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Abstract

Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by (1) where dZL is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot be priced using the standard Black–Scholes–Merton (BSM) arbitrage-free pricing criterion. Therefore, we develop a pricing theory based on a more general pricing criterion: that the expected return of all diversifiable portfolios is at the risk-free rate. We show that even though derivatives based on Lévy or tempered Lévy-driven assets cannot be hedged to make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option prices are given by (2) where the expected value uses the “real-world” probability measure, but with the asset price X(t) replaced by the alternative “pricing process” (3) which grows at the risk-free rate. We analyze these models to obtain explicit asymptotic formulas for European option prices. This analysis shows that as the time to expiry decreases, eventually all these Lévy and tempered Lévy-based models reduce to the same canonical model, and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These similarity solutions are then examined to assess the mishedging that arises from Brownian-based models in a Lévy world.
ON BEYOND BLACK
ANDREA KARLOVÁ
AND
PATRICK S. HAGAN
Abs tract. Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded
markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in
the real world by
dX =a(t, ·)dt +b(t, X)dZ
L
,
where dZ
L
is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk-free by dynamic hedging,
so these derivatives cannot be priced using the standard Black/Scholes/Merton arbitrage free pricing criterion. Therefore we
develop a pricing theory based on a more general pricing criterion, that the expected return of all diversifiable portfolios is at
the risk free rate. We show that even though derivatives based on Lévy or tempered Lévy driven assets cannot be hedged to
make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option
prices are given by
V(t, x) = e
r(Tt)
EV(T, ˆ
X(T))
ˆ
X(t) = xfor any t < T ,
where the expected value uses the “real world” probability measure, but with the asset price X(t)replaced by the alternative
“pricing process,”
dˆ
X=rˆ
Xdt +b(t, ˆ
X)dZ
L
,
which grows at the risk free rate.
We analyze these models to obtain explicit asymptotic formulas for Europ ean option prices. This analysis shows that as
the time-to-expiry decreases, eventually all these Lévy and tempered Lévy based models reduce to the same canonical model,
and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These
similarity solutions are then examined to assess the mishedging that arises from Brownian based mo dels in a Lévy world.
1. Introduction. The risks of European options are usually analyzed and managed using stochastic
volatility models, such as the SABR[1] or Heston models[2]. Both of these models can usually be fitted
so that they closely match the implied volatility smiles observed in the marketplace. Yet matching the
pronounced smiles of very short dated options requires enormous volatilities-of-volatility, often exceeding
400% for one week options. For larger time-to-expiries, less extreme vol-of-vols are needed; a vol-of-vol of
100% may suffice to generate the more moderate smiles of three month options. For options with a year or
more to expiry, vol-of-vols of 30% may suffice to match their modest implied volatility smiles. See figures
1.1 and 1.2. The intensification of the smile as the time-to-expiry decreases is an innate feature of observed
volatility surfaces. Surely it should also be an innate feature of any reasonable theoretical description of
volatility surfaces, a feature which can be explained without forcing the model parameters to change by an
order of magnitude.
Consider options with a specific expiry date. As time advances, the time-to-expiry decreases, so larger
and larger vol-of-vols are needed to match the increasingly pronounced smile of these options, especially
for the last few months before expiry. Thus, using stochastic volatility models to calculate the hedges of
vanilla options may not be sound, especially in the last few months before expiry, since changing the model
parameters each day to match the increasingly pronounced smiles, means that we’re actually using a different
model each day. In addition, the inability to explain market data at different expiries without markedly
changing parameters, makes stochastic volatility models ill-suited for managing the risks of instruments like
Bermudans and knock-outs, instruments which are sensitive to volatilities at different expiry dates as well
as at different strikes.
To address these problems we need models for volatility surfaces, models that simultaneously predict
both the pronounced smiles of short-dated options and the moderate smiles of longer-dated options. At
first glance, it would seem that local volatility or stochastic/local volatility models would work[3]-[6], since
they can match almost any current volatility surface. However, in these models, the implied volatility smile
andrea.karlova@gmail.com
pathagan1954@yahoo.com
1
0%
10%
20%
30%
40%
50%
60%
0.5 1.0 1.5 2.0
K/F
Im plied No r m al Vols
1w
2w
1m
3m
6m
12m
 .Volatility smiles fo r the EUR/USD exchange rate on Sept 26, 2014. Shown are the im plied normal volatilities
for European options as a function of K/F a t different expiry dates, where Kis the op tion strike and Fis the forw ard FX
rate for the expiry date.
0%
100%
200%
300%
400%
1 10 100 1000
Days to expiry
VolVol
0%
4%
8%
12%
1 10 100 1000
Days to expiry
alpha
-75%
-50%
-25%
0%
25%
50%
75%
1 10 100 1000
Days to expiry
rho
 .The SABR volatility α, correlation ρ, and vol-of-vol νas a function of the time-to-expiry, obta ined by fitting
the SABR model to each smile in Fig. 1 .1. The vol-of-vol νis very high for short dated options, and fa lls steadily as the
time-to-expiry increases, while the vola tility αand correlation ρare relatively consta nt, even though the time-to-expiry goes
from 1 week to 3 years.
for options with expiry date T
ex
is a term volatility, and is essentially an average of the local volatility
smiles for each date between today and T
ex
. When calibrated to typical implied volatility surfaces, with
pronounced short dated smiles and modest long dated smiles, the local volatily smiles are also intense for
small tand become flatter and flatter as tincreases. Consequently, these models predict that for any expiry
date T
ex
, the option smiles become flatter and flatter as time tadvances and the time-to-expiry decreases,
since the average from tto T
ex
is continually losing the shortest — and thus steepest — local smiles. This is
opposite of the observed behaviour, that smiles become more intense as time-to-expiry decreases. I.e., the
forward volatility predictions of local volatility and stochastic/local volatility models are opposite to market
observations. Accordingly, we shall not consider these models further.
2
To us, the intensification of the smile suggests that some sort of jump model is needed: For a one week
option, a jump in the underlying asset price would have a profound impact on the option value; accounting
for this possibility would require increasing the option values — and thus the volatilities — in the wings. For a
one year option, a jump in the asset price should have a much smaller impact, since one anticipates several
such jumps occurring during the year.
Many investigators have modeled volatility surfaces by adding Poisson jump processes to conventional
volatility and stochastic volatility models. See, e.g., [7]-[10]. Yet adding jumps to stochastic volatility models
adds at least three more parameters (jump frequency, mean jump size, and the variance in the jump size).
This leads to models with more parameters than can reasonably be pinned down by calibrating to market
data. To obtain a more parsimonious approach to managing our risks, we need to ask whether there is an
underlying regularity to the jump processes, a regularity that we can exploit to obtain relatively simple,
robust models. We believe that there is an underlying regularity, and that this regularity leads to asset
prices being driven by Lévy flights and tempered Lévy flights.
Consider a model
(1.1) dX =a(t, ·)dt +b(t, ·)dZ
T L
for the market value X(t)of a financial asset in the real world. In section 2 we present the theoretical reasons
and empirical evidence for believing that dZ
T L
should be a Lévy flight or a tempered Lévy flight, instead
of Brownian motion. This brings up the question of how to price derivatives and contingent claims based
on such assets X(t). No finite set of hedges can hedge out all of the risks of a non-trivial derivative based
on X(t)when dZ
T L
is a Lévy and/or a truncated Lévy flight. Since one cannot create a risk-free portfolio
by hedging, one cannot price derivatives using the standard Black-Scholes-Merton criterion that risk free
portfolios grow at the risk free rate. We are unwilling to pretend these portfolios are risk-free, and use the
BSM criterion anyway, since hedged derivative portfolios are decidely not risk-free in the real world.
We also cannot simply postulate that equation 1.1 represents “the dynamics of X(t)in the risk neutral
measure,” and then price derivatives using a Martingale pricing formula,
(1.2) V(t, x) = e
r(Tt)
E{V(T, X(T)) |X(t) = x},
where the expected value is over the probability measure implied by equation 1.1. First, the process 1.1 has
been obtained from arguments and observations about asset prices in the real world. It is not clear that
a “risk neutral measure” exists for these processes, and if it does exist, we have no theoretical arguments
which enable us to connect the risk neutral measure with real-world measures based on Lévy flights. Second,
for forward contracts and other delta 1 products, one can hedge away all the risk. Standard arguments show
that if the asset pays no dividends or coupons, then the value of the forward contract is
(1.3) V
fc
(t, x) = xKe
r(T
m
t)
,
where x=X(t)is the current asset value, and where Kand T
m
are the strike and maturity of the forward
contract. For general drift terms a(t, ·)dt,
e
r(T
m
t)
E{V(T
m
, X(T
m
)) |X(t) = x} ≡ e
r(T
m
t)
E{(X(T
m
)K)|X(t) = x}(1.4)
=e
r(T
m
t)
E{X(T
m
)|X(t) = x} − Ke
r(T
m
t)
=xKe
r(T
m
t)
,
so the value of the forward contract is does not equal the expected value of the payoff discounted by the risk
free rate r. I.e., the Martingale pricing formula gives the wrong price. Intriguingly, the correct price can be
obtained from the Martingale pricing formula
(1.5) V
fc
(t, x) = e
r(T
m
t)
Eˆ
X(T
m
)K)ˆ
X(t) = x=xKe
r(T
m
t)
,
3
where ˆ
X(t)) is an alternative process in which the real world drift term a(t, ·)dt has been replaced with
growth at the risk free rate,
(1.6) dˆ
X=rˆ
Xdt +b(t, ·)dZ
T L
.
