Using some results from risk theory on stop-loss order and comonotone risks, we show in this paper that the price of an arithmetic Asian option can be bounded from above by the price of a portfolio of European call options.

Dhaene, Jan

Goovaerts, Marc

Simon, S

English

Research Papers in Economics

KATHOLIEKE 
UNIVERSITEIT 
LEUVEN 
DEPARTEMENT TOEGEPASTE 
ECONOMISCHE WETENSCHAPPEN 
RESEARCH  REPORT 9950 
AN EASY COMPUTABLE UPPER  BOUND 
FOR THE  PRICE  OF AN ARITHMETIC ASIAN 
OPTION 
by 
S.  SIMON 
M. GOOVAERTS 
J. OHAENE 
0/1999/2376/50 An Easy Computable Upper Bound for  the 
Price of an Arithmetic Asian Option 
S.  Simon*  M.J.  Goovaertst  J.  Dhaene+ 
November 1999 
Abstract 
Using some results from risk theory on stop-loss order and comonotone 
risks, we show in this paper that the price of an arithmetic Asian op-
tion can be bounded from above by the price of a portfolio of European 
call options. 
Keywords:  Asian options, stop loss order, comonotonicity 
1  Introduction 
We consider a securities market consisting of one risky asset S(t) and a risk-
less money-market account in which money can be invested at a fixed spot-
rate r.  The risky asset S(t) is assumed to be defined on a filtered probability 
space (n, Ft , P)  with Ft  the filtration generated by S(t).  Furthermore, we 
assume there exists a single equivalent martingale measure Q, i.e.  that we are 
dealing with a  complete and arbitrage free market, see Harrison and Kreps 
(1979) and Harrison and Pliska (1981).  An arithmetic Asian call option with 
exercise date T  , n  averaging dates and exercise price K  generates a  pay-off 
[~  ~~':Ol S(T - i) - K] + at T,  and will as such trade at t  against a  price 
given by: 
AA(t, S(t), n, K, T, r)  =  e-(T-t)r  EQ ([  ~  ~  S(T -i) - K LIFt )  (1) 
*K.U.Leuven 
tK.U.Leuven, U.v.  Amsterdam 
+K.U.Leuven, U.v.  Amsterdam, Universiteit Antwerpen 
1 However,  in most  cases  this pricing formula  is  hard to evaluate.  For 
instance, the distribution of  L:~==-J S(T - i)  is  not known when the price 
process S(t) is an exponential Brownian motion.  One can use Monte-Carlo 
simulation techniques to obtain a numerical estimate of the price, see Kernna 
and Vorst (1990) and F.J. Vazquez-Abad (1998), or one can numerically solve 
a  parabolic partial differential equation,  see  Rogers  (1995).  But as both 
approaches are rather time consuming, it would be more than interesting to 
have an accurate, easily computable approximation of this price. 
A very accurate lower bound was obtained in Rogers (1995).  In Jacques 
(1996)  an approximation is  obtained by approximating the distribution of 
L:~==-J S(T - i)  by a more tractable one.  We will follow this last approach, 
using results from  actuarial risk theory on comonotone risks to obtain an 
accurate upper bound for the price of the Asian option. 
2  Options and Stop-loss Transforms 
In actuarial science one often compares risks, i.e.  nonnegative random vari-
ables,  by means of their stop-loss premiums/stop-loss transforms.  We will 
use stop-loss transforms of distribution functions that are concentrated on 
the positive half line.  For the sake of completeness, we give the following 
definition. 
