Valuation of Barrier Options in a Black Scholes Setup with Jump Risk

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Discussion Pap er No. B{446

Valuation of Barrier Options in a

Black{Scholes Setup with Jump Risk

Dietmar P.J. Leisen

This version: January 12, 1999.

Abstract

This pap er discusses the pitfalls in the pricing of barrier options using approx-

imations of the underlying continuous pro cesses via discrete lattice mo dels.

These problems are studied rst in a Black{Scholes mo del. Improvements re-

sult from a trinomial mo del and a further mo di ed mo del where price changes

o ccur at the jump times of a Poisson pro cess. After the numerical diculties

have b een resolved in the Black{Scholes mo del, unpredictable discontinuous

price movements are incorp orated.

Keywords

binomial mo del, option valuation, lattice{approach, barrier option

JEL Classi cation

C63, G12, G13

Stanford University, Ho over Institution, Stanford, CA 94305, U.S.A., and University

of Bonn, Department of Statistics, Adenauerallee 24{42, 53113 Bonn, Germany, email:

[email protected]

We are grateful to Dieter Sondermann and Holger Wiesenb erg for comments and assistance.

Financial supp ort from the Deutsche Forschungsgemeinschaft, Sonderforschungsb ereich 303

at Rheinische Friedrich-Wilhelms-Universitat Bonn is gratefully acknowledged. 1

1 Intro duction

Over time, barrier options have b ecome increasingly p opular to reduce the

cost of plain vanilla options while incorp orating individual views of market

participants concerning the asset evolution in an easy way: the payo of barrier

options dep ends on whether the asset price path will not cross a presp eci ed

b oundary. Of course, similar to standard options, they are also used to insure

against price drops b elow the barrier. Standard barrier options do then either

pay o a call or a put at the maturity date. For all these eight standard barrier

options with single barrier, closed form solutions are available in the Black{

Scholes setup see Rubinstein and Reiner 1991 and Carr 1995. For double

barrier options, however, analytical approximations are known only in some

sp eci c cases see Kunitomo and Ikeda 1992. Numerical pro cedures haveto

be applied to come up with prices, esp ecially in those generalizations of the

Black{Scholes setup, in which jumps are p ossible.

In this pap er we are interested in the \ecient" pricing of barrier options us-

ing approximations of the underlying pro cess. It is well known that binomial

mo dels su er from numerical de ciencies with barrier options: increasing the

re nement, prices converge erratically in a saw{to oth manner to the contin-

uous time price and even high re nements do not ensure adequate accuracy.

Many authors addressed this problem and suggested adjustments. Ritchken

1995, and Cheuk and Vorst 1996 constructed trees where the no des lie on

the barrier. Figlewski and Gao 1998 re ne the tree further at the barrier.

Our improvements are related to the nite di erence approach see Boyle and

Tian 1998: we start from a discretization of the asset space, instead of dis-

cretizing time rst, as previous approaches did. With barrier options this is a

straightforward way to have no des on critical levels by construction. We re-

cover binomial mo dels with the sp eci c re nementofBoyle and Lau 1994. To

incorp orate our approach in full generalitywe make use of trinomial mo dels.

Since discretizing the asset space breaks up the time discretization, we al-

low for random trading, similar to Leisen 1998b for American put options,

and Rogers and Stapleton 1998 for barrier options. However, the latter did

not recognize the numerical de ciencies resulting from the strike at maturity,

which will be prevented in our trinomial mo del. We also di er from their

approach by assuming that trading dates are the jump times of a Poisson

pro cess, which results in simple valuation formulas. Our parameters are set

in accordance with a convergence theorem of the pro cesses which also en-

sures convergence of prices. The mo del smo othes the convergence structure

even b etter. The order of convergence increases from 1=2 to 1 and removes

the wavy patterns. Using extrap olation we are able to obtain even quadratic

order. 2

After the numerical de ciencies have b een removed in the Black{Scholes setup,

we turn to the framework of Merton 1976, adding a comp ound Poisson pro-

cess to the Black{Scholes mo del. This mo dels the empirical observation of

sudden strong price changes. An extension of binomial mo dels to this setup

was prop osed by Amin 1993, which unfortunately inherits from there its

poor convergence prop erties, esp ecially with barrier options. Our mo del with

random jump times driven by a Poisson pro cess allows us in a simple way

and intuitive way to incorp orate jumps. It inherits the extraordinary conver-

gence prop erties from the trinomial mo del in its randomized version. Again,

a theorem ensures convergence to the continuous time solution.