We also cannot price derivatives on X(t)by switching to an “equivalent Martingale measure.” Even
though equivalent Martingale measures may exist for X(t)when dZ
T L
is a tempered Lévy distribution[17],
changing to this measure involves more than changing the drift term. It also involves changing the truncation
used to temper the Lévy distribution[17]; i.e., to changing the frequency of large jumps. This is troublesome,
as there is no financial justification for pricing options using jump frequencies different than the actual
real-world jump frequencies, especially for large jumps.
Finally, we cannot find the market price of a derivative by optimizing the expected value of a utility
function[19]. Utility functions necessarily relate to a particular trading party, and are generally different for
different trading parties[20]. We are seeking a pricing criterion for market prices, a pricing criterion which
is necessarily independent of any one party or market view. Even if one could use the utility functions for
individual market participants to create an over-arching utility function for the market itself, it is unclear
which properties the “market utility function” inherits from individual utility functions.
1
Similarly, we
cannot use super- or sub-replication arguments to obtain the market price: Since neither counterparty can
hedge away all the risk, this would imply that the market places all the risk on one of the two counterparties,
which seems unlikely.
To obtain a general pricing principle, in section 3 we note that when asset prices X(t)are driven by
Lévy flights and tempered Lévy flights, they are not Itö processes. Instead they belong to a wider class of
processes, let’s call them dark noise processes, for which the key requirement is that for any constant M, no
matter how large,
(1.7) E{|dX|} ≡ E{|X(t+dt)X(t)|} > Mdt for all dt small enough.
I.e., their risks decrease slower than dt as dt 0. Dark noise processes include all risky Itö processes, for
which E{|dX|} ∼ O(dt)
1/2
, processes driven by Lévy flights and tempered Lévy flights, and processes which
combine both Itö processes and Lévy flights. When asset prices are dark noise processes, the market is
generally not complete, and, as in real markets, one cannot eliminate all the risks of non-trivial derivative
contracts by hedging.
But suppose that by hedging the derivative we can construct a portfolio Π(t)which eliminates the worst
of the risk. Specifically, suppose that we can construct a portfolio Π(t)which has some constant Csuch that
(1.8) E{|dΠ|} ≡ E{|Π(t+dt)Π(t)|} < Cdt for all dt small enough.
Let’s call portfolios which satisfy this criterion diversifiable portfolios. Since we are unable to create risk-free
portfolios, we postulate that all diversifiable portfolios earn an expected return at the risk free rate. This is
our pricing criterion for obtaining mid-market derivative prices.
Note that if we were to restrict asset prices to be only Itö processes, then diversifiable portfolios would
be the same as risk free portfolios, since any risky Itö process Π(t)would have E{|dΠ|} ∼ O(dt)
1/2
. Thus,
our pricing criterion is identical to the Black/Scholes/Merton criterion when we restrict assets to be Itö
processes. The new criterion is simply an extension of the BSM criterion to a larger class of assets, dark
noise assets.
Also note that this pricing criteria is linear: if Aand Bare both diversifiable portfolios, then a portfolio
of massets of type Aand nassets of type Bis diversifiable, and its value is just mtimes the value of Aplus
ntimes the value of B.
1
Utility functions also lead to nonlinear pricing. As a trading party (or “the market”) buys an instrument, the utility
function’s marginal price for the instrument generally decreases. Upon selling the instrument, the unit price usually increases.
Thus, the value assigned by a utility function to massets of type Ais usually not mtimes the value of 1 unit of type A.
4
In section 3 we use this criterion to price options on an asset ˜
X(t), where
(1.9) d˜
X=a(t, ·)dt +b(t, ˜
X)dZ
T L
.
Here, dZ
T L
is a general class of stochastic processes which include Lévy flights and tempered Lévy flights.
By adapting the original Black-Scholes hedging arguments[18], we show that delta hedging the option leads
to a diversifiable portfolio. Requiring this portfolio to grow at the risk free rate then implies that the option
value satisfies
(1.10a) V(t, x) = e
r(Tt)
EV(T, ˆ
X(T)) ˆ
X(t) = xfor any t < T,
where the original “real world” process ˜
X(t)has been replaced by the alternative “pricing process” ˆ
X(t),
defined by
(1.10b) dˆ
X=rˆ
Xdt +b(t, ˆ
X)dZ
T L
.
As we shall see, this means that the option price is the solution of the backwards equation
(1.11) V
t
+rxV
x
+
−∞
K(y){V(t, x +b(t, x)y)V(t, x)b(t, x)yV
x
(t, x)}dy =rV,
where K(y)is the “jump kernel” of dZ
T L
.
So to price derivatives based on dark noise processes, we keep the real-world measure, and directly replace
the real world process ˜
X(t)with an alternative process ˆ
X(t)which grows at the risk free rate. In contrast,
in standard pricing theory one evaluates options on Itö processes by changing from real-world probabilities
to a risk neutral measure, a measure constructed so that assets grow at the risk free rate rXdt instead of
at their real-world rate a(t, ·)dt. Only when the dark noise process X(t)happens to be an Itö process, are
these two procedures equivalent.
In section 4 we use singular perturbation methods to analyze the model 1.9, where dZ
T L
is taken to be
tempered Lévy distribution, obtaining explicit formulas for call and put prices in the different asymptotic
regimes of 1.11. There we find that if the time-to-expiry is long enough, then the option prices are given by
the solution of
(1.12a) V
t
+rxV
x
+
1
2
b
2
(t, x)M
2
V
xx
=rV for 0< t < T
ex
,
to leading order, where
(1.12b) M
2
=y
2
K(y)dy.
(M
2
is finite due to the tempering of dZ
T L
). These are the same prices that could be obtained from a
Brownian motion based model, and the jumps of the Lévy process visibly effect the pricing only in the
extreme wings, where the time values are transcendentally small.
For shorter times-to-expiry, there is a transition regime in which a diffusion-dominated region near-the-
money co-exists with a hybrid region in the wings. As the time-to-expiry decreases, the diffusion-dominated
region shrinks and then disappears. The shortest regime is the Lévy regime, representing options with just
a few months to expiry. In this regime, we shall find that all the models in 1.9 reduce to basically the same
canonical model. Analyzing this canonical model shows that the European call and put prices are given by
similarity solutions,
V
call
c
γ
[T
ex
t]
γ
I
call
Kx
c
γ
[T
ex
t]
γ
,(1.13a)
V
put
c
γ
[T
ex
t]
γ
I
pui
Kx
c
γ
[T
ex
t]
γ
,(1.13b)
5
in this regime, where
1
2
< γ < 1. As the time-to-expiry decreases, these similarity solutions become increas-
ingly accurate. Moreover, the intensification of the smile as the time-to-expiry decreases is an innate feature
of these solutions.
For any European option, eventually the time-to-expiry gets short enough so that it is governed by the
canonical model. So in section 5 we examine the similarity solutions more closely. There we note that the
implied normal volatility is also in similarity form,
(1.14) σ
N
(t, x) = c
γ
[T
ex
t]
γ
1
2
ΘKx
c
γ
[T
ex
t]
γ
.
There we also compare the new hedges, based on Lévy or tempered Lévy processes, with the conventional
hedges, based on implied normal volatilities.
Using Lévy flights and tempered Lévy flights to model asset prices is an active area of research, and
arguments and models similar to ours can be found in [11]-[16] and the references cited therein. We believe
that the pricing criterion for diversifiable portfolios,
2
the valuation of derivatives by replacing the asset price
process by an alternative process which grows at the risk free rate (instead of using alternative measures),
and the similarity solution for short dated options are new.
Throughout we use implied normal (also called absolute or Bachelier) volatilities. We find log normal
volatilities less useful, since they introduce an extra length scale into the pricing.
2. Motivation.
2.1. Lévy flights. The following argument
3
is why we believe that asset prices are driven by Lévy
flights and tempered Lévy flights[22]. Suppose that we wish to develop a model for the market value X(t)
of a financial asset,
(2.1) dX =a(t, ·)dt +b(t, ·)dZ.
The first question is which stochastic process should we use for dZ? Surely not Brownian motion, because
market processes surely have fatter tails than the transcendentally thin, Gaussian tails of Brownian motion.
Instead of trying to guess the correct stochastic process, let us pin it down by setting out the basic properties
that the noise process Z(t)ought to have:
1. Markovian. For simplicity, we’d like to ignore any history dependence in Z(t), and insist that it is
Markovian. Then the value of Z(t
)for any t
> t depends only on the current value of Z(t), and not on
how Z(t)got to its current value. With Z(t)being Markovian, we can define the transition density
(2.2) p(t
1
, z
1
;t
2
, z
2
) = E{δ(Z(t
2
)z
2
)|Z(t
1
) = z
1
}.
2. Stationarity and homogeneity. We assume that Z(t)is stationary in t, so that the transition density
depends only on the time difference t
2
t
1
, and not on t
1
and t
2
seperately. Any dependence of X(t)on
absolute time will have to be accounted for through explicit tdependence of the coefficients a(t, ·)and b(t, ·).
Similarly, we assume that the transition density depends only on the relative difference z
2
z
1
, and not on
the particular values of z
1
and z
2
. Again, any dependence on absolutes will have to be accounted for through
the coefficients aand b. With these assumptions, the transition density simplifies to
(2.3) p(t
1
, z
1
;t
2
, z
2
) = g(t
2
t
1
, z
2
z
1
)
for some function g(τ, z ).
3. Mean zero. Since any average motion can be modeled via the drift terms a(t, ·)dt, we assume that
dZ has mean zero.
2
But see [21].
3
Similar reasoning can also be found in [12]-[16] and the references cited therein.
