Analytic properties of American option prices under a modified Black-Scholes equation with spatial fractional derivatives

0.12 MB·

by  Wenting Chen

Analytic properties of American option prices under a modified
Black–Scholes equation with spatial fractional derivatives
WentingChena, Kai Dub, XinziQiub
aSchoolofBusiness,JiangnanUniversity,Wuxi,JiangsuProvince,214100,China.
bSchoolofMathematicsandAppliedStatistics,UniversityofWollongongNSW2522,Australia.
7 Abstract
1
0 This paper investigates analytic properties of American option prices under the finite
2
moment log-stable (FMLS) model. Under this model the price of American options is

n
characterised by the free boundary problem of a fractional partial differential equation
a
J (FPDE) system. Using the techniqueof approximationwe provethat the American put

6
price under the FMLS model is convex with respect the underlying price, and specify

] theimpactofthetail indexon optionprices.
P
C Keywords: American option,FMLSmodel,fractionalpartial differentialequation,
. convexityofoptionprices, impactoffat tails
n
i
f
-
q
1. Introduction
[

1 The celebrated Black–Scholes (BS) model is based on the assumption that the un-
v
5 derlying asset price followsa geometricBrownian motion[1]. However,it is well doc-
1 umented in the literature that the BS model usually underestimates the probability of
5
the appearance of jumps or large movements of stock prices over small time steps [2];
1
0 forexample,whenanalyzingtheS&P500data,a“leptokurticdistribution”isobserved,
.
1 which has a higher peak and two heavier tails than those of the normal distribution.
0
Manyeffortshavebeenmadetodevelopmathematicalmodelsthatcapableofcapturing
7
1 theleptokurticfeatureobservedinfinancialmarketdata. Afeasibleapproachistoadopt
:
v a Le´vy process extendingBrownian motionfor the descriptionof the price, such as the
i Press model [17], Merton’s jump diffusion model [16], and so on. This idea allows us
X
to model large price changes due to sudden exogenous events or information, and can
r
a
explainsomesystematicempiricalbiaseswithrespect to theBS model.
Among all the Le´vy process models, the finite moment log-stable (FMLS) pro-
cess model, proposed by Carr and Wu [2], can not only successfully capture the high-
frequency empirical probability distribution of the S&P 500 data, but also fit simulta-
neously volatility smirks at different maturities. Most importantly, in contrast to many
other Le´vy process models, the FMLS model ensures the finiteness of all moments of
the underlying index level and the existence of an equivalent martingalemeasure. This
PreprintsubmittedtoElsevier January9,2017
paperis carried out undertheframework oftheFMLSmodel. Theextensionofourap-
proach and results to other Le´vy processes models (e.g., KoBol and CGMY mentioned
in [3])isquitepromising.
Mathematically, the key tool to characterize the non-locality induced by the pure
jumps under the FMLS model is the fractional partial differential equation (FPDE),
whichisasubsetoftheclassofpseudo-differentialequations. InthenewFPDEassoci-
ated with theFMLSmodel, thesecond-orderspatial derivativeinvolvedin thestandard
BS equation is replaced by an α-order spatialderivativewithα ∈ (1,2]. In comparison
to the derivative of integer order, the fractional-order derivative at a point not only in-
volves properties of the function at that particularpoint, but also the information of the
function in a certain subset of the entire domain of definition. This global dependency
of the fractional derivativehas added an additional degree of difficulty when either an-
alyticalmethodsorthenumerical simulationsare attempted.
Under the FMLS model, many techniques have been developed to compute option
values,as summarizedin[18]. Cartea and del-Castillo-Negrete[3] considered thepric-
ing of barrier options under the FMLS model purely numerically, by using a finite dif-
ference method. Recently, Chen et al. [6] derived closed-form analytical solutions for
European-style options under the same model, which is one of the main tools in our
approach. In contrast, pricing American options is more complicated even in the clas-
sical BS model[13, 14, 15, 20], with challengemainlystemmingfrom thenonlinearity
originated from the inherent characteristic that an American option can be exercised at
any time during its lifespan. This additional right casts the American option pricing
problem into a free boundary problem, which is highly nonlinear and far more difficult
to deal with. Several numerical schemefor pricing American optionswere proposed in
[7, 4, 5]undertheFMLSmodel.