Definition 1  For a distribution function F(x) with a support D  ~  jR+,  the 
stop-loss transform W  F (r)  is given by: 
wF:jR+f-+jR+:r--+WF(r)=  (  (x-r)dF(x)  (2) 
J[r,+oo[nD 
Now,  we  can define  the stop-loss order for  distribution functions,  and 
therefore as well of random variables, as: 
Definition 2  Of distribution functions F(x) and G(x), both with their sup-
port in jR+, F(x) is said to precede G(x) in stop-loss order,  written F  ~81 G, 
if: 
(3) 
2 If we  combine equalities 1  and 2,  we  see that the pay-off of an arith-
metic Asian option can be written using the stop-loss transform of Wn(T) = 
2:~~J S(T - i). First we can rewrite equation 1 as: 
-(T-t)r  [n-l  1 
AA(t,S(t),n, K, T,r) =  e  n  E~  ~S(T  - i) - nK 1Ft 
,~  + 
(4) 
For a given value s of S(t), we have immediately: 
e-(T-t)r 
AA(t,s,n,K,T,r) =  Wp:B  (nK)  n  Wn(T) 
With:  F~n(T)(x) = Q  (Wn(T)  :::;  x IS(t) = s) 
The problem of pricing an arithmetic Asian option therefore turns out to 
be equivalent to calculating the stop-loss transform of a sum of dependent 
risks.  And as such we  can apply results on bounds for stop-loss transforms 
to the option pricing problem. 
3  Bounds for Stop-loss Thansforms 
In this section we will discuss some results from actuarial science on bounds 
for  stop-loss transforms of sums of dependent stochastic variables.  Let us 
first return to the pricing of an arithmetic Asian option. As explained in the 
introduction, the main problem we are confronted with in pricing this type 
of options, is that we,  in general, do not know the distribution of the sum 
2:~~  S(T - i). However, we do know the distribution of every term in this 
sum, i.e.  of every S(T - i).  The stop-loss bounds that we  will introduce 
in this section will  be based on these marginal distributions.  Let us first 
introduce the concept of a Frechet class. 
Definition 3  The  F'rechet  class  Rn (Fl, ... , Fn) determined  by  n  (monovari-
ate)  distribution functions,  Fl, ... Fn'  is the  class  of all n-variate distribution 
functions F  with FI, ... Fn  as  marginal distributions. 
Notation: 
By (Xl, ... ,Xn)  rv Rn(FI , ... , Fn ), we mean that the marginal distributions 
of this random vector are given by FI to Fn, i.e.:  Fx E Rn (FI' ... , Fn) 
3 We will now  introduce the concept of comonotone risks, which we  will 
need to construct the upper bound for  an Asian option.  For more results 
on comonotone risks in the actuarial field,  we  refer the interested reader to 
Dhaene et al.(1997)  and Wang and J.Dhaene (1997).  We use the following 
definition from Dhaene et al.  (1997): 
Definition 4  A positive random vector (XI, ... , Xn)  is said to be comonotone 
(the positive random variables XI, ... , Xn  are said to be mutually comonotone) 
if any of the following equivalent conditions hold: 
1.  For the n-variate distribution function we  have: 
2.  There exist a random variable Z  and non-decreasing functions gl, ... , gn 
on lR,  such that: 
3.  For any random variable U uniformly distributed on [0,1],  we have: 
From the first condition in the above definition it is clear that for a given 
Ftechet class R,. (Fl' ... , Fn)  the distribution of a comonotone random vector 
(XI, ... ,Xn) '" R,. (Fl' ... , Fn)  is uniquely defined. 
Remark 1  For a given  Fh~chet class R,. (Fl' ... , Fn),  the n-variate distribu-
tion in the right-hand side of equation 5 is called the upper Prechet bound of 
R,. (Flo""  Fn),  we  write: 
Definition 5  Let a Prechet class R,.(Fl' ... , Fn)  be given and let (Xl, ... , Xn)  '" 
R,. (Flo ... ,F n )  be a given comonotone vector.  We define the distributionfunc-
tion FR  as: 
(6) 
4 Note that as the distribution of the comonotone vector (Xl, ... ,Xn) is 
uniquely defined, equality 6 uniquely defines the distribution FR(X). 