6
At this point we can deduce a surprising amount about the transition density g(τ, z ). As a probability
density with mean zero, it satisfies
(2.4a)
+
−∞
g(τ , z)dz = 1,
+
−∞
zg(τ , z)dz = 0 for all τ > 0.
Since Z(t)is Markovian, the Chapman-Kolmogorov relation states that for any t > 0, we have
(2.5a) g(t, z) =
+
−∞
g(τ , y)g(tτ, z y)dy for all 0< τ < t.
After all, at any intermediate date τ,g(t, z)must be in some state y. Clearly g(τ , z)also satisfies
(2.5b) g(t, z)δ(z)as t0.
The Chapman-Kolmogorov relation is a convolution in z, which suggests taking the Fourier transform,
(2.6) G(t, k)
+
−∞
e
ikz
g(t, z)dz.
Since probability densities g(t, z)are real, taking complex conjugates yields
(2.7a) G
(t, k) =
+
−∞
g(t, z)e
+ikz
dz =G(t, k)for all real k.
We also have
G(t, 0) =
+
−∞
g(t, z)dz = 1,(2.7b)
G
k
(t, 0) = i
+
−∞
zg(t, z )dz = 0,(2.7c)
from eq. 2.4a.
Taking the Fourier transform of the Chapman-Kolmogorov relation yields
(2.8) G(t, k) = G(τ , k)G(tτ, k)for all 0< τ < t.
At each kthis is a functional equation in t. The only non-pathological solutions are e
const ·t
, where the
constant can depend on k[23]. So,
(2.9a) G(t, k) = e
(k)
for each k, for all t > 0,
where conditions 2.7a-2.7c yield
φ(0) = 0, φ
(0) = 0,(2.9b)
φ(k) = φ
(k)for all real k.(2.9c)
To go further, we need to add a final, powerful assumption.
4. dZ has no intrinsic time scale. We are assuming that any time scales in the market can be accounted
for in the coefficients a(t, ·)and b(t, ·), and are not part of the basic noise process. Later we will find that
this assumption is too strong, and we will have to back away from it slightly to match empirical data.
Without an intrinsic timescale, the transition density must be a similarity solution,
(2.10) g(t, z) = 1
t
γ
h(z/t
γ
).
7
For, if the shape of the transition density varies with time, then one could obtain a characteristic time scale
from the changing shape. Taking the Fourier transform of eq. 2.10 yields
(2.11) G(t, k) =
+
−∞
e
ikz
h(z/t
γ
)
t
γ
dz =
+
−∞
e
ikt
γ
q
h(q)dq H(kt
γ
).
Putting this together with eq. 2.9a shows that G(t, k) = e
(k)
with
(2.12a) φ(k) = ˜ck
α
for k > 0
˜c
|k|
α
for k < 0,
where
(2.12b) α1/γ.
These seemingly innocuous assumptions have nearly pinned down the distribution, or at least its Fourier
transform. These stochastic processes are known as Lévy flights[24], or α-stable motion, and are used
extensively to model, e.g., anomalous diffusion[25]. Since the magnitude of ˜ccan be absorbed by re-scaling
t, Lévy flights are determined by the Lévy exponent αand by the phase of ˜c:
(2.13) ˜c=ce
iπαs
0
.
Here cis real and positive, and the phase has been written as iπαs
0
for later convenience. All that remains
is to invert the Fourier transform to find the transition density,
(2.14) g(t, z) = 1
2π
+
−∞
e
ikz(k)
dk.
Here, the reality condition 2.9c requires the branch cut(s) of k
α
to be interpretted so that ce
iπαs
0
(k)
α
=
ce
iπαs
0
k
α
for all real and positive k.
The transition density g(t, z)cannot be written in closed form, so in Appendix A we set out an efficient
method for calculating g(t, z)numerically. There we also determine the properties of g(t, z)by analyzing
the inversion integral 2.14. For example, evaluation of the integral for |z|/t
γ
1shows that,
g(t, z)Γ(1 + α)sin παp
1
π
ct
|z|
1+α
for z/ (ct)
γ
1,(2.15a)
g(t, z)Γ(1 + α)sin παp
2
π
ct
|z|
1+α
for z/ (ct)
γ
1,(2.15b)
where the asymmetry parameters p
1
, p
2
are
(2.15c) p
1
=
1
2
+s
0
, p
2
=
1
2
s
0
.
These probability densities for Lévy flights decay algebraically instead of exponentially for large z. As αp
1
or αp
2
increase to values above 1,sin παp
1
or sin παp
2
becomes negative, and so g(t, z)becomes negative
for large positive and/or large negative z. In fact, it has been shown[26] that g(t, z)is positive for all zonly
when both 0< αp
1
<1and 0< αp
2
<1. That is, when 11/α < p
1
, p
2
<1. So we exclude all other
cases. In addition, when
(2.16) α=α(p
1
+p
2
)1,
the transition density g(t, z)decays so slowly that |z|g(t, z)const /|z|
α
is not absolutely integrable. I.e.,
we cannot properly define the mean of the distribution. So we exclude these exponents as well. Thus, we
only consider Lévy flights with
1< α < 2(2.17a)
p
1
+p
2
= 1,11/α < p
1
, p
2
<1/α,(2.17b)
8
and re-write the phase of ˜cas
(2.17c) παs
0
=
1
2
πα (p
1
p
2
).
We can gain insight into the nature of Lévy flights by noting that
(2.18) G
t
=φ(k)G=φ(k)
k
2
k
2
G.
Since φ(0) = 0, φ
(0) = 0, the function φ(k)/k
2
is integrable, so inverting the Fourier transform yields
(2.19a) g
t
(t, z) =
−∞
˜
K
L
(y)g
zz
(t, z y)dy =
−∞
˜
K
L
(y)[g(t, z y)g(t, z ) + yg
z
(t, z)]
yy
dy,
with
(2.19b) ˜
K
L
(y)1
2π
−∞
φ(k)
k
2
e
iky
dk.
Integrating by parts twice yields
(2.20a) g
t
(t, z)
−∞
K
L
(y)[g(t, z y)g(t, z ) + yg
z
(t, z)]dy,
with
(2.20b) K
L
(y) = ˜
K
′′
L
(y) = c
πΓ (α+ 1)
sin παp
1
y
α+1
for y > 0
sin παp
2
|y|
α+1
for y < 0.
Equation 2.20a shows that Lévy flights can be thought of as a jump model where all jumps are inde-
pendent, and where jumps of size yoccur at a rate of K
L
(y)per unit time, summed over all jump sizes
y. The term K
L
(y)g(t, z y)accounts for the probability increase due to jumps into state z, and the term
K
L
(y)g(t, z)accounts for the probability decrease due to jumping out of state z. The compensation term
K
L
(y)yg
z
(t, z)is needed to ensure that Z(t)has no net drift. It also ensures that the integral converges
about y= 0. Note that the jump frequency K
L
(y)becomes infinite as the jump size ygoes to zero, so
infinitesimal jumps occur infinitely often, as behooves a process replacing Brownian motion.
2.2. Market data. Eqs. 2.15a, 2.15b show that the Lévy density g(t, z)decays so slowly that the
seconds moment z
2
g(t, z)const /z
α1
is not integrable, even with 1< α < 2. So Lévy flights have an
infinite variance. This aspect of Lévy flights does not seem to match the behavior of real asset prices, so let
us look at market data to see how it may all fit together.
Figure 2.1 presents the transition density g(τ, z)measured for the S&P 500 for a τ= 1 minute time
difference[31]. In this figure, the parabola is the best fit Gaussian (Brownian motion); the other curve is the
transition density g(t, z)for the symmetric Lévy flight (p
1
=p
2
=
1
2
) with exponent α= 1.4and the best
fit constant c. Brownian motion is clearly a poor fit to the observed probability density. The vertical axis
is log base 10, so the density from Brownian motion is too low by a factor of 3 at the distribution’s center.
Careful inspection shows that the density from Brownian motion is too high by a factor that reaches ten
in the near wings (|Z/σ| ∼ 15). In the far wings, the Gaussian probability density is too low by orders
of magnitude. In contrast, the density from the Lévy flight fits the measured distribution extremely well
in the center and near wings of the distribution. In the far wings, the empirical distribution decays more
rapidly than the algebraic decay of the Lévy flight, and appears to be decaying exponentially. Remarkably,
at the point where the empirical density first deviates from the Lévy flight density, the density has already
fallen to around 10
3
of the density at the center. So the deviation truly occurs in the far wings. Similar
results have been measured in many other markets. Empirically, transition densities are expressed well by
Lévy flights, usually with exponents αbetween 1.3 and 1.75, except in the extreme wings, where exponential
decay appears to prevail. See [28] - [31] and the references cited therein.
9
 .Histogram of 1.2 mil lion o bservations in the change in the S& P500 over one m inute for 1984. The histogram
provides the empirical ly observed transition density g(τ , z)for τ= 1 min . The parabola is the best fit Brownia n motion. The
other curve is the symmetric Levý flight with α= 1.4. Reproduced from [31].
2.3. Tempered Lévy flights. Based on these empirical results, we will model asset prices using noise
processes dZ which are “truncated” or “tempered” Lévy flights, Lévy flights which have been modified
slightly so that their transition densities decay exponentially in the far wings. These modified processes
possess all moments, and are similar to processes used by others to model financial assets. See [32] - [34]
and the references cited therein.
One approach is to replace φ(k)in 2.12a with a function that is very near to φ(k), but is analytic around
k= 0, and then derive the new jump kernel K
T L
(y)from K
T L
(y) = ˜
K
′′
T L
(y), where
(2.21) ˜
K
T L
(y)1
2π
−∞
φ(k)
k
2
e
iky
dk.