The purpose of this paper is to investigate analytic properties of American options
under the FMLS model. This topic has been extensivelystudied in the literature under
thestandardBSmodel. Forexample,Ekstro¨m[10]provedthatthepriceofanAmerican
put is convex with respect the underlying price (see also El Karoui et al. [11] and
Hobson [12] for related results), and convexity of the optimal exerciseboundary for an
American put was obtained by Ekstro¨m [9] and Chen et al. [8]. Those properties can
helpusunderstandtheasymptoticbehavioroftheoptimalexerciseboundarynearexpiry
[8]. The first main result of this paper (see Theorem 4.3 below) is an extension of the
above results, which shows that the American put price under the FMLS model is also
convex with respect the underlying price. Regarding the difference between the FMLS
model and the BS model, our second main result (see Theorem 5.2 below) proves that
thepriceofanAmericanputismonotonicallydecreasing withrespecttotheindexα in
the FMLS model. This phenomenon was first observed in [6, 7] from their numerical
examples, for which we providea rigorous analytic proof in this paper. As the index α
measures the fatness of the tail of the return distribution and the FMLS model reduces
2
totheBS modelwhen α ↑ 2,oursecondresultactuallyrevealsthattheBS modeltends
tounderpriceputoptions,andthepricingbiasbecomeslargerasthetailbecomesfatter,
whichcorresponds tosmallerα values.
This paper is organised as follows. In Section 2 we introduce theFMLS model and
present the mathematical formulation of pricing American options under this model.
In Section 3 we approximate an American put by a sequence of Bermudan options. A
Bermudan option is an American-style option with a restricted set of possible exercise
dates, being a combination of American and European options. The approximation
result provides us with another main tool to prove our main theorems in the next two
sections. In Section 4 we obtain the convexity results for American options under the
FMLS models, and in Section 5 we discussthe influence ofthe tail indexon the option
prices. Concludingremarksare givenin thelast section.
2. Americanoptions under FMLSmodels
An American option is an instrument which gives the owner the right to buy/sell
one asset of a certain stock at a fixed price at any time prior to some pre-determined
expiration time T. In this paper we consider a market consisting of a risk-free asset
with constant return r ≥ 0, and a risky asset with risk-neutral price process (S ) .
t t≥0
UndertheriskneutralmeasureQ,thepriceofanAmericanoptionisgivenasafunction
ofthecurrent timet ≤ T and thecurrent underlyingprice S = S > 0 by
t
V(S,t) = sup EQ e−r(γ−t)g(S ) , (1)
γ
γ∈[t,T]
(cid:2) (cid:3)
where g(·) is the pay-off function and γ is a stopping time with respect to the given
filtration. As the American call on an asset with no dividends usually has the same
value of the corresponding European call [19], in this paper we focus on the American
putforwhich thepay-offfunctionhas theform
g(S) = (K −S)+ = max{K −S,0} (2)
withagivenstrikepriceK > 0.
The FMLS model assumes that, under the risk neutral measure Q, the log value of
the underlying, i.e., x = lnS , follows a stochastic differential equation of the maxi-
t t
mallyskewedLSprocess:
dx = (r −ν)dt+σdLα,−1, (3)
t t
where ν = σαsec απ is aconvexityadjustment[7]. In anotherwords, theFMLSmodel
2
adoptstheLe´vyprocessLα,−1insteadoftheBrownianmotioninthestandardBSmodel.
t
In general, Lα,β denotes the Le´vy-Stable (LS) process with α ∈ (0,2] being the
t
tail index describing the deviation from Brownian motion and β ∈ [−1,1] being the
3
skew parameter. The tail index α is usually restricted to (1,2] such that the underlying
return has the support on the whole real line [2]. In the maximally skewed LS process,
namely β = −1, the random variable x is maximally skewed to the left, meaning that
t
therighttailofthedistributionisfastdecayingsothatexponentialmomentsexit. When
α = 2theFMLSmodelbecomestheBSmodel,whilewhenα < 2thesituationismuch
more complicated since fractional partial differential equations (FPDE) and non-local
operators areinvolved. Market observationsshowthatα isusuallyaround1.4 [2].