The remainder of the pap er is organized as follows. In section 2 we present the

jump{di usion setup as well as the barrier option contract. Section 3 explains

the pitfalls in discretizing according to CRR. In section 4 these diculties are

circumvented using a suitable trinomial mo del. This mo del is then random-

ized. Section 5 incorp orates \strong" jumps. Section 6 discusses barrier option

valuation and the price accuracy in our framework. Section 7 concludes the

pap er.

2 The Setup

On a probability space ; F ;P we study an economy B; S , consisting of

the Bond B and the sto ck S . The interest rate r is assumed to b e constantover

time B = exp frtg and the sto ck price evolves under the ob jective measure

P according to

S = S G J ; 1

t 0 t t

where G = exp ft + W g ; 2

t t

V : 3 and J =

i t

i=1

N isaPoisson pro cess with constant parameter ,V a sequence of non{

t t i i

negative iid random variables, 2 IR, 2 IR .

The pro cess G is the continuous pro cess known as geometric Brownian

t t

motion and has stationary, Gaussian returns. It was suggested by Samuelson

1965 to mo del the evolution of a sto ck. Since Black and Scholes 1973 it is

used as one of the standard nancial mo dels. S evolves according to G until

the next jump time of the Poisson pro cess at which N changes from, say,

i to i +1. We then observe a p er{cent change V 1, i.e., the sto ck changes

value from S b efore the jump to S V .

i 3

So, the two parts can be interpreted as follows: G mo dels the \typical"

t t

evolution of the sto ck under the \normal" arrival of information, whereas

J mo dels jumps in the sto ck prices, due to some rare strong information

t t

sho ck. Since the Poisson pro cess is \memoryless," the exp ected time until

the next sho ck o ccurs is equal to 1=, indep endent of current time. Merton

1976 studied this mo del under the assumptions that the V are lognormally

i i

distributed random variables and and N are sp eci c for the rm under

t t

consideration.

The Black{Scholes setup is a so{called complete market, i.e., any contingent

claim on the sto ck can be hedged see Due 1992. The no{arbitrage prin-

ciple is sucient to price any claim. In the language of Harrison and Kreps

1979 this means that there is a probability measure Q, equivalent to P ,

and under which G =B is a martingale. Such a measure gives rise to a

t t t

linear pricing op erator; it is completely sp eci ed for the B; G market by

G 2

= r =2.

In our setup, however, we are in an incomplete market; the unforeseeable jump

can not hedged. Under the assumption of no{arbitrage there isamultiplicity

of equivalent martingale measure EMM under which agents do evaluate this

risk. The mathematical nance literature suggests the use sp eci c EMM, de-

rived from hedging criteria see Follmer and Sondermann 1986 and Follmer

and Schweizer 1991. When our setup is used to mo del \market crashes,"

i.e., J represents jumps in the aggregate or market p ortfolio, the pro ce-

t t

dure taken in the mathematical nance literature seems appropriate. If the

asset J would be traded in the market, the three assets B ; S; J would

t t

constitute a complete market. As it is, however, not traded, under the EMM

Q chosen by the market for valuation, only S has to be a martingale, but

t t

J not; the description of J under P and the choice of Q together with

t t t t

the Girsanov{Theorem are equivalent. Our view on this is that of sp ecifying

under an appropriate measure the pro cess J by

t t

E [V 1]t

J = J e V ; 4

t 0 i

i=1

for some 2 IR, which will be explained b elow. Using the description of the

sto chastic exp onential we derive E [J =J ] = e see Protter 1990 or Jaco d

t 0

and Shiryaev 1987. Using the Girsanov{Theorem as in the Black{Scholes

setup gives then the EMM for the B ; S; J market as a change in the Wiener

S 2

measure for which = r =2 . For a detailed discussion, we refer to

Wiesenb erg 1998.