See 2.18-2.20b. The analyticity of φ(k)around 0then ensures that the new kernel decays exponentially for
large y, although one still has to establish that K
T L
(y)remains non-negative. See [32] and [27]. We prefer
the alternative of directly specifying the jump kernel K
T L
(y). We then define our truncated Lévy flight as
the process in which all jumps are independent, and whose transition density is given by
g
t
(t, z)
−∞
K
T L
(y)[g(t, z y)g(t, z ) + yg
z
(t, z)]dy,(2.22a)
g(t, z)δ(z)as t0.(2.22b)
Should we need the φ(k)corresponding to a particular jump kernel K
T L
(y), we can infer it via Fourier
transforms.
10
3. General option pricing. For the purpose of developing the hedging and pricing theory, we assume
that the jump kernels K(y)satisfy
K(y)>0for all y= 0,(3.1a)
K(y)is smooth for all y= 0,(3.1b)
|y|
2+η
1
K(y)> δ > 0for some constant δfor all small enough |y|,for some 0< η
1
<1,(3.1c)
|y|
2+η
2
K(y)is bounded as y0for some 0< η
2
<1,(3.1d)
|y|
2+η
3
K(y)is bounded as y→ ±∞ for some 0< η
3
<1.(3.1e)
This class of kernels K(y)includes Lévy flights, as well as truncated Lévy flights. When we seek explicit
European option prices in sections 4 and 5, we will narrow our focus to the tempered Levy distribution
whose kernel is
(3.2) K
T L
(y)c
π
Γ (α+ 1)
|y|
α+1
sin παp
1
e
y/λ
1
for y > 0
sin παp
2
e
−|y|
2
for y < 0,
since this usually matches the data closely. Note that as λ
1
, λ
2
→ ∞, this kernel approaches the jump kernel
for Lévy flight.
3.1. Backwards Kolmogorov equation. We model financial asset prices ˜
X(t)by
(3.3) d˜
X=a(t, ˜
X)dt +b(t, ˜
X)dZ,
where the noise processes Z(t)are the truncated Lévy processes defined in 2.22a, 2.22b, and where the jump
kernel K(y)satisfies 3.1a-3.1e above. To obtain the backwards equation for ˜
X(t), define p(t, x;T, X )as the
transition density,
(3.4) p(t, x;T, X ) = Eδ˜
X(T)X|˜
X(t) = x.
In [26] it is shown that the transition density p(t, x;T, X )satisfies the “backwards Kolmogorov” equation,
(3.5a) p
t
+a(t, x)p
x
+
−∞
K(y){p(t, x +b(t, x)y)p(t, x)b(t, x)yp
x
(t, x)}dy = 0
for t < T , with
(3.5b) p(t, x)δ(xX)as tT.
Here we are omitting the forward arguments T, X for clarity. A brief, informal derivation of the backwards
equation is sketched in Appendix B; a rigorous proof can be found in [26].
A Feynman-Kacs formula follows immediately. Define Φ (t, x)as
(3.6) Φ(t, x) = e
r(T
ex
t)
EH(˜
X(T)) |˜
X(t) = x.
Clearly,
(3.7a) Φ
t
+a(t, x
x
+
−∞
K(y){Φ(t, x +b(t, x)y)Φ(t, x)b(t, x)yΦ
x
(t, x)}dy =rΦ,
for t < T , with
(3.7b) Φ(t, x)H(x)as tT.
The backwards equation and Feynman-Kacs formula show that ˜
X(T)can be interpretted as an infinite
collection of jump processes, but now jumps of size b(t, x)yoccur with frequency K(y).
11
3.2. Option pricing. Let Vbe the value of a European option on ˜
X(t)with expiry date T
ex
. Since
˜
X(t)is composed of a collection of independent jump processes, hedging out all the option’s risk would
require hedging Vfor jumps of size yfor each y. Unless Vwas a constant product (i.e., a forward
contract), we cannot expect any finite number of hedges to eliminate all the risk. Excluding these trivial
derivatives,we cannot create the risk-free portfolios needed to use the B/S/M arbitrage free pricing criterion
(that riskless portfolios grow at the risk free rate) to price this derivative.
Since ˜
X(t)is Markovian, the option value must also be Markovian, so V=V(t, ˜
X(t)). We assume
that V(t, X)is a smooth function tand X. Specifically that θ(t, x) = V
t
(t, x),∆(t, x) = V
x
(t, x), and
Γ(t, x) = V
xx
(t, x)exist and are bounded for all tbounded away from the expiry date, t
1
< t < t
2
< T
ex
.
It is easily seen that ˜
X(t), and thus the option price V(t, ˜
X(t)),must be a dark noise process. From 3.1c
there is a constant C > 0and a y
0
>0such that
y
K(y)dy > C
|y|
1+η
1
for all 0< y < y
0
,(3.8a)
y
−∞
K(y)dy > C
|y|
1+η
1
for all y
0
< y < 0.(3.8b)
Thus, jumps larger than |y|occur more frequently than O(1/|y|
1+η
1
), and so
(3.9) Ed˜
XE˜
X(t+dt)˜
X(t)> C
(dt)
1/(1+η
1
)
=C
(dt)
η
1
/(1+η
1
)
dt
for some constant C
and all small enough dt > 0. Consequently, for any constant Mwe have
(3.10) Ed˜
XE˜
X(t+dt)˜
X(t)>C
(dt)
η
1
/(1+η
1
)
dt > Mdt
for all small enough dt. Note that for Lévy flights, we have 1 + η
1
=α, so eq. 3.7 shows that d˜
X
(dt)
1
= (dt)
γ
, as expected.
3.2.1. The hedged portfolio. Pricing. The option value V(t, ˜
X(t)) is a dark noise process, so we
need to construct a diversifiable portfolio. Since the infinite number of infinitesimal jumps is causing the
problem, we consider a portfolio which is delta hedged:
(3.11) Π(t) = V(t, x)xwith ∆ = V
x
(t, x).
In appendix C we show that in the next instance dt of time, that
(3.12) E{|Π(t+dt)Π(t)|} < Cdt
for some constant C, so the hedged portfolio is diversifiable. Since the porfolio is diversifiable, our pricing
critersion states that the expected return of the portfolio should increase at the risk free rate,
(3.13) EΠ(t+dt, ˜
X(t+dt)) ˜
X(t) = x= (1 + rdt) Π(t, x) = e
rdt
Π(t, x).
Applying the backwards equation shows that the portfolio value satisfies the equation
(3.14) Π
t
+a(t, x
x
+
−∞
K(y){Π(t, x +b(t, x)y)Π(t, x)b(t, x)yΠ
x
(t, x)}dy =rΠ.
Substituting Π = V(t, x)xand ∆ = V
x
(t, x)yields the pricing equation,
(3.15) V
t
+rxV
x
+
−∞
K(y){V(t, x +b(t, x)y)V(t, x)b(t, x)yV
x
(t, x)}dy =rV.
12
This option pricing formula is analogous to the Black-Scholes equation, except that the diffusion term
has been replaced by an integral operator in the final pricing equation. This should not be surprising since
we are replacing a local process (Brownian motion) with a nonlocal one. This pricing formula can also
be interpreted as a backwards equation. Consider the function V(t, x)defined as the discounted expected
value,
(3.16a) V(t, x) = e
r(Tt)
EV(T, ˆ
X(T)) ˆ
X(t) = xfor all t < T,
where the original “real world process” ˜
X(t)has been replaced by the “pricing process”
(3.16b) dˆ
X=rˆ
Xdt +b(t, ˆ
X)dZ.
The only difference between the “real world process” ˜
X(t)and “pricing process” ˆ
X(t)is that the original
drift term a(t, ˜
X)has been replaced by the risk free growth rate. The backwards equation for eqs. 3.16a,
3.16b is identical to the pricing formula 3.15. Since the expected value must satisfy the backwards equation,
it must be the solution of 3.15 as well. I.e., it must be the option price.
It is easily seen that this pricing formula ensures call-put parity. Define
(3.17) V
diff
(t, x)V
call
(t, x)V
put
(t, x).
On expiry,
(3.18) V
diff
(T
ex
, x)[xK]
+
[Kx]
+
=xK.
The solution of eq. 3.15 with this initial condition is V
diff
=xKe
(T
ex
t)
, as can be verified by direct
substitution. So
(3.19) V
call
(t, x)V
put
(t, x)xKe
(T
ex
t)
for any t < T
ex
.
The hedge has a key place in the analysis, since the hedged portfolio reduces to a diversifiable portfolio
only if the portfolio uses the hedge. This does not necessarily mean that one should use the hedge in
practice. For example, one may prefer to maximize one’s own utility function over a set time horizon to
determine the hedge.
Finally, these arguments can be generalized to derivatives that depend on multiple assets. Consider the
example
(3.20) d˜
X
i
=a
i
(t, ˜
X)dt +
j
B
ij
(t, ˜
X)dZ
j
,
where the noise processes Z
j
(t)are independent truncated Lévy processes. Generalizing the arguments leads
to the same pricing formula
(3.21a) V(t, x) = e
r(Tt)
EV(T, ˆ
X(T)) ˆ
X(t) = x,
where the drift terms of all market assets have been replaced with the risk free growth rate,
(3.21b) dˆ
X
i
=rˆ
X
i
dt +
j
B
ij
(t, ˆ
X)dZ
j
.
That is,
V
t
+
i
rx
i
V
x
i
(3.22)
+
−∞
···
−∞
K(y){V(t, x+B(t, x)y)V(t, x)B(t, x)y·
x
V(t, x)}dy
1
···dy
n
=rV.
