Download Lecture 4 From Binomial Trees to the Black

Lecture 4
From Binomial Trees to the Black-Scholes Option
Pricing Formulas
In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees,
as an important model for a single risky security. It has been extensively used by practitioners
in pricing various kinds of derivatives of stocks or bonds. Historically, the model was proposed
independently by Cox/Ross/Rubinstein (1979, J. Fin. Econ. 7, 229-263) and Rendleman/Bartter
(1979, J. Fin. 34, 1093-1110), although it was often referred to as the CRR model. Furthermore,
we will show that the celebrated Black-Scholes formulas in option pricing can be derived from the
binomial option pricing formulas through an asymptotic argument, provided the parameters in the
binomial model are set appropriately.
4.1
The basic binomial tree model
The evolution of a risky security, say stock, is represented by S = {S(t), t = 0, 1, . . . , T }. Starting
from an initial (positive) price S(0), assume in each time period the stock price either goes up by
a factor u > 1 with probability p, or goes down by a factor 0 < d < 1 with probability 1 − p. The
moves over time are iid Bernoulli random variables. For each t, S(t) = S(0)unt dt−nt , where nt
represents the number of up moves up to t.
The bank account process B is deterministic with B(0) = 1 and a constant interest rate 0 <
r < 1. Hence B(t) = (1 + r)t .
The filtration FF is taken as the one generated by the history of S. The sample space Ω contains
K = 2T different paths. The underlying probability P is defined by P (ω) = pU (ω) (1 − p)T −U (ω) ,
where U (ω) represents the total number of up moves in the path ω. We assume 0 < p < 1 so that
P (ω) > 0 ∀ω ∈ Ω.
As for EMMs, we have the following
Proposition 4.1 There exists a unique EMM Q ⇐⇒ d < 1 + r < u. In this case,
Q(ω) = q U (ω) (1 − q)T −U (ω) ,
Proof
with q =
1+r−d
.
u−d
Let ξt = nt − nt−1 . Then for every t, S ∗ (t) = S ∗ (t − 1) (1 + r)−1 uξt d1−ξt . Therefore,
EQ [S ∗ (t) | Ft−1 ] = S ∗ (t − 1)
⇐⇒
⇐⇒
u Q(ξt = 1 | nt−1 ) + d [1 − Q(ξt = 1 | nt−1 )] = 1 + r
1+r−d
Q(ξt = 1 | nt−1 ) =
,
u−d
1
(4.1)
where Q(ξt = 1 | nt−1 ) denotes the conditional probability (under Q) that the next move is up given
nt−1 up moves up to time t − 1. We can denote this (constant) conditional probability by q since
it does not depend on t or nt−1 . This implies that ξ1 , . . . , ξT are iid Bernoulli random variables,
and the martingale measure Q is given by (4.1). Note that 0 < Q(ω) < 1 for every ω if and only if
0 < q < 1 if and only if d < 1 + r < u. The above argument also shows such an EMM Q is unique.
Corollary 4.1 The binomial tree model is a complete market.
Exercise 4.1 Show in the binomial tree model, the return R(t) = ∆S(t)/S(t − 1) has the (risk
neutral) expectation EQ R(t) = r (interest rate for the money market) for all t ≥ 1. This is true in
general (later).
4.2
Option pricing using binomial trees
A European option is a contingent claim such that the owner of the option may choose (but with
no obligation) to exercise it at an expiry or expiration time T and receive the payment Y from
the writer of the option. Naturally, the option should be exercised if and only if the payment is
positive.
In the simplest case, the contingent claim is expressed as Y = g(S(T )) with some function
g. Using (3.1) in the binomial tree model, the pricing formula for a European option at time
t = 0, 1, . . . , T − 1 is given by
T
−t
X
1
V (t) =
(1 + r)T −t k=0
Ã
!
T −t k
q (1 − q)T −t−k g(S(t)uk dT −t−k ).
k
(4.2)
Here are some examples.
Example 4.1 Call options. g(S(T )) = (S(T ) − c)+ where c > 0 is called the exercise price or
strike price. A special case was given in Lecture 2. Note that S(t)uk dT −t−k − c > 0 ⇐⇒ k >
log (c/(S(t)dT −t ))
. Let k ∗ be the smallest k such that this inequality holds. If k ∗ > T − t, then
log (u/d)
V (t) = 0. If k ∗ ≤ T − t, then (4.2) becomes
V (t) = S(t)
T
−t
X
k=k∗
Ã
!
