10 Jump–Diffusion Processes and Market Completeness Revisited
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14.10 JUMP–DIFFUSION PROCESSES AND MARKET COMPLETENESS
501
crucial,17 together with the risk aversion of the investor. Valuing the option by itself
(without the partial hedging strategy we have chosen) no longer makes sense, since the
perfect hedging portfolio ‘somewhere in the back of our minds’ no longer exists. Similarly, simply looking at the variance of returns obtained by holding the ‘naked’ option
does not make a lot of sense either, because some hedging, albeit imperfect, is certainly
possible, and better than none. (See the results in Section 14.11 and Cochrane’s (2001)
quote that closes Section 1.5.) At the same time, since there is no such thing as the perfect
hedging strategy, there is no single correct hedging portfolio. We have therefore reached
the ﬁrst conclusion, namely that the trader will obtain one overall variance of return for
every imperfect hedge she might dream up. The ﬁnancial subtleties, however, are not
over yet.
Since we have been ‘spoiled’ by the familiar Brownian processes with perfect hedges,
we have become accustomed to identifying absence of arbitrage with the existence of a
unique price for the option. This is, however, a special case that applies only when a
perfectly replicating strategy can be put together, and the returns from the total portfolio
(i.e. the one made up of the option and of the parallel perfect hedging strategy ‘in the
back of our minds’) can therefore be made purely deterministic. In general, however,
given a set of possible prices for the securities traded in the market in different states of
the world, three different cases can arise (see Pliska (1997) for a very clear discussion):
1. a pricing measure exists and is unique;
2. there are inﬁnitely many pricing probability measures;
3. there are no pricing measures.
Case 1 applies when markets are complete, but case 2 is true if not every contingent
claim is replicable (i.e. when markets are not complete). Arbitrage, however, should not
be allowed to exist not only in the restrictive case 1, but also in case 2. If the problem
is looked at in this manner, obtaining a unique price from a no-arbitrage argument, as
is possible in case 1, is the exception rather than the rule, and the lack of uniqueness of
option prices in the case of incomplete markets is very fundamental.
How can we observe in an incomplete market a single quoted price for an option, if
that is the case? Because, out of the inﬁnitely many possible pricing measures that could
coexist in an incomplete market, ‘the market’ chooses a single one. The pricing measure
is chosen among the inﬁnitely many possible by the process of market clearing. This,
incidentally, is the meaning of Bjork’s (1998) often repeated quote: ‘Who chooses the
measure? The market!’
This observation has an interesting corollary. If option payoffs can be perfectly replicated, option prices contain no information about risk aversion. To the extent, however,
that perfect replication is impossible, some information about risk aversion can be recovered from the observed market prices of options. Whether, to what extent, and under
what conditions the investors’ risk aversion can be recovered in practice, by itself or
in conjunction with other statistically accessible quantities, from observations of option
prices in incomplete markets is a complex topic. We can however make some qualitative
observations. Let us assume for a moment that we have found an ‘implied’ methodology
17 Actually, it is not just the variance that matters, unless one wants to impose some additional and strong
conditions on the investor’s utility function, but the full distribution of returns from the total portfolio.
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CHAPTER 14 JUMP–DIFFUSION PROCESSES
by means of which, from the prices of plain-vanilla options, we manage to gain access
to the investors’ risk aversion (see, however, the discussion at the end of this section).
Note in passing that, even if we could gain access to this jump risk aversion, what we
could obtain from the market prices of options would only be related to the ‘average’
utility function of the market agents. For all its intrinsic interest, this quantity would not
tell the individual trader how an equity option should be priced given her risk aversion
(which will, in general, be different from the market’s). To use Bjork’s words again, in
incomplete markets, the market itself, not the trader, chooses the pricing measure.
Let us none the less make for the moment another heroic assumption: let us suppose
that the trader will adopt as her own the distilled market’s risk aversion. Even in this
case she would still have to evaluate at least the (non-zero) variance of return from the
option and her chosen hedging strategy. As discussed, the relevant quantity for her is not
the variance of returns from the (exotic) option in isolation, but the variance of returns
from the option itself plus its proposed hedging strategy. As for the latter, not only can
there be no such thing as a perfectly replicating strategy (by the very deﬁnition of market
incompleteness); but there is not even an a priori general agreement about (or knowledge
of) the ‘best’ hedging strategy. As mentioned above, the resulting overall variance of
returns of the combined (exotic option + hedging strategy) portfolio, would therefore
depend on the particular hedge chosen by the trader.