Then ~ =lnE [J =J ]=t r = r can b e interpreted as the excess return on

t 0

the risky pro cess J over the riskless rate. With exogenously xed param-

t t

eters and V , sp eci es the risk{premium. As this is a \free" parameter,

i 4

we can treat the choice problem of an EMM as the sp eci cation problem of

an appropriate risk{premium in the market. Hamilton 1995, Bates 1996,

and Trautmann and Beinert 1995 discussed the implementation, i.e., howto

infer the prop erties of the jump comp onentJ in the sto ck pro cess S .

t t t t

Similar to Merton 1976 we are interested in the case where jumps are rm

sp eci c, and therefore uncorrelated with the market as a whole. In the sense

of the CAPM this is non systematic risk, has a of zero and therefore the

premium is zero. Bates 1996 gives statistical evidence that the risk premium

is non{zero, i.e., the jump{risk is correlated to the market as a whole, and

\crashes" need to be taken into account. However we use the Merton 1976

mo del as a starting p oint for our presentation; our approach is general and can

easily b e generalized to a non{zero . Section 6 studies numerical simulations

in the case of a rm allowing for ruin V 0.

Barrier options are a typ e of exotic options where the payo dep ends on the

crossing of predetermined levels, which may be either discretely or continu-

ously monitored. In the event of a crossing, dep ending on the sp eci cation, a

lump{sum called \rebate" mightbepayed, another asset might b e activated

called \kno ck{in" or deactivated \kno ck{out". Typical examples are those

where the secondary asset is a plain vanilla option, e.g., a down up and in

out barrier option pays this o , if some barrier H < S H > S is not

0 0

crossed and nothing otherwise.

We are interested here in continuously monitored constant barrier options,

where the nal payo is that of a plain vanilla Europ ean option. We explicitly

allow for multiple barrier options. The barrier option payo dep ends on a

\choice variable" as follows: The set contains all those paths ! , where the

terminal payo will b e activated. Then the terminal payo is 1 f S , where

! 2 T

h i

f is the payo function at maturity, and its price is E e 1 f S .

! 2 T

3 Binomial Pitfalls

This section studies the numerical de ciencies with barrier option pricing in

the Black{Scholes setup, i.e., the mo del S = S G , using binomial mo dels.

t 0 t

Starting with CRR many authors presented so called binomial mo dels for the

asset evolution. We recall here brie y the CRR mo del to present the main

diculties. The sp eci cation of a re nement n of the time axis [0;T] yields a

discretization set T = f0= t

n;0 n;1 n;n

dates, i.e., t t =t = . The logarithmic asset price evolves in some

n;i+1 n;i n

xed grid, with grid p oints at distance x . More sp eci cally it is supp osed

that from one date to the next the asset can jump only to the next adjacent 5

no de, i.e., we mo del the p er{p erio d return by

+x ; q

n n

R ; 5

n;i

x ;1 q

n n

and the sto ck pro cess by

G := R ; 6

n;i

i=1

: 7 with N :=

n n

Let us denote by X = ln G X = ln G the discrete logarithmic pro-

t t

cess. To match the continuous p er{p erio d variance, we set

t : 8 x =

n n

For pricing, only the evolution of the pro cesses under the risk{neutral probabil-

ity measures Q matters. It corresp onds here to the risk{neutral probability

Q of the continuous pro cess, and is represented by the probability q for an

up{move, such that E [ R ] = exp fr t g.

n;1 n

h h i i

Theorem 1 If E ln t , then E R R = r = exp fr t g +

n n;1 n;1 n

R ] = O t . On the other hand, if the martingale measure condition E [

n;1 n

i h

3=2

= r R exp fr t g holds, then E ln t + O t .

n;1 n n

In both cases we have S = S .

PROOF. The condition exp fr t g = E [R ] is equivalent to

n n;1

1+ r t + O t

= exp fr t g = q exp x +1 q exp x

n n n n n

2 2 3=2

= q 1+x +x =2 +1 q 1 x +x =2 + O t

n n n n

n n n

3=2 2

=2+O t = 1+x 2q 1+x

n n

3=2

which is equivalentto r t =2q 1 t +O t = E [ln R ]+

n n n n;1

3=2

O t . This and the converse | which follows similarly | prove the rst

two assertions. Then Donsker's theorem proves X = X and the pro of

concludes, observing that the exp onential function is continuous. 6

According to theorem 1 there are two alternativeways of sp ecifying q . First,

lo oking at the pro cesses G and G we can match their risk{neutral drift

i h

= r E [R ] = exp fr t g. Second, we can take E ln R t , to

n;i n n;1 n

X . In the limit the same match the drift of the logarithmic pro cesses X and

pro cesses and thus the same prices will result. We will use the second one

h i

to sp ecify the risk{neutral probability, and require E R = t , where

n;1 n

= r . Easy calculations reveal that

t 1

n n

9 + q =

2 2x

= t : 10 +

2 2

Figure 1 presents a pricing example for a down{and{out call with strike

8.1 CRR 8 continuous price

7.9

7.8

7.7

price 7.6

7.5

7.4

7.3

7.2 20 40 60 80 100 120 140 160 180 200

refinement n

Fig. 1. price picture for the barrier option, depicting the convergence structure

