In their seminal paper, Gerber and Shiu (1994) introduced the concept of the Esscher transform for option pricing. As examples they considered the shifted Poisson process, the random walk, a shifted gamma process and a shifted inverse Gaussian process to describe the logarithm of the stock price. In the present paper it is shown how upper and lower bounds in convex order can be obtained when we use these types of models to describe the financial stochasticity for a given cash-flow.Cash flow; Pricing; Processes; Models; Model;

De Schepper, A

Dhaene, Jan

Goovaerts, Marc

Kaas, Robert

Vyncke, David

English

Research Papers in Economics

KATHOLIEKE 
UNIVERSITEIT 
LEUVEN 
OEPARTEMENT TOEGEPASTE 
ECONOMISCHE WETENSCHAPPEN 
RESEARCH  REPORT 0132 
ON THE DISTRIBUTION OF CASH-FLOWS 
USING ESSCHER TRANSFORMS 
by 
D.  VVNCKE 
M. GOOVAERTS 
A.  DE SCHEPPER 
R.  KAAS 
J.DHAENE 
0/2001/2376/32 On the distribution of cash-flows 
using Esscher transforms 
D.  Vyncke1, M.J. Goovaerts1,2, 
A.  De Schepper3,  R.  Kaas1,2, J. Dhaene1,2 
Abstract 
In their seminal paper, Gerber and Shiu (1994) introduced the concept of 
the Esscher transform for  option pricing.  As  examples they considered  the 
shifted Poisson process,  the random  walk,  a  shifted gamma process  and  a 
shifted inverse Gaussian process to describe the logarithm of the stock price. 
In the present paper it is shown how upper and lower bounds in convex order 
can be obtained when we  use these types of models to describe the financial 
stochasticity for a given cash-flow. 
1  Introduction 
In their seminal paper, H.  Gerber and E. Shiu (1994) advocated the Esscher 
transform as a tool to deal with stock-price movements for  a family of pro-
cesses.  With M(h) denoting the moment generating function of a  random 
variable X, i.e. 
M(h) = E [ehX]  (1) 
the Esscher transform (with parameter h) of the density f(x) is  obtaiued in 
case the function 
ehx f(x) 
f(x, h) =  M(h) 
1  University of Leuven, Belgium 
2  University of Amsterdam, the Netherlands 
3  University of Antwerp, Belgium 
1 
(2) is  a  density.  In Goovaerts et al.  (1984)  it is  shown  how the Esscher  trans-
form evolves from utility theory in measuring the price of a random variable. 
Indeed, one has the following theorem: 
Theorem 1.  Assume an ins1J,rer has an  exponential utility function with risk 
aversion Q.  If he  charges  a premium of the form  E[cp(X)XI  where  cp(-)  is  a 
continuous increasing function with  E[cp(X)1  =  1,  his utility is  rrw,x1:m'ized  1:/ 
cp( x) ex:  eax,  i. e.  if he  uses the  Esscher premium principle with pammeier Ct. 
For a proof of this theorem, we refer to the Appendix.  If the utility function 
u is an exponential one, i.e. u(x) = k  (1 - e-hx), then 
(3) 
so the Esscher transform of the risk X evolves.  If  u(x) is quadratic, hence e.g. 
u(x) = ax2 + bx,  we get cp(x)  0::  2ax +  b,  and E[cp(X)] = 1 gives 
E[Xcp(X)]  =  E[X(2aX +  b)/  E[2aX +  b]]  =  2a~~~~~  ~~[X] 
1 
E[X] + E[X] + b/2a  VadX], 
which is  a premium of type variance premium, if risks with a given expecta-
tion are considered. 
These rather simple results indicate a relationship between the actuarial ap-
proach of premium principles and the financial approach of pricing risks  by 
means of a  measure transformation.  Gerber and Shiu  (1994)  considered  a 
stochastic process  {X(t)h2:o  with  stationary and  independent  increments, 
X(O)  = 0,  such that 