With thechange of variables x = lnS, the American put price V as a function of x
and t (we will write V = V(x,t;α) if no confusion occurs) satisfies the free boundary
problem
∂V 1 ∂V 1
+(r + ν) − νDαV −rV = 0 ifx > x(t;α), (4)
∂t 2 ∂x 2 x
where {x = x(t;α)} is the logarithm of the optimal exercise boundary, and Dα is the
x
fractional differentialoperatoroforderα, interpretedin theCaputo sense, namelyfora
C2 functionf,
1 x f′′(y)
Dαf(x) := dy, 1 < α < 2.
x Γ(2−α) (x−y)α−1
Z−∞
Fromtheabovedefinitiononecanobservethatthefractionaldifferentiationisnon-local,
and it involves the option price in the exercise region (−∞,x(t)]. Along with (4), we
have the far-field boundary condition and the terminal condition for the American put
price:
lim V(x,t;α) = 0,
x→∞ (5)
V(x,T;α) = (K −ex)+,
In order to properly close the FPDE system, we impose the following two moving
boundaryconditions:
V(x,t;α) = K −ex ifx = x(t,α)
∂V (6)
(x,t;α) = −ex ifx = x(t,α).
∂x
The above system (4)–(6) is the free boundary problem satisfied by the American
put price V(t,x;α) and the optimal exercise boundary x(t,α) under the FMLS model.
Onecan seethat whenα → 2 itapproaches totheclassicalBS equation.
3. Approximatingan AmericanoptionwithBermuda options
The Bermudan option is an option where the buyer has the right to exercise at a set
(always discretely spaced) numberof times,which is intermediatebetween a European
optionandan American option.
4
Let g = g(·) be a pay-off function (later on, g(S) is chosen to be (K −S)+ with a
strike price K in our cases). With predetermined times 0 = t < t < ··· < t = T
0 1 M
thepriceofaBermudan optionis
B(S,0) := sup EQ[e−rγg(S )], (7)
γ
γ∈{t0,t1,···,tM}
where γ is a stopping time. Using the dynamic programming principle, the family of
prices can becalculated as follows:
(1) ThepriceB(S,t ) at t = T isg(S);
M M
(2) GiventhepriceB(·,t ), for1 ≤ i ≤ M, thepriceat t is
i i−1
B(S,t ) = max EQ [e−r(ti−ti−1)B(S ,t )], g(S) .
i−1 s,ti−1 ti i
(cid:8) (cid:9)
From the above induction, one can see that the Bermudan option is a kind of link
from European options to American options. In fact, the price B(S,t ) of a Bermu-
m−1
dan option at t = t can be calculated inductively as the maximum of g(S) and
m−1
the price of a European option with expiry t and contract function B(S,t ). On the
m m
otherhand, wecan showinthenextlemmathatthepriceofan Americanoptioncanbe
approximatedbya sequenceofBermudan option. Tothisend, welet
A := {0,T2−N,2T2−N,3T2−N,··· ,T},
N
and
B (S,0) = sup EQ[e−rγg(S )]. (8)
N γ
γ∈AN
Lemma 3.1. Assume that g is a non-negative function. As the possible exercise times
of theBermudanoptiongets denser, theBermudanoptionpriceconverges to thecorre-
spondingAmerican optionpriceundertheFMLS models,i.e.
B (S,0) → V(S,0),
N
as N → ∞, whereV(S,0) = sup E[e−rγg(S )] istheAmericanoptionprice.
0≤γ≤T γ
Proof. Under the standard Black-Scholes model, this convergence was proved in [10].
Here,ourprooffollowsfromtherebyusingthedominatedconvergencetheorem. Given
astoppingtimeγ ∈ [0,T],let
γ := inf{τ ≥ γ : τ ∈ A }.
N N
One can see that γ ∈ A for any N, γ → γ almost surely as N → ∞. By the
N N N
dominatedconvergencetheorem,
|EQ[e−rγg(S )]−EQ[e−rγNg(S )]| ≤ EQ|e−rγg(S )−e−rγNg(S )| → 0,
γ γN γ γN
5
as N → ∞. Sinceg ≥ 0, theaboveinequalityimpliesthat
liminf B (S,0) ≥ V(S,0).