13
4. Explicit option prices. In this section we use Green’s functions to obtain leading order asymptotic
formulas for option prices in the different asymptotic regimes. Because of call-put parity (see eq. 3.19), we
will usually work with the most convenient of V
call
,V
put
, or the time value
(4.1) V
tv
(t, x) = V
call
(t, x)[xKe
(t
ex
t)
]
+
=V
put
(t, x)[Ke
(t
ex
t)
x]
+
.
We model real-world asset prices as
(4.2a) d˜
X=a(t, ˜
X)dt +b(t, ˜
X)dZ
T L
,
where dZ
T L
is the tempered Lévy flight with jump kernel
(4.2b) K
T L
(y) = cΓ (α+ 1)
π|y|
α+1
sin παp
1
e
y/λ
1
for y > 0
sin παp
2
e
−|y|
2
for y < 0.
The prices of calls on ˜
X(t)are then
(4.3a) V
call
(t, x) = e
r(T
ex
t)
Eˆ
X(T
ex
)K
+
ˆ
X(t) = x,
where the pricing process ˆ
X(t)is
(4.3b) dˆ
X=rˆ
Xdt +b(t, ˆ
X)dZ
T L
,
regardless of the orginal drift term a(t, ˜
X)dt. The backwards equation for eqs. 4.3a-4.3b is
(4.4a) V
t
+rxV
x
+
−∞
K
T L
(y){V(t, x +b(t, x)y)V(t, x)b(t, x)yV
x
(t, x)}dy =rV
for t < T
ex
, with
(4.4b) V
call
= [xK]
+
at t=T
ex
.
We change variables to
T=T
ex
t, x
new
xe
r(T
ex
t)
=xe
rT
,(4.5a)
V
new
(T, x) = e
rT
V(t, x), b
new
(T, x
new
) = e
rT
b(t, x),(4.5b)
to simplify the analysis. Tis now the time to expiry, x
new
is the forward value of the asset, and V
new
is the
forward option price. After changing the integration variable to b(t, x)y, the backwards equation becomes
an initial value problem
V
T
=b
α
(T, x)
−∞
˘
K
T L
(y){V(T, x +y)V(T , x)yV
x
(T, x)}dy for T > 0,(4.6a)
V
call
= [xK]
+
at T= 0,(4.6b)
in the new variables, with
(4.6c) ˘
K
T L
(y) = cΓ (α+ 1)
π|y|
α+1
sin παp
1
e
y/λ
1
b(T,.x)
for y > 0
sin παp
2
e
−|y|
2
b(T,x)
for y < 0.
As always, we drop the “new” superscripts for clarity. Call-put parity is now
(4.7) V
call
(T, x)V
put
(T, x) = xK.
14
Define the Green’s function g(T, x;x
0
)by
g
T
=b
α
(T, x)
−∞
˘
K
T L
(y){g(T, x +y;x
0
)g(T, x;x
0
)yg
x
(T, x;x
0
)}dy for T > 0,(4.8a)
g(T, x)δ(xx
0
)as T0.(4.8b)
The call value is then
(4.9) V
call
(T, x) =
K
[x
0
K]g(T, x;x
0
)dx
0
,
so clearly the Green’s function is also the probability density for the process.
4.1. Scaling. We assume that asset prices have been scaled so that xand Kare both O(1) quantities,
and that chas been chosen so that b(T, x)is also O(1). Based on the market data presented earlier, we
assume that
(4.10) λ
1
, λ
2
O(1) with O(λ
1
) = O(λ
2
)O(λ),
so that the exponential truncation of the jump frequency occurs on length scales λwhich are still very small
compared to the overall length scale of the option. We re-scale the problem by defining
(4.11) τ=b
α
(0, K)
η
α
cT b
α
0
η
α
cT, χ =Kx
ξ,
where our choices for η1and ξ1will define each asymptotic regime. We also define
(4.12) g
new
(τ , χ;χ
0
) = ξg
old
(T, x;x
0
),
so that
(4.13)
−∞
g
new
(τ , χ;χ
0
)=
−∞
g
old
(T, x;x
0
)dx = 1.
Note that when τO(1), we have (cT )
γ
=O(η), so we expect the Lévy flights to spread the density over
O(η)distances on this timescale.
We focus on three sets of time scales. For the shortest time scales, we choose
(4.14a) ηλ1.
By selecting the space scale ξ=η, we will find that the bulk of the probability distribution spreads out
according to the Lévy flight’s similarity solution,
(4.14b) g(τ, z ) = 1
τ
γ
h(z/τ
γ
),
in this regime. Consequently, the option prices are also given by a similarity solution,
(4.14c) V
call
=τ
γ
I
call
(χ/τ
γ
),
to leading order. By re-selecting the space scale as ξ=λ, we can examine the far wings, where we find an
outer solution in which the probability densities (and time values) are dominated by single large jumps.
We investigate the transition regime by choosing the time scale
(4.15a) η=λO(1)
15
and space scale ξ=η. In this regime, the probability distribution has spread over distances O(λ), and is
increasingly constricted by the exponential truncation factors. We will find that in a region
(4.15b) ν
diff
min
τ < χ < ν
diff
max
τ,
the random process resembles diffusion, and that at any point χin this region, the spread of the probability
distribution slows down as τincreases, until eventually it spreads out like τ
1/2
instead of τ
γ
. Exterior to the
region 4.15b, the process and the option’s time value resembles diffusion plus a single large jump.
For the last regime we choose
(4.16) ληO(1)
and ξ=η. On these time scales we find that the process resembles diffusion, with the density spreading
out like τ
1/2
over distances large compared to λ, and that the leading order option prices can be obtained
by replacing the jump process with an appropriate Brownian motion-based model. Even so, in the extreme
wings the transcendentally small transition probabilities and time values are still dominated by discrete
jumps.
4.2. Green’s function. Changing variables to τ,χ, and g
new
(τ , χ;χ
0
), we obtain
g
τ
=η
ξ
α
b
b
0
α
−∞
˜
K
T L
(y){g(τ, χ y;χ
0
)g(τ , χ;χ
0
) + yg
χ
(τ , χ;χ
0
)}dy,(4.17a)
g(τ , χ)δ(χχ
0
)as τ0,(4.17b)
where
(4.18) ˜
K
T L
(y) = Γ (α+ 1)
π|y|
α+1
sin παp
1
e
ξy/bλ
1
for y > 0
sin παp
2
e
ξ|y|/bλ
2
for y < 0.
and where we are droppng the “new” superscript. Note that
(4.19) η1, ξ 1with ηξ
in all the asymptotic limits we investigate. Thus,
(4.20) b(T, x)
b
0
=b([η/b
0
]
α
τ , K ξχ)
b(0, K)= 1 + · ·· ,
so to leading order,
g
τ
=η
ξ
α
−∞
ˆ
K
T L
(y){g(τ , χ y;χ
0
)g(τ , χ;χ
0
) + yg
χ
(τ , χ;χ
0
)}dy for τ > 0,(4.21a)
g(τ , χ)δ(χχ
0
)as τ0,(4.21b)
with
(4.22a) ˆ
K
T L
(y) = Γ (α+ 1)
π|y|
α+1
sin παp
1
e
κ
1
y
sin παp
2
e
κ
2
|y|
.
Here, the constants κ
1
and κ
2
are
(4.22b) κ
1
=ξ
b
0
λ
1
, κ
2
=ξ
b
0
λ
2
.
Equation 4.21a is autonomous, so g(τ , χ;χ
0
)g(τ , χ χ
0
)g(τ , z). European option prices are thus
(4.23a) V
call
(τ , χ) = ξ
χ
(zχ)g(τ, z)dz,
16
to leading order, where
g
τ
=η
ξ
α
−∞
ˆ
K
T L
(y){g(τ, z y)g(τ , z) + yg
z
(τ , z)}dy for τ > 0,(4.23b)
g(τ , z)δ(z)as τ0.(4.23c)
Being autonomous suggests using Fourier transforms, so we introduce the transform pair
(4.24) G(τ, k)
−∞
e
ikz
g(τ , z)dz;g(τ , z)1
2π
−∞
e
ikz
G(τ , k)dk.
To transform g(τ, z), we first integrate eq. 4.23b by parts twice and obtain
g
τ
=η
ξ
α
−∞
˜
K
T L
(y)g
zz
(τ , z y)dy,(4.25a)
g(τ , z)δ(z)as τ0,(4.25b)
with
(4.25c) ˜
K
T L
(y) =
y
(y
1
y)ˆ
K
T L
(y
1
)dy
1
for y > 0
y
−∞
(yy
1
)ˆ
K
T L
(y
1
)dy
1
for y < 0.
Taking the transform of 4.25a, 4.25b yields
G
τ
=η
ξ
α
φ(k)
k
2
k
2
Gfor τ > 0,(4.26a)
G(0, k) = 1,(4.26b)
where φ(k)/k
2
is defined by
(4.26c) φ(k)
k
2
=
−∞
e
ikz
˜
K
T L
(z)dz.
Using ˜
K
′′
T L
(z) = ˆ
K
T L
(z), and integrating by parts twice, we obtain
(4.27) φ(k) =
0
e
ikz
1 + ikzˆ
K
T L
(z)dz
0
e
ikz
1ikzˆ
K
T L
(z)dz,
and working out the integrals yields
φ(k) = sin παp
1
sin πα (k
1
)
α
e
iπα/2
κ
α
1
iακ
α1
1
k
(4.28)
+sin παp
2
sin πα (k+
2
)
α
e
iπα/2
κ
α
2
+iακ
α1
2
k.
