ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL 1. Introduction The concept of arbitrage is ...

13 Pages·2013·0.15 MB·English

ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL
FATMAHABAANDANTOINEJACQUIER
Abstract. InthecontextoftheHestonmodel,weestablishapreciselinkbetweenthesetofequivalent
martingale measures, the ergodicity of the underlying variance process and the concept of asymptotic
arbitrageproposedinKabanov-Kramkov[13]andinF(cid:127)ollmer-Schachermayer[8].
1. Introduction
The concept of arbitrage is the cornerstone of modern mathematical (cid:12)nance, and several versions of
the so-called fundamental theorem of asset pricing have been proved over the past two decades, see for
instance[5]foranoverview. Aversionofitessentiallystatesthatabsenceofarbitrageisequivalenttothe
existence of an equivalent martingale measure under which discounted asset prices are true martingales.
Thisthenallowstheuseof‘martingalemodels’(eithercontinuousorwithjumps)asunderlyingdynamics
for option pricing. In practice, should short-term arbitrages arise|due to some market discrepancies|
they are immediately exploited by traders, and market liquidity therefore acts as an equilibrium agent,
to prevent them occurring signi(cid:12)cantly. It can be argued, however, that one may generate long-term
riskless pro(cid:12)t, when the time horizon tends to in(cid:12)nity. This turns out to hold in most models used in
practice. The existence and nature of such in(cid:12)nite horizon asymptotic arbitrage opportunities have been
studied in a handful of papers, for example [7, 14, 19].
Among the plethora of models used and analysed both in practice and in theory, stochastic volatility
models have proved to be very (cid:13)exible and suitable for pricing and hedging. Due to its a(cid:14)ne structure,
the Heston model [11] has gained great popularity among practitioners for equity and FX derivatives
modelling,seeinparticular[10,9]foradetailedaccountofthisfame. Becauseofthecorrelationbetween
the asset price and the underlying volatility, the market is incomplete, and the Heston model admits an
in(cid:12)nity of equivalent martingale measures. Its a(cid:14)ne structure allows us to study precisely the existence
(or absence) of asymptotic arbitrage. Speci(cid:12)cally, we shall endeavour to understand how the parameters
of the model in(cid:13)uence the nature|such as its speed and existence|of the asymptotic arbitrage. Of
particular interest will be the link between asymptotic arbitrage and the ergodicity of the underlying
variance process. In [8] the authors proved under suitable regularity conditions that price processes with
a non-trivial market price of risk (see De(cid:12)nition 2.3) allow for asymptotic arbitrage (with linear speed).
Using the theory of large deviations, we shall show that S may allow for such arbitrage even if it does
not admit an average squared market price of risk.
The organisation of this paper is as follows: all the notations and de(cid:12)nitions are given in Section 2.
Asymptotic arbitrage in the Heston model is studied in Section 3; the main contribution of this paper
is Theorem 3.7, which identi(cid:12)es su(cid:14)cient (and sometimes necessary) conditions on the set of equivalent
Date:February26,2013.
Key words and phrases. Stochasticvolatilitymodel,Hestonmodel,asymptoticarbitrage,largedeviations.
1
2 FATMAHABAANDANTOINEJACQUIER
martingale measures under which asymptotic arbitrage occur with linear speed. These conditions are
di(cid:11)erent from those in Proposition 3.11 in which we study how the ergodicity of the variance process
plays an important role to prove the existence of asymptotic arbitrage with slower speed.
2. Notations and definitions
Let ((cid:10);F;F;P) be a (cid:12)ltered probability space where the (cid:12)ltration F = (Ft)t(cid:21)0 satis(cid:12)es the usual
conditions and let S = eX model a risky security under an equivalent martingale measure. Let H
denote{the class of predict}able, S-integrable admissible processes. We de(cid:12)ne for each t > 0 the sets
∫
Kt := 0tHsdSs :H 2H and Met(S) := fQ(cid:24)P such that (Su)0(cid:20)u(cid:20)t is a local Q-martingaleg. We
shall always assume that Me(S) is not empty for all t (cid:21) 0. Furthermore, for any set A in (cid:10), we shall
t
denote by Ac :=(cid:10)nA its complement.
2.1. Asymptotic arbitrage. The following de(cid:12)nition of a long-term arbitrage is taken from [8]:
De(cid:12)nition 2.1. The process S admits an (" ;" )-arbitrage up to time t if for (" ;" ) 2 (0;1)2, there
1 2 1 2
exists X 2K such that
t t
(i) X (cid:21)(cid:0)" P-almost surely;
t 2
(ii) P(X (cid:21)1(cid:0)" )(cid:21)1(cid:0)" .
t 2 1
Thismeansthatthemaximallossofthetradingstrategy,yieldingthewealthX attimet,isbounded
t
by " and with probability 1(cid:0)" the terminal wealth X equals at least 1(cid:0)" . We shall be interested
2 1 t 2
hereinthefollowingcharacterisationoflong-termarbitrage,namelythenotionofasymptoticexponential
arbitrage with exponentially decaying failure probability, (cid:12)rst proposed in [8] and later in [3] and [7].
De(cid:12)nition 2.2. The process S allows for asymptotic exponential arbitrage with exponentially decaying
failure probability if there exist t 2 (0;1) and constants C;(cid:21) ;(cid:21) > 0 such that for all t (cid:21) t , there is
0 1 2 0
X 2K satisfying
t t
(i) Xt (cid:21)(cid:0)e(cid:0)(cid:21)2t P-almost surely;
(ii) P(Xt (cid:20)e(cid:21)2t)(cid:20)Ce(cid:0)(cid:21)1t.
Asymptotic exponential arbitrage with exponentially decaying failure probability can be interpreted
as a strong and quantitative form of long-term arbitrage. In particular, let (cid:21)1;(cid:21)2 > 0, "2 := e(cid:0)(cid:21)2t and
"1 := Ce(cid:0)(cid:21)1t, then De(cid:12)nitions 2.1 and 2.2 are equivalent In [8], F(cid:127)ollmer and Schachermayer showed
that this strong form of asymptotic arbitrage was actually a consequence, under some assumptions (see
Theorem 1.4 therein), of the following concept:
De(cid:12)nition 2.3. Let f : R(cid:3)+ ! R(cid:3)+ be a smooth function such that limt%+1f(t) = +1. The process
S is said to hav(e an av∫erage squared m)arket price of risk (cid:13)i (i = 1;2) above the threshold ci > 0 with
speed f(t) if P f(t)(cid:0)1 t(cid:13)2(s)ds<c tends to zero as t tends to in(cid:12)nity.