Looking at the problem in this light, the trader would seem to arrive at a price not only
by taking into account her risk aversion, but also by carrying out a portfolio-varianceminimizing search over all the possible hedging strategies that can be put together using
the instruments traded in the economy. This is less surprising than it might at ﬁrst sound
if one thinks of the pricing behaviour of an exotic trader who is faced with a new
complex product; who begins to experiment with some tentative hedging strategies and
quotes some rather ‘defensive’ prices; and who reﬁnes and becomes progressively more
conﬁdent about her hedging strategy, and consequently ‘tightens her price’.
The degree of portfolio variability (i.e. the distribution of slippages) that the trader
can expect when employing a very simple hedging strategy and the underlying process
is a jump–diffusion, is analysed in the following section where I bring to the forefront
the imperfectly replicating portfolio that can safely lie in the deterministic-volatility case
in ‘the back of our minds’. The results should therefore be analysed in the light of
the discussion of this section, and of the ‘robustness’ of the Black-and-Scholes pricing
approach already examined in Chapters 4 and 13.
14.11
Portfolio Replication in Practice:
The Jump–Diffusion Case
The treatment mirrors what we did in the stochastic-volatility case: the real-world (jump–
diffusive) evolution of the stock price is simulated and an imperfectly replicating portfolio
is constructed and rebalanced in parallel. At expiry (time T ) we examine the slippages,
deﬁned in Chapter 4, between the terminal option payoff and the time-T value of the
portfolio. The reader might want to re-read Section 13.6 and the observations made there
about the non-self-ﬁnancing nature of the trading strategy.
Once again, it is assumed that the trader does not know the true nature of the process
of the underlying, but attempts to parameter-hedge an option purely on the basis of her
14.11 PORTFOLIO REPLICATION WITH JUMPS
503
knowledge of the average realized quadratic variation (including the jumps) and of the
expectation of the jump amplitude ratio, Y0 :
Y = Y0
(14.109)
The trader will recognize the value of the option as if it were given by the Black-andScholes formula with the associated average square root volatility as input. She will then
attempt to protect the value of the option so calculated either by using only the stock,
or by using the stock and another option. In either case she will have to decide how to
translate into Black-and-Scholes-related quantities the hedge ratios obtained in Section
14.5 above. The next section explains how this can be done.
14.11.1 A Numerical Example
The Set-Up
I present in this section the results obtained when the initial stock price is $100 and the
option maturity is one year. To each path there will in general correspond a different
realized root-mean-squared volatility, σ , due both to the diffusive component (the same
for each path as the trading frequency approaches inﬁnity) and to the jumps. For the case
study analysed below, the average root-mean-squared volatility, obtained computationally
using a very large number of paths, turned out to be 17.10%. I assume that the trader knows
this quantity exactly. For the purpose of future comparison, the cost of the (imperfectly)
replicating portfolio obtained by using as input to the Black formula the average rootmean-squared volatilities above turned out to be $9.37.
As mentioned, I will employ two hedging strategies, one based on the stock only, and
the other based on the stock and another hedging option. This second option, h, will also
be chosen to be approximately at-the-money, but to have a maturity of two years. With
the ﬁrst (stock-only) hedging strategy the amount of stock to hold will be assumed to be
given by the Black-and-Scholes formula for the delta with the (perfectly known) average
root-mean-squared volatility used as input.
For the second hedging strategy we have to choose a ‘vega’ amount of option to hold.
We know that, even if the trader knew the true process and its parameters, no hedging
strategy based on the stock and one option could provide an exact hedging. Recall that
this is because in Equation (14.111) the quantity Y S is stochastic (there is no single
jump amplitude ratio). None the less, one can hope that an acceptable hedging might be
achieved by ‘pretending’ that a single jump amplitude ratio existed, and that it was equal
to its expectation. If this were the case the hedge ratios for the amounts to hold of the
stock, α, and of the second option, γ , would be given by (see Equations (14.110) and
(14.111) and (14.87)):
∂C
∂h
−γ
∂S
∂S
C(Y0 S) − C(S)
∂C
−
∂S
SY0 − S
γ =−
∂h h(Y0 S) − h(S)
−
∂S
SY0 − S
α=−
(14.110)
(14.111)
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Evaluating the correct amount of hedging option would require knowing the correct
∂h
derivatives of the two options prices with respect to the stock ( ∂C
∂S and ∂S ). The trader,
however, does not know these quantities, and therefore decides to approximate them with
the derivatives of the Black-and-Scholes option prices with respect to the stock using as
input the root-mean-squared volatility, σ . Note that this latter quantity also incorporates
the contributions from the jumps:
∂C(σ )
∂S
∂h(σ )
∂S
∂C(σ )
∂S
∂h(σ )
∂S
(14.112)
(14.113)
As for the terms
C(Y0 S) − C(S)
(14.114)
h(Y0 S) − h(S)
(14.115)
and
the trader will simply use the Black-and-Scholes formula evaluated at St and at St Y0 ,
again with volatility σ , and with the appropriate strikes and maturities. During the course
of each simulated path, the trader will have to buy and sell different amounts γ (σ ) of the
hedging option, h. I assume for simplicity that the ‘market’ prices for these transactions
are given by the Black formula with the root-mean-squared volatility σ .
