Lecture Notes In Mathematical Finance

114 Pages·1996·0.535 MB·English

by  Lin X.S.

LECTURE NOTES IN MATHEMATICAL
FINANCE
X. Sheldon Lin
Department of Statistics & Actuarial Science
University of Iowa
Iowa City, IA 52242
Phone: [phone]
Fax: [phone]
Email: [email protected]
Comments are welcome!
c X. Sheldon Lin, 1996.
(cid:13)
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Contents
I Discrete-Time Finance Models 4
1 Basic Concepts and One Time-Period Models 5
1.1 The Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Characterisation of No-Arbitrage Strategies . . . . . . . . . . . . . . . . . 8
1.4 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Discrete-Time Stochastic Processes and Lattice Models 16
2.1 Discrete-Time Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 General Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 No-Arbitrage Valuation 28
3.1 No-Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Risk-Neutral Probability Measures . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Binomial Models of Option Pricing . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Binomial Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Multinomial/Multifactor Interest Rate Models . . . . . . . . . . . . . . . . 46
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II Continuous-Time Finance Models 51
4 Stochastic Calculus 52
4.1 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Wiener Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Re(cid:13)ection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Stochastic(Ito) Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Stochastic Di(cid:11)erential Equations and Ito’s Lemma . . . . . . . . . . . . . . 67
4.6 Feynman-Kac Formula and Other Applications . . . . . . . . . . . . . . . . 72
4.7 Option Pricing: Dynamic Hedging Approach . . . . . . . . . . . . . . . . . 75
4.8 Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Multi-Dimensional Ito Processes . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Continuous-Time Finance Models 89
5.1 Security Markets and Valuation . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Digital and Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4 Swaps and Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A Probability Theory 106
B Functional Analysis 109
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Part I
Discrete-Time Finance Models
4
Chapter 1
Basic Concepts and One Time-Period
Models
1.1 The Basic Setup
We consider a security market with the following conditions:
There are only two consumption dates: the initial date t = 0 and the terminal date
(cid:15)
t = T. Trading takes place at t = 0 only.
There are (cid:12)nite number of states of economy
(cid:15)
(cid:10) = !1;!2; ;!J
f (cid:1)(cid:1)(cid:1) g
with the probability at state !j being P(!j):
Hence((cid:10); ;P)consistsofaprobabilityspacewiththe(cid:27)-algebrabeingallthesubsets
F
of (cid:10).
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There are N primitive securities. The n-th security has price pn at time 0 and
(cid:15)
terminal payo(cid:11)
dn(!1)
0 1
dn(!2)
dn = B C
B .. C
B . C
B C
B C
B C
B dn(!J) C
B C
@ A
Thus, we have a price system
p = (p1;p2; ;pN)0;
(cid:1)(cid:1)(cid:1)
where 0 denotes the corresponding transpose, and the payo(cid:11) matrix
d1(!1) dN(!1)
0 . (cid:1)(cid:1)(cid:1) . 1
D = .. ..
B C
B C
BB d1(!J) dN(!J) CC
B (cid:1)(cid:1)(cid:1) C
@ A
Investorsarepricetakersandhavethehomogeneousbelief P = (P(!1);P(!2); ;P(!J)).
(cid:15) (cid:1)(cid:1)(cid:1)
There is only one perishable consumption good.
(cid:15)
1.2 Trading Strategies
If an investor possesses (cid:18)n shares of security n, the portfolioof the securities of the investor
N
has the payo(cid:11) n=1(cid:18)ndn at time T. Let e(0);e(T) be the initial endowment and the
P
terminal endowment for the investor, respectively. Thus, the investor’s consumptions are
N
c(0) = e(0) (cid:18)npn; (1.1)
(cid:0)n=1
X
N
c(T) = e(T)+ (cid:18)ndn: (1.2)
n=1
X
We call (cid:18) = ((cid:18)1;(cid:18)2; ;(cid:18)N)0 a trading strategy. The set (e;p) containing all consumption
(cid:1)(cid:1)(cid:1) B
processes c = (c(0);c(T)) over all (cid:18) is called the budget set with respect to the endowment
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process e = (e(0);e(T)) and the price system p. Mathematically, a budget set is an a(cid:14)ne
J+1
space of R .
A consumption process is said to be attainable if its terminal consumption can be
expressed as the payo(cid:11) of a portfolio, i.e.
N
c(T) = (cid:18)ndn:
n=1
X
It is easy to see that a consumption process is attainable if and only if
Rank(D) = Rank(D;c(T)):
It is also easy to see that the terminal consumption of any attainable consumption process
isintheimageofthematrixD, regardedasalinearmap. Thus, every consumptionprocess
is attainable if and only if Rank(D) = J, therefore, if and only if there are J independent
securities. In this case, we say the market is complete. Otherwise, we say the market is
incomplete. We will see later that if the market is complete, any consumption process can
be priced uniquely.
When the market is not complete, there is a need to create new securities in order
to complete the market. One approach is to create derivative securities on the existing
securities such as European-type options.
A European call option written on a security gives its holder the right( not obligation)
to buy the underlying security at a prespeci(cid:12)ed price on a prespeci(cid:12)ed date; whilst a
European put option written on a security gives its holder the right( not obligation) to
sell the underlying security at a prespeci(cid:12)ed price on a prespeci(cid:12)ed date. The prespeci(cid:12)ed
price is called the strike price and the prespeci(cid:12)ed date is called the expiration or maturity
date.
(cid:22) (cid:22) (cid:22)
Given a security with terminal payo(cid:11) d = (d(!1); ;d(!J))0, the payo(cid:11) of a European
(cid:1)(cid:1)(cid:1)
call option with strike price K then is
(cid:22)
max d K; 0 :
f (cid:0) g
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Similarly, the payo(cid:11) of a European put option with strike price K then is
(cid:22)
max K d; 0 :
f (cid:0) g
Example 1.1 Consider two securities with payo(cid:11) d1 = (1;2;4)0; d2 = (2;0;1)0; respec-
tively. Since the number of the states is 3 and the number of securities is 2, the market
is not complete. Write a European call option on the (cid:12)rst security with strike price 1.
Then its payo(cid:11) is d3 = (0;1;3)0: These three securities are algebraically independent and
therefore complete the market.
We now consider no arbitrage strategies. A trading strategy (cid:18) = ((cid:18)1;(cid:18)2; ;(cid:18)N)0 is said
(cid:1)(cid:1)(cid:1)
to admit arbitrage if either
N N
(cid:18)npn = 0; and (cid:18)ndn 0 (1.3)
n=1 n=1 (cid:21)
X X
N
with n=1(cid:18)ndn(!j) > 0 for some j,
P
or
N N
(cid:18)npn < 0; and (cid:18)ndn 0 (1.4)
n=1 n=1 (cid:21)
X X
These conditions imply that with the zero endowment process, we will be able to obtain a
nonzero nonnegative consumption process.
1.3 Characterisation of No-Arbitrage Strategies
We are now lookingfor the necessary and su(cid:14)cient condition under which the price system
does not admit arbitrage.
We (cid:12)rst recall the Hahn-Banach Theorem which will be used to derive the condition.
Theorem 1.1(Hahn-Banach) Let A and B be two disjoint convex sets in a Hilbert
space . Assume that there exist a A and b B such that d(A;B) = a b ; where
H 2 2 k (cid:0) k
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d(A;B) is the distance between A and B de(cid:12)ned by
d(A;B) = inf x y ; for any x A and y B :
fk (cid:0) k 2 2 g
Then, there exists a z and a scalar h such that for any x A; x z > h, and for any
2 H 2 (cid:15)
y B; y z < h: See Appendix B for a proof.
2 (cid:15)
It is easy to see that the price system admits arbitrage if and only if some consumption
J+1
processwithzeroendowmentprocessliesinthesetR+ 0 :Thus, noarbitragecondition
(cid:0)f g
J+1
is equivalent to the condition that the sets (0;p) and R+ 0 are separate. Suppose
B (cid:0)f g
J+1 1
this is the case. Let A = x R+ ;x0 + + xJ 2 . Then it can be shown (see
f 2 (cid:1)(cid:1)(cid:1) (cid:21) g
Appendix B) that there exist a A and b (0;p) such that d(A; (0;p)) = a b :
2 2 B B k (cid:0) k
By the Hahn-Banach Theorem, there is a z = (z0;z1; ;zJ)0 and a scalar h such that for
(cid:1)(cid:1)(cid:1)
any x A; x0z > h, and for any y (0;p); y0z < h: Since (0;p) is a linear space, y0z
2 2 B B
is either 0 or unbounded from above on (0;p). Thus, y0z = 0 for all y (0;p). This
B 2 B
means that
(cid:18)0D0z(cid:22)= z0(cid:18)0p
for any (cid:18), where z(cid:22)= (z1; ;zJ)0. That (cid:18) is arbitrary implies
(cid:1)(cid:1)(cid:1)
D0(cid:11) = p; (1.5)
z1 z2 zJ
where (cid:11) = (z0; z0; ; z0)0: It is easy to see that h > 0: Let sj be the vector whose
(cid:1)(cid:1)(cid:1)
(j +1)-th component is 1 and others 0. Then sj A and hence s0jz > 0, which implies
2
zj > 0; j = 0;1; ;J. Thus, (cid:11) > 0:
(cid:1)(cid:1)(cid:1)
Therefore, the price system does not admit arbitrage implies that there is a vector (cid:11),
all of whose components are positive, such that the equation (1.5) holds.
Conversely, if there is a positive vector (cid:11) such that the equation (1.5) holds, there will
^ ^
be no arbitrage. Otherwise, let (cid:18) be an arbitrage trading strategy. Multiplying (cid:18)0 both
sides of the equation from the left gives an inequality, which is a contradiction.
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Summerizing the above arguments, we conclude that
Theorem 1.2 The price system does not admitarbitrageif and only if there isa positive
vector (cid:11) such that
D0(cid:11) = p: (1.6)
Let us now consider the case that one of these securities is a riskless bond, say the (cid:12)rst
security. Denote r the rate of return of the bond. Thus, d1 = (1+r)p1: The (cid:12)rst equality
in equation (1.6) gives (1+r)((cid:11)1+(cid:11)2+ +(cid:11)J) = 1: Let Q(!j) = (1+r)(cid:11)j; j = 1; ;J.
(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
Then, Q = (Q(!1); ;Q(!J))0 is a probability measure on ((cid:10); ) and the equation ((1.6)
(cid:1)(cid:1)(cid:1) F
becomes
D0Q = (1+r)p: (1.7)
The nth scalar equation in (1.7) gives
J
dn(!j)Q(!j) = (1+r)pn:
j=1
X
Thus,
EQ(dn)
pn =
1+r
and
K
1
EQ(Rn) = dn(!j)Q(!j) 1 = r;
pn j=1 (cid:0)
X
dn
where Rn = pn 1:
(cid:0)
Hence, the price system does not admit arbitrage if and only if there is a probabil-
ity measure Q on ((cid:10); ) such that under this measure, the price of each security is the
F
discounted value of its expected payo(cid:11) and all securities have the same expected rate of
return. The probability measure Q is often referred to as a risk-neutral probability mea-
sure. If the market is complete it uniquely exists under no arbitrage condition. However,
it the market is not complete, there are more than one risk-neutral probability measure.
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