0 i i
2.2. Stochastic volatility models. We consider here the Heston stochastic volatility model, namely
the unique strong solution to the stochastic di(cid:11)erential equations (2.1) below. As is well-known (see [12]
for example), there may not be a unique risk-neutral martingale measure for such models. The following
ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL 3
SDEs are therefore understood under one such risk-neutral measure Q.
p ( √ )
dS =S =(cid:22)dt+ V (cid:26)dW (t)+ 1(cid:0)(cid:26)2dW (t) ; S =1
t t t 1 2 0
(2.1) p
dV =(a(cid:0)bV )dt+ 2(cid:27)V dW (t); V >0;
t t t 1 0
where W and W are independent Q-Brownian motions, (cid:22);a;(cid:27) > 0, b 2 R and j(cid:26)j < 1. The class of
1 2
equivalent martingale measures Q can be considered in terms of the Radon-Nikodym derivatives
(2.2) (cid:12) { (∫ ∫ ) (∫ ∫ )}
Zt = ddQP(cid:12)(cid:12)(cid:12) =exp (cid:0) t(cid:13)1(s)dW1(s)+ t(cid:13)2(s)dW2(s) (cid:0) 12 t(cid:13)12(s)ds+ t(cid:13)22(s)ds :
jF 0 0 0 0
t p ( √ )
The condition (cid:22)(cid:0)r = V (cid:26)(cid:13) (t)+ 1(cid:0)(cid:26)2(cid:13) (t) is necessary for an equivalent local martingale
t 1 2
measure to exist, and ensures that the discounted stock price is a local martingale. Since Z is a positive
localmartingalewithZ =1, itisasupermartingale, andatruemartingaleifandonlyifE(Z )=1. For
0 t
the Heston stochastic volatility model we obtain, for any any real constant (cid:21),
( )
√ 1 (cid:22)(cid:0)r √
(2.3) (cid:13) (t)=(cid:21) V and (cid:13) (t)= √ p (cid:0)(cid:21)(cid:26) V :
1 t 2 1(cid:0)(cid:26)2 Vt t
3. Main results
For any ((cid:11);(cid:12);(cid:14))2R3, we introduce the process (Xt(cid:11);(cid:12);(cid:14))t(cid:21)0 de(cid:12)ned (pathwise) by
∫ ∫
t t
(3.1) X(cid:11);(cid:12);(cid:14) :=(cid:11)V +(cid:12) V ds+(cid:14) V(cid:0)1ds; for any t(cid:21)0;
t t s s
0 0
where V is the Feller di(cid:11)usion for the variance in (2.1). De(cid:12)ne the real interval D by
(cid:12);(cid:14)
 [ ]
 (((cid:0)a14(cid:0)(cid:27);(cid:14)(cid:27)()a2(cid:0);4(cid:27)b(cid:27)2)(cid:12)2 ^; b2 ]; iiff (cid:12)(cid:12) >>00;; (cid:14)(cid:14) <>00;;
(3.2) D = [ 4(cid:27)(cid:14) ] 4(cid:27)(cid:12)
(cid:12);(cid:14)  [4(ab(cid:27)2(cid:12)(cid:0);(cid:27)()a24(cid:0)(cid:27)_(cid:14)(cid:27))b22 ;;+1); iiff (cid:12)(cid:12) <<00;; (cid:14)(cid:14) ><00;:
4(cid:27)(cid:14) 4(cid:27)(cid:12)
Whenever (cid:12)(cid:14) =0, we de(cid:12)ne D(cid:12);(cid:14) by taking the limits of the interval (a closed bound(becoming]open if it
becomes in(cid:12)nite), where we use the slight abuse of notation "1=0=1", i.e. D = (cid:0)1; b2 if (cid:12) >0
[ ) (cid:12);(cid:14) 4(cid:27)(cid:12)
and (cid:14) = 0, D = b2 ;+1 if (cid:12) < 0 and (cid:14) = 0, and D = R if (cid:12) = (cid:14) = 0. Let us further de(cid:12)ne the
(cid:12);(cid:14) 4(cid:27)(cid:12) (cid:12);(cid:14)
function (cid:3)(cid:12);(cid:14) :D !R by
(cid:12);(cid:14)

√ √
 ba (cid:0) 1 ((a(cid:0)(cid:27))2(cid:0)4(cid:27)(cid:14)u)(b2(cid:0)4(cid:27)(cid:12)u)(cid:0) 1 b2(cid:0)4(cid:27)(cid:12)u; if (cid:14) 6=0;
2(cid:27) 2(cid:27) 2
(3.3) (cid:3)(cid:12);(cid:14)(u)=
 a (b(cid:0)√b2(cid:0)4(cid:27)(cid:12)u); if (cid:14) =0:
2(cid:27)
In the case (cid:14) 6=0 above, we further impose the condition a>(cid:27) for the de(cid:12)nition of the function (cid:3)(cid:12);(cid:14).
Remark 3.1. Itmaybesurprisingat(cid:12)rstthatthefunction(cid:3)(cid:12);(cid:14) related|insomesensede(cid:12)nedprecisely
below|does not depend on (cid:11). This function actually describes the large-time behaviour of the process
X(cid:11);(cid:12);(cid:14). Since the variance process V is strictly positive almost surely (by the Feller condition imposed
∫
above),theterm tV dsclearlydominatesV foranyt,whichexplainswhy(cid:11)bearsnoin(cid:13)uenceon(cid:3)(cid:12);(cid:14).
0 s t
4 FATMAHABAANDANTOINEJACQUIER
The condition a>(cid:27) imposed above in the case (cid:14) 6=0 should not surprise the reader since this is nothing
else than the Feller condition, ensuring that the variance process never touches the origin almost surely.
We further de(cid:12)ne the Fenchel-Legendre transform (cid:3)(cid:3) :R!R of (cid:3)(cid:12);(cid:14) by
(cid:12);(cid:14) +
(3.4) (cid:3)(cid:3) (x):= sup fux(cid:0)(cid:3)(cid:12);(cid:14)(u)g:
(cid:12);(cid:14)
u2D(cid:12);(cid:14)
Notation. Whenever (cid:12) =0 or (cid:14) =0, we shall drop the subscript and write respectively (cid:3)(cid:14) or (cid:3)(cid:12). The
same rule will be followed for the domains and the Fenchel-Legendre transforms.
In the general case, (cid:3)(cid:3) does not have a closed-form representation. In the particular case where (cid:14) is
(cid:12);(cid:14)
null|which shall be of interest for us|it actually does, and a straightforward computation shows that
(bx(cid:0)a(cid:12))2
(3.5) (cid:3)(cid:3)(x)= ; for all x2R(cid:3):
(cid:12) 4(cid:27)j(cid:12)xj
In that case, the function (cid:3)(cid:3) is strictly convex on R(cid:3) (respectively on R(cid:3)) with a unique minimum
(cid:12) + (cid:0)
attained at ja(cid:12)=bj (resp. at (cid:0)ja(cid:12)=bj). In particular on R(cid:3), if b(cid:12) (cid:20) 0 then (cid:3)(cid:3) is strictly decreasing and
+ (cid:12)
strictlypositiveonR(cid:3). Otherwise,ifb(cid:12) >0,then(cid:3)(cid:3)(ja(cid:12)=bj)=0and(cid:3)(cid:3)(x)>0forallx2R(cid:3) nfja(cid:12)=bjg.