X = 110 and a barrier at H = 90 written on a sto ck when to day's sto ck price

S is 100, the volatilityis =0:2 and the interest rate is r =0:1. We observea

unregular convergence to the continuous time solution. The structure exhibits

the typical \saw{to oth" pattern well known in the literature. We observe also

the \o dd{even ripple." Figure 2 depicts the error on a log{log scale. Since the

function 1= n is an appropriate upp er b ounding line, the order of convergence

is 1=2. Compared to the pricing of plain vanilla options, where the order is 1 we

lose 1=2. We see that wehave quite high errors in the worst case. Furthermore

we deduce that we need at least a re nementof1= n =0:01 , n = 10000 to

ensure \p enny{accuracy."

The convergence is slower than in a standard Europ ean call and put option

case, since in binomial mo dels the whole probability mass is concentrated in 7 1 continuous price 1/sqrt(n)

0.1

0.01 error

0.001

0.0001 10 100

refinement n

Fig. 2. error picture for the barrier option

the tree{no des. The di erence in probability between two adjacent no des is

known to b e of the order O 1= n see Feller 1966. Now, if, resulting from

an increase in the re nement by one, a no de \jumps" over the barrier layer,

taking the corresp onding probability mass with, we will observe a similar

numerical e ect. To prevent this for barrier options, Ritchken 1995 p ointed

out that it has to b e ensured that there lies a no de exactly on the barrier for

any re nement. Wecharacterize all for double or even multi barrier options

barrier lines as a \critical line." Similarly Boyle and Lau 1994 argued to take

dep ending on m 2 IIN only the re nements

7 6

2 2

7 6

m T

7 6

: 11

5 4

These re nements are exactly the one's b efore an entire layer jumps over the

barrier, again. Pricing errors are reduced to a size comparable to those of call

options.

4 How to discretize prop erly

This section discusses the construction of a trinomial mo del in a rst approach

to remove the pitfalls of the previous section. Our approach to this problem is

as follows: Discretizing time, and then studying the resulting X {grid, runs into

problems. We are discretizing the wrong thing; the X{axis by a re nement m

needs to b e discretized rst, yielding x , and then the re nement n of the

m m

t{axis should b e set appropriately, rewriting equation 8 as 8

t = : 12

: 13 n = T

For a down{and{out call option to place grid p oints on the barrier H requires

ln S =H

. Interestingly, this way we recover formula 11 derived by x =

Boyle and Lau 1994 as the resulting re nement n in equation 13. In the

symmetrical case where jS X j = jS H j this improves the accuracy and

0 0

reduces the oscillations to a size comparable to Europ ean call options. To

encounter this problem in the general case we need to ensure that no des also

lie on the strike.

We will now present a rst approach to resolve this diculty in full generality.

Later then, we explain it on a concrete example. We call critical layer all

barrier lines p ossibly many, the strike and the current asset price. Let us

denote by L the ordered set l < ::: < l of critical layers, and by L =

1 L

Lnfl ;l g the inner p oints. First we de ne variables for 0 < i < L 1,

1 L

l l

i+1 i

x = , and then x =x and x =x , and for x 2

m;i m;L m;L1 m;0 m;1

u d

[l ;l [: x x= and for x 2]l ;l ]: x x= . This de nes a

i1 i m;i1 i1 i m;i

m m

discrete time{homogeneous grid for the sto ck values over time. Moreover we

de ne the minimum x = min x of all. We mo del the return, dep ending

m i m;i

on some x 2 IR, by a trinomial random variable R of the form:

m;i

x x ; p x

R x : 14

0 ;1 p x q x

m;i

m m

x x ; q x

For t ;n de ned by equation 12,13, we call the pro cesses

n m

m n

X = R X 15

m;i

t t

i=1

m n

= exp 16 G X

t t

17 N =

the Trinomial Adjusted TA mo del.

u d

The choice of x x; x x will give us sucient degrees of freedom to

m m

place no des on critical \lines," like the barrier and the strike price. Here, we

deal with a trinomial mo del in order to get the variance right despite the

complication that discretizations are not constant and change at critical lines.