Overall the strategy is rather crude and could certainly be reﬁned. It is however interesting to study how even this simple strategy will perform. This is presented below.
Exercise 3 What would you do if the expectation of the jump amplitude ratio was Y =
Y0 = 1?
Exercise 4 We assume that the trader (somehow) knew exactly the average quadratic
variation. How could you make use of the results in Section 12.3.3 to obtain a theoretically
incorrect (why?) but useful ‘market estimate’ of this quantity?
Exercise 5 Repeat the procedure described above, but without assuming that during the
course of each simulation the trader ﬁnds the market prices of the option h at the Black
prices with the root-mean-squared volatility σ . Assume instead that the market prices of
the option h are given by the jump–diffusion formula presented above (see Section 14.6).
Note carefully that the jump frequency and jump amplitude ratio in the pricing formula
and in the simulation should not be the same.
14.11.2 Results
In the computer experiments reported in this section I used 2048 paths of a ‘smart’ Monte
Carlo, with a variable number of steps to the ﬁnal maturity, as discussed below.
Looking at the results, the ﬁrst observation is that ‘optically’ a hedging portfolio made
up of a bond and the stock only will perform either very well (see Figure 14.33), or very
14.11 PORTFOLIO REPLICATION WITH JUMPS
505
16
14
12
10
Opt
8
Port
6
PortSmpl
4
2
400
379
358
337
316
295
274
253
232
211
190
169
148
127
85
64
43
106
−2
22
1
0
Figure 14.33 The behaviour of the call to be hedged (curve labelled ‘Opt’) and of the hedging
portfolios when no jump occurs to the expiry of the ﬁrst option (Mat1 ). The hedging portfolio
with the stock only is labelled ‘PortSimpl’, and the portfolio with the stock and an option is
labelled ‘Port’. The simulation parameters were S0 = 100, Strike1 = 100, σ = 17.00%, ν = 0.2,
RevLev = 100, RevSpeed = 0.2, Mat1 = 1, r = 5%, JAmpl = 0.90, JVol = 0.1, λ = 0.40,
Mat2 = 2, Strike1 = 100 and Nsteps = 440.
poorly (see Figure 14.34), according to whether a jump has occurred or not during the life
of the option. The good performance of the stock-and-bond-only hedge in the absence of
jumps is remarkable, given the number of crude approximations made in arriving at the
hedge parameters (above all, the association of the average root-mean-squared volatility
with the Black constant volatility). It is even more remarkable if one considers that, when
no jumps occur, the estimate of the root-mean-squared volatility used to determine the
Black hedging ratios is not only crude, but also biased. The trader in fact knows the
correct average quadratic deviation of the process, but this average is over all the paths,
i.e. the paths with no jumps and the fewer paths when jumps do occur. The average
quadratic variation is therefore associated with neither type of path in isolation, and when
jumps do not occur the realized quadratic will almost certainly be lower than what was
assumed in setting up the hedges.
The same two ﬁgures display the performance of the hedging portfolio when vega hedging is undertaken. (Recall that, in order to vega-hedge, an option with the same strike but
double maturity has been chosen.) When no jumps occur along the path the hedging performance is again optically almost perfect, but not signiﬁcantly better than what was obtained
using a stock-and-bond-only hedge. The difference in performance with and without vega
hedging changes radically when a jump does occur (see Figure 14.34). Visually the quality
of the tracking of the call value is remarkable, especially if we consider how simplistically
the vega amount of stock had been calculated. See Equations (14.112)–(14.115). The ﬁrst
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CHAPTER 14 JUMP–DIFFUSION PROCESSES
15
10
5
Opt
Port
PortSmpl
153
145
137
129
121
113
105
97
89
81
73
65
57
49
41
33
25
9
17
1
0
−5
−10
Figure 14.34 Same as Figure 14.33 when a jump does occur.