Ã
!
T
−t
X
T −t k
c
T −t k
T −t−k
b (1 − b)
−
q (1 − q)T −t−k ,
(1 + r)T −t k=k∗
k
k
(4.3)
where b = qu/(1 + r) ∈ (0, 1) (why?). The nice thing about this formula is that it involves two
sums of T − t − k ∗ + 1 binomial probabilities.
Example 4.2 Put options. Set g(S(T )) = (c − S(T ))+ . The owner of this option normally
chooses to sell the stock at T for the strike price c if S(T ) < c (thus make the profit c − S(T )), or
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chooses not to exercise the option if S(T ) ≥ c. A pricing formula similar to (4.3) can be derived
easily.
Note: Denote by ct and pt respectively, the time t values of the European call and put options
with the same expiry T and exercise price c. Since (S(T ) − c)+ − (c − S(T ))+ = S(T ) − c, we have
the following put-call parity
ct − pt = S(t) −
c
.
(1 + r)T −t
(4.4)
Example 4.3 Chooser options. A chooser option is an agreement that the owner of the option
has the right to choose at a fixed decision time T0 < T whether the option is to be a call or a put
with a common exercise price c and remaining time to expiry T − T0 . To determine the time t value
of the chooser option (t ≤ T0 ), notice that the payoff at T is
(S(T ) − c)+ IA + (c − S(T ))+ IAc = (c − S(T ))+ + IA (S(T ) − c),
where the event A = {cT0 > pT0 }, Ac is the complement of A, and IA is the indicator of A. By the
put-call parity, cT0 − pT0 = S(T0 ) − c (1 + r)−(T −T0 ) , which leads to A = {S(T0 ) > c (1 + r)−(T −T0 ) }.
Therefore, the time T0 value of the chooser option is given by
(1 + r)−(T −T0 ) EQ [(c − S(T ))+ + IA (S(T ) − c) | FT0 ]
¯
·
¸
S(T ) − c ¯¯
= pT0 + IA EQ
FT0
(1 + r)T −T0 ¯
·
¸
c
= pT0 + IA S(T0 ) −
(1 + r)T −T0
·
¸+
c
= pT0 + S(T0 ) −
.
(1 + r)T −T0
Introducing the notation C(t, T, c) (resp. P (t, T, c)) for the time t value of a call (resp. put) option
with the expiry T and exercise price c, then for any t = 0, 1, . . . , T0 , the time t value Vch (t) of the
chooser option can be represented as
³
´
Vch (t) = P (t, T, c) + C t, T0 , c (1 + r)−(T −T0 ) ,
(4.5)
or equivalently (why?) as
³
´
Vch (t) = C(t, T, c) + P t, T0 , c (1 + r)−(T −T0 ) .
Exercise 4.2 Verify (4.5) and (4.6).
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(4.6)
4.3
The Black-Scholes Option Pricing Formulas
Fix T > 0, a real number. For a positive integer n, partition the interval [0, T ) into
[(j − 1)T /n, jT /n), j = 1, . . . , n. The previous notation S(j) in the binomial model now represents
the stock price at time jT /n. Similarly, B(j) represents the bank account at time jT /n. Let rn =
rT /n be the interest rate, where r > 0 is thought of as the q
instantaneous rate with the continuous
n
rT
compounding, since limn→∞ (1 + rn ) = e . Let an = σ Tn where σ > 0 is interpreted as the
instantaneous volatility. Set the up and down factors by un = ean (1 + rn ) and dn = e−an (1 + rn ).
Note that dn < 1 for sufficiently large n.
The risk neutral probability, as n → ∞, has the asymptotic expression
qn =
=
=
=
1 + rn − dn
un − dn
1 − e−an
ean − e−an
an − 12 a2n + o(a2n )
2an + 13 a3n + o(a3n )
1
1
− an + o(an ),
2
4
where the notation o(²) with ² > 0 means o(²)/² → 0 as ² → 0.