Therefore,
(4.29a) G(τ, k) = e
(η/ξ)
α
φ(k)τ
,
and the Green’s function is given by the inverse transform:
(4.29b) g(τ, z)1
2π
−∞
e
ikz(η/ξ)
α
φ(k)τ
dk.
17
In particular, φ(0) = φ
(0) = 0, so
(4.30) G(τ, 0) =
−∞
g(τ , z)dz = 1,iG
k
(τ , 0) =
−∞
zg(τ , z)dz = 0.
The call price can be obtained by integrating the payoff against the Green’s function g(t, z). Alterna-
tively, eq. 4.23a shows that the call price is a convolution. So, writing
(4.31a) V
call
(τ , χ) = lim
δ0
ξ
−∞
Q(χz)g(τ, z)dz,
where
(4.31b) Q(z) = [z]
+
e
δz
,
\ccccand taking the Fourier transform
(4.32)
−∞
Q(z)e
ikz
dz =1
(k+)
2
,
allows us to express the call price as an inverse transform
(4.33a) V
call
(τ , χ) = lim
δ0
ξ
2π
−∞
1
(k+)
2
e
ikχ(η/ξ)
α
φ(k)τ
dk.
Similarly,
(4.33b) V
put
(τ , χ) = lim
δ0
ξ
2π
−∞
1
(k)
2
e
ikχ(η/ξ)
α
φ(k)τ
dk.
This provides a method for evaluating option prices directly.
4.3. The Lévy regime.
4.3.1. Inner solution. To analyze the Lévy regime, we select
(4.34a) ηO(λ)O(1) .
On time scales of (cT )
γ
=O(η), the Lévy process can spread out over length scales of O(η),so we select
the space scale
(4.34b) ξ=η,
so that χ= (Kx). Then
(4.35) κ
1
=η
b
0
λ
1
1, κ
2
=η
b
0
λ
2
1,
and g(τ , z)is
(4.36a) g(τ, z ) = 1
2π
−∞
e
φ(k)τ+ikz
dk,
where φ(k)is now
(4.36b) φ(k) = sin παp
1
sin πα ki0
+
α
e
iπα/2
+sin παp
2
sin πα k+i0
+
α
e
iπα/2
18
to leading order.
In this regime, the drift term a(t, X), the volatility factor b(t, X), and the truncation parameters λ
1
,
λ
2
have all dropped out. To leading order, all the models encompassed by eqs. 4.2a, 4.2b have reduced to
the same canonical model,
(4.37) dX =dZ
L
,
where dZ
L
is pure (untruncated) Lévy flight. Using the integration variable k
new
=τ
1
k=τ
γ
kshows that
the transition density for this model is the similarity solution
(4.38a) g(τ, z ) = 1
τ
γ
h(z/τ
γ
),
to leading order, where
(4.38b) h(y) = 1
2π
+
−∞
e
ikyφ(k)
dk.
Thus, European option prices are also given by a similarity solution
(4.39a) V
call
(τ , χ) = ητ
γ
y
(y
1
y)h(y
1
)dy
1
=ητ
γ
I
call
(y)
to leading order, where
(4.39b) y=χ
τ
γ
=Kx
b
0
(cT )
γ
.
In Appendix A we analyze the Fourier inversion integral to obtain expansions for the transition density
h(y)and the option prices. There we find that for |y| ≪ 1,
h(y) =
k=1
Γ(γk + 1)
πk!sin (πkp
2
)y
k1
,(4.40a)
I
call
(y) = p
1
y+ Γ(1 γ)sin (p
2
π)
π+
k=1
γΓ(γk)
(k+ 1)!
sin kπp
2
πy
k+1
.(4.40b)
These power series converge uniformly over all compact regions in the complex yplane, and thus h(y)and
I
call
(y)are entire functions. Call-put parity yields I
put
(y) = I
call
(y) + y.
For large |y|, Appendix A shows that
h(y)
k=1
()
k1
Γ(+ 1)
πk!
sin kπαp
1
|y|
+1
for y1,(4.41a)
I
call
(y)
k=1
()
k1
Γ(1)
πk!
sin kπαp
1
|y|
1
for y1,(4.41b)
and that
h(y)
k=1
()
k1
Γ(+ 1)
πk!
sin kπαp
2
|y|
+1
for y1,(4.42a)
I
call
(y)∼ −y+
k=1
()
k1
Γ(1)
πk!
sin kπαp
2
|y|
1
for y1.(4.42b)
19
In Appendix A, we also present an efficient method for calculating the transition densities h(y)and
option prices I
call
(y)numerically.
The Lévy regime represents the last few months before expiry. As the time to expiry decreases, these
similarity solutions become increasingly accurate. We emphasize that all options of the form
(4.43) d˜
X=a(t, ˜
X)dt +b(t, ˜
X)dZ
T L
,
are governed by the similarity solutions when we approach the expiry date closely enough. We examine these
similarity solutions further in section 5, comparing the effectiveness of Lévy based hedging with hedges
derived using standard Brownian motion-based models. In section 5 we also find that the implied normal
vol (see Appendix D) is a similarity solution in the Lévy regime.
4.3.2. Outer solution. The outer solution for the Lévy regime can be obtained by choosing the time
scale ηO(λ)O(1) as before. Even though the transition density is predominantly within the region
|xx
0
|=O(η), we choose the much larger space scale ξ=λη, so we can observe the wings of the
distribution. See eq. 4.11. With this scaling,
(4.44a) g(τ, z )1
2π
−∞
e
ikz(η/λ)
α
φ(k)τ
dk,
where φ(k)is given by 4.28, but now the constants κ
1
and κ
2
are O(1) ,
(4.44b) κ
1
=λ
b
0
λ
1
, κ
2
=λ
b
0
λ
2
.
To obtain the density g(τ , z)for z > 0, we deform the contour around the upper branch cut and onto
the path k=
1
+qe
i3π/2
for > q > 0and k=
1
+qe
+iπ/2
for 0< q < . This yields
(4.45) g(τ, z)ie
κ
1
z
2π
0
e
qz(η/λ)
α
φ
+
(q)τ
dq +cc,
where φ
+
(q) = φ
1
+qe
iπ/2
. Expanding g(τ, z )in powers of η/λ yields
(4.46a) g(τ, z )
j=1
()
j1
j!η
λ
J
j
(z)τ
j
,
where
(4.46b) J
j
(z) = e
κ
1
z
2π
0
e
qz
iφ
+
(q)
j
+iφ
+
(q)
j
dq for z > 0.
Note that J
0
(z)0. Similarly, for z < 0we deform around the lower branch cut. This again yields the
power series 4.46a, but now
(4.46c) J
j
(z) = e
κ
2
z
2π
0
e
qz
iφ
(q)
j
iφ
(q)
j
dq for z < 0,
where φ
(q) = φ
2
+qe
iπ/2
.
The j
th
term is proportional τ
j
, and so represents the probability of ending up at zafter jlarge jumps.
Since η/λ 1, the power series in 4.46a is dominated by the first term. I.e., the leading order outer solution
is determined by the probabilty of single large jumps. Working out J
1
(z), we obtain
g(τ , z) = τη
λ
α
Γ (α+ 1)
π|z|
α+1
sin παp
1
e
κ
1
z
for z > 0
sin παp
2
e
κ
2
|z|
for z < 0+···(4.47)
=τη
λ
α
ˆ
K
T L
(z) + ···
20
to leading order. The corresponding option values are
V
call
(τ , χ) = λτ η
λ
α
C
1
π
χ
zχ
π|z|
α+1
e
κ
1
z
dz for χ > 0,(4.48a)
V
put
(τ , χ) = λτ η
λ
α
C
2
π
χ
−∞
χz
π|z|
α+1
e
κ
2
|z|
dz for χ < 0,(4.48b)
to leading order, with C
1,2
= Γ (α+ 1) sin παp
1,2
. Thus, in the Lévy regime, the probability density and the
time values of European options both grow linearly in τin the far wings.
4.4. Transition regime. In the transition regime, (cT )
γ
=O(λ), so we expect the Lévy process to
spread out over the O(λ)length scale and the truncation factors e
−|y|/b
0
λ
to become significant. To analyze
this regime, we scale
(4.49) η=λ1, ξ =η.
From eqs. 4.29b and 4.33a, we find that the probability density and option prices are given by the Fourier
inversion integrals
(4.50a) g(τ, z ) = 1
2π
−∞
e
τ[φ(k)ikz/τ]
dk,
(4.50b) V
call
(τ , χ) = lim
δ0
η
2π
−∞
1
(k+)
2
e
τ[φ(k)ikχ/τ]
dk,
where
φ(k) = sin παp
1
sin πα (k
1
)
α
e
iπα/2
κ
α
1
iακ
α1
1
k
(4.50c)
+sin παp
2
sin πα (k+
2
)
α
e
iπα/2
κ
α
2
+iακ
α1
2
k.
However, now
(4.50d) κ
1
=λ
b
0
λ
1
=O(1) , κ
2
=λ
b
0
λ
2
=O(1) .