Table 14.12 Summary information of the hedging tests. See the text for a detailed explanation.
Nsteps
10
20
40
80
160
320
440
440
440
Avg(P-O)
StDev
(P-O)
Avg
(PSmpl-O)
StDev
(PSmpl-O)
0.072
0.014
0.054
0.054
0.075
0.018
0.036
0.059
0.063
1.184
0.957
0.573
0.525
0.826
0.553
0.686
0.482
0.548
−0.853
−0.674
−0.703
−0.808
−0.771
−0.727
−0.847
−0.717
−0.769
4.2202
3.1938
2.7375
2.6074
2.1738
1.9762
2.2092
2.1002
2.0189
StDev(Opt)
9.4094
9.1206
9.1651
9.7038
8.3864
10.1018
9.6713
8.8426
9.5102
Corr(Port)
Corr(Smpl)
0.0946
0.1291
0.3285
0.3182
0.1816
0.0245
0.1267
0.3196
0.2374
−0.6518
−0.6516
−0.7390
−0.7472
−0.8266
−0.8278
−0.8533
−0.8145
−0.8219
qualitative conclusion is therefore that, even if the trader does not know the nature of the
true process, let alone its exact parameters, but does know the average quadratic variation,
an acceptable replication along the path can still be achieved by means of an approximate
vega hedging.
Let us look at the quality of the hedging performance more quantitatively. Table 14.12
reports a wealth of information regarding the hedging simulations conducted as described
in the previous section. The various columns display (starting from the left-most):
• Column 1: the number of re-hedging steps along the life of the option (one year).
• Column 2: the average of the differences (slippages) between the values at expiry of
the replicating vega-hedged portfolio and of the option to be hedged. Note that this
14.11 PORTFOLIO REPLICATION WITH JUMPS
507
average does not appear to depend on the number of steps, and that it is a rather
small, but consistently positive, number.
• Column 3: the standard deviation of the quantity whose average is reported in column
2. Note that the standard deviation decreases when the hedging frequency is increased
from 20 times a year to approximately 80 times a year, but then appears to have
reached an asymptotic value: we cannot reduce it any more, no matter how ﬁnely
we re-hedge.
• Column 4: the average of the differences between the values at expiry of the replicating delta-hedged portfolio and of the option to be hedged. Unlike the vega-hedged
case, the bias is now large (and strongly negative), and approximately independent
of the re-hedging frequency.
• Column 5: the standard deviation of the quantity whose average is reported in column
4. The dispersion of the results is much greater than for the vega-hedged case. It
also soon ceases to improve as a function of the re-hedging frequency.
• Column 6: the standard deviation of the payoffs from the naked (unhedged) option.
This is reported in order to give an idea of the degree of replication brought about
by the vega- and delta-hedging strategies. It is apparent that vega hedging reduces
the slippages by up to a factor of 20, but delta hedging only by a factor of about
4. This result should be compared with the ﬁndings in the stochastic-volatility case
(where introducing an option for hedging purposes was less effective in reducing
the standard deviation).
• Column 7: the correlation between the deviation of the realized quadratic variation
from its average level and the slippages in the case of the vega-hedged portfolio.
If one engages in vega hedging it appears that the explanatory power of the realized quadratic variation to account for the (much smaller) slippages is rather poor.
In other words, if we vega-hedge, the effectiveness of our strategy is no longer
strongly dependent on the realized quadratic variation. Also in this case the results
are different from the stochastic-volatility case. See Figure 14.35, which shows that
in most of the cases the approximate vega hedging is successful in producing very
small slippages for a variety of realized quadratic variations, but occasionally fails
rather dramatically. This tends to occur when the underlying stock experiences a
jump that brings the option to be hedged very close to the strike when there is little
time to expiry (very high gamma).
• Column 8: the correlation between the deviation of the realized quadratic variation
from its average level and the slippages in the case of the stock-only-hedged portfolio. The correlation is now very high, and becomes higher as the hedging frequency
is increased. Whether we manage to replicate well or not now does depend on the
random realized quadratic variation. Looking at Figure 14.36, one can recognize a
high density of relatively small (and positive) slippages, and a low density of very
large (and very negative) slippages, associated with low realized quadratic variation
(no jumps) and high realized quadratic variation (jumps), respectively.