Recall the iid Bernoulli random variables ξj , j = 1, . . . , n introduced in Section 4.1, with
Q(ξj = 1) = qn . The stock price at T is represented as
1 +···+ξn )
S(n) = S(0) uξn1 +···+ξn dn−(ξ
.
n
Hence the value of the put option at time 0 is given by
(n)
p0
µ
= (1 + rn )−n EQ (c − S(n))+ = EQ
c
− S(0) eYn
(1 + rn )n
¶+
,
(4.7)
where
Yn =
n
X
Yn,j =
j=1
n
X
j=1
Ã
log
ξ
1−ξ
unj dn j
1 + rn
!
.
(4.8)
Note that for fixed n, Yn,1 , . . . , Yn,n are iid random variables with
EQ Yn,j = qn log
un
dn
−1 2
+ (1 − qn ) log
=
a + o(a2n ),
1 + rn
1 + rn
2 n
2
EQ Yn,j
= a2n ,
(4.9)
(4.10)
4
and
EQ |Yn,j |m = o(a2n ) ∀ m = 3, 4, . . . .
(4.11)
Using characteristic functions [see the note after (4.14)], it follows that Yn converges in distribution
to N (−σ 2 T /2, σ 2 T ) as n → ∞. It is noteworthy that the family {Yn,j } is a triangular array, hence
the asymptotic distribution of Yn need not always belong to the Gaussian distribution family. In
other words, the argument here goes somewhat beyond the basic form of “Central Limit Theorem”.
Since
³
(n)
|p0 − EQ
c e−rT − S(0) eYn
´+
| ≤ c |(1 + rn )−n − e−rT |,
(why?)
(4.12)
we have
(n)
lim p
n→∞ 0
=
=
lim EQ
n→∞
³
c e−rT − S(0) eYn
Z ∞ −z 2 /2 "
e
−∞
√
2π
´+
Ã
−rT
ce
− S(0) exp
√
σ2T
+ σ Tz
−
2
!#+
dz
= c e−rT Φ(−v2 ) − S(0) Φ(−v1 ),
2
/2) T
√
where v1 = log(S(0)/c)+(r+σ
,
σ T
distribution function of N (0, 1).
√
v2 = v1 − σ T =
2 /2) T
log(S(0)/c)+(r−σ
√
,
σ T
and Φ is the cumulative
This is the Black-Scholes pricing formula for a European put option. We choose to consider put
options first since their payoff (or loss) functions are bounded which make the asymptotic argument
easier. The following pricing formula for a call option can be derived using put-call parity:
(n)
lim c
n→∞ 0
= S(0) Φ(v1 ) − c e−rT Φ(v2 ).
Furthermore, by changing 0 to any t ∈ (0, T ) and T to T − t, the same argument goes through,
which provides the Black-Scholes formulas for pricing the time t value C(t, T ) of a (European) call
option:
C(t, T ) = S(t) Φ(v1 ) − c e−r(T −t) Φ(v2 ),
(4.13)
and the time t value P (t, T ) of a (European) put option:
P (t, T ) = c e−r(T −t) Φ(−v2 ) − S(t) Φ(−v1 ),
where v1 =
2 /2) (T −t)
log(S(t)/c)+(r+σ
√
σ T −t
√
and v2 = v1 − σ T − t =
5
(4.14)
2 /2) (T −t)
log(S(t)/c)+(r−σ
√
.
σ T −t
Note: To verify that Yn converges in distribution to N (−σ 2 T /2, σ 2 T ) as n → ∞, consider the
√
characteristic function EQ eiwYn of Yn where w ∈ IR and i = −1 (imaginary unit in complex
analysis). Following the fact that Yn,1 , . . . , Yn,n are iid, and (4.9) — (4.11), we have the Taylor
expansion
EQ e
iwYn
=
n
Y
EQ eiwYn,j
j=1
Ã
=
1 + iwEQ Yn,j
w2
iθ3
2
3
−
EQ Yn,j
−
EQ Yn,j
2
3!
!n
−→ exp (−iwσ 2 T /2 − w2 σ 2 T /2)
as n → ∞, where θ satisfies |θ| ≤ |w|. Note that exp (−iwσ 2 T /2−w2 σ 2 T /2) is just the characteristic
function of N (−σ 2 T /2, σ 2 T ).
Exercise 4.3 Derive the formula (4.13).
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