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10 Jump–Diffusion Processes and Market Completeness Revisited

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.



502



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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CHAPTER 14 JUMP–DIFFUSION PROCESSES



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,



508



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).



5



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,

511



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