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2 The Financial Model: Smile Tale 2 Revisited

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



450



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)



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