+ (cid:12) (cid:12) +
Symmetric statements hold on R(cid:0).
3.1. The large deviations case. Inthissection, weshallbeinterestedinprovingasymptoticarbitrage
results for the stock price process when the speed is linear. We shall in particular observe that the
ergodicity of the variance process plays a key role. We (cid:12)rst start though with the following two technical
lemmas, which will be used heavily in the remaining of the paper, and the proofs of which can be found
in Appendix A.
Lemma 3.2. For any ((cid:11);(cid:12))2R2, the family (t(cid:0)1X(cid:11);(cid:12);0) satis(cid:12)es a large deviations principle on R(cid:3)
t t>0 +
if (cid:12) >0 and on R(cid:3) if (cid:12) <0 with speed t(cid:0)1 and rate function (cid:3)(cid:3) characterised in (3.5).
(cid:0) (cid:12)
Lemma 3.3. The family (t(cid:0)1Xt(cid:11);(cid:12);(cid:14))t(cid:21)0 satis(cid:12)es
(i) a full LDP (on R) if (cid:12)(cid:14) <0;
( p )
(ii) a partial LDP on 2 (cid:14)(cid:12);+1 if (cid:12) >0 and (cid:14) >0;
( p )
(iii) a partial LDP on (cid:0)1;(cid:0)2 (cid:14)(cid:12) if (cid:12) <0 and (cid:14) <0;
(iv) a partial LDP if (cid:12) =0 or (cid:14) =0 on the domain given by taking the limit in (ii) or (iii).
In each case, the rate function is (cid:3)(cid:3) and the (partial) LDP holds with speed t(cid:0)1.
(cid:12);(cid:14)
In [8, Theorem 1.4], F(cid:127)ollmer and Schachermayer proved that if the stock price process has an average
market price of risk above a threshold then asymptotic arbitrage holds. Using the large deviations
principleprovedabove, we(cid:12)rstshowthatS doesnotalwaysadmitanaveragemarketpriceofriskfor(cid:13)
1
(Proposition 3.4) or (cid:13) (Proposition 3.5) above any threshold. This is in particular so when the variance
2
processisnotergodic(b(cid:20)0). Thishowever|asprovedinTheorem3.7below|doesnotprecludeabsence
of asymptotic arbitrage.
Proposition 3.4. Fix (cid:21) (cid:21) 0 and c > 0. The stock price process does not satisfy an average squared
market price of risk (cid:13) above the threshold c if either (i) b(cid:20)0 or (ii) b>0 and c>a(cid:21)2=b.
1
ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL 5
( )
∫
Proof. Note (cid:12)rst that (cid:21) = 0 implies (cid:13) (cid:17) 0 and hence P t(cid:0)1 t(cid:13)2(s)ds<c = 1 for all t > 0, so
1 0 1
that the proposition is trivial. Assume from now on that (cid:21) 6= 0 and let c be an arbitrary strictly
∫ ∫
positive real number. The de(cid:12)nition of (cid:13) in (2.3) implies P(t(cid:0)1 t(cid:13)2(s)ds (cid:21) c) = P(t(cid:0)1 tV ds (cid:21)
1 0 1 0 s
c=(cid:21)2)=P(t(cid:0)1Xt0;1 (cid:21)c=(cid:21)2). From Lemma 3.2, the family (t(cid:0)1Xt0;1)t(cid:21)0 satis(cid:12)es a LDP on R(cid:3)+ with rate
function (cid:3)(cid:3) . Hence
1;0
{
( )
1 c (cid:0)(cid:3)(cid:3) (c=(cid:21)2)<0; if c>a(cid:21)2=jbj;
limsup logP X0;(cid:12) (cid:21) (cid:20)(cid:0) inf (cid:3)(cid:3) (x)= 1:0
t%+1 t t (cid:21)2 fx(cid:21)c=(cid:21)2g 1;0 0; if c(cid:20)a(cid:21)2=jbj;
When b (cid:20) 0,(cid:3)(cid:3) (c=(cid:21)2) is strictly positive for all c > 0. Thus P(t(cid:0)1X0;1 (cid:21) c=(cid:21)2) converges to zero as t
1:0 ∫ t
tends to in(cid:12)nity, which in turn implies that P(t(cid:0)1 t(cid:13)2(s)ds < c) converges to 1 as t tends to in(cid:12)nity,
0 1
andstatement(i)inthepropositionfollows. Whenb>0,considerthecasec>a(cid:21)2=b. Thereexistst(cid:22)>0
such that for all t(cid:21)t(cid:22), P(t(cid:0)1X0;1 (cid:21)c=(cid:21)2)(cid:20)exp((cid:0)(cid:3)(cid:3) (c=(cid:21)2)t), and hence P(t(cid:0)1X0;1 (cid:21)c=(cid:21)2) converges
t 1:0 t
to zero as t tends to in(cid:12)nity, which again proves statement (ii) in the proposition. (cid:3)
Proposition 3.5. Fix (cid:21) (cid:21) 0 and let c > 0. The stock price process does not satisfy an average squared
market price of risk (cid:13) above the threshold c if any of the following conditions hold:
2
(i) (cid:21)(cid:26)((cid:22)(cid:0)r)>0;
(ii) (cid:21)(cid:26)((cid:22)(cid:0)r)<0 and c>(cid:0)4(cid:21)(cid:26)((cid:22)(cid:0)r)=(1(cid:0)(cid:26)2);
(iii) (cid:21)(cid:26)6=0, (cid:22)=r and b(cid:20)0;
(iv) (cid:21)(cid:26)6=0, (cid:22)=r, b>0 and c>a(cid:21)4(cid:26)2=(b(1(cid:0)(cid:26)2));
(v) (cid:21)(cid:26)=0;
Remark 3.6. Notethatthecaseofacompletemarket((cid:26)=0)isincludedincase(v)oftheproposition.