4.4.1. Diffusion region. In appendix E we use the method of steepest descents to analyze these
inversion integrals in the limit τ1with z/τ fixed. This analysis shows that if
(4.51a) v
diff
min
< z/τ < v
diff
max
,
where
v
diff
min
=(κ
1
+κ
2
)
α1
αsin παp
1
|sin πα|1w
α1
2
+αsin παp
2
|sin πα|w
α1
1
,(4.51b)
v
diff
max
= (κ
1
+κ
2
)
α1
αsin παp
1
|sin πα|w
α1
2
+αsin παp
2
|sin πα|1w
α1
1
,(4.51c)
with w
1
= 1 w
2
=λ
1
/(λ
1
+λ
2
), then the integrands are dominated by a saddle point. By deforming the
integration contours to take advantage of the saddle point, we obtain
(4.52a) g(τ, z ) = e
τφ(iq
0
)q
0
z
2πφ
kk
(iq
0
)τ{1 + ···},
21
where φ(k)and φ
kk
(k)are evaluated at the saddlepoint k=iq
0
. The saddle point q
0
(z/τ )is defined
implicitly by
(4.52b) z/τ ≡ −αsin παp
1
|sin πα|(κ
1
q
0
)
α1
κ
α1
1
+αsin παp
2
|sin πα|(κ
2
+q
0
)
α1
κ
α1
2
,
and is between the two branch cuts,κ
2
< q
0
(z/τ )< κ
1
. So
φ(iq
0
) = sin παp
1
|sin πα|(κ
1
q
0
)
α
κ
α
1
+ακ
α1
1
q
0
(4.52c)
sin παp
2
|sin πα|(q
0
+κ
2
)
α
κ
α
2
ακ
α1
2
q
0
,
Inspecting φ(iq
0
)shows that the probability density depends on jump rates for jumps of both signs.
Thus, we identify this as a diffusive region, where the dominant probabilities are made up of paths with
many relatively small left and right jumps. Indeed, at any fixed z, as τincreases until z/τ becomes small,
we have
(4.53) q
0
=1
φ
kk
(0)z/τ {1 + · ··} for |z|1,
so g(τ , z)reduces to the Gaussian density,
(4.54a) g(τ, z ) = 1
2πτφ
kk
(0)e
z
2
/2τφ
kk
(0)
{1 + ··· } for |z|1,
with
(4.54b) φ
kk
(0) = α(α1)
|sin πα|sin παp
1
κ
2α
1
+sin παp
2
κ
2α
2
.
So on these time and space scales, the truncated Lévy processes resembles Brownian motion.
Option prices. In appendix E, the method of steepest descents is also used to obtain European option
prices. For the diffusion region
(4.55) v
diff
min
< χ/τ < v
diff
max
,
the option values are
(4.56a) V
call
(τ , χ) = λ[χ]
+
+λe
τφ(iq
0
)q
0
χ
q
2
0
2πτφ
kk
(iq
0
){1 + ··· },
where χ=Kx. Here, φand it’s derivatives are evaluated at the saddle point k=iq
0
(χ/τ), where
q
0
(χ/τ)is defined by 4.52b as before. At each χ, when τincreases enough so that χ/τ 1, then q
0
=
{χ/φ
kk
(0) τ}{1 + ···}, and the option’s time value becomes
(4.57) V
call
(τ , χ) = λ[χ]
+
+λ[τφ
kk
(0)]
3/2
χ
2
2πe
χ
2
/2τφ
kk
(0)
{1 + ···}
to leading order. This matches the value of a far out of the money option under the normal (Bachelier)
model with implied normal vol σ
norm
=λφ
1/2
kk
(0). See Appendix D.
22
4.4.2. Hybrid regions. When z/τ > v
diff
max
or z/τ < v
diff
min
, the integrand no longer has a saddle point.
Instead, the method of steepest descents wraps the contour around the branch cut at k=
1
if z/τ > v
diff
max
and around the branch cut at k=
2
if z/τ < v
diff
min
. In appendix E we carry out this analysis, and find
that if
(4.58a) z/τ > v
diff
max
,
then
(4.58b) g(τ, z) = Γ (α+ 1) sin παp
1
π
τe
κ
1
zτφ(
1
)
zv
diff
max
τ
α+1
{1 + ···} for τ1,
where
(4.58c) φ(
1
) = (α1) sin παp
1
|sin πα|κ
α
1
sin παp
2
|sin πα|(κ
1
+κ
2
)
α
ακ
1
κ
α1
2
κ
α
2
.
This probability density can be written as
(4.59) g(τ, z) = τ K (zv
dff
max
τ)e
τ
(
φ(
1
)+κ
1
v
diff
max
){1 + ···}
to leading order, where K(y)is the underlying jump kernel of our process. This density is roughly the
probability density of being near v
diff
max
τ, times the probability of taking an additional jump of size zv
diff
max
τ
beyond v
diff
max
τ. This suggests that in this region, the distribution is dominated by paths which have one
large jump plus diffusion, and that the time scale is too short for the region to be subsumed by diffusion
from the center, and too short for a second large jump to have significantly affected the probability.
In appendix E, the method of steepest descents also shows that the option prices are given by
(4.60) V
call
(τ , χ) = Γ (α+ 1) sin παp
1
πκ
2
1
λτe
κ
1
χτφ(
1
)
χv
diff
max
τ
α+1
{1 + ···} for τ1,
when χ/τ > v
diff
max
.
The analysis for z/τ < v
diff
min
is identical, mutatis mutandi. Since v
diff
min
<0, if
(4.61a) z/τ < v
diff
min
,
then
(4.61b) g(τ, z) = Γ (α+ 1) sin παp
2
π
τe
κ
2
zτφ(
2
)
v
diff
min
τz
α+1
{1 + ···} for τ1,
when z/τ < v
diff
min
, where
(4.61c) φ(
2
) = sin παp
1
|sin πα|(κ
1
+κ
2
)
α
κ
α
1
ακ
α1
1
κ
2
(α1) sin παp
2
|sin πα|κ
α
2
,
and this yields
(4.62) V
call
(τ , χ) = λχ + Γ (α+ 1) sin παp
2
πκ
2
2
λτe
κ
2
|χ|−τφ(
2
)
v
diff
min
τχ
α+1
{1 + ···} for τ1,
for χ/τ < v
diff
min
<0.
23
4.5. Diffusion regime. We now analyze larger timescales, selecting ηλ. On these timescales, we
anticipate that the random process will have dispersed the probability density over distances that are large
compared to the truncation length λ. So we choose
(4.63) λη1, ξ =η.
With this scaling, 4.23a-4.23c become
(4.64a) V
call
(τ , χ) = η
χ
(zχ)g(τ, z)dz,
with
g
τ
=
−∞
ˆ
K
T L
(y){g(τ , z y)g(τ , z) + yg
z
(τ , z)}dy,(4.64b)
ˆ
K
T L
(y) = Γ (α+ 1)
π|y|
α+1
sin παp
1
e
κ
1
y
for y > 0
sin παp
2
e
κ
2
|y|
for y < 0,(4.64c)
and
(4.64d) κ
1
=η
b
0
λ
1
, κ
2
=η
b
0
λ
2
.
Note that ˆ
K
T L
(y)decays exponentially on the short 1=O(η/λ)1length scale, so we expand
(4.65) g(τ, z y) = g(τ , z)yg
z
(τ , z) +
1
2
y
2
g
zz
(τ , z)
1
6
y
3
g
zzz
(τ , z) +
1
24
y
4
g
zzzz
(τ , z) + ···
Substituting this into 4.64b-4.64c yields
(4.66a) g
τ
=
1
2
M
2
g
zz
1
6
M
3
g
zzz
+
1
24
M
4
g
zzzz
+·· ·
where
M
2
−∞
y
2
ˆ
K
T L
(y)φ
kk
(0) = α(α1)
|sin πα|sin παp
1
κ
2α
1
+sin παp
2
κ
2α
2
,(4.66b)
M
3
−∞
y
3
ˆ
K
T L
(y)≡ −
kkk
(0) = α(α1) (2 α)
|sin πα|sin παp
1
κ
3α
1
sin παp
2
κ
3α
2
.(4.66c)
Each moment M
j+1
is a factor 11smaller than the preceding moment, so to leading order we can drop
all moments except for M
2
, obtaining
(4.67a) g(τ, z ) = 1
2πτ φ
kk
(0)e
z
2
/2τφ
kk
(0)
{1 + ···},
whence
(4.67b) V
call
(τ , χ) = ηχN χ
(φ
kk
(0) τ)
1/2
+η(φ
kk
(0) τ)
1/2
Gχ
(φ
kk
(0) τ)
1/2
.
In the original units, this is
(4.68a) V
call
(τ , χ) = (xK)NxK
σ
N
T
1/2
+σ
N
T
1/2
GxK
σ
N
T
1/2
,
where
(4.68b) σ
N
=b(0, K)(α1) λ
2α
1
sin παp
1
+λ
2α
2
sin παp
2
|sin πα|
1/2
.
See eqs. 4.11, 4.63, 4.64d, and 4.66b. Clearly the European option prices in this regime are equivalent to the
option prices that could have been obtained from Brownian motion-based models, at least to leading order.
24
4.5.1. Outer solution. The jump rate K(y)decays like e
const |y|
,while Gaussians decay like e
const y
2
,
so even in the diffusion regime ηλ, the probability density must be dominated by discrete jumps for large
enough y. This region can be found and analyzed by choosing a larger space scale,
(4.69) λη1,with ξ=ηη
b
0
λ
α1
η.
The method of steepest descents can be used to analyze the Fourier inversion integral for g(τ , z). We
do not repeat the arguments here, since the calculation and results are identical to the results obtained for
the transition regime. In a nutshell, when
(4.70a) v
diff
min
< z/τ < v
diff
max
,
the transition density appears diffusive, and for z/τ outside this interval, the density looks like the result of
a single large jump plus diffusion. Here,
v
diff
min
=αλ
α1
|sin πα|λ
α1
1
sin παp
1
(λ
1
+λ
2
)
α1
λ
α1
2
λ
α1
2
+ sin παp
2
λ
α1
1
λ
α1
2
,(4.70b)
v
diff
max
=αλ
α1
|sin πα|λ
α1
2
sin παp
1
λ
α1
2
λ
α1
1
+ sin παp
2
(λ
1
+λ
2
)
α1
λ
α1
1
λ
α1
1
.(4.70c)
to leading order.