Finally, Figure 14.37 displays the histogram of the realized root-mean-squared volatilities. The very fat right tail is a direct consequence of the existence of jumps, and explains,
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CHAPTER 14 JUMP–DIFFUSION PROCESSES
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−5
0
5
10
15
20
25
Figure 14.35 Realized root-mean-squared volatility vs slippages (vega hedging).
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−25
−20
Figure 14.36
−15
−10
−5
0
Realized root-mean-squared volatility vs slippages (stock-only hedging).
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14.11 PORTFOLIO REPLICATION WITH JUMPS
509
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.15
0.2
0.25
0.3
0.35
0.4
0.45
RMS volatility
Figure 14.37 Histogram of the realized root-mean-squared volatilities obtained during the course
of the simulation.
on the one hand, the existence of a pronounced smile, and on the other, the origin of the
very large slippages in the delta-hedged case.
14.11.3 Conclusions
These results can be proﬁtably compared with the stochastic-volatility case. In that case
we found that, if we assumed that we knew the average quadratic deviation, for reasonable
values of the volatility of the volatility and of the reversion speed of the volatility, vega
hedging was producing a noticeable, but not dramatic, improvement. Again, it is difﬁcult to
generalize, but the magnitude of the improvement brought about by using a second option
was comparable with what obtained in the deterministic-volatility case if the volatility is
imperfectly known. When it comes to jump–diffusions, however, the situation changes
and vega hedging becomes crucial. This is a direct consequence of the discontinuous
nature of the paths. In the presence of jumps, being correctly delta hedged ‘on average’
is no longer adequate. Also, in our computer experiment we chose to hedge with a samestrike, different-maturity option. This was probably a good choice, because jumps produce
the largest slippages when they suddenly bring an option close to at-the-money, thereby
increasing its gamma. If the two options have a similar strike, they are more likely to
experience a similar change in gamma.
Finally, it must be stressed that we are fully in parameter-hedging territory: the strategy
cannot be guaranteed to be self-ﬁnancing and, even if we put in place the ‘cleverest’ hedge,
there will in general be some slippage at the end of any given path. Therefore the quantity
to monitor is the distribution of the slippages (its variance, its tails, etc.) and only on the
basis of this analysis will the trader be able to make a price. As I have argued throughout
this book, however, while this situation is theoretically different from the deterministicvolatility (or, to some extent, the stochastic-volatility) case, in practice the existence of
a ﬁnite-variance distribution of slippages is the norm and not the exception. Cochrane’s
(2001) remarks at the end of Chapter 1 are particularly valid also in this context.
Chapter 15
Variance–Gamma
15.1
Who Can Make Best Use of the Variance–Gamma
Approach?
In Chapter 1 I drew a distinction between models that attempt to reproduce as accurately as
possible the evolution of the future smile surface, and models that focus on recovering the
dynamics of the underlying. In a ‘perfect world’ there should be no difference between the
two sets of models: if the market is perfectly efﬁcient in distilling the statistical properties
of the process for the underlying into prices of plain-vanilla options, the model that best
accounts for the properties of the underlying will also give rise to the best description of
the present and future smiles (i.e. of the present and future prices of plain-vanilla options).
I also argued in Chapter 1 that in the real world the possibility should be at least
entertained that the dynamics of the underlying and the price-making mechanism for
the plain-vanilla options might be imperfectly coupled. If this was correct, the prices
quoted by traders might not be compatible with any process displaying the properties that
we expect of a reasonable description of the underlying (e.g. time homogeneity, smooth
dependence of the option prices on the strike, perhaps, but not necessarily, unimodal
risk-neutral price density, etc.).
I stated clearly in Chapter 1 what my views are: on the basis of the historical evolution of pricing practice (e.g. the sudden appearance of certain types of smiles) and of the
difﬁculty of enforcing the pseudo-arbitrage that could make the option market more efﬁcient, I strongly endorse the second (‘imperfect’) view of the world. The joint practices
of vega re-hedging and model re-calibration together produce a logically inconsistent,
but empirically very robust, approach to pricing complex derivatives (i.e. derivatives that
take not only the stock process, but also the plain-vanilla options, as their underlyings).
Simplifying greatly, one can reconstruct the practice of complex-derivatives trading as
follows: the vega hedge is put in place on the trade date at the current market prices for
hedging the complex option. Because of this the model should recover correctly the prices
of these hedging instruments, hence the emphasis on the price recovery of the current
plain-vanilla options. If the hedging instrument has been ‘intelligently’ chosen, the net
residual deltas from the complex instrument and from the hedging portfolio will largely
cancel out. If static replication is possible, the future prices of options would be irrelevant,
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