We have the following result, a prove can be found in Dennenberg (1994): 
Proposition 1  For any given Prechet class Rn (Fl, ... , Fn) , we  have: 
n 
FRl(X) = LF;-l(x)  (7) 
;=1 
Goovaerts and Dhaene (1999) prove a more general version of the follow-
ing result on Fh§chet classes: 
Proposition 2  For a given Prechet class Rn(Fl, ... , Fn)  we have: 
n 
'\Ir  E ~+  WFR(r)  = L  WFi  [F;-l (FR (r))]  (8) 
;=1 
Note that we  only have to calculate FR  for  one value r,  and that this 
can be done using expression 7.  Combining these two results, we obtain an 
upper bound for the stop-loss transform of any sum I:~=l X;,  when only the 
marginal distributions are known. 
Proposition 3  Let  a Prechet  class  Rn(Fl, ... , Fn)  be  given.  For  all  non-
negative random vectors (XI, ... , Xn)  rv Rn(Fl' ... , Fn)  we have: 
(9) 
(10) 
With: 
(11) 
Proof: 
Because of formula 8 the right-hand side of inequality 10 is indeed equal 
to WFR(d).  And from formula 11  and formula 7 we have that  I:~=l d;  = d, 
and therefore inequality 10 holds. 
•  Moreover, the choice of the retention levels as in formula 11 turns out to 
be optimal, as stated in the following theorem. 
5 Theorem 1  Given  a Prechet  class Rn(H, ... ,  Fn)  and  a non-negative ran-
dom vector (Xl, ... , Xn)  rv Rn (FI, ... , Fn)  we have that for any retention level 
d  E  ffi.+  and for  any  choice  of the retention levels dl , ..• , dn  E  ffi.+  such that 
I:~=l di  =  d we  have: 
n  n 
"E[K  -d~] <"  E[K -d·]  L-t  'l,  'l,+-L..J  ~  '1,+  (12) 
i=l  i=l 
with: 
Proof: 
For dl , ... , dn  E ffi.+  such that I:~=l di  =  d we have: 
(13) 
However, under the assumption of comonotonicity, which leaves the right-
hand side unchanged, the left hand side is given by: 
n 
2: E [Xi - d;]+ 
i=l 
Which proves inequality 12. 
• 
4  Application to Asian Options, the General 
Case 
Here we will apply the results of the previous section to the option pricing 
formula 1 for  the price AA(t, s, n, K, T, r) at time t  of an arithmetic Asian 
option with exercise date T, exercise price K  and n  averaging dates when 
S(t) = s, with the securities market as described above.  By F(X2, t2, Xl, tl ) 
we  will  denote the conditional distribution of S(t2)  under the equivalent 
martingale measure Q,  i.e.: 
6 Let us first assume that at time t the averaging has not yet started.  In 
this case the n variables S(T -n+  1), ... , S(T) are still unknown, i.e.  random 
and the n  distribution functions F(x, T  - n +  1, s, t), ... , F(x, T, s, t)  deter-
mine a Frechet class Rn (F(x, T - n +  1, s, t), ... , F(x, T, s, t)).  As discussed 
in section 2 we have: 
e-(T-t)r 
AA(t,s,n,K,T,r) =  WFw  (nK)  n  n 
By proposition 3 we obtain: 
e-(T-t)r 
AA(t,S(t),n,K,T,r)::;  wFR(nK) 
n 
With  F  R  being  defined  with  respect  to  the  Frechet  class 
Rn (F(x, T - n + 1, s, t), ... , F(x, T, s, t))  along definition 5.  By proposition 
3 we obtain the following inequality: 
e-(T-t)r n-l 
AA(t, n, K, T, r)  ::;  n  L W  F(.,T-i,8,t) [F-1 (FR (K) ,T - i, s, t)] 
i=O 
Until now, we assumed that t < T - n +  1.  We will now turn to the case 
that t  ?:  T - n + 1.  Let i*  be such that:  T - i*  ::;  t  < T  - i* + 1, then we 
know the first n - i*  prices and only the last i*  prices:  S(T - i*), ... , S(T), 
remain random. Therefore we obtain: 
AA(t, n, K, T, r) 
e-(:-t)r EQ  [~S(T  - i) - (nK - ~  S(T - i)) 1Ft 1  + 
c':-'"  EQ  [~S(T  - i) - K,. IF,l + 
With:  Kj  = nK - '£~~/ S(T - i) for j  < n, and Kn = nK. 