N→∞ N
On the other hand, by definitions (7) of Bermudan option prices, B (S,0) ≤ V(S,0)
N
forall N. Therefore, thelemmais proved.
4. Convexityofoptionprices
In this section we aim to prove that the American put price V(S,t) is convex with
respect to the current underlying S under the FMLS model. Our starting point is the
explicit closed-form analytical solution for European options under the FMLS model
obtained by therecent work [6]. With the pay-offfunction g(·) the price of a European
optionisgivenby
V (S,t) = EQ[e−r(T−t)g(S )].
E T
Define
τ = ν(T −t),
whereforsimplifyingthecomputation,weassignν = −1σαsec απ withanegativesign
2 2
to that of the convexity adjustment in (3). This transformation changes the backward
problem into a forward problem. Also define γ = r/ν as the relative interest rate of
thevolatilitywith fractional order α to therisk-free interest rate r. Onehas theexplicit
analyticalexpression
∞
1
V (x,τ;α) = e−γτΠ(x−(1−γ)τ −ταm)f (|m|)dm, (9)
E α,0
Z−∞
wherex = lnS and
Π(x) = (K −ex)+ (10)
isthepay-offfunctioncorrespondingto(2),andf istheLe´vystabledensitygivenby
α,0
|z| 1 |z| (1− 1, 1) (1, 1)
f = H1,1 α α 2 2 ,
α,0 τ1/α α 2,2 τ1/α (0,1) (1, 1)
(cid:18) (cid:19) (cid:20) (cid:12) 2 2 (cid:21)
(cid:12)
whereH(x) istheFox function[6],whosegen(cid:12)eral formsare defined by
(cid:12)
(a ,A ) (a ,A ) ··· (a ,A )
Hm,n z 1 1 2 2 p p
p,q (b ,B ) (b ,B ) ··· (b ,B )
1 1 2 2 q q
(cid:20) (cid:12) (cid:21) (11)
= 1 (cid:12)(cid:12)(cid:12) Πmj=1Γ(bj +Bjs) Πnj=1Γ(1−aj −Ajs) z−sds
2πi Πq Γ(1−b −B s) Πp Γ(a +A s)
ZL (cid:0)j=m+1 j (cid:1)j(cid:0) j=n+1 j j(cid:1)
where L is a certain(cid:0)contour separating the po(cid:1)l(cid:0)es of the two factors(cid:1)in the numerator.
FordetailedcomputationviaFouriertransformandtechniqueissueswereferthereader
6
to[6]. Butwewouldliketopointoutthatthedependencyofxintheintegrandof(9)is
onlyon thepay-offfunctionΠ.
Once having the above closed form analytical solutions for European options, we
can derivetheirconvexitieswithrespect to variousparameters. Then viatheBermudan
option,theconvexitycan pass overtoAmerican optionscorrespondingly.
Lemma 4.1. With the pay-off function g given in (2), the European option price V is
E
convex inthecurrentunderlyingS.
Proof. Recalling that S = ex, the function Π(x) = g(S) = (K − S)+ is convex
with respect to S, where g(·) is defined in (2). In the integrand of the above analytical
expression (9) for V under the FMLS model, the only part involvingthe underlying S
E
is
1
Π(x−(1−β)τ −ταm) = (K −C(m)S)+,
1
where C(m) = e−(1−β)τ−ταm is a positivefactor for all m ∈ R. Therefore, by integra-
tionV is convexin S.
E
Next, let’s pass the above convexity over to the Bermudan option price B(s,t) de-
fined in (7). Since the Bermudan option price is inductively constructed by the maxi-
mumofEuropeanoptionprices, thefollowinglemmais immediate.
Lemma 4.2. With the pay-off function given in (2), the Bermudan option price B(S,t)
isconvex intheunderlyingS foranyfixed t.
