrated model. (See, however, also the discussion in Chapter 17.) Therefore,
the deceptively simple extension of the canonical binomial construction presented in
Chapter 2 to the case of three possible branching states actually brings us into completely
different, and far more complex, territory.
Exercise 1 Following the reasoning just presented, build a suitable tree to explain why
also in the case of a fully stochastic-volatility model knowledge of the process of the
hedging option is required to price derivatives in a unique preference-free manner.
Generalizing: not just knowledge of the prices of as many hedging options as possible
jump amplitudes is required, but also of their full process. This, however, is a strongly
model-dependent piece of information, and, unlike the situation encountered in the oneperiod setting, two traders might disagree about the future price of the hedging options,
and, therefore, ultimately about the price of the option O2 today. Unless the very strong
assumption is made that we know the prices of the hedging options in all possible future
states, even in the presence of a ﬁnite number of possible jump amplitudes the option
market therefore remains incomplete, and perfect replication is not possible.
14.4
Analytic Description of Jump–Diffusions
14.4.1 The Stock Price Dynamics
Since jump processes are less widely used than Brownian diffusions in ﬁnancial applications, I brieﬂy describe them in this section. See also Merton (1990) for one of the
earliest (and clearest) descriptions of jump processes in the context of option pricing. In
this chapter I largely follow Merton’s approach and notation. Many of the applications
of mixed jump–diffusion processes are in the credit derivatives area. The literature is
immense, but one interesting paper dealing with the necessary change in measure when
jumps are possible is Schoenbucher (1996). In the interest-rate area, a detailed discussion
of the change of measure can be found in Glasserman and Kou (2000) and Glasserman
and Merener (2001).
We will assume that the stock behaviour is described in the real (‘econometric’) world
by a mixed jump–diffusion process of the form5
dS(t)
= (µ − λk) dt + σ (t) dz(t) + dq
S(t)
4 In
(14.10)
other words, the stock price process is an adapted process. See Chapter 5.
ease of reference, in this chapter I have tried to use, as much as possible, the same notation as in
Merton (1990). Some of the symbols, however, have been changed in order to keep consistency with the
notation employed elsewhere in this book.
5 For
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CHAPTER 14 JUMP–DIFFUSION PROCESSES
In Equation (14.10) above, S(t) denotes the stock price, µ its drift in the real world, σ (t)
its percentage volatility,6 dz(t) is the increment of a standard Brownian motion and dq
is the increment of a Poisson process. As for the latter, λ is the mean number of events
(jumps) per unit time in the real world. If Y − 1 is the percentage change in the stock
price before and after the jump, one can deﬁne
k = E[Y − 1]
(14.11)
to be the expected size of the percentage jump amplitude. Note that, by this deﬁnition, if
we want the stock price to remain strictly positive at all times, so must the quantity Y :
Y = 1 corresponds to no jump, Y = 0 represents a jump of negative amplitude equal to
the stock price at the time of the jump, and any value of Y greater than 1 gives an ‘upward’
jump. These observations suggest some of the desirable distributional properties of the
random quantity Y , discussed below. Finally, in the standard treatment the increments of
the Brownian and Poisson processes are assumed to be independent.
The intuitive interpretation of Equation (14.10) is that the stock price follows a diffusive
behaviour, characterized by a time-dependent percentage volatility, σ (t), on top of which
discontinuous jumps, of random magnitude and sign, are superimposed with a known
deterministic frequency, λ. The stock price path is therefore continuous (but nowhere
differentiable) between jumps, and discontinuous when a jump occurs. If a jump does not
occur, the process for the stock price has the usual diffusive form
dS(t)
= (µ − λk) dt + σ (t) dz(t)
S(t)
(14.12)
If the jump takes place, the stock price instantaneously changes from S to SY , and the
overall percentage change in the stock price (due both to the diffusive and to the jump
components) is given by
dS(t)
= (µ − λk) dt + σ (t) dz(t) + (Y − 1)
S(t)
(14.13)
The Counting Process and the Compensator
Let n(t) be the number of jumps from time 0 to time t. This quantity is called a ‘counting
process’.7 The process n(t), R → I , takes on integer values, starts at zero, n(0) = 0, and
increases by one every time there is a jump. If the jump process is Poisson in nature, one
can directly obtain from the deﬁning properties of Poisson processes (see, for example,
6 The percentage volatility at time t is therefore the square root of the instantaneous variance contingent on
no jump events occurring at time t.