8(t) = 8(0) . eX(t)  (t 2: 0).  (4) 
To make sure that the stock prices of the model are internally consistent, they 
seek for a h = h*  so that the discounted price process 
(.5) 
is  a martingale with respect to the probability measure corresponding to h*. 
In particular, 
8(0) = e-tit E* [8(t)]  (G) 
2 Table 1:  Esscher transforms for some types of stochastic processes 
Stock-price mode!  TN",  t· h*)  ...  \..u"  .  h* 
Wiener process  N(x; (J-L+ h*rr2)t, rr2t)  0= (J-L + h*rr2) + !rr2 
Shifted Poisson process  A (xtd;>.eh*kt)  0=  >.eh*k (e-k - 1) - e 
Random walk  B ( X-(J.t, t  7r(h*))  (l  *)  eli-e"  7r  /,  =--::r;--::a  b-a' "  e  -e 
Shifted Gamma process  G (x +  et; at, (3 - h*)  d  ({-J-h*  r"  -c  e  =  (J-h*-l  e 
Shifted inverse Gaussian  J (x + ct; at, b - h*)  o  = a (  Vb - h* - Vb - h* - 1) - e 
where 0 denotes the constant risk-free force of interest.  Based on the theorem 
above,  the price of the risk is  calculated by choosing the coefficient  of risk 
aversion such that the premium coincides with the market price. 
The results presented by Gerber and Shiu  (1994)  can be summarized as  ill 
Table 1,  where F(x, t; h*) is  the cumulative distribution function of the Es-
scher transform of the process X(t).  A definition of the stochastic processes 
as well as an overview of the notations for the functions in the second column 
can be found in the Appendix. 
In this paper, we are interested in deriving upper and lower bounds in convex 
order for a  (discrete version of the) cash-flow 
C(t) = lot e(s)eX(t)-X(s)ds,  (7) 
where  X(t)  is  assumed  to be  one  of the stochastic price  models  given  in 
Table 1,  using the approach described by Kaas, Dhaene & GooV'derts  (2000). 
Note that the problem of pricing an Asian option under the stochastic proces-
ses of Table 1 can be solved by using the same approach, because the bounds 
will have higher, resp. lower, stop-loss premiums. 
3 In the following section, we explain the concept of convex order and describe a 
methodology to obtain upper bounds.  As we will also construct lower bounds, 
the results in this paper extend the results in Goovaerts et al.  (2000).  To cal-
cwate the lO,Xfer  bound,  and  to improve the upper  bOUIld,  the methodology 
requires the knowledge of the conditional distribution of the process {X  (1;) }, 
conditionally on some random variable Z.  A potential conditional distribu-
tion is  derived  in  section 3.  Finally,  in  section 4  we  apply  the techniques 
to the problem at hand and in  section 5 we  illustrate the obtained bounds 
graphically. 
2  Convex order and comonotonicity 
The distribution function of (7)  is  very hard, or even  impossible,  to obtain 
due to the dependency structure among the different random variables.  The-
refore, instead of calculating the exact distribution, we will look for  bounds, 
in the sense of "more favourable/less dangerous" and  "less favourable/more 
dangerous", with a simpler structure.  This technique is common practice in 
the actuarial literature. When lower and upper bounds are close to each oLher, 
together they can provide reliable information about the original and more 
complex variable. The notion "less favourable" or "more dangerous" variable 
will be defined by means of the convex order. 
Definition 1.  A  random variable V  is smaller than a random variable !tV  in 
convex order if 
E [n(V)1  :S  E [n(W)],  (8) 