The pro cess seems to be path{dep endent; however this is only conditionally 9

on the actual state. It remains recombining and therefore computationally

simple, as it can b e handled as any standard trinomial mo del for calculations.

The dep endence on x is easy to resolve. We will drop it in the sequel to present

the main ideas.

Let us explain our approach in detail for a down{and{out call option in gure

3 where a strike price is at X = 120 and a barrier is at H = 90. We see that

m;3

strike

* X

m;2

current sto ck

price

m;1

barrier

m;0

t t t

2;0 2;1 2;2

Fig. 3. Example dynamics

we can distinguish L = 3 critical layers and four di erent ranges: one b elow

the barrier, one between the barrier and the current asset price, one between

the strike and the current asset price and one ab ove the strike. In each of the

two inner ranges we need a di erent x . To place no des on critical layers,

we takebetween the barrier and the current asset price x = j ln H=S j=m

m;1 0

and between the strike and the current asset price x = j ln K=S j=m.

m;2 0

This de nes our grid for the asset evolution. As there is no clear choice for

x x , our approach takes x x for simplicity. Please note

m;3 m;0 m;2 m;1

that although we fo cus here on a down{and{out call for ease of exp osition,

our approach is general and can easily be generalized, e.g. to double barrier

options.

A consistency requirement is = G. Since the exp onential function is G

X , according = X . So, setting = r continuous, this is equivalentto

to Donsker's theorem the following two conditions are sucient:

h i

E R

m;i

! ; 18

h i

Var R

m;i

and ! : 19

We require them to hold with equality: 10

d u

q = t ; 20 p x x

m n m

m m

u 2 d 2 2

x p +x q = t ; 21

m m n

m m

which can be resolved easily

d 2

t x +

p = ; 22

u u d

x x +x

m m m

u 2

x + t

: 23 and q =

d u d

x x +x

m m m

Theorem 2 For suciently high re nements, al l probabilities p ;q ; 1 p

m m m

m m

d d

X G = X and = G. q are positive and

PROOF. Positivity of the probabilities follows from t = x = , and

n n

equations 22, 23 since

x x

0 < < 1 ;

u u d

x x +x

m m m

x x

and ! 0 :

u u d

x x +x

m m m

The observation that the barrier option with a put as payo {function is contin-

uous and b ounded, the ab ove theorem and put{call parity ensure convergence

of approximate values calculated to their continuous time counterpart. Please

note, that this price consistency holds whenever we know that pro cesses are

consistent in the limit.

Table 1 gives a pricing example for a barrier option using CRR, TA, and the

mo di ed Richardson extrap olation rule see Leisen 1998a

=n n =n n ; 24

m+1 m+1 m m m+1 m

iterating m = 1;::: ;8. The RT mo del will be intro duced in the following

section. Figure 4 contains the error picture for the same parameter constel-

lation as in table 1. Here we take only sp eci c re nements; we see that the

convergence structure is much smo other than in gure 2. However it is not suf-

ciently smo oth to apply extrap olation, which do therefore not depict. For TA,

in comparison to CRR, errors are drastically reduced and extrap olation gives

quickly \p enny{accuracy." We do also observe that the convergence structure

is fairly smo oth. 11

e e

m n

m CRR TA RT

TA RT

1 8 8.44693 7.89878 7.93419 6.60928 8.14613

2 34 8.17859 7.92586 7.92367 7.78452 7.97775

3 78 8.04017 7.92463 7.98542 7.89352 7.97775

4 140 7.97567 7.95155 7.98644 7.93082 7.97837

5 220 8.03844 7.96423 7.97371 7.94811 7.97904

6 316 8.02729 7.96711 7.97778 7.95751 7.97889

7 430 7.97612 7.96994 7.98570 7.96318 7.97884

8 562 8.0086 7.97364 7.97786 7.96686 7.97883

Table 1

Pricing example with S=100, X=110, H=85, T=1, r=0.1, =0:2. The continuous time price is 7.97888.