First, let us assume that Ki •  > O.  Under this assumption we can apply 
the same method to obtain upper bounds as in the case t < T - n + 1.  But 
7 we are now working in the Frechet class:  R;,. (F T - i., ...  , FT )  and that K has 
been replaced by the adjusted exercise price Ki•.  As such we obtain: 
AA(t, s, n, K, T, r) 
e-(T-t)r i'-l 
<  n  LWF(-,T-i,8,t) [F-1(FR(Ki.),T-i,s,t)] 
i=O 
If  we define J),i  as: 
K,i  = F-1 (FR (Ki.) ,T - i, s, t) , i = 0, ... i*. 
Then we obtain: 
e-(T-t)r i*-l 
AA(t, s,n, K,T,r) :::;  L  WF(o,T-i,8,t)  (K,i) 
n 
(14) 
We will now look at what happens when Ki •  is equal to or smaller then 
zero.  In this case,  we  can not use the bounds from the previous section. 
However we have: 
EQ [Wi'  - Ki.]+ 
EQ [Wi']  - Ki• 
i*-l 
L EQS(T - i) - Ki* 
i=O 
As the discounted price process e-trS(t)  is  a  martingale under Q,  the 
above expression is equal to: 
i*-l 
S(t) L  e(T-i-t)r - Ki• 
i=O 
If  we combine this result with the option pricing formula 1, we obtain, in 
case of a negative Ki.: 
AA(t, s, n, K, T, r)  e-(:-t)r (S(t) ~  e(T-i-t)r - Ki*) 
;  [S(t) ~  e-ir + e-(T-t)r ~  S(T - i)]- e-(T-t)r K 
8 If  we denote with EC (t, T, K, r) the price of a European call option with 
exercise date T  and exercise price K, we obtain the following result: 
Theorem 2  In a securities  market as  described  above,  the following  result 
holds for AA(t, s, n, K, T, r) 
1.  If K i •  > 0: 
AA(t, s,n, K, T,r) 
<  W  F(.,T-i,.,t) (h;i) 
1 i'-1  .  -;;: 2:= e-tT EC (t, s, h;i, T - i, r) 
i=O 
(15) 
Where:  h;i = F-1 (FR (Ki.) , T - i, s, t), i = 0, ... i*.  With  the  exercise 
prices ni,  i = 0, ... i*  being  optimal in the sense of theorem  1. 
2.  If Ki •  ::::;  0: 
AA(t, s, n, K, T, r) 
;  [S(t) ~  e-iT + e-(T-t)T ~  S(T - i)]- e-(T-t)T K  (16) 
As such,  we  have shown  (for  K i•  > 0)  that the price of an arithmetic 
Asian option is bounded from above by the price of a portfolio of European 
call options.  And we have found the exercise prices for the European options 
that optimize this bound. 
5  Application in a Black and Scholes Setting 
Here we  assume that the price process of the underlying asset S(t) follows 
an exponential Brownian motion with constant coefficients, i.e.: 
dS(t) = S(t) [f.£dt + (TdB(t)]. 
Which enables us to use the well known Black and Scholes pricing formula 
for  European call options.  Combining the Black and Scholes formula with 
formula 15 yields: 
9 i*-l 
AA(t,s,n,K,T,r)  <  ~  2::,>-irEC(t,s,r;l(Ki*,s),T-i,r) 
n  i=O 
i*-l 
~  L e-ir  [sN (di,l) - "'i e-r(T-i-t)N (di,2)] 
n  i=O 
where dl  and d2  are given by: 
d,  _log(sj"'i)+(r+0'2j2)(T-i-t) 
.,1  - O'VT - i - t 
and: 
d, 2 = d'l - O'VT - i - t.  '2.,  '2., 
As such, the main difficulty is to calculate the different strike prices "'i. 