for all convex fnnctions 11,  : IR -+ R : X  I-t n(  x), provided the expectations eJ;'isi. 
This is denoted as 
V  :Sc.T.  w.  (9) 
Since convex functions  are functions that take on  their largest values  i  TI 
the tails, the variable W  is  more likely to take on extreme values  than  j;lJe 
variable V, and thus W  is more dangerous. 
The convex order can also be interpreted in terms of utility theory.  Indeed, 
if V  :S= W, then V  is  preferred to W  by all risk averse decision  makers, see 
e.g.  [41.  This means that replacing the unknown distribution function of the 
4 variable V by the distribution function of the variable W  is a prudent strat.egy. 
Since  the functions  u(x)  =  x,  u(x)  =  -x and u(x)  =  x 2  are  all  convex 
functions;  it follows  immediately that V  :::;e;L  W  implies  E[V]  =  E[W]  aud 
Var[V]  :::;  Var[W]. 
The following lemma provides an interesting and  useful  characterizat.ioll of 
convex order, a proof of which can be found in  [6]  : 
Lemma 1.  For any two  random variables V  and W,  we  have  the following 
equivalence: 
{  E[(V - k)+]  :::;  E[(W - k)+] 
V  :::;cr,  W {:}  E[V] = E[W] 
for all k, 
(10) 
where (x)+ = max{O, x}. 
Now, if V  consists of a  sum of random variables Xl, ... , X n ,  then replacing 
the copula of (Xl, ... ,X n )  by the comonotonic copula yields an upper bound 
for  V  in the convex order.  On the other hand, applying Jensen's inequality 
to V  provides us witb a lower bound.  Finally, if we combine both ideas, then 
we end up with an improved upper bound.  This is formalized in the followilJg 
proposition. 
Proposition 1.  Cons1:der  an arbitrary sum of random variables 
and define the related stochastic quantities 
~Yjl (U) +  ~y;  (U) + ... + F,y2 (  U) 
Vi,u  F,;;j1IZ(U) + FX::IZ(U) + ... +  ~;;~IZ(U) 
E[XIIZ] + E[X2IZ] + ... + E[XnIZ]' 
(11) 
(12 ) 
(] ;-l) 
(J 4) 
with U  an arbitrary random variable,  uniformly distributed on [0,1]'  and with 
Z  an  arbitrary random variable,  independent of U.  The  following  relations 
then hold: 
(] 5) 
5 Proof: see [1]  and [5]. 
For each j  =  1, ... ,n,  the terms in  the original variable V  and  the corre-
sponding terms in the upper bOlLTlds  Vu  and Viu  are all mutually identieally 
distributed, i.e. 
(Hi) 
For the lower bound,  the equalities of the distributions of Xj and  EIXjlZj 
only hold in case all Xj, given Z = z, are constant for each z. 
These results can be extended to the case where V  consists of a  surn of mo-
notonic functions  cPj  of random variables Xj, simply by substituting Yj  for 
cPj(Xj )  and applying Proposition 1, see  [4,  5,  7]. 
3  The conditional Esscher transform of a 
process with stationary and independent in-
crements 
Because we consider stationary and independent increments, the unconditio-
nal Esscher transform of the process {X  (s ) }o::;s::;t  equals 
M[z, s; h*] = E* [ez xes)]  =  M[z, 1; h*]".  (17) 
The application of Jensen's inequality requires the knowledge of the conditi-
onal distribution of the process {X  (t)  - X (s )}o::;s::;t,  conditionally on some 
random variable Z.  To simplify the computations, we will choose Z = X(t). 
Then, we have for the conditional Esscher transform 
Melz, s; h*]  E* [ez(X(t)-X(s»  I  X(t) = c]  :c PTob* (X(t) s::  c)  (18) 
E* [ez(X(t)-X(s»  .  J(X(t) = c)]  ,  (19) 
where  J ( .)  denotes  the indicator function,  i.e.  J (A)  =  1  if  A  is  true and 
J (A) = 0 otherwise.  Hence, 
j
+OO 
Me[z, s; h*]  = eze  -00  dx e-zx f(x, t - s; h*)f(c - x, s; h*) ,  (20) 
6 where  f(x, s; h*)  =  d~ F(x, s; h*).  Inversion  with  respect  to z  gives  us  the 
density of the conditional random variable X(t) - X(s)IX(t) = c : 
and of course 
d 
-d  Prob* (X(t) - X(s) .s xIX(t) = c)  x  . 
f(x, t - s; h*)f(c - x, s; h*) 
f(c, t; h*) 
F.Y(t)-X(s)IX(t)=c(x, s; h*) 
j
.x  f(y, t - s; h*)f(c - y, s; h*)  dy. 
-00  f(c,t;h*) 
(21) 
(22) 
(23) 
(24) 
Consequently, for given s the inverse conditional distribution can be calculated 
by solving 
j
F C-1(U,S;h*) f(x, t - s; h*)f(c - x, s; h*) d 
U=  .  x. 
-00  f(c, t; h*) 
(2;5) 
Example:  Shifted inverse Gaussian process 
The conditional distribution can be calculated for any of the above distributi-
ons with the right Gerber-Shiu parameterization. As an example, we  consi(lf~r 
the case of the shifted inverse Gaussian process 
X(t) = Y(t) - at  (2G) 
where {Y(t)} is an inverse Gaussian process with cumulative probability flll1c-
tion 
Prob[Y(t) ::; yj  J(y;a,b)  (y>O)  (27) 
<»  ( -a + V2bY) +  e2av'b<»  (  -a  - -/2bY)  (28) 
V2Y  V2ii 
and with probability density function 
d 
dy Prob[Y(t) ::; yj  j(y;a,b)  (y > 0)  (29) 
(30) 
7 This gives 
F(x, t; h*) = J(x +  at; at, b*)  (x> -at)  (31) 
and 
f(x, t; h*) = j(x +  at; at, b*)  (x> -at),  (32) 
with b* = b - h*. 
For the conditional distribution, applying (25) yields, for 0 ~  11,  ~  1, 
j
F;-l(U,S;h*) j(x + a(t - s); a(t - s), b*)  . j(c - x + as; as, b*)  (3.'~) 
11,=  ~  J 
-o(t-s)  j(c+ at; at, b*) 
where, taking into account the support of j(.), the following restriction applies 
(34) 
or 
o  ~ F c- 1 (11"  S; h*) + a(t - s)  ~  c + at.  (35) 
Hence, 
11,  =  rFc- 1(u,s;h*)+o(t-s) j(x; a(t - s), b*)  .  j(c - x +  at; as, b*)  dx.  (36) 
10  j(c +  at; at, b*) 
4  Bounds 
We now derive the upper and lower bounds in convex order for  the discret;e 
cash-flow (the continuous cash-flow arises by taking appropriate limits) 
n  n-l 
C(t) = L cjeX(t)-X(tj )  = en + L cjeX(t)-X(tj ) ,  (37) 
j=l  j=l 
with t = tn,  using the approach described in section 2.  Henceforth,  we  will 
assume that Cj  ~  0, (j = 1, ... ,n), merely to facilitate notation. 
8 4.1  Upper bound 
Applying (12) yields 
with 
D  r {fI{./-\  1. \  1./ Dr  {fl  (./-\  1. \  1 
D  l\V\~1 - "'I+J  .:::::  D  l\v'u\~1 - "'J+J 
n-l 
Cu(t) =  en + I:  CjeF-1(U,t-tj;h*). 
j=l 
(38) 
(39) 
Since  d~E[(Y  -k)+l = Fy(k) -1 for any random vdfiable Y  and any reteIltion 
k, the distribution of the upper bound follows as 
F~(x) ~ 1-11 duI (c,.1  ~CjeF-,(.,t_tj;hO)geqx)  (40) 
where Ie) is  the indicator function, i.e. I(A) =  1 if A  holds, I(A) =- 0 if not. 
Hence, let u'" be defined as the value for which 
n-l 
C n + I:  cjeF- 1  (ux,t-tj;h*) = X, 
j=l 
then 
4.2  Improved upper bound 
Applying (13) with Z  =  X(t) yields 
(41 ) 
(42) 
E[(C(t) -k)+l:s E[(C"u(t) -k)+l:s E[(Cu(t) -k)+l  (43) 
with 
E I( O;.(t) - k h I  ~  E x(t)Eu [ ( c,. + ~  cji'a,l(U,tj;hOIX(t)) - k) J,  (44) 
where the distribution function Fco(u, tj; h*IX(t)) is defined by its realizati-
ons 
Fco(u, s; h*IX(t) = c) = Fc(u, s; h*). 
9 Since the stop-loss premium for the improved upper bound can be written as 
E [(Ci'u(t)  - k)+l =  j~:oo  dc  f(c; t, h*) 
.1  /  n-l  \  (46) 
x ),  du ( Cn  I ~  CjeF~'  (tl,'j;h") - k )1 
the distribution of the improved upper bound follows as 
j
+OO 
Fi·u(x)  =  -00  dc  f(c; t, h*) ux(c), 
where u,,,(e)  is defined as the root of 
n-1 
en + I::  cjeFC-1 (-u:z; (c),tj;h*)  =  x. 