Proof. Let c be an (arbitrary strictly p)ositive real number. Note (cid:12)rst that if (cid:21)(cid:26) = 0, and (cid:22) = r, then
∫
(cid:13) (cid:17)0 and hence P t(cid:0)1 t(cid:13)2(s)ds<c =1 for all t>0. If (cid:22)6=r, then
2 0 2
( ∫ ) ( ∫ ) ( )
1 t 1 t ds 1(cid:0)(cid:26)2 X0;0;1 1(cid:0)(cid:26)2
P (cid:13)2(s)ds(cid:21)c =P (cid:21) c =P t (cid:21) c ;
t 2 t V ((cid:22)(cid:0)r)2 t ((cid:22)(cid:0)r)2
0 0 s
and Lemma A.1 implies that (cid:3)(cid:3) is strictly positive, so that (v) follows. Assume now that (cid:21)(cid:26) 6= 0 and
0;1
(cid:22)6=r. The de(cid:12)nition of (cid:13) in (2.3) implies that
2
( ∫ ) ( ∫ ∫ )
1 t ((cid:22)(cid:0)r)21 t ds (cid:21)2(cid:26)2 1 t 2(cid:26)(cid:21)((cid:22)(cid:0)r)
P (cid:13)2(s)ds(cid:21)c =P + V ds(cid:21)c+
t 2 1(cid:0)(cid:26)2 t V 1(cid:0)(cid:26)2 t s 1(cid:0)(cid:26)2
0 ( 0 s ) 0
X0;(cid:12);(cid:14) 2(cid:26)(cid:21)((cid:22)(cid:0)r)
=P t (cid:21)c+ ;
t 1(cid:0)(cid:26)2
where (cid:12) = ((cid:22)(cid:0)r)2 > 0, (cid:14) = (cid:21)2(cid:26)2 > 0, and where X0;(cid:12);(cid:14) is de(cid:12)ned in (3.1). By Lemma 3.3, the family
1(cid:0)(cid:26)2 1(cid:0)(cid:26)2 p
(X0;(cid:12);(cid:14)=t) satis(cid:12)es a large deviations principle on (2 (cid:14)(cid:12);+1) with rate function (cid:3)(cid:3) , i.e.
t t>0 (cid:12);(cid:14)
( ∫ ) { }
1 t 2(cid:26)(cid:21)((cid:22)(cid:0)r)
limsupt(cid:0)1logP (cid:13)2(s)ds(cid:21)c (cid:20)(cid:0)inf (cid:3)(cid:3) (x):x(cid:21)c+ :
t%+1 t 0 2 (cid:12);(cid:14) 1(cid:0)(cid:26)2
[ ) ( p )
When (cid:21)(cid:26)((cid:22)(cid:0)r) > 0, c+ 2(cid:26)(cid:21)((cid:22)(cid:0)r);+1 is a subset of 2 (cid:12)(cid:14);+1 , and (i) follows immediately from
1(cid:0)(cid:26)2 [ ) ( p )
Lemma A.1. When (cid:21)(cid:26)((cid:22)(cid:0)r) < 0, the interval c+ 2(cid:26)(cid:21)((cid:22)(cid:0)r);+1 is a subset of 2 (cid:12)(cid:14);+1 if and
1(cid:0)(cid:26)2
only if c > (cid:0)4(cid:26)(cid:21)((cid:22)(cid:0)r) > 0. Since (cid:12)(cid:14) = (cid:21)2(cid:26)2((cid:22)(cid:0)r)2 > 0, Lemma A.1 implies that (cid:3)(cid:3) (x) > 0 for any
1(cid:0)(cid:26)2 (1(cid:0)(cid:26)2) (cid:12);(cid:14)
6 FATMAHABAANDANTOINEJACQUIER
( )
p
x > 2 (cid:12)(cid:14) = 2j(cid:21)(cid:26)((cid:22)(cid:0)r)j. Therefore, P X0;(cid:12);(cid:14)=t(cid:21)c+ 2(cid:26)(cid:21)((cid:22)(cid:0)r) converges to zero as t tends to in(cid:12)nity.
∫ 1(cid:0)(cid:26)2 t 1(cid:0)(cid:26)2
Then P(t(cid:0)1 t(cid:13)2(s)ds<c) converges to one as t tends to in(cid:12)nity. Assume that (cid:21)(cid:26)6=0 and (cid:22)=r. The
0 2
de(cid:12)nition of (cid:13) in (2.3) implies that
2
( ∫ ) ( ∫ ) ( )
1 t 1 t 1(cid:0)(cid:26)2 X0;1;0 1(cid:0)(cid:26)2
P (cid:13)2(s)ds(cid:21)c =P V ds(cid:21) c =P t (cid:21) c ;
t 2 t s (cid:21)2(cid:26)2 t (cid:21)2(cid:26)2
0 0
and (iii) and (iv) then from Proposition 3.4. (cid:3)
We can now move on to our main theorem.
Theorem 3.7. Let " 2 (0;1), (cid:13) > 0 and de(cid:12)ne the set A := fZ (cid:21) e(cid:0)(cid:13)tg 2 F . Then S allows
(cid:21);t t t
for strong asymptotic arbitra(ge (wipth s)peed t) wi(th exponentially dec)aying probability (in the sense of
De(cid:12)nition 2.2) with C =exp (cid:21)V = 2(cid:27) , (cid:21) =(cid:0) pa(cid:21) +(cid:13)+(cid:3)(cid:11);(cid:12)(1) and (cid:21) =(cid:13) if
0 1 2
( ) p 2(cid:27)
(i) (cid:21)2Rn (cid:0)pb (cid:0) (cid:13) ;(cid:0)pb + (cid:13) , when 2(cid:27) >1;
2(cid:27) a(cid:16)+ p2(cid:27) a(cid:16)(cid:0)
(ii) (cid:21)<(cid:0)pb (cid:0) (cid:13) , when 2(cid:27) (cid:20)1,
2(cid:27) a(cid:16)+ p p
where we de(cid:12)ne (cid:16)(cid:6) := 2(cid:27)(cid:6)1= 2(cid:27).
Remark 3.8.
(cid:15) Note that the su(cid:14)cient condition is not necessary. Consider for instance (cid:21)=0 and (cid:22)=r. Then
clearly Z =1 almost surely for all t(cid:21)0, and P(Z (cid:21)e(cid:0)(cid:13)t)=1 for any (cid:13) >0.
t t
(cid:15) Let f :R !R is a continuous function such that t=f(t) tends to in(cid:12)nity as t tends to in(cid:12)nity,
+ + ( )
then for any (cid:13) >0 and t large enough, e(cid:0)(cid:13)f(t) (cid:21)e(cid:0)(cid:13)t. Therefore P Z (cid:21)e(cid:0)(cid:13)f(t) tends to zero
t
as well as t tends to in(cid:12)nity; We cannot however conclude that Theorem 3.7 holds, i.e. that S
allows asymptotic arbitrage with speed f(t), since this does not give us any information about
( )
the behaviour of Q Z (cid:21)e(cid:0)(cid:13)f(t) .
t
Proof. Let (cid:13) > 0 and de(cid:12)ne the set A(cid:21);t := fZt (cid:21) e(cid:0)(cid:13)tg 2 Ft. Since the(processes W2 and V are ind)e-
∫ ∫
pendent,thetowerpropertyforconditionalexpectationimpliesE(Zt)=E e(cid:0) 0t(cid:13)1(s)dW1(s)(cid:0)21 0t(cid:13)12(s)ds .