The distribution function F(X2, t2, xl, t l )  =  Q (S(t2)  ~  x2IS(tl) =  Xl) is now 
given by: 
Where LN(x; j.t, 0')  is the lognormal distribution with mean j.t  and stan-
dard deviation 0'.  Using formula 7 we can rewrite equation 14 as: 
i*-l 
Ki * = L F-l  (F ("'i' T - i, S, t) ,T - k, S, t). 
k=O 
It is easily verified that this is equivalent to: 
'*  1 
~  s(l-li.=~~n  J~=~~:  ~(V(T-k-t)(T-i-t)-(T-k-t») - K 
L.-;  ~i  e  - i*· 
k=O 
If  we define O!k(i)  and f3k(i, s) as: 
O!k(i)  = e~(  V(T-k-t)(T-i-t)-(T-k-t»)  and f3k(i)  = ,IT  - ~ - t, 
V  T-z-t 
then we can rewrite equation 17 as: 
10 
(17) (18) 
where: 
i"'-l 
fi(x, S) = L ak(i) Sl-i3k(i)~k(i).  (19) 
k=O 
6  Numerical Example and testing 
In this section we  will give  a  numerical example of the Black and Scholes 
case.  We will compare our results with those of Jacques (1996)  where the 
distribution of the arithmetic Asian option was approximated by means of 
a lognormal (LN) and the inverse Gaussian (IG) distribution. Therefore the 
parameters that were used to generate the results given in table 1 were given 
the same values as in Jacques (1996)  : an initial stock price 8(0) =  100, an 
annual interest rate of 9%, i.e.  the daily rate r = In(1+0.09)/365, a maturity 
of 120 days, an averaging period of 30  days and three values:  0.2,  0.3 and 
0.4 for the volatility b. 
6.1  Evaluating the Upper Bound 
Here we will look how sharp the upper bound is.  Table 1 gives the values 
obtained by  Jacques and compares them with our results.  The last two 
columns give  the relative difference  between the price obtained by means 
of the bound and the lognormal  and the inverse  Gaussian approximation 
respectively. 
We see that the prices obtained by means of the comonotone approxi-
mation are relatively close to the values obtained by either the lognormal as 
well as the inverse Gaussian approximation. Furthermore, the results suggest 
that the upper bound is  sharper for  options for which the exercise price K 
is smaller than the initial stock price.  The effect of the volatility level is less 
clear as this seems to be intertwined with the effect of the exercise price. 
11 TABLE 1 
Comparing copula results with LN and IG 
b  K  LN  IG  Upper Bound  Upper Bound vs LN  Upper Bound vs IG 
0.2  90  12.68  12.68  12.72  0.32% 
100  5.46  5.46  5.56  1.83% 
110  1.63  1.63  1.71  4.91% 
0.3  90  13.85  13.85  13.95  0.72% 
100  7.48  7.49  7.62  1.87% 
110  3.48  3.49  3.62  4.02% 
0.4  90  15.36  15.37  15.51  0.98% 
100  9.51  9.53  9.70  2.00% 
110  5.48  5.49  5.67  3.47% 
In table 2 we compare the upper bound with a price obtained by gener-
ating 10000 paths.  This was done for three different options:  the first with 
a maturity of 120  days and 30  averaging dates, the second with a maturity 
of 60  days and 30  averaging dates and the third one with again a maturity 
of 120 days but only 10  averaging dates.  In each case we  considered the 5 
following exercise prices:  80, 90,  100, 110 and 120. 