Proof. It is well known in convex analysis that the supremum of a family of convex
functions is still convex. Under the FMLS model, the Bermudan option price B(S,t)
in (7) is inductively constructed by the maximum of European option prices V . From
E
Lemma4.1weknowthatV isconvexinS. Henceforanyfixedttakingthesupremum
E
itis readilyto seethatB(S,t) is convexwithrespect toS, asinherited fromV .
E
We arenowin apositiontoprovethefirstmain resultinthispaper.
Theorem 4.3. TheAmericanoptionpriceV(S,t)givenby(1)withthepay-offfunction
g(S) = (K −S)+ isconvex intheunderlyingS foreach t.
Proof. ToprovetheconvexityofV = V(S,t)withrespecttoS,onecanfixatimetwith
t ≤ T. Without loss of generality we may regard t = 0, otherwise make a change τ →
τ − t. From Lemma 3.1 we know the approximating sequence {B (S,0)} converges
N
to V(S,0) as N → ∞. And from Lemma 4.2, for each N the Bermudan option price
B (S,0)isconvexinS. Sincethepoint-wiselimitofaconvergentsequenceofconvex
N
functions is again convex, we have the limit V(S,0) is convex in S. Therefore, the
theoremis proved.
7
5. The impactoffattailsonoptionprices
OneofthemajoradvantagesoftheFMLSmodeloverthestandardBSmodelisthat
itcapturesthefattailfeature(leptokurtosis)observedinrealworldmarkets[2]. Indeed,
comparedtotheGaussiandensityoftheunderlyingpricesundertheBSmodel,theLe´vy
density increases the probability of the stock price exhibiting large moments or jumps
oversmalltimesteps,andhasfattertailsatbothendsthanthelognormaldistributionof
theBSmodel.
How the fat tails impact on European option prices under the FMLS model has
beenillustratedin[6]bytheirnumericalexperiments,butwithoutrigorousproof. They
observed that once the tail index α increases up to 2, the option prices are gradually
decreasingtotheBSprice. Inotherwords,theBSformulatendstounderpriceEuropean
puts with underlying following a Le´vy process. Moreover, the pricing bias of the BS
formula gets larger as α becomes smaller. They also gave a plausibleexplanation from
afinancial pointofviewforthisphenomenon.
In the following we give a rigorous analytical proof for the above observation for
European optionsand alsoforAmerican options.
Proposition 5.1. As α approaches to 2, the price of a European put V computed in
E
(9) decreasingly converges to its price determined under the standard BS model in a
sufficientlylargeunderlyingregionx ≥ x .
0
Proof. Recall the analytical expression for the European put option price in (9), or
equivalently
V (x,τ;α) = Ke−γτ ∞f (|m|)dm−ex ∞e−τ−τα1mf (|m|)dm, (12)
E α,0 α,0
Zd1 Zd1
where d = x−lnK−(1−γ)τ. It suffices to show that ∂VE < 0, as α → 2 and x ≥ x
1 τα1 ∂α 0
sufficientlylarge. Since(12) isquiteinvolved,weneed to computederivativesforeach
functionofα separately.
Adoptingthenotationsfrom[6],
1 απ
ν = − σαsec
2 2
istheconvexityadjustment. By differentiating,
∂ν 1 απ ∂σα 1 ∂sec απ
= − sec − σα 2 ,
∂α 2 2 ∂α 2 ∂α
(cid:18) (cid:19)
and
∂σα ∂eαlnσ α∂σ
= = σα lnσ + .
∂α ∂α σ∂α
(cid:18) (cid:19)
8
From the normalisationof volatilitiesto σ = 0.25, one can see that ∂σ/∂α → 0 and
BS
lnσ ∼ −1.38, as α → 2. Hence, one has ∂σα/∂α < 0, and furthermore as α → 2,
sec απ → −1,
2
∂ν π sin απ
< − σα 2 < 0. (13)
∂α 4 cos2 απ
(cid:18) 2 (cid:19)
Therelativeinterestrateγ and thebackward timeτ are defined by
−2r r
γ = = ,
σαsec(απ/2) ν
and
1 απ
τ = − σα(sec )(T −t) = ν(T −t),
2 2
respectively, where r is the risk free interest. It is readily to see that γτ = r(T −t) is
independentofα.