Markov’s inequality therefore yields
[ ( )]
∫ ∫
P(A )(cid:20) E(Zt) = E exp (cid:0) 0t(cid:13)1(s)dW1(s)(cid:0) 12 0t(cid:13)12(s)ds
(cid:21);t exp((cid:0)(cid:13)t) e(cid:0)(cid:13)t
( ) [ ( ( )∫ )]
(cid:21)V a(cid:21)t (cid:21)V b(cid:21) (cid:21)2 t
=exp p 0 + p +(cid:13)t E exp (cid:0)p T (cid:0) p + V ds
2(cid:27) 2(cid:27) 2(cid:27) 2(cid:27) 2 s
[ ( ) ] 0
(cid:21)V a(cid:21)
=exp p 0 + p +(cid:13) t (cid:3)(cid:11);(cid:12)(t);
t
2(cid:27) 2(cid:27)
where (cid:11) = (cid:0)p(cid:21) and (cid:12) = (cid:0)pb(cid:21) (cid:0) (cid:21)2. From the proof of Lemma 3.2, we know that t(cid:0)1log(cid:3)(cid:11);(cid:12)(t)
2(cid:27) 2(cid:27) 2 t
converges to (cid:3)(cid:11);(cid:12)(1). This implies that for any (cid:14) >0 there exists t~>0 such that for any t>t~, we have
e((cid:3)(cid:11);(cid:12)(1)(cid:0)(cid:14))t (cid:20)(cid:3)(cid:11);(cid:12)(t)(cid:20)e((cid:3)(cid:11);(cid:12)(1)+(cid:14))t:
t
We then deduce that for any t>t~,
[ ( ) ] [ ( ) ]
(cid:21)V a(cid:21) (cid:21)V a(cid:21)
exp p 0 + p +(cid:13)+(cid:3)(cid:11);(cid:12)(1)(cid:0)(cid:14) t (cid:20)P(Z (cid:21)e(cid:0)(cid:13)t)(cid:20)exp p 0 + p +(cid:13)+(cid:3)(cid:11);(cid:12)(1)+(cid:14) t :
t
2(cid:27) 2(cid:27) 2(cid:27) 2(cid:27)
ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL 7
Since (cid:14) can be chosen as small as desired, we simply need to prove that pa(cid:21) +(cid:13)+(cid:3)(cid:11);(cid:12)(1)<0. Now,
2(cid:27)
√
√ ( ) (cid:12) p (cid:12)
ab ab b (cid:21) ab (cid:12) (cid:12)
(cid:3)(cid:11);(cid:12)(1)= (cid:0)a b2(cid:0)4(cid:27)(cid:12) = (cid:0)a b2+4(cid:27)(cid:21) p + = (cid:0)a(cid:12)(cid:21) 2(cid:27)+b(cid:12);
2(cid:27) 2(cid:27) 2(cid:27) 2 2(cid:27)
(cid:12) p (cid:12)
which is always well de(cid:12)ned. Therefore, we are left to prove that (cid:12)(cid:21) 2(cid:27)+b(cid:12)> (cid:13) + b + p(cid:21) . This is a
a 2(cid:27) 2(cid:27)
piecewise linear inequality in (cid:21), which is clearly satis(cid:12)ed if and only if
( ) p
(i) (cid:21)2Rn (cid:0)pb (cid:0) (cid:13) ;(cid:0)pb + (cid:13) , when 2(cid:27) >1;
2(cid:27) a(cid:16)+ p2(cid:27) a(cid:16)(cid:0)
(ii) (cid:21)<(cid:0)pb (cid:0) (cid:13) , when 2(cid:27) (cid:20)1.
2(cid:27) a(cid:16)+ ( [ ( ) ] )
In the (cid:12)rst case, the interval is never empty. Let ("1;"2):= exp p(cid:21)V0 + pa(cid:21) +(cid:13)+(cid:3)(cid:11);(cid:12)(1) t ;e(cid:0)(cid:13)t .
2(cid:27) 2(cid:27)
De(cid:12)ne now the probability measure Q via the Radon-Nikodym theorem by Q(B) := E(BZ ), for any
t
B 2 (cid:10). Clearly Q 2 Me(S) and therefore there exists t > 0 such that for any t (cid:21) t , A satis(cid:12)es
t 0 0 t;(cid:21)
P(A )(cid:20)" and Q(A )(cid:21)1(cid:0)" . Proposition 2.1 in [8] implies that S allows for (" ;" )-arbitrage in
t;(cid:21) 1 t;(cid:21) 2 1 2
the sense of De(cid:12)nition 2.1. Arbitrage with decaying failure as in the theorem immediately follows. (cid:3)
3.2. Case t=f(t) tends to in(cid:12)nity as t tends to in(cid:12)nity. Let b > 0, in which case the variance
process is ergodic and its stationary distribution (cid:25) is a Gamma law with shape parameter a=(cid:27) and scale
∫ ∫
parameter (cid:27)=b; namely t(cid:0)1 th(V )ds converges to h(x)(cid:25)(dx) almost surely for any h2L1((cid:25)) (see [4]
0 s R
and [15]). In this section, we consider a continuous function f : R(cid:3) ! R such that t=f(t) tends to
+ +
in(cid:12)nityas t tendsto in(cid:12)nity. Weshallprovebelowthat(under someconditions onthe risk parameter(cid:21))
the ergodicity of the variance ensures that S allows an asymptotic arbitrage with sublinear speed f(t).
Proposition 3.9. ThestockpriceprocessS in (2.1)hasanaveragesquaredmarketpriceofrisk(cid:13) above
1
the threshold a(cid:21)2=b with speed f(t). If furthermore a > (cid:27) and (cid:21)(cid:26)((cid:22)(cid:0)r) (cid:20) 0, then there exists c > 0
2
such that S has an average squared market price of risk (cid:13) above the threshold c with speed f(t).
2 2
Remark 3.10. As the proof shows, we can actually be more precise regarding the threshold c :
2
(cid:15) if (cid:22)=r, then c = a(cid:21)2(cid:26)2 ;
2 b(1(cid:0)(cid:26)2)
(cid:15) if (cid:22)6=r and (cid:26)(cid:21)<0, then no further condition on c is needed;
2
(cid:15) if (cid:22)6=r and (cid:26)(cid:21)=0, then c = ((cid:22)(cid:0)r)2b ;
2 (a(cid:0)(cid:27))(1(cid:0)(cid:26)2)
ItisratherinterestingtocomparethisresultwiththoseofProposition3.4andProposition3.5. Indeed,
when b > 0, if f(t) (cid:17) t then the stock price process does not satisfy an average squared market price of
risk(cid:13) abovethethresholda(cid:21)2=b. However,whent=f(t)tendstoin(cid:12)nity,thenS hasanaveragesquared
1
market price of risk (cid:13) above the threshold a(cid:21)2=b. When b > 0, (cid:21)(cid:26) 6= 0 and (cid:22) = r, if f(t) (cid:17) t then
1
the stock price process does not satisfy an average squared market price of risk (cid:13) above the threshold
2
a(cid:21)4(cid:26)2 ,butdoessoabovethethreshold a(cid:21)2(cid:26)2 whent=f(t)tendstoin(cid:12)nity. Finally,whenb>0,(cid:21)(cid:26)=0
b(1(cid:0)(cid:26)2) b(1(cid:0)(cid:26)2)
and (cid:22) 6= r the stock price process never satis(cid:12)es an average squared market price of risk (cid:13) with speed
2
f(t)(cid:17)t, but does above the threshold b((cid:22)(cid:0)r)2 whenever t=f(t) tends to in(cid:12)nity.
(1(cid:0)(cid:26)2)(a(cid:0)(cid:27))
Proof of Proposition 3.9. Letf beasstatedintheproposition. Forb>0,thevarianceprocessisergodic
and its stationary distribution is a Gamma law with shape parameter a=(cid:27) and scale parameter (cid:27)=b
∫
(see [15]). In particular, t(cid:0)1 tV ds converges in probability to a=b as t tends to in(cid:12)nity, and hence for
0 s
8 FATMAHABAANDANTOINEJACQUIER
any c 2(0;a(cid:21)2=b),
1
( ∫ ) ( ∫ )
1 t 1 t
(3.6) lim P (cid:13)2(s)ds<c =0; and hence lim P (cid:13)2(s)ds<c =0;
t%+1 t 0 1 1 t%+1 f(t) 0 1 1
which proves the (cid:12)rst part of the proposition.
√
Consider now (cid:13) . When (cid:22)=r, the de(cid:12)nitions (2.3) implies that (cid:13) =(cid:0)(cid:26)(cid:13) = 1(cid:0)(cid:26)2, and hence
2 2 1
( ∫ ) ( ∫ )
1 t 1 t (1(cid:0)(cid:26)2)c
lim P (cid:13)2(s)ds<c = lim P (cid:13)2(s)ds< 2
t%+1 f(t) 0 2 2 t%+1 f(t) 0 1 (cid:26)2
( )
is equal to zero if and only if 1(cid:0)(cid:26)2 c =(cid:26)2 2(0;a(cid:21)2=b), and the proposition follows.
2 ∫
We now assume that (cid:22) 6= r. If a > (cid:27) we further know that (see proposition 4 in [1]) t(cid:0)1 tV(cid:0)1ds
0 s
converges in probability to b=(a(cid:0)(cid:27)) as t tends to in(cid:12)nity. Therefore for any c2(0;b=(a(cid:0)(cid:27))) we have
( ∫ ) ( ∫ )
1 t ds 1 t ds
(3.7) lim P <c =0; and hence lim P <c =0:
t%1 t 0 Vs t%+1 f(t) 0 Vs
Letc ;c0;c0 bethreestrictlypositivenumberssuchthatc =c0 +c0. Thede(cid:12)nitionof(cid:13) in(2.3)implies
2 1 2 2 1 2 2
( ∫ ) ( ∫ ∫ )
1 t 1 ((cid:22)(cid:0)r)2 t ds 2(cid:26)(cid:21)((cid:22)(cid:0)r) t 1 (cid:21)2(cid:26)2 t
P (cid:13)2(s)ds<c =P (cid:0) + V ds<c
f(t) 2 2 f(t) 1(cid:0)(cid:26)2 V 1(cid:0)(cid:26)2 f(t) f(t)1(cid:0)(cid:26)2 s 2
0 ( ∫ 0 s ) ( ∫ 0 )
1 (cid:21)2(cid:26)2 t 1 ((cid:22)(cid:0)r)2 t ds 2(cid:26)(cid:21)((cid:22)(cid:0)r) t
(cid:20)P V ds<c0 +P (cid:0) <c0
f(t)1(cid:0)(cid:26)2 s 1 f(t) 1(cid:0)(cid:26)2 V 1(cid:0)(cid:26)2 f(t) 2
( ∫ 0 ) ( ∫ 0 s[ ])
1 t 1 t ds 1(cid:0)(cid:26)2 2(cid:26)(cid:21)((cid:22)(cid:0)r) t
=P (cid:13)2(s)ds<c +P < c0 +
f(t) 1 1 f(t) V ((cid:22)(cid:0)r)2 2 1(cid:0)(cid:26)2 f(t)
0 0 s
with c0 = (cid:26)2 c >0. As long as c 2(0;a(cid:21)2=b), the (cid:12)rst probability tends to zero as t tends to in(cid:12)nity
1 1(cid:0)(cid:26)2 1 1
by (3.6). Now, when (cid:26)(cid:21)((cid:22)(cid:0)r) < 0, then since t=f(t) tends to in(cid:12)nity, the second probability tends to
zero (as t tends to in(cid:12)nity) by (3.7) because c0 + 2(cid:26)(cid:21)((cid:22)(cid:0)r) t tends to (cid:0)1 (and because the variance
2 1(cid:0)(cid:26)2 f(t)
process is non-negative almost surely). No condition on c0 is needed here.
2
When (cid:26)(cid:21)=0, then the (cid:12)rst line of the equation above simpli(cid:12)es to
( ∫ ) ( ∫ )
1 t 1 ((cid:22)(cid:0)r)2 t ds
P (cid:13)2(s)ds<c =P <c :
f(t) 2 2 f(t) 1(cid:0)(cid:26)2 V 2
0 0 s
From (3.7), it tends to zero as t tends to in(cid:12)nity when 0 < c < ((cid:22)(cid:0)r)2b , and hence the proposition
2 (1(cid:0)(cid:26)2)(a(cid:0)(cid:27))
follows from De(cid:12)nition 2.3. (cid:3)
Wenowstateandproveour(cid:12)nalresult, namelyastrongasymptoticarbitragestatementforthestock
price process when the speed is sublinear. However it is not as clear as in Theorem 3.7 how to prove the
exponentially decaying failure probability.
Proposition 3.11. Assume that a>(cid:27) and (cid:21)(cid:26)((cid:22)(cid:0)r)(cid:20)0, and let "2(0;1), (cid:13) >0. Then S satis(cid:12)es a
strong asymptotic arbitrage with speed f(t) and with (" ;" )=(";e(cid:0)(cid:13)f(t)),
[ √ √ 1]2
(cid:15) if and only if (cid:21)2Rn (cid:0) 2b(cid:13)(1(cid:0)(cid:26)2); 2b(cid:13)(1(cid:0)(cid:26)2) when (cid:22)=r and (cid:26)2 (cid:20)1=2;
a(cid:26)2 a(cid:26)2
[ √ √ ]
(cid:15) if and only if (cid:21)2Rn (cid:0) 2b(cid:13)=a; 2b(cid:13)=a when (cid:22)=r and (cid:26)2 (cid:21)1=2;
(cid:15) if (cid:22)6=r and (cid:26)(cid:21)<0;
(cid:15) if (cid:22)6=r, (cid:26)(cid:21)=0 and (cid:13) < ((cid:22)(cid:0)r)2b .
2(a(cid:0)(cid:27))(1(cid:0)(cid:26)2)
ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL 9
Proof. Recall that we are in the framework of Proposition 3.9, so that c > 0 and c > 0 are the
1 2
thresholds for (cid:13) and (cid:13) above which S has an average squared market price of risk. In this proof,
1 2
we follow steps similar to those in [8]. For any " > 0, (cid:12)(x 0 < (cid:13)∫ < (cid:13)(cid:22) < (cid:13)0 < c)1=2 = a2(cid:21)b2 and t0 >
8(cid:13)0=[((cid:13)0 (cid:0)(cid:13) +(cid:13)(cid:22))2"] such that for any t (cid:21) t we have P f(t)(cid:0)1 t(cid:13)2(s)ds(cid:20)2(cid:13)0 < "=4. De(cid:12)ne the
{ ∫ 0 } 0 1 ∫
stopping time (cid:28) :=t^inf s2[0;t]: s(cid:13)2(u)du(cid:21)2(cid:13)0f(t) . Using the fact that (cid:28)1(cid:13)2(s)ds(cid:20)2(cid:13)0f(t),
1 0 1 0 1
Chebychev’s inequality implies
((cid:12)∫ (cid:12) )
P (cid:12)(cid:12)(cid:12) (cid:28)1(cid:13)1(s)dW1(s)(cid:12)(cid:12)(cid:12)(cid:21)((cid:13)0(cid:0)(cid:13)+(cid:13)(cid:22))f(t) (cid:20) ((cid:13)0(cid:0)(cid:13)2+(cid:13)0(cid:13)(cid:22))2f(t) < 4":
0
( ∫ ∫ )
For Z :=exp (cid:0) (cid:28)1(cid:13) (s)dW (s)(cid:0) 1 (cid:28)1(cid:13)2(s)ds , we then obtain
(cid:28)1 0 1 1 2 0 1
( ) ( ∫ )
(cid:28)1 1
P Z (cid:21)e((cid:13)(cid:22)(cid:0)(cid:13))f(t =P (cid:0) (cid:13) (s)dW (s)(cid:0) (cid:13)2(s)ds(cid:21)((cid:13)(cid:22)(cid:0)(cid:13))f(t)
(cid:28)1 1 1 2 1
((cid:12)∫ 0 (cid:12) ) ( ∫ )
(cid:20)P (cid:12)(cid:12)(cid:12) (cid:28)1(cid:13)1(s)dW1(s)(cid:12)(cid:12)(cid:12)(cid:21)((cid:13)(cid:22)(cid:0)(cid:13)+(cid:13)0)f(t) +P 12 (cid:28)1(cid:13)12(s)ds(cid:20)(cid:13)0f(t)
0 0
" " "
(cid:20) + = :
4 4 2
( )
∫
Takenow0<(cid:13)(cid:22) <(cid:13)00 <c =2,andt > 8(cid:13)00 suchthat,fort(cid:21)t ,P f(t)(cid:0)1 t(cid:13)2(s)ds(cid:20)2(cid:13)00 <"=4.
2 1 ((cid:13)00(cid:0)(cid:13)(cid:22){)2" ∫ 1 } 0 2
De(cid:12)ne the stopping time (cid:28) by (cid:28) := t^inf s2[0;t]: s(cid:13)2(u)du(cid:21)2(cid:13)00f(t) and the random variable
( ∫ 2 2 ∫ ) 0 2 ( )
Z := exp (cid:0) (cid:28)2(cid:13) (s)dW (s)(cid:0) 1 (cid:28)2(cid:13)2(s)ds . We then have P Z (cid:21)e(cid:0)(cid:13)(cid:22)f(t) (cid:20) "=2. The two sets
(cid:28)2 { 0 1 } 2 2 0 2{ } (cid:28)2
A := Z (cid:21)e((cid:13)(cid:22)(cid:0)(cid:13))f(t) 2F and B := Z (cid:21)e(cid:0)(cid:13)(cid:22)f(t) 2F . clearly satisfy the following inequalities:
t (cid:28)1 t t (cid:28)2 t
P(A )(cid:20)"=2; Q(Ac)(cid:20)e((cid:13)(cid:22)(cid:0)(cid:13))f(t);
t t
P(B )(cid:20)"=2; Q(Bc)(cid:20)e(cid:0)(cid:13)(cid:22)f(t);
t t
where again we de(cid:12)ne the probability Q(B):=E(BZ ), for any B 2(cid:10). Combining these inequalities, we
t
conclude that there exist t ;t > 0 such that for t (cid:21) t _t , we have P(A [B ) (cid:20) " and Q(Ac\Bc) (cid:20)
0 1 0 1 t t t t
(e(cid:0)(cid:13)f(t). Usi)ng [8, Proposition 2.1], we can now introduce the random varia(ble Yt = (cid:0)e(cid:0)(cid:13)f()t)11At[Bt +
1(cid:0)e(cid:0)(cid:13)f(t) 11Ac\Bc. Clearly Yt 2 Kt and satis(cid:12)es Yt (cid:21) (cid:0)e(cid:0)(cid:13)f(t) and P Yt (cid:21)1(cid:0)e(cid:0)(cid:13)f(t) (cid:21) 1 (cid:0) ".
t t
Letting t(cid:22):=t0_t1, the proposition follows. [ √ √ ]
Note that the constraint a(cid:21)2=b = c > 2(cid:13) reads (cid:21) 2 Rn (cid:0) 2b(cid:13)=a; 2b(cid:13)=a . The constraints on c
1 2
depend on the sign of (cid:21)(cid:26)((cid:22)(cid:0)r), as explained in Remark 3.10:
[ √ √ ]
(cid:15) if(cid:22)=rand(cid:26)2 <1=2,thenc >c ;thenc =2>(cid:13)ifandonlyif(cid:21)2Rn (cid:0) 2b(cid:13)(1(cid:0)(cid:26)2); 2b(cid:13)(1(cid:0)(cid:26)2) ;
1 2 2 a(cid:26)2 a(cid:26)2
[ √ √ ]
(cid:15) if (cid:22)=r and (cid:26)2 >1=2, then c <c ; then c =2>(cid:13) if and only if (cid:21)2Rn (cid:0) 2b(cid:13)=a; 2b(cid:13)=a ;
1 2 1
(cid:15) if (cid:22)6=r and (cid:26)(cid:21)<0, no further assumption on (cid:21) is needed;
(cid:15) if (cid:22)6=r and (cid:26)(cid:21)=0, then the following constraint has to hold: 0<(cid:13) < ((cid:22)(cid:0)r)2b .