TABLE 2 
Comparing the upper bound with the simulated price, M  =  120, N =  30 
K  Bound  Simulation  Relative Error 
80  21.9269  21.9315  0.02% 
90  12.7207  12.6907  0.24% 
100  5.5563  5.4971  1.08% 
110  1.7077  1.5839  7.82% 
120  0.3675  0.3155  16.48% 
12 
0.32% 
1.83% 
4.91% 
0.72% 
1.74% 
3.72% 
0.91% 
1.78% 
3.82% TABLE 3 
Comparing the upper bound with the simulated price, M  = 60, N = 30 
K  Bound  Estimate  Relative Error 
80  20.7845  20.7334  0.25% 
90  11.0601  10.9259  1.23% 
100  3.3452  3.1357  6.68% 
110  0.4084  0.3487  17.12% 
120  0.0185  0.0066  178.58% 
TABLE 4 
Comparing the upper bound with the simulated price, M  = 120, N = 10 
K  Bound  Estimate  Relative Error 
80  22.1735  22.1013  0.33% 
90  13.0233  13.0363  0.10% 
100  5.8934  5.8885  0.08% 
110  1.9442  1.8853  3.12% 
120  0.4666  0.4292  8.72% 
Again, we see that the upper bound is close to the estimated true price 
for  options that are in  the money.  The price  estimate even exceeds  the 
upper bound for  K  =  90  in two  cases:  for  M  =  120, N  =  10  and  for 
M  =  120, N  =  30.  However,  for  options that are out of the money,  the 
upper bound becomes less accurate. 
Note that the upper bound performs best for  the option with M  =  120 
and N  = 10  and worst when M = 60  and N  = 30.  This is probably due to 
the fact that the upper Frechet bound is a better model for  the dependency 
between the averaging values S(T - i), for small values of N and large values 
ofT. 
7  Conclusion 
Using some results from risk theory on comonotone risks and stop-loss order, 
we  were able to show that the price of an arithmetic Asian option can be 
bounded from above by the price of a portfolio of European call options. The 
13 upper bound appears to be rather sharp if compared to simulated prices. 
Furthermore,  we  showed that the exercise prices we  derived optimize this 
portfolio of European options. 
References 
D. DENNENBERG, 'Non-Additive Measure and Integral', Kluwer Academic 
Publishers, Boston, 1994 
J.  DHAENE, S.  WANG,  V.  YOUNG,  M.  GOOVAERTS,  Comonotonicity 
and Maximal Stop-loss Premiums, submitted 
M.J. GOOVAERTS and J. DHAENE, Supermodular Ordering and Stochas-
tic Annuities,  accepted for  publication in Insurance  Mathematics  and Eco-
nomics 
M.J. GOOVAERTS, J. DHAENE and A. DE SCHEPPER, Stochastic Bounds 
for  Present Value Functions, Research Report 9914,  Department of Applied 
Economics K.U.Leuven 
J. HARRISON and D.  KREPS, Martingales and Arbitrage in Multiperiod 
Securities Markets,  Journal of Economic Theory 20 (1979), 381-408 
J. HARRISON and R.PLISKA, Martingales and Stochastic Integrals in the 
Theory of Continuous Trading,  Stochastic  Processes  and their Applications 
11 (1981), 215-260 
M.  JACQUES, On the Hedging Portfolio of Asian Options, ASTIN Bulletin 
26 (1996), 165-183 
A.G.Z. KEMNA and A.C.F. VORST, A Pricing Method for Options Based 
on Average Asset Values, Journal of Banking and Finance 14 (1990), 113-129 
D.  LAMBERTON and B.  LAPEYRE,'Introduction to Stochastic Calculus 
Applied to Finance' Chapman & Hall, London, 1996 
L.C.G.  ROGERS  and Z.  sm,  The Value  of an Asian  Option,  Journal  of 
Applied Probability  32 (1995), 1077-1088 
F.J.  VAZQUEZ-ABAD  and D.  DUFRESNE,  Accelerated  Simulation  For 
Pricing Asian  Options,  Research Paper Nr 62,  Centre for  Actuarial Stud-
ies, The University of Melbourne 
S.  WANG  and J.  DHAENE,  Comonotonicity,  Correlation Order and Pre-
mium Principles 
14 

Non-Additive Measure and Integral',

An easy computable upper bound for the price of an arithmetic Asian option.

https://lirias.kuleuven.be/bitstream/123456789/118653/1/OR_9950.pdf

An easy computable upper bound for the price of an arithmetic Asian option.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics