Available online at
http://acad.pvamu.edu/content/mathdept/AAMJournal.html
Vol.1, Issue 1 (June 2006), pp. 25 - 35
(Previously, Vol. 1, No. 1)
Applications and
Applied Mathematics
(AAM):
An International Journal
A Discrete Time Counterpart of the Black-Scholes Bond Replication Portfolio
Andrzej Korzeniowski
Department of Mathematics
University of Texas, Arlington, Texas 76019-0408, USA
korzeniowski@uta.edu
Received October 13, 2005; revised received April 22, 2006; accepted April 27, 2006
Abstract
We construct a discrete time self-financing portfolio comprised of call options short and stock shares long which is
riskless and grows at a fixed rate of return. It is also shown that when shorting periods tend to zero then so devised
portfolio turns into the Black-Scholes bond replication. Unlike in standard approach the analysis presented here
requires neither Ito Calculus nor solving the Heat Equation for option pricing.
Keywords: Binomial Model, Call Replication, Discrete Time Black-Scholes Hedging
1. Introduction
Black and Scholes (1973) discovered that a portfolio of one call option short and
∂ C (t , S )
∂S
stock
shares long, where C (t , S ) and S is the option and stock price at time t respectively, has no risk
and consequently it replicates a bond portfolio earning a fixed rate of return,
provided C (t , S ) satisfies certain partial differential equation. Since then a great deal of studies
circled around discrete time option pricing originated by Cox -Ross-Rubinstein (1979),
Rubinstein and Leland (1981), which focused entirely on no-arbitrage option replication
portfolio determined from a martingale measure corresponding to the discounted stock price on
binomial tree. Meanwhile no attention has been given to the fact that the bond portion of such
hedging portfolio, in contrast to its continuous time counterpart, is never riskless! In fact, the
bond portion is a unique random process whose distribution at each time step is determined by
terminal stock prices under the martingale measure. The above raises a question of whether it is
possible to construct riskless portfolio in discrete time analogous to continuous time BlackScholes bond replication portfolio.
The aim of this article is to answer the question in the positive and explain how the findings
bridge the long-standing differences between the two models. Namely, we first construct a selffinancing bond replication portfolio by augmenting a combination of suitably chosen number of
stock shares long and one call option short with a third position in carry-on cash flow (excess or
25
26
Andrzej Korzeniowski
shortage of funds resulting from creating the stock-option position) and then we convert the cash
flow position into the risk free combination of stocks and options. The key feature of this
discrete time portfolio lies in the fact that it does not suffer from the well-known shortcomings
present in the continuous model: an impossible to implement continuous time trading coupled
with prohibitive(infinite) transaction costs associated with it. The model presented here applies
to both small and large time period situations and works particularly well in the latter case. For
example, when the time intervals are days, months or years and the underlying security can be
modeled by assuming only two different outcomes at the end of each time period (e.g., success
or failure with a price tag) then over the multi-step period the model provides a hedging strategy
that has no risk and for which the transaction costs may in fact be assumed negligible, whereas in
the continuous Black-Scholes model the costs cannot be ignored unless the realities of the actual
security trading are being rejected!
2. Notation, assumptions and binomial model facts
probability=
space Ω {(ω1 ,..., ωN ) | ωi ∈ {U , D},1
=
≤ i ≤ N }, 2Ω ,
1+ r − d
P((ω1 ,...,=
ωN )) p #U ' s (1 − p) # D ' s , with 0=
<p
< 1 subject to 0 < d < 1, 1 + r < u where
u−d
r is a fixed risk-free rate of return per one time period, 
n = σ ((ω1 ,..., ωn )) and
is defined by P(stock
 = {∅, Ω}, n = 1,..., N . Then the risk-neutral stock process S n ∈ 
n
Let
goes
(Ω,  , P) be
up)
a
= P( S n −1  S=
S n −1u=
) P (ω=
U=
) p
n
n
and
P(stock
goes
down)
=
Here
S n −1 = S n −1 (ω1 ,..., ωn-1 ) and
P( S n −1  S n = S n −1d ) = P (ωn = D) = 1 − p .
S n = S n (ω1 ,..., ωn-1 , U ) , or S n = S n (ω1 ,..., ωn-1 , D) , respectively, and S n is binomial with
k n−k
P( S n S0u=
d )
=
n k
  p (1 −
k 
p ) n − k . We emphasize that the probability distribution of the actual
stock process is irrelevant as it does not play any role in determining no-arbitrage option pricing,
aside from the fact that it is rarely known! The only thing the two stock processes have in
common is the identical sample space {S0 , S1 ..., S N } which constitutes the binomial binary tree.
We make the following assumptions about discrete time portfolio modeling:
(p1) prices of stocks, options and bonds change at n = 0,..., N and stay constant throughout
[n − 1, n), n =
1,..., N ;
(p2) =
portfolio values are declared at n 0,..., N and stay constant during [ n − 1, n), n = 1,..., N ;
(p3) positions (rebalanced holdings) are decided based on time n -1 prices, take effect
instantaneously after time n -1 and are held unchanged throughout ( n -1, n], n = 1,..., N ;
(p4) portfolio is self-financing in the sense that its value at time n − 1 and after rebalancing are
equal, n = 1,..., N ; and
(p5) portfolio transaction costs are zero, borrowing along with short and fractional sales are
allowed, option pricing assumes no-arbitrage.
AAM: Intern. J., Vol. 1, Issue 1 (June 2006), [Previously, Vol. 1, No. 1]
27
The significance of (p3) stems from the fact that different portfolio positions must necessarily be
held over non-overlapping time intervals until the securities undergo price change in order to
unambiguously define portfolio holdings and value when rebalancing. Moreover, (p3)
incorporates the fact that security prices must be known first before the portfolio adjustments can
be made and consequently one way to implement this is to consider the discrete time sequence as
a subset of the continuous time with a proviso that rebalancing takes effect instantaneously after
the prices have changed.
Allowing the portfolio to undergo content and value changes periodically at discrete time
instances only, while keeping the portfolio intact at all other times, considerably simplifies the
analysis of the portfolio dynamics. In addition, it offers a practical alternative to a continuous
time model while retaining its validity for both large and small scales of portfolio inactivity
period.
A basic fact from binomial asset pricing model Karatzas & Shreve (1998) or Williams (2001)
states that a European call option can be replicated through a hedging strategy by holding a
portfolio of uniquely determined combination of stocks and bonds. Since the option replication
serves as basis for our bond replication construction we first recall all necessary facts needed for
further analysis.
Given a strike price K > 0 the value of the call option at time N is defined by
C=
Max( S N − K , 0) ≡ ( S N − K ) +
N
and we use the following notation:
S n is the stock price at time n
Cn is the call option price at time n
=
Bn B0 (1 + r ) n is the bond price at time n
an is the number of stock shares held during (n − 1, n]
bn is the number of bonds shares held during (n − 1, n]
r is a fixed risk-free return rate per period
Then by (p3)-(p4) portfolio values{ X n , 0 ≤ n ≤ N } satisfy
X 0 = a0 S0 + b0 B0 = a1 S0 + b1 B0
X n −1 =
an −1 S n −1 + bn −1 Bn −1 =
an S n −1 + bn Bn −1 and X n =
an S n + bn Bn ,
n=
1,..., N
(1)
Remark. Since the initial assets allocation (a0 , b0 ) for X 0 is limited to time n = 0 and then
instantaneously replaced by (a1 , b1 ) we may assume without loss of generality that
=
a0 a=
b1 because by X 0 = a0 S0 + b0 B0 = a1 S0 + b1 B0 this has no bearing on the initial
1 and b0
portfolio value. Therefore (a1 , b1 ) is taken as the initial allocation throughout [0,1] (not just
(0,1] ) which renders the choice of (a0 , b0 ) as a technical assumption and allows
identifying X n allocations with a sequence (an , bn ), n = 1,..., N .
28
Andrzej Korzeniowski
A call option replication portfolio
=
Π c { X n , (an , bn ), 0 ≤ n ≤ N } is determined by
Xn =
an S n + bn Bn =
Cn =+
(1 r ) − ( N − n ) E[( S N − K ) + | 
n ],
Cn ( S n −1u ) − Cn ( Sn −1 d )
=
an
∈ n −1 , =
n 1,..., N
S n −1u − S n −1 d
C − an S n Cn −1 − an S n −1
bn = n
=
∈ n −1 ,
n = 1,..., N
Bn
Bn −1
n=
0,..., N
(2)
(3)
(4)
One checks that X n , an ≥ 0 but bn can be negative in which case an investor owes −bn Bn or
equivalently the investor sold bn shares of bond short. Cn is determined by backward recursion
that starts at the terminal time N of the binomial tree according to
1
(5)
)
[ pCn (S n −1u ) + (1 − p )Cn (S n −1 d )],
Cn −1 (S n −1=
=
n N , N − 1,..., 2,1
1+ r
3. Construction of bond replication portfolio
The first step is to built the option replication portfolio
=
Π c { X n , (an , bn ), 0 ≤ n ≤ N } satisfying
(1)-(5). The second step amounts to recursive construction of a portfolio with values
{Yn , 0 ≤ n ≤ N } subject to (p1)-(p5) which is based on Π c and grows risk free.
The interval [0,1]. Set Y0 = a0 S0 − C0 ≡ a1 S0 − C0 as the initial capital. Define
Y0 = a1 S0 − C0 = a1 S0 − C0 + (Y0 + C0 − a1 S0 ) ≡ a1 S0 − C0 + g1 during (0,1) .
Then
Y1 = a1S1 − C1 = Y0 (1 + r ),
since a1 S0 − C0 becomes a1 S1 − C1 =
−b1 B1 =
−b1 B0 (1 + r ) = (a1 S0 − C0 )(1 + r ) =
Y0 (1 + r ) due to
C0 =X 0 =a0 S0 + b0 B0 =a1 S0 + b1 B0 and C
=
X=
a1 S1 + b1 B1 while g1 ≡ 0.
1
1
The interval (1,2]. Use the first step and define
Y1 = Y0 (1 + r ) = a2 S1 − C1 + (Y1 + C1 − a2 S1 ) ≡ a2 S1 − C1 + g 2 during (1, 2)
Then
Y2 = a2 S 2 − C2 + g 2 (1 + r ) = Y0 (1 + r ) 2 ,
since a2 S1 − C1 becomes a2 S 2 − C2 =
−b2 B2 and g 2 earns interest r during (1,2) yielding
g 2 (1 + r ) = Y1 (1 + r ) + (C1 − a2 S1 )(1 + r ) = Y1 (1 + r ) + b2 B1 (1 + r ) = Y0 (1 + r ) 2 + b2 B2 due to
C1 =X 1 =a1 S1 + b1 B1 =a2 S1 + b2 B1 .
The interval (n − 1, n], n =
3,..., N . Use the n − 1 step and define
n −1
Yn −1 = Y0 (1 + r ) = an S n −1 − Cn −1 + (Yn −1 + Cn −1 − an S n −1 ) ≡ an S n −1 − Cn −1 + g n on (n-1,n)
Then
Yn = an S n − Cn + g n (1 + r ) = Y0 (1 + r ) n , g n ∈ 
n −1
AAM: Intern. J., Vol. 1, Issue 1 (June 2006), [Previously, Vol. 1, No. 1]
29
by an argument similar to the case n = 1, 2.
Summing up, given {S n −1 , an , Cn −1 , Yn −1 } ∈ 
n −1 create portfolio positions for n = 1,..., N as
follows:
Open a position taking effect instantaneously after time n − 1 according to
● sell the entire holdings to receive Yn −1
● sell one option short to receive Cn −1
● buy an shares of stock to spend an S n −1
● hold the cash flow g n = Yn −1 + Cn −1 − an S n −1 earning interest r over ( n − 1, n)
Close the position at time n
● declare the portfolio value Yn
Theorem 1:
Let Π c { X n , (an , bn ), 0 ≤ n ≤ N } be the option replication portfolio. Then there exists a bond
=
replication =
portfolio Π {Yn , (an , g n ), 0 ≤ n ≤ N } of an shares of stock long, one call option
short and a carry-on cash flow g n such that
(i) Yn −1 = an S n −1 − Cn −1 + g n
during the time period (n-1,n)
(ii) Yn = an S n − Cn + g n (1 + r ) = Y0 (1 + r ) n
at time n ,
n = 1,..., N
(6)
Proof: By construction, it is a consequence of unique Y0 ≡ a0 S0 − (1 + r ) − N E[( S N − K ) + ] ,
a0 ≡ a1 , an , Cn found from (3) - (5) and g n =Yn −1 + Cn −1 − an S n −1 =(bn − b1 ) Bn −1 , because
an S n − Cn =
Y0 (1 r ) n −1 =
X n −1
−bn Bn and Yn −1 =+
−b1 B0 (1 + r ) n −1 =
−b1 Bn −1 thanks to Cn −1 =
=an −1 S n −1 + bn −1 Bn −1 =an S n −1 + bn Bn −1 .
Below we provide concrete calculations in the case N = 3.
Example 1. Let S=
K= 40, u=
0
5
,
4
d=
4
,
5
=
r
1
40
which gives p=
1+ r − d
1
= .
2
u −d
A special choice
of ud 1=
, r and consequently p 12 is only for keeping calculations simple but there are no
such restrictions in general, other than as specified in section 2. Solving (5) backward for Cn and
using (3) we obtain
=
C0 7.90,
=
a1 .636 and Y0 = a1 S0 − C0 = 17.54 .
a2 ( D) .338},{a=
a=
Then=
{a2 (U ) .826,=
1, a=
.555,
3 (U , U )
3 (U , D )
3 ( D, U )
=
a=
C1 (U ) 13.83,=
C=
0},{
C1 ( D) 2.38},{C=
23.47, C=
3 ( D, D )
2 (U , U )
2 (U , D )
2 ( D, U )
=
C2 ( D, D) 0},{C=
C=
C=
4.88,
38 18 , C=
10,
3 (U , U , U )
3 (U , U , D )
3 (U , D, U )
3 ( D, U , U )
C=
3 (U , D, D )
C=
C=
C=
0} . For the stock process
3 ( D, U , D )
3 ( D, D, U )
3 ( D, D, D )
=
{S1 (U ) 50,
=
S1 ( D) 32},{S=
62 12 , S=
S=
40, S=
25 53},
2 (U , U )
2 (U , D )
2 ( D, U )
2 ( D, D )
{S=
78 18 , S=
S=
S=
50, S=
S3 ( D, U , D)
3 (U , U , U )
3 (U , U , D )
3 (U , D, U )
3 ( D, U , U )
3 (U , D, D )
30
Andrzej Korzeniowski
.
= S=
32, S=
20 12
3 ( D, D, U )
3 ( D, D, D )
25 }
Calculations were carried out in several decimal places but only two or three digits are displayed
here as needed. We remark that the bond position b3 in the call option replication
portfolio Π c { X n , (an , bn ), 0 ≤ n ≤ N } is a random variable with a distribution corresponding to
=
binomial probabilities and thus not riskless! Indeed, by a3 (U , U ) = 1 it follows that
P
=
(b3
C3 (U ,U ,U ) − S3 (U ,U ,U )
=
)
(1+ r )3
2
P=
( S 2 S=
p2 , P
=
(b3
0u )
C3 (U , D ,U ) − S3 (U , D ,U )
=
)
(1+ r )3
P ( S 2 =S0 ud ∪ S 2 =S0 du ) =2 p (1 − p ), P (b3 =0) =P( S2 =S0 d 2 ) =(1 − p ) 2 .
We illustrate the actual portfolio trading and rebalancing on a single stock path which first goes
up then down and then up again, i.e.,
=
S0 40,
=
S1 (U ) S0 u , S 2 (U , D) = S0 ud ,
S3 (U , D
=
, U ) S=
50. Open a position on (0,1] by selling one option short and
0 udu
=
=
=
receive C0 7.90, which combined with
initial Y0 17.54
buys a1 .636 shares of stock
at the price S0
=
40
and leaves the cash flow g1 0. Close the position at time 1 with
=
=
= Y0 (1 + r ). Open a position on (1,2] by
Y1 a1 S1 (U ) − C1 (U
holdings valued
) 17.98
=
=
selling one
call option short to receive C1 (U ) 13.83,
which combined with initial Y1 17.98
requires to borrow g 2 (U ) = Y1 + C1 (U ) − a2 (U ) S1 (U) = − 9.49 to buy a2 (U ) = .826 shares
Y1 a2 (U ) S1 (U ) − C1 (U) + g 2 (U ). Close the
of stock at the price S1=
(U ) 50 and arrive at=
position at time 2 with holdings=
valued Y2 a2 (U ) S 2 (U , D) − C2 (U , D) + g=
2 (U )(1 + r )
= Y0 (1 + r ) 2 . Open a position on (2,3] by selling one call option short to receive
18.43
4.88,
=
which combined with initial Y2 =18.43 buys a3 (U , D) .555 shares of stock
C2 (U , D)
=
at the price S 2 (U , D) 40 and leaves g3 (U , D) = Y2 + C2 (U , D) − a3 (U , D) S 2 (U , D) = 1.11
to invest at rate r. Close the position at time 3 with holdings valued Y3 = a3 (U , D) S3 (U , D, U )
=
−C3 (U , D, U ) + g3 (U , D)(1 + r ) 18.89 which again matches the desired Y0 (1 + r )3 . The
analysis of the other seven cases is similar and shows that in each time step the porfolio so
devised matches the fixed rate of return given the initial investment Y0 .
Theorem 2: There exists a bond replication portfolio
=
Π b {Yn* , (an* , d n* ), 0 ≤ n ≤ N } of an*
shares of stock long and d n* call options short which replicates the bond exactly in each time
interval according to
(i)=
Yn*−1 an* S n −1 − d n*Cn −1
(ii) Y =a S n − d Cn =Y0 (1 + r )
*
n
where
*
n
*
n
during the time period (n − 1, n)
n
at time n ,
n = 1,..., N
(7)
AAM: Intern. J., Vol. 1, Issue 1 (June 2006), [Previously, Vol. 1, No. 1]
=
an*
b1
=
a , d n*
bn n
b1
bn
31
with (an , bn ) determined by Π c =
{ X n , (an , bn )}
(8)
Proof: It follows from Theorem1 because by C
=
X=
an S n −1 + bn Bn −1 we have
n −1
n −1
−bn Bn −1 = an S n −1 − Cn −1 which multiplied by
bn − b1
−bn
and consequently Yn −1= an S n −1 − Cn −1 + g n= an
gives g n = an (
b1
bn
S n −1 −
b1
bn
b1
− 1) S n −1
bn
−(
b1
− 1)Cn −1
bn
Cn −1= Yn*−1 as claimed.
To avoid the phrase instantaneously after time n the working assumption below is to use the term
at time n whenever referring to buy or sell in reference to opening or closing any given position.
Example 2. As before we illustrate the hedging for the up-down-up case. Solving for b1 ,
b
b2
b
b3
=
b2 b=
b3 (U , D) =
=
gives 1 .654
and 1 1.0627, whence=
(a1* , d1* ) (a1 ,1),
2 (U ), b3
=
=
(a2* , d 2* ) (.5405,
.654), (a3* , d3* ) (.5903, 1.0627). The first step is the same as in Example1
because g1 = 0. That=
is, start at time 0 with Y0 17.54,
=
short one call at C0 7.90 and buy
*
a=
a=
.636 shares of stock at S=
40, then at time 1 sell the stock at S1 (U=
) 50 and buy
1
1
0
17.98, short
back call at C1=
(U ) 13.83 to arrive =
at Y1 17.98
= Y0 (1 + r ). At time 1, with Y1 =
d 2* (U ) = .654 calls at C1 (U ) 13.83
=
and buy a2* (U )=.5405 shares of stock at S1 (U ) 50, then
at=
time 2 sell the stock at S 2 (U , D) 40
=
and buy back calls at C2 (U , D) 4.88 to arrive at Y2 =
18.43 =
18.43, short d3* (U , D) = 1.0627 calls at C2 (U , D) =
Y0 (1 + r ) 2 . At time 2, with Y2 =
4.88 and buy a3* (U , D) .5903
shares of stock at S 2 (U , D) 40, then at time 3 sell the stock
=
=
=
) 50 and buy back calls at C3 (U , D, U
=
) 10 to arrive at=
Y3 18.89
= Y0 (1 + r )3 .
at S3 (U , D, U
4. Convergence to Black-Scholes bond replication portfolio
Continuous time modeling assumes the actual stock price to follow exponential Brownian
motion process St = S0 e
( µ − 12 σ 2 ) t +σ Wt
with ESt = S0 e µ t . In the special case of µ = r it turns into the
( r − 1 σ 2 ) t +σ W
t
risk-neutral process St = S0 e 2
which corresponds to the discrete time risk-neutral
process S n discussed earlier. Here{Wt , 0 ≤ t ≤ T } is the standard Brownian motion, µ is the
average return rate on stock, r is the bond return rate and σ is the stock volatility. Notice that
the expected return on the risk-neutral process ESt = S0 e r t and when volatility is set to zero then
the risk-neutral process turns into the bond process itself. The Black-Scholes no-arbitrage option
price Ct e − r (T − t ) E[( ST − K ) + | St ] , where the conditional expected value is taken with respect to
=
the martingale measure corresponding to the discounted risk-neutral process, says that given the
actual stock price St = S0 e
C=
e
t
where
− r (T − t )
+
( µ − 12 σ 2 ) t +σ Wt
E[( ST − K ) | S=
e
t]
the option price at time t is calculated as follows:
− r (T − t )
E[( St e Z − K )1[ln[ K ],∞ ) ( Z )]
S
(9)
32
ST
Andrzej Korzeniowski
St e
( r − 12 σ 2 )(T −t ) +σ (WT −Wt )
≡ St e Z , Z is normal ~ N ((r − 12 σ 2 )(T − t ), σ 2 (T − t ))
(10)
Furthermore, the Black-Scholes bond replication portfolio
=
Π B − S {Yt , (at ,1), 0 ≤ t ≤ T } of at
shares of stock long and 1 call option short satisfies the following
a0 S0 − e − rT E[( ST − K ) + ]
Yt = at St − Ct = Y0 e r t with Y0 =
(11)
with at is determined by
∂Ct
|S
at =
∂S
∞
− r (T − t )
z
S=
Z
S St
t
ln[ KS ]
e
=
d
S∫
( Se − K ) dF ( z ) |
e − r (T − t ) E[e Z 1[ln[ K ], ∞ ) ( Z )]
=
(12)
St
Evaluating the above expectations gives Ct =St Φ (α ) − e − r (T −t ) K Φ ( β ) and at = Φ (α )
where
α
S
ln t + ( r + 1 σ 2 )(T −t )
2
,β
=K
σ T −t )
α σ T − t and Φ ( x) is the standard normal cumulative
=−
distribution.
To see how our previously constructed riskless portfolio Π b on binomial tree becomes the
celebrated Black-Scholes portfolio Π B − S in the limit, additional notation with certain particular
choices for underlying variables are in order.
Let in the discrete time model
T
1
N
t
n m, r ∆t be the return rate per period, =
u eσ ∆t , =
d
,=
,
= n, the time period ∆=
u
n
p=
er ∆t − d
u− d
Furthermore, let for any n = 1, 2,.., U n1 ,..., U nn be a sequence of independent
.
identically distributed random variables with P(U ni = u ) = p, P (U ni = d ) = 1 − p .
1 ≤ m ≤ n,
we
have
σ ∆t ( ε n1 +...+ε nm )
S m= S0U n1 ⋅ ⋅ ⋅ U nm= S0 e
Then for
and ε ni 's are i.i.d. P(ε ni= 1)= p,
P(ε ni =−1) =−
1 p. In the sequel, whenever invoking basic facts about convergence in
distribution, Billingsley (1999) is a standard reference without further mention. Denoting by [x]
the integer part of x we have the following convergence in distribution
d
m
(13)
m [n Tt ] → ∞
Z=
σ ∆t (ε n1 + ... + ε nm ) 
→ Z1 ~ N ((r − 12 σ 2 )t , σ 2 t ) ,=
1
and
d
m [n Tt ] → ∞ . (14)
Z mn=
σ ∆t (ε nm +1 + ... + ε nn ) 
→ Z ~ N ((r − 12 σ 2 )(T − t ), σ 2 (T − t )) ,=
+1
This follows because Z1m =W1 + ... + Wm + mEσ ∆tε ni , Wi =σ ∆tε ni − Eσ ∆tε ni , EWi = 0 ,
Eσ ∆tε ni = (r − 12 σ 2 )∆t + o(∆t ), Var(Wi )= σ 2 ∆t + o(∆t ), E | Wi |3 ≤ σ 3 ∆t 3 , m∆t → t and
m o(∆t ) → 0 as n ≥ m → ∞ . Therefore from Lindeberg’s Central Limit Theorem
W1+...+ Wm
Var (W1+...+ Wm )

→ N (0,1) , thanks to Lyapunov’s
d
E |W1|3 +...+ E |Wm |3
2σ 2 T
≤
n2
( Var (W1+...+ Wm ))3
3
condition, as n ≥ m → ∞ concludes the proof of (13). The proof of (14) is similar.
→0
AAM: Intern. J., Vol. 1, Issue 1 (June 2006), [Previously, Vol. 1, No. 1]
33
Lemma:
Let X n (ω ), Yn (ω '), n 1, 2,..., be independent defined on (ΩxΩ', x
=
', PxP ')
d
d
such that sup n Ee 2Yn < M < ∞ . If X n 
=
→ X X (ω ), Yn 
→ Y and=
F ( y ) P '(Y ≤ y )
is continuous then
Yn
Y
P(ω | lim Ee
=
1[ X n (ω ), ∞ ) (Yn ) Ee
=
1[ X (ω ), ∞ ) (Y )) 1
(15)
n →∞
Proof: Due to Skorokhod’s representation theorem one can assume without loss of generality
that X n , X are chosen in such a way that P(ω | lim X=
(ω )) 1 .
X=
n (ω )
n →∞
By assumption Fn ( y=
) P '(Yn ≤ y ) → F ( y ) for every real number y. Then denoting by
=
X n ∧ X min( X n , =
X ), X n ∨ X max( X n , X ) we have
EeYn 1[ X n (ω ), ∞ ) =
(Yn )
y
y
∫e 1
[ Xn ∧ X ,Xn ∨ X )
y
( y )dFn ( y ) + ∫ e 1[ X ,∞ ) ( y )dFn ( y ) → ∫ e 1[ X ,∞ ) ( y ) dF ( y )
because by Cauchy-Schwarz inequality the first integral above is dominated by
∫e
2y
dFn ( y ) ∫ 1[ X n ∧ X , X n ∨ X ) ( y )dFn ( y ) ≤ M ( Fn ( X n ∨ X ) − Fn ( X n ∧ X − 1n )) → 0
whence X n ∨ X → X , X n ∧ X − 1n → X and continuity of F implies Fn (yn ) → F ( y ) , whereas the
second integral
Yn
e 1[ X ,∞ ) (Yn ) are
EeYn 1[ X ,∞ ) (Yn ) converges to
uniformly
integrable
by
EeY 1[ X ,∞ ) (Y )
assumption
for X = X (ω )
sup n Ee
2Yn
fixed because
< M < ∞,
and
h( y ) = e 1[ X ,∞ ) ( y ) is continuous except when y = X which has probability zero due to continuity
of F ( y ) .
y
Theorem 3:
The discrete
time
bond
replication
portfolio
=
Π b {Yn* , (an* , d n* ), 0 ≤ n ≤ N}
with
Y= a S m − d C=
Y (1 + r ∆t ) , converges to the Black-Scholes bond replication portfolio
m
*
m
*
m
*
m
*
0
m
=
Π B − S {Yt , (at ,1), 0 ≤ t ≤ T }
with
Y=
at St − C=
Y0 e r t , 0 ≤ t ≤ T .
t
t
The
convergence
=
Π b {Y , (a , d )} =
→ Π B − S {Yt , (at ,1), 0 ≤ t ≤ T } means
*
n
*
n
*
n
(i) S m → St , Cm → Ct , am* → at , d m* → 1, Ym* → Y0 e rt
d
(ii) S n 
=
→ S {S0 e
( r − 12 σ 2 ) t +σ Wt
, 0 ≤ t ≤ T} in C [0, T ]
as =
m [n Tt ] → ∞
as n → ∞
Proof: Since part (ii) is standard, e.g., it follows from Prohorov’s generalization of Donsker’s
Invariance Principle for Brownian motion in Billinglsely (1999), it suffices to show (i). As
m
d
before, since by (13)
=
S m S0 e Z1 
→=
S0 e Z1 St , we may assume without loss of generality that
with probability one. By (2) with the following substitutions,
S m → St
(9)-(10),
Lemma
applied
to
N ===
n, n m, r : r ∆t and m =
[n Tt ] ,
d
=
Yn Z mn +1 
→ Z ,=
X n ln SKm → ln SKt and (15) we obtain with probability one
34
Andrzej Korzeniowski
Cm = (1 + r ∆t ) − ( n − m ) E[( S n − K ) + | S m ]
(16)
= (1 + r ∆t ) − ( n − m ) E[( S m e Zm+1 − K )1[ ln[ K ], ∞ ) ( Z mn +1 )] → e − r (T − t ) E[( St e Z − K )1[ ln[ K ], ∞ ) ( Z )] = Ct
n
Sm
St
thanks to continuity of Z and the fact that sup n Ee
Ee
2 Z mn +1
2Yn
< ∞ which follows from
= [ Ee 2σ∆t ]n − m = [e 2σ∆t p + e −2σ∆t )(1 − p )]n − m = [1 + (2r + σ 2 )∆t + o(∆t )]n − m
→ e(2 r +σ
2
)(T −t )
To see that am* → at , d m* → 1 it suffices to check that g m =
(bm − b1 ) Bm −1 → 0
and am → at because then
b1
*
=d
m
bm
→ 1.
Namely,
convergence
of
portfolio
values
Ym of Theorem 1 to Yt implies convergence Ym* of Theorem 2 to Yt provided g m → 0. To show
that g m → 0 it suffices to verify Cm → Ct , am → at , and
Y0 e r t
Ym= Y0 (1 +=
r ∆t ) m (a0 (Yn ) S0 − (1 + r ∆t ) − n E[( S n − K ) + ])(1 + r ∆t ) m → Yt =
= (a0 (YT ) S0 − e − rT E[( ST − K ) + ]) e r t .
The latter convergence follows from am → at
d
(i.e., a0 (Yn ) → a0 (YT ) for =
t=
0 ), S n S0 e Z1 + Zm+1 
=
→ S0 e Z1 + Z ST by (13)-(14) and the fact that
S n are uniformly integrable. Consequently, by (16), the proof will be concluded if we show that
am → at . Given S m −1 , by (2)-(3) applied to S m = S m −1u and S m = S m −1d we have
m
am =
n
Cm ( S m −1u ) − Cm ( S m −1d )
=
S m −1u − S m −1d
(1 + r ∆t ) − ( n − m ){E[( S m −1ue Zm+1 − K )1[ln[
n
n
K ], ∞ )
Sm −1u
( Z mn +1 )] − E[( S m −1de Zm+1 − K )1[ln[
K
Sm −1d
], ∞ )
( Z mn +1 )]}
S m −1u − S m −1d
(1 + r ∆t ) − ( n − m ) {E[( S m −1u − S m −1d )e Zm+1 1[ln[
n
n
K ], ∞ )
Sm−1u
( Z mn +1 )] + E[( S m −1de Zm+1 − K )1[ln[
K ], ln[ K
Sm−1u
Sm−1d
])
=
( Z mn +1 )]}
S m −1u − S m −1d
(1 + r ∆t ) − ( n − m ) E[( S m −1de Zm+1 − K )1[ln[
n
= (1 + r ∆t ) − ( n − m ) {E[e
Z mn +1
1[ln[
K ], ∞ )
Sm−1u
( Z mn +1 )] +
K ], ln[ K
Sm−1u
Sm−1d
])
( Z mn +1 )]
S m −1u − S m −1d
→ e − r (T − t ) E[e Z 1[ln[ K ],∞ ) ( Z )] = at , by Lemma and (12), as=
m [n Tt ] → ∞
St
Zn
since | ( S m −1de m +1 − K ) | ≤ S K u ( S m −1u − S m −1d )
m −1
E1[ln[
K
], ln[ S K d
Sm −1u
m −1
])
on the interval
[ln[
K ], ln[ K ]) while
Sm −1u
Sm −1d
d
→F
( Z mn +1 )] ≤ Fm (ln[ SmK−1d ]) − Fm (ln[ SmK−1u ] − m1 ) → 0, u= u (m) → 1 , Fm 
where Fm and F is the distribution function of Z mn +1 and Z respectively.
AAM: Intern. J., Vol. 1, Issue 1 (June 2006), [Previously, Vol. 1, No. 1]
35
Acknowledgements. The author thanks David Diltz of the UTA Finance Department for helpful
discussions and comments and the referees for suggestions that enhanced the original exposition.
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