0.1

0.01

0.001 price error

CRR 0.0001 TA extrapolated TA RT 1e-05 extrapolated RT 10/n 3/(n*n) 1e-06 10 100

refinement n

Fig. 4. error picture for the barrier option

5 Randomizing and Jump{di usions

We will now randomize the previous mo del. This will b e an easy and straight-

forward way to incorp orate the additional jumps whichcharacterize the jump{

di usion mo del in di erence to the Black{Scholes setup. It turns out, in ac-

cordance to Leisen 1998b, that such a randomized mo del yields even b etter

convergence results.

m m

We start with a sequence of Poisson pro cesses N =N where N

t t

has parameter . N is describ ed byinterarrival times , indep endent

m m m;i

= exp onentially distributed random variables with parameter 1= and N

t g. In the previous section we approximated the pro cess max fn j

m;i

i=1 12

X between two trading dates t ;t 2 T by iid random variables R .

n;i n;i+1 m;i

Similarly here we will now approximate it by random variables R between

m;i

two interarrival times ; , which of the same form. All x l =

m;i m;i+1 m;l

1;::: ;L are de ned as in the last section describ ed to place no des on critical

\lines," like the barrier and the strike price. Then study the discrete pro cesses

m m

and , de ned similar to 15{17 by X G

t t

= X R ;

m;i

i=1

m m

= exp : G X

t t

and we require, since 1= = E [ ],

m m;i+1 m;i

i h

! r R ; 25 E

m m;i

h i

and Var R ! : 26

m;i m

m m

d d

X = X and G = Theorem 3 Under conditions 25 and 26 we have

PROOF. Let us de ne the two sequences dep ending on m of pro cesses

m m

M and A by

t t

M = R r t ;

m;i

i=1

and A = t:

m m m

Then for each m, the pro cesses M and M A are martin-

t t t

gales. As the jump sizes are of order x and vanish in the limit, we deduce

from the Martingale Central Limit Theorem as stated in Ethier and Kurtz

d d

= G, since the 1986 that M G = W . This is also sucient for

exp onential function is continuous.

We require equations 25 and 26 to b e ful lled with equality, i.e., 13

u 2 d 2

x p +x q = ;

m m

u d

; x p x q =

m m

or equivalently

2 u

+ r x

p = ;

u 2 u d

x +x x

m m m

d 2

x r

and q = :

d 2 u d

x +x x

m m m

Similarly to Leisen 1998b we take

: =

This choice is justi ed from the assumptions and since E [ ]=1= which

m;1 m

corresp onds to t in equation 13. Then p ;q 2 [0; 1];p + q 1 which

n m m m m

makes them feasible transition probabilities and the Poisson pro cess N

is stationary, which will make valuations in the next section esp ecially easy

to p erform. We call this mo del the Randomized Trinomial RT mo del. The

only di erence to the pro cess studied in the last section is the driving pro-

cess. Whereas there it was bt=t c, here it is the Poisson pro cess N with

parameter .

This mo del can also b e used easily to construct an approximation in the jump{

di usion setup. Startin with a sequence N ; R of the ab ovetyp e, where

m m

E [V 1], we de ne the pro cess here = r

N = N + N ; 27

which isaPoisson pro cess with intensity = + , the sequence of random

m m

variables

V ;

R ; 28

m;i

R ;

m;i

and the pro cesses

m 0

Y = R ; 29

t m;i

i=1

m m

S = exp Y : 30 and

t t 14

The counterpart to theorem 3 is:

Theorem 4 For the sequence of discrete models Y de nedbyequations

m m

d d

Y = Y and S = S , where S is the process 27{30 above we have

de ned in equation 1 and Y =lnS .