7 A note on terminology: the arrival time, τ , of a jump is a stopping time. A collection of stopping times is
called a point process: {τ1 , τ2 , . . . , τk }. Despite the name, this is not a process that, loosely speaking, should
be a random variable indexed by time. A point process can be turned into a ‘real’ process ﬁrst by deﬁning an
indicator process, N (t), for each stopping time:
N (t) = 1{τ ≤t}
(14.14)
14.4 ANALYTIC DESCRIPTION OF JUMP–DIFFUSIONS
451
Ross (1997)) that, if λ is the frequency (intensity) of the jumps, the probability of k jumps
having occurred out to time t is given by
P [n(t) = n] = exp[−λt]
(λt)n
n!
(14.16)
This expression will be made use of when calculating the pricing formula for calls in the
jump–diffusion case.
In order to obtain the expected value of the increment, dn(t), of the counting process,
recall that, for a Poisson process, the occurrence of a jump is independent of the occurrence
of previous jumps, and that the probability of two simultaneous jumps is zero. Let us
then assume that, out to time τ , j jumps have occurred. Then, over the next small time
interval, dt, one extra jump will occur with probability λ dt, and no jumps will occur
with probability 1 − λ dt. The change in the counting process will therefore be 1 with
probability λ dt and 0 with probability 1 − λ dt. Therefore
E[dn(t)] = 1 ∗ λ dt + 0 ∗ (1 − λ dt) = λ dt
(14.17)
If the jump frequency is constant, the expected number of jumps from time 0 to time t
is therefore given by
E[n(t)] = λt
(14.18)
It is important to observe that, from Equations (14.12) and (14.18), it follows that the
instantaneous expected return from the stock is µ: during ‘normal’ times (no jumps),
the stock deterministically grows at a rate µ − λk; occasionally jumps occur, altering
its expected growth rate by the expectation of the jump amplitude ratio, k, times the
probability of occurrence of the jump, λ dt. For this reason one sometimes introduces a
compensated counting process, M(t), deﬁned by
dM(t) = dn(t) − λ dt
(14.19)
Given the deﬁnitions above the expectation of dM(t) is, by construction, zero:
E[dM(t)] = E[dn(t) − λ dt] = E[dn(t)] − λ dt = λ dt − λ dt = 0
(14.20)
It is easy to check that the process M(t) satisﬁes the conditions for being a martingale.
The process M(t) is called the compensated counting process because it is constructed
so as to ensure that on average it exactly ‘compensates’ for the increase in the number
of jumps accumulated in n(t). See Figure 14.4.
Then one can create the sum of the indicator processes associated with each stopping time in the set
{τ1 , τ2 , . . . , τk }:
n(t) =
1{τi ≤t}
(14.15)
i=1,k
This quantity is called the counting process. See Schoenbucher (2003) for a simple and clear introduction.
452
CHAPTER 14 JUMP–DIFFUSION PROCESSES
7
5
3
N
M
1
−1
0
10
20
30
40
50
60
70
−3
Figure 14.4 A realization of a counting process n(t) and its associated compensated process M(t).
Note that the expectation of M(t) is zero, whilst the expectation of n(t) grows with time (at the
rate λ).
Closed-Form Expression for the Realization of the Stock Price
If the volatility, σ , the jump frequency, λ, the expectation of the jump amplitude ratio,
k, and the instantaneous drift, µ, are all constant, it is possible to write a closed-form
expression for the realization of the stock price after a ﬁnite time t, given its value, S(0),
at time t0 as
S(t) = S(0) exp (µ − 12 σ 2 − λk)t + σ Z(t) Y (n)
(14.21)
In Equation (14.21) the ﬁrst part of the RHS has the same expression as in the diffusive
case, and therefore
t
Z(t) =
dz(s)
(14.22)
0
The interesting term is the multiplicative factor Y (n). The argument n denotes different
realizations of the counting process, i.e. the number of jumps that have occurred between
time 0 and time t. Y (n) is therefore given by
Y (0) = 1
Y (1) = Y1
Y (2) = Y1 Y2
...
Y (n) =
Yj
j =1,n
(14.23)