j=l 
4.3  Lower bound 
Finally, applying (14) with Z = X(t) yields 
E [(Ce(t)  - k)+l ~ E [(C(t) - k)+l 
with 
E [(Ce(t)  - k)+l = 
E(x{I)) [ ( Cn + ~  cjE' [eX(I)-X('j) I  X (t)]-k) J  . 
The stop-loss premium for the lower bound equals 
j
.+OO 
E[(Ce(t)-k)+I=  -00  dcf(c;t,h*) 
(  n-l  j'+oo  ) 
X  en + I::  Cj  dx eX  f~(x, tj; h*)  - k 
j=l'  -00 
so the distribution of the lower bound follows as 
Fe(x) =  j~  de  f(c; t, h*)  , 
10 
-I 
(47) 
(48) 
(49) 
(50) 
(51 ) 
( 52) where G c IR  is defined as the collection of all V'dlues  C for which 
5  Numerical illustration 
In this section,  we  illustrate the upper and lower bounds by  plotting their 
distribution functions.  We assume that the process {X(t)} is a shift;ed inverse 
Gaussian process with parameters a = 3v'l.2, b = 7.5  and a  = 0.5  (see  r2j, 
p. 118). The parameter b* corresponding to a risk-free force of interest t5 = 0.1, 
equals 961/120. 
The distribution functions for Cu,  Ciu and Ce  corresponding to a  cash-flow 
Cj = 10,  j  = 1, ... ,10, are depicted in Figure 1.  Since the upper and lower 
bounds appear to be rather close to each other, they prove to be quite good 
approximations for the unknown distribution of C(tn)'  The improved upper 
bound Ci'a  indeed improves the upper bound Ca ,  albeit slightly. 
In order  to assess  the influence  of the cash-flow,  we  change  it  to  Cj  '-,  .1 
and Cj = 11 - j, j  =  1, ... ,10, in Figures 2 and 3 respectively.  Taking into 
account the scale of the price axis, the bounds appear to behave very similarly 
in both cases. 
Corresponding author : 
David Vyncke 
University of Leuven, Department of Applied Economics 
Minderbroedersstraat 5,  B-3000 Leuven 
tel: +32-16-32.37.45, fax:  +32-16-32.37.40 
e-mail: david.vyncke@econ.kuleuven.ac.be 
References 
[1]  Dhaene J.,  Denuit M.,  GooV'derts  M.J., Kaas R..  & Vyncke  D.  (2001). 
"The Concept of Comonotonicity in Actuarial Science and Finance: The-
ory", North American Actuarial Journal,  to appear. 
11 [2]  Gerber H.  &  Shiu  E.  (1994).  "Option pricing by  Esscher transforms", 
Transactions of the society of actuaries,  vol. 46 , p.99-139. 
[3]  Goovaerts M.,  De  Vylder F., & Haezendonck J.  (1984).  In8umnce  Pn;-
miums, North-Holland, pp.XI+406. 
[4]  Goovaerts M.,  Dhaene J.,  &  De  Schepper  A.  (2000).  "Stochastic  up-
per bounds for  present value functions",  Journal of R1:sk  and In8v,ra,nce, 
vo1.67(1), p.1-14. 
[5]  Kaas R, Dhaene J., & Goovaerts M.  (2000).  "Upper and lower bouJ\ds 
for sums of random variables", Insurance:  Mathematic8 and Econornic8, 
vo1.27(2), p.151-168. 
[6]  Shaked M. & Shanthikumar J.G. (1994). Stochastic order8  and their op-
plications, Academic Press, pp.545. 
[7]  Vyncke  D., Goovaerts M.  & Dhaene J.  (2001).  "Convex upper and  lo-
wer bounds for  present value functions",  Applied Stochastic Models  for 
Business and Industry,  17, 149-164. 
C! 
--------- --
..... - /_  .. 
/.-... 
<Xl 
ci  A/:.~· 
to 
ci 
'6 
() 
..,. 
ci 
'"  ci 
0 
ci 
100  200  300  400 
price 
gure 1:  Distribution functions of the lower bound Ce (--), the improved upper 
mnd C,.u  ( ... ) and the upper bound Cu (-) for  Cj  =  10  (j = 1, ... ,10). 
12 :1 
--------------
ci 
<0 
ci 
'6 
u 
'"  0 
'"  ci 
0 
ci 
40  60  80  100  120  140  160  180 
price 
Figure 2:  Distribut.ion funct.ions of Oe  (--) and Cu  (-) for  Cj  =:::.  1, ... , 10. 
CO! 
<Xl 
ci  & 
<0 
ci 
'5 
u 
.... 
ci 
'"  ci 
0 
ci 
50  100 
/. 