PROOF. Denote by h the function h : x 7! x + U on D . Since

i=1

R = r t + W ;

i=1

and the latter is continuous, we conclude using VI.1.23 and VI.3.8 ii of Jaco d

and Shiryaev 1987:

0 1

! !

m 0

@ A

= h r Y = h R t + W = Y :

m;i

i=1

6 Implementing barrier option valuation in the randomized mo dels

Since the structure in the randomized mo dels di ers only in the sp eci cation

of the return variable, we will treat pricing in those mo dels for a general R

m;i

R orR . The value V of a barrier option is given which is then either

m;i m

m;i

h i

V = e E 1 f S

m T

S jN ] = e E E [1 f

m m

S jN N = e = n P = n : E 1 f

T T

n=0

This splits up the valuation task by conditioning rst on N and then av-

eraging over all p ossible values. This is feasible, due to the indep endence of

m 00

N and the random variables in the sequence R . Let us now denote by

the \choice variable" corresp onding to in the n{step tree with return

mo deled by , and de ne R

m;i

= E 1 f S jN = n

n T

S jN = n = E 1 f

T 15

Therefore

P 1 1

- -

P P 1

P P

m m m

P P

P P 2 2 0

P P

- Pq Pq

Fig. 5. A trinomial grid

r + T m

V = e 31

n=0

We cut o the in nite sum in equation 31 at an appropriate such that

lim P [N 2 f0;::: ; g] = 1. Due to the Central Limit Theorem for

m!1 m

renewals,

N T

m d

=N0; 1 ;

setting =2b T c is one appropriate choice. In the sequel we adopt as our

m m

cut{o

2b T c

m r + T

: 32 V e

n=0

The value can be interpreted as the value calculated by backward{

induction in an n{step tree grid with return exactly as in the CRR R

m;i

mo del, if we do not p erform discounting see gure 5 for R = R . Please

m;i

m m

note that for any~n calculating gives us for any n =0;::: ;n~ as inter-

mediate calculations. Thus we can calculate prices as intermediate calculations

in an 2b T c step tree and computing prices in our mo del is comparable to

a trinomial mo del and therefore to the CRR mo del in terms of the compu-

tational burden. In order to compare b oth approaches prop erly dep ending on

its complexity,we index calculations in the RT by 2b T c.

As explained on gure 3 we adopt the discretizations x = x =

m;0 m;1

j ln H=S j=m and x =x = j ln K=S j=m to place al l no des on critical

0 m;2 m;3 0 16

e e

m n

m RT RT

RT RT

1 8 11.0891 13.6483 2.2053 0.3539

2 34 13.0461 13.2948 0.2483 0.0004

3 78 13.1864 13.2930 0.1080 0.0015

4 140 13.2336 13.2937 0.0608 0.0008

5 220 13.2555 13.2945 0.0390 0.0001

6 316 13.2673 13.2943 0.0271 0.0001

7 430 13.2745 13.2942 0.0200 0.0002

8 562 13.2791 13.2942 0.0153 0.0002

Table 2

Ruin pricing example with S=100, X=110, H=85, T=1, r=0.1, =0:2, =0:1

\lines." Table 1 presents prices and errors. We see that RT yields even b etter

price approximations than TA. By extrap olation we get extremely accurate

price approximations. This b ecomes even more apparent in the error picture

4. We see that prices converge with order one and extrap olated prices seem

to converge even with order two. Moreover we observe a gain in accuracy by

extrap olation in comparison to the extrap olated TA mo del.

0.1

0.01

0.001 price error

0.0001 RT extrapolated RT 1/n 1e-05 1/(n*n)

1e-06 10 100

refinement n

Fig. 6. error picture for the barrier option

We now discuss the accuracy in a ruin setup V 0 with = 0:1. Table 2

and gure 6 presentvalues calculated for this case. We also depict errors and

e e

extrap olated prices errors , calculated according to equation 24. We

calculated 13.2944 as the price in the CRR mo del with a re nement of 100000.

This is an estimation of the continuous time price. In gure 2 wesaw that 1= n

was an appropriate upp er b ound for the error. If we assume a similar error

b ound here, then the error of our estimate is less than 0:0031. Here we see 17

very impressively the slow convergence of the CRR mo del. Immediately with

m =2we fall b elow this level. We do not p erform a graphical analysis of the

order, since we can not calculate suciently accurate values; iterating the RT

higher than m = 2 to compare the accuracy is even doubtful. It is astonishing

that the remarkable convergence prop erties carry over from the Black{Scholes

setup to the jump{di usion case.

7 Conclusion

This pap er constructs a randomized trinomial mo del. This is a natural way

to approximate jump{di usions. The parameters are set such as to get consis-

tency with the continuous{time pro cesses. We discuss an easy approximation

to jump{di usions and how ecientnumerical approximations for barrier op-

tions result in the Black{Scholes and the jump{di usion setup.

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