2(a(cid:0)(cid:27))(1(cid:0)(cid:26)2)
(cid:3)
10 FATMAHABAANDANTOINEJACQUIER
Appendix A. Large deviations results
( )
Proof of Lemma 3.2. Recall from [16, Chapter 6, Proposition 2.5] that for any t (cid:21) 0, logE eXt(cid:11);(cid:12);0 =
(cid:0)a(cid:30)(cid:0)(cid:11);(cid:0)(cid:12)(t)(cid:0) (cid:0)(cid:11);(cid:0)(cid:12)(t)V0, where
( )
1 2(cid:31)et(b(cid:0)(cid:31))=2
(cid:30) (t):=(cid:0) log ;
(cid:11);(cid:12) (cid:27) 2(cid:27)(cid:11)(1(cid:0)e(cid:0)(cid:31)t)+((cid:31)(cid:0)b)e(cid:0)(cid:31)t+((cid:31)+b)
(cid:11)[((cid:31)+b)e(cid:0)(cid:31)t+((cid:31)(cid:0)b)]+2(cid:12)(1(cid:0)e(cid:0)(cid:31)t)
(t):= ;
(cid:11);(cid:12) 2(cid:27)(cid:11)(1(cid:0)e(cid:0)(cid:31)t)+((cid:31)(cid:0)b)e(cid:0)(cid:31)t+((cid:31)(cid:0)b)
√
with (cid:31) := b(2+4(cid:27)(cid:12). )For any t > 0, the moment(generating] function of Xt(cid:11)[;(cid:12);0=t the)refore reads
(cid:3)(cid:11)t;(cid:12)(u) := E euXt(cid:11);(cid:12);0=t , for u 2 Dt(cid:12) where Dt(cid:12) = (cid:0)1;4b(cid:27)2(cid:12)t if (cid:12) > 0 and 4b(cid:27)2(cid:12)t;+1 if (cid:12) < 0.
Straightforward computations yield
( √ )
a
(cid:3)(cid:12)(u):= lim t(cid:0)1log(cid:3)(cid:11);(cid:12)(ut)= b(cid:0) b2(cid:0)4(cid:27)(cid:12)u ;
t%+1 t 2(cid:27)
as given in (3.3), for u2D(cid:12) :=limt%+1Dt(cid:12), de(cid:12)ned in (3.2). We further have
(cid:12)a 2(cid:12)2(cid:27)a
@ (cid:3)(cid:12)(u)= √ and @ (cid:3)(cid:12)(u)= ;
u b2(cid:0)4(cid:27)(cid:12)u uu (b2(cid:0)4(cid:27)(cid:12)u)3=2
( )
for any u 2 Do, and hence @ (cid:3)(cid:12) Do = R(cid:3) if (cid:12) > 0 and R(cid:3) if (cid:12) < 0. Therefore (cid:3)(cid:12) is convex
(cid:12) u (cid:12) + (cid:0)
on D , and hence the G(cid:127)artner-Ellis theorem (see [6]) applies, albeit only on subsets of @ (cid:3)(cid:12)(Do). We
(cid:12) u (cid:12)
now characterise the rate function (cid:3)(cid:3). Recall that the Fenchel-Legendre transform of (cid:3) is de(cid:12)ned by
{ }
(cid:3)(cid:3)(x):=sup ux(cid:0)(cid:3)(cid:12)(u):u2D . Letus(cid:12)rstconsiderthecase(cid:12) >0. Sincethefunction(cid:3)0 isstrictly
(cid:12) (cid:12)
increasingonDo and@ (cid:3)(cid:12)(Do)=R(cid:3),thenforanyx>0,theequation@ (cid:3)(cid:12)(u)=xhasauniquesolution
( (cid:12) )u ( (cid:12) ) + u
u(cid:3)(x)= b2x2(cid:0)(cid:12)2a2 = 4(cid:27)(cid:12)x2 , and we deduce (cid:3)(cid:3)(x)=u(cid:3)(x)x(cid:0)(cid:3)(cid:12)(u(cid:3)(x))=(bx(cid:0)a(cid:12))2=(4(cid:27)(cid:12)x), for
(cid:12)
any x>0. For x(cid:20)0, the de(cid:12)nition of the Fenchel-Legendre transform implies (cid:3)(cid:3)(x)=+1. In the case
(cid:12)
(cid:12) < 0, an analogous analysis holds: the G(cid:127)artner-Ellis theorem applies on subsets of (cid:3)0(Do) = R(cid:3) with
(cid:12) (cid:0)
rate function (cid:3)(cid:3) given in (3.5) on R(cid:3) and in(cid:12)nity on R . (cid:3)
(cid:12) (cid:0) +
Proof of Lemma 3.3. We (cid:12)rst start with the case b6=0. The moment generating function of the random
variable X(cid:11);(cid:12);(cid:14)=t is given by (see [2, proposition 2]),
t
( ( ∫ ∫ ))
(cid:11)u (cid:12)u t (cid:14)u t
(cid:3) (u)=E exp V + V ds+ V(cid:0)1ds
t t t t s t s
{0 0 ( )}
(cid:0)((cid:20)+(cid:23)=2+1=2) b AV At
= exp (at+V )(cid:0) 0 coth
(cid:0)((cid:23)+1) 2(cid:27) 0 2(cid:27) 2
( ) (( ) )
AV (cid:23)=2+1=2(cid:0)(cid:20) 2(cid:27)(cid:11)u sinh(At=2) (cid:0)(cid:23)=2(cid:0)1=2(cid:0)(cid:20)
(cid:2) 0 b(cid:0) +cosh(At=2)
2(cid:27)sinh(At=2) t A
( )
(cid:23)+1 A2V
(cid:2) F (cid:20)+ ;(cid:23)+1; ( 0 )
1 1 2 2(cid:27)sinh(At=2) (b(cid:0) 2(cid:27)(cid:11)u)sinh(At=2)+cosh(At=2)
t
√ √
where (cid:20) := a , A := b2(cid:0) 4(cid:27)(cid:12)u, (cid:23) := 1 (a(cid:0)(cid:27))2(cid:0) 4(cid:27)(cid:14)u. The con(cid:13)uent hypergeometric function is
2(cid:27) ∑ t (cid:27) t
de(cid:12)nedby F (u;v;z)= u(n)zn,withv(n)denotingtherisingfactorialv(n) :=v(v+1):::(v+n(cid:0)1).
1 1 n(cid:21)0 v(n) n!