..... -.,.,...-
/,/ 
-----~-~~~-=-=-=-=-=-=-=-====~ 
150  200  250  300 
price 
Figure 3:  Distribution functions of Oe (--) and Ou  (--) for  Cj  ~~ 10, ... ,1. 
13 Appendix 
Overview of the stochastic processes of Table 1 
= Vliener process : 
X(t) = o-Z(t) + Id 
where {Z(t)} is a standard Brownian motion. 
•  Shifted Poisson process : 
X(t) =  kN(t) - ct 
where {N(t)} is a Poisson process with parameter A,  k and c are positive 
constants. 
•  Random walk: 
X(t) = Xl + X 2 + ... + X t 
where Xj is such that P(Xj =  b)  =  p =  1 - P(Xj = a), a < 6"  < b. 
e  Shifted Gamma process: 
X(t) = Y(t) - ct 
where  {Y(t)}  is  a  Gamma process  with  parameters  0:  and  (3;  c  is  a 
positive constant. 
•  Shifted inverse Gaussian process: 
X(t) = Y(t) - ct 
where {Y(t)} is an inverse Gaussian process with parameters a and  b;  c 
is  a positive constant. 
Overview of the functional notations of Table 1 
•  N (x; fL,  0-2) =  <J?  ("'::1=') 
A(  e)  ,,",'£  e-O(}k  (  ) 
•  X;.  =  Dk=O k!  x  2':  0 
•  B(x; n, e)  = 2:%=0 (  ~ ) ek(l - ey,·-k  (x 2':  0) 
•  G(x' 0:  (3)  =  (.p.  f:I;yo.-le-fjYdy  (x 2':  0) 
"  '.  f(o.) Jo 
•  J(x; a, b) =  <J?  (~  + V2bx) + e2aVb<J?  (~  - V2bx)  (x> 0) 
Proof of Theorem 1 
The proof of Theorem 1 is based on the technique of variational calculus and 
adapted from Goovaerts et al. (1984).  Let u(·) be a convex increasing utility 
14 function, and introduce Y = ip(X). Then, because ip(') increases continuously, 
we  have  X  =  ip-l  (Y).  Write fey)  =  ip-l (y).  To  derive  a  condition  for 
E[u(  - f(Y)+E[f(Y)Yj)] to be maximal for all choices of continuous increasing 
functions v/hen E[Y] = 1, consider a function  f(y) + Eg(y) for some arbitrary 
continuous function g(.).  A little reflection  will  lead to the conclusioIl tbat 
the fact that fey) is  optimal, and this new function is not, must mean tbat 
!  E[u( - fey) + E[f(Y)Y] +  E{ -g(Y) + E[g(Y)YI})] 1"'=0  =  o. 
But this derivative is equal to 
E[u'( - fey) + E[f(Y)Y] +  c:{ -g(Y) + E[g(Y)Y]}) 
{-g(Y) + E[g(Y)Y]}]. 
For E = 0,  this derivative equals zero if 
E [u'( - fey) + E[f(Y)Yj)g(Y)] = 
E [u'( - fey) + E[f(Y)Y])] E[g(Y)Y]. 
Writing c = E [u'( - fey) + E[f(Y)YJ)], this can be rewritten as 
E [{u'(-f(Y) + E[f(Y)Y]) - cY}{g(Y)}] = O. 
Since the function g(.) is  arbitrary, by a well-known theorem from variatiollal 
calculus we find that necessarily 
u'(  - fey) + E[f(Y)Y]) - cy = o. 
Using x =  fey)  and y = ip(x), we  see that 
ip(x) ex: u'( -x + E[Xip(X)]). 
Now, if u(x) is  exponential(a), so It(X)  = -ae-ax , then 
ip(x)  ex:  e-a(-x+E[X<p(X)j) ex:  eax. 
Since E[ip(X)]  =  1,  we  obtain ip(x)  = eax jE[eaX] for  the optimal staIl(lar-
dized weight  function.  The resulting premium is  an Esscher  premium with 
parameter h =  a.  D 
15 

Convex upper and lower bounds for present value functions&quot;, Applied Stochastic Models for Business and

Option pricing by Esscher transforms&quot;,

Stochastic order8 and their opplications,

Stochastic upper bounds for present value functions&quot;,

The Concept of Comonotonicity in Actuarial Science and Finance: Theory&quot;,

Upper and lower bouJ\ds for sums of random variables&quot;,

On the distribution of cash-flows using Esscher transforms.

https://lirias.kuleuven.be/bitstream/123456789/118649/1/OR_0132.pdf

On the distribution of cash-flows using Esscher transforms.

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Research Papers in Economics