4.6. The macro regime. Finally, we look for option prices that vary on the O(1) space scale. This
case is essentially a repeat of the diffusion regime. Starting from eqs. 4.4a, 4.4b,
(4.71a) V
t
+rxV
x
+
−∞
K
T L
(y){V(t, x +b(t, x)y)V(t, x)b(t, x)yV
x
(t, x)}dy =rV
for t < T
ex
, with
(4.71b) V
call
= [xK]
+
at t=T
ex
,
we note that K
T L
(y)decays exponentially on the much shorter O(λ)length scale. So we expand
(4.72) V(t, x +b(t, x)y) = V(t, x) + b(t, x)yV
x
(t, x) +
1
2
b
2
(t, x)y
2
V
xx
(t, x) +
1
6
b
3
(t, x)y
2
V
xxx
(t, x) + ·· ·
as before. Substituting into eq. 4.71a now yields
(4.73a) V
t
+rxV
x
+
1
2
b
2
M
2
V
xx
(t, x) + ···=rV
to leading order, where the local volatility is
(4.73b) b
2
(t, x)M
2
=(α1) λ
2α
1
sin παp
1
+λ
2α
1
sin παp
2
|sin πα|b
2
(t, x).
5. Similarity solution. Hedging. Conclusions. We investigated European option prices using a
very general, very realistic, class of models for asset prices in the real world,
(5.1a) d˜
X=a(t, ˜
X)dt +b(t, ˜
X)dZ
T L
,
where dZ
T L
is the tempered Lévy flight with the jump kernel
(5.1b) K
T L
(y) = cΓ (α+ 1)
π|y|
α+1
sin παp
1
e
y/λ
1
for y > 0
sin παp
2
e
−|y|
2
for y < 0.
25
Pricing these options required us to extend the Black/Scholes/Merton pricing criterion from arbitrage-free
portfolios to diversifiable portfolios. Our extended “diversifiable portfolio” criterion, along with delta hedging
arguments, then showed that option prices are the expected value of the payoff,
(5.2a) V(t, x) = e
r(T
exp
t)
EV(T
exp
,ˆ
X(T
exp
)ˆ
X(t) = x,
but with the actual process ˜
X(t)replaced by a process ˆ
X(t)growing at the risk free rate:
(5.2b) dˆ
X=rˆ
Xdt +b(t, ˆ
X)dZ
T L
.
Here dZ
T L
is the same truncated Lévy process as before.
In the preceding section we used this theory to evaluate option prices. We found that if time-to-expiry is
long enough, then the option values are essentially the same as option prices that could have been obtained by
using Brownian motion based processes instead of Lévy processes. The only real differences are in the extreme
tails, where the Lévy-based models have larger time-values to account for the possibility of a large jump. As
time increases, and the time-to-expiry decreases, the model enters a transition regime. In this regime there
is a diffusion dominated region near the money, where the prices could have been obtained from Brownian
motion, co-existing with a jump dominated region in the wings. As time advances and the time-to-expiry
decreases further, the diffusion region disappears and we enter a regime in which (cT)
γ
O(λ)O(1).
Then the truncation factors e
−|y|
j
and any explicit Xdependence of the model becomes increasingly
irrelevent.
4
In this regime, which represent the last few months of the option’s life, all of the above models
reduce to the same canonical model,
(5.3a) V(t) = e
r(T
exp
t)
E{V(T
exp
, X (T
exp
)},
where the forward value X(t)of the asset is
(5.3b) dX =b
0
dZ
L
,
and where dZ
L
is a pure Lévy flight with jump kernel,
(5.3c) K
L
(y) = cΓ (α+ 1)
π|y|
α+1
sin παp
1
for y > 0
sin παp
2
for y < 0.
This canonical “Bachelier-Lévy” model becomes increasingly accurate as the time-to-expiry decreases.
It is worth briefly exploring the practical consequences of the Bachelier-Lévy model, since all options are
eventually governed by it. For this model, we found that European option prices are similarity solutions,
V
call
(T, K) = b
0
(cT )
γ
I
call
Kx
b
0
(cT )
γ
,(5.4a)
V
put
(T, K) = b
0
(cT )
γ
I
put
Kx
b
0
(cT )
γ
,(5.4b)
where
(5.4c) I
call
(y) = p
1
y+ Γ(1 γ)sin (p
2
π)
π+
k=1
γΓ(γk)
(k+ 1)!
sin kπp
2
πy
k+1
,
I
call
(y)
k=1
()
k1
Γ(1)
πk!
sin kπαp
1
|y|
1
for y1,(5.4d)
I
call
(y)∼ −y+
k=1
()
k1
Γ(1)
πk!
sin kπαp
2
|y|
1
for y1,(5.4e)
4
Even if we dropped our assumption that O(λ)O(1), the model would reduce to the canonical model, and the solution
would b e give by the similarity solution when both (cT )
γ
O(λ)and (cT )
γ
O(1) hold.
26
and where I
put
(y) = I
call
(y) + y.
The implied normal volatility σ
N
(T, X )can be obtained by equating these Bachelier-Lévy option prices
to the option prices obtained from the normal model,
(5.5) dX =σ
N
dW.
Under the normal model, option prices are,
(5.6a) V
N
call
(T, K )σ
N
T
1/2
I
N
call
Kx
σ
N
T
1/2
,
where
(5.6b) I
N
call
(ψ) =
1
2
ψ+1
2π1
2π
j=1
()
j
ψ
2j
2
j
(2j1) j!
See Appendix D. Selecting σ
N
(T, X )to equate the normal price in eqs. 5.6a, 5.6a with the Lévy price in
eqs. 5.4a-5.4c shows that the implied normal vol is also given by a similarity solution,
(5.7a) σ
N
(T, X ) = b
0
c
γ
T
γ
1
2
ΘKx
b
0
(cT )
γ
.
Near the money,
Θ (y) = 2πsin πp
1
πΓ (1 γ) +
1
2
p
1
y(5.7b)
+1
4Γ (1 γ) sin πp
1
2γsin
2
πp
1
sin πγ 1y
2
+···.
In appendix D we derived a rough formula,
(5.8) σ
N
(T, K )|xK|
2Tlog
1/2
|xK|
2πV
tv
for the implied volatility of far-from-the-money options in terms of the option’s time value V
tv
. See eq. D.6b.
Using this with eqs. 5.4d, 5.4e yields the implied normal volatility
Θ (y) = |y|
2 log
1/2
π/2y
α
Γ (α1) sin παp
1
for y=Kx
b
0
()
γ
1,(5.9a)
Θ (y) = |y|
2 log
1/2
π/2|y|
α
Γ (α1) sin παp
2
for y=Kx
b
0
()
γ
1(5.9b)
So in the wings of the distribution,
(5.10) σ
N
=b
0
c
γ
T
γ
1
2
|y|
2 log
1/2
(···)=|Kx|
2Tlog
1/2
(···),
and we see that the implied volatility increases roughly linearly with the strike K, and decreases like the
square root of the time to expiry, closely matching observed market behavior. Morevover, since
1
2
< γ < 1,
27
0
1
2
3
4
5
6
-6 -4 -2 0 2 4 6
y
Implied vols
1.3
1.5
1.6
1.9
 .Implied normal vol σ
N
(T, K )again st y= (Kx)/b
0
(cT )
γ
for th e sym metric Lévy mode l (p
1
=p
2
=
1
2
) for
various α. As α2, th e smile flattens out since the Lévy flight reduces to Brow nian motion.
0
1
2
3
4
5
-6 -4 -2 0 2 4 6
y
Implied vols
1/3
1/2
7/12
2/3
 .Implied normal vol σ
N
(T, K )against y= (Kx)/b
0
(cT )
γ
for th e Lévy model with exponent α= 1.5for
different symmetry param eters p
1
. T he sm iles w ith p
1
=
1
3
and
2
3
represent the ma ximally skewed smiles because these are the
largest and smal lest allowed p
1
with α= 1.5.
the at-the-money implied vol decreases as the time-to-expiry decreases, again matching observed market
behavior.
Figures. 5.1 and 5.2 illustrate the implied normal vol Θ (y). Note that y→ ±∞ as T0, so the
smile gets increasingly steep as the time-to-expiry decreases. Thus, the increasing severity of the smile as T
decreases is an innate feature for all the models in 5.1a, 5.1b.
In practice, risks and hedges are the key concern of trading desks. Often the most important reason for
pricing deals is to understand the risks and hedges, since prices are usually readily available from the market
itself. Figure 5.3 compares the delta hedges obtained from the symmetric Bachelier-Lévy model with the
delta hedges obtained from the normal model,
∂V
call
(T, X)
∂X ,V
N
call
(T, X )
∂X .
28
We see that the conventional delta from the normal model underhedges far out-of the money options. This is
hardly surprising since the normal model underpredicts the tail probabilities. The conventional delta hedge
also substantially overhedges options struck nearer to the money. The Lévy-based models recognize that
some of the risks are unhedgeable, while the less realistic the conventional model is predicated on being able
to hedge all the risk, which apparently results in overhedging.Similarly, figure 5.4 compares the delta hedges
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
y
Delta
normal
1.3
1.6
1.9
 .Delta hedge for a cal l graphed against y= (Kx)/b
0
(cT )
γ
for the symmetric Lévy m odel (p
1
=p
2
=
1
2
) for
various α. On ly y > 0is sh own