TheLe´vy densitysatisfies theinversepower-lawasymptoticallyat large underlying
values,thusforsufficientlylargex,
1 ∂f
f (x) ∼ , and α,0 < 0. (14)
α,0 |x|1+α ∂α
Nextwecomputethepartialderivativeofd inα. Notethatγτ isindependentofα,
1
wehave
1
∂d 1 ∂τ ∂τ−α
1
= (x−lnK −(1−γ)τ) − +
∂α "τα1 ∂α ∂α #
(cid:18) (cid:19)
=: (x−lnK −(1−γ)τ)(I +I ),
1 2
wherethecoefficient(x−lnK−(1−γ)τ) > 0whenxissufficientlylarge. From(13),
1 ∂τ 1 ∂ν
I = − = τ−α(T −t) − > 0;
1 τα1 ∂α ∂α
(cid:18) (cid:19) (cid:18) (cid:19)
and noting that as τ → 0, τ lnτ → 0, which means the speed of 1 goes to infinity is
τ
muchfaster thanthatof|lnτ|, thus
1
∂τ−α 1 lnτ 1 ∂τ
I = = τ−α + (− ) > 0.
2 ∂α α2 ατ ∂α
(cid:18) (cid:19)
Therefore, weobtain
∂d
1 > 0. (15)
∂α
9
Now wecan computepartialderivativesoftheintegralsin (12).
∂V ∞ ∂f ∂d 1 ∂d
E =Ke−γτ α,0dm−f (d ) 1 +ex−τ−ταd1f (d ) 1
α,0 1 α,0 1
∂α ∂α ∂α ∂α
(cid:20)Zd1 (cid:21)
∞ 1 ∂f ∂τ ∂τα1
−ex e−τ−ταm α,0 +f (− − m) dm
α,0
∂α ∂α ∂α
"Zd1 ! #
∞ ∂f ∞ 1 ∂f
= Ke−γτ α,0dm−ex e−τ−ταm α,0dm
∂α ∂α
(cid:20) Zd1 Zd1 (cid:21)
∂d 1
+ f (d ) 1 ex−τ−ταd1 −Ke−γτ
α,0 1
∂α
(cid:20) (cid:21)
(cid:16) (cid:17)
∞ 1 ∂τ ∂τα1
−ex f e−τ−ταm − − m dm
α,0
∂α ∂α
Zd1 !
=:Q +Q +Q .
1 2 3
Observing that K ≥ ex, and γτ = r(T − t) → 0 as t → T. Hence, from (13)–(15),
1
we have Q < 0 and Q < 0. Last, since ∂τ < 0 and ∂τα < 0, it is obvious that
1 2 ∂α ∂α
Q < 0 as well. Finally, we prove that ∂VE < 0, namely as α → 2, V decreasingly
3 ∂α E
approachestotheEuropeanputoptionpriceunderthestandardBlack-Scholesmodelfor
largeunderlyingx, and thusverifythemonotonepropertyas observedinthenumerical
examplesin[6].
With the help of the above result and Bermudan options we can prove the second
maintheorem inthispaperas follows.
Theorem5.2. ForsufficientlylargeunderlyingthepriceofanAmericanputdetermined
undertheFMLS modeldecreasinglyconverges toitsBlack–Scholespriceas α ↑ 2.
Proof. The proof is via an approximation to the American option price by a sequence
ofBermudanoptionprice(8). BytheinductivedefinitionofBermudanoption,theprice
B (S,t ;α) is the maximumof g(S) the the price of a European option V (S,t;α)
N m−1 E
with expiry t and contract function B (S,t ;α). And by the proof of Lemma 3.1,
m N m
B (S,0;α) → V(S,0;α)as N → ∞.
N
Since V (S,t;α) → V (S,t) decreasingly as α → 2 for large S, B (S,t;α) →
E BS N
B (S,t;2) decreasingly for large S, where B (S,t;2) is the Bermudan option corre-
N N
sponding to the standard BS model. Then we take N → ∞ to make the Bermudan
option denser, and t → T to obtain V(S,0;α) → V(S,0;2) decreasingly as α → 2,
for large S. Note that V(S,t;2) = V(S,t) is the American put option price under the
standard BSmodel. Therefore, thetheorem isproved.
10