t + dt)S t+dt + nb(S t+dt, t + dt)B t+dt cfw_ns(S t , t)S t+dt + nb(S t , t)B
t+dt(3.10) = h cfw_nb(S t+dt, t + dt)S t+dt + nb(S t+dt, t + dt)B t+dt cfw_nb(S
t , t)S t + nb(S t , t)B t i +ns(S t , t) h S t S t+dti + nb(S t , t) h B t B
t+dti = dV nsdS nb
earlier conclusions. This makes the subject exciting, both intellectually
and in practice. Once subtleties of multi-period investment are
understood, the reward in terms of enhanced investment performance
can be substantial. This chapter shows how to desi
Suppose we use the utility function U(x) = 2Mx x 2 and have a number
of assets a1, , am to invest upon for a total capital V0. Assume the
return of asset ai is Ri . Then for a portfolio with weight w = (w1, ,
wm), Pm i=1 wi = 1, on assets (a1, , am), its
needed, then step 1 is totally unnecessary. That is, only step 2 is the real
proof that the price of derivative security satisfies the Black-Scholes
equation. We present step 1 here is to let the reader see how BlackScholes equation is first formally deri
becomes zero coupon bond in the financial market, it has the simplest
case flow cfw_(0, 0),(1, 0), ,(19, 0),(20, 1000). Of course, each coupon in
the above example can also be regarded as a zero-coupon bond. The
mathematical usage of a zero-coupon bond is
functions that are popular: 1. Exponential U(x) = e x, > 0. This
utility function has negative values, but it does not matter, as long as it
is strictly increasing. 2. Logarithmic U(x) = ln x, x > 0, U(x) = x 6 0.
Note that this function has a severe pena
SdW + nbrBdt = cfw_nsS + nbrBdt + nsSdWt (3.7) after we plug in the
assumed dynamics for prices of the stock and bond. Note that dV t is a
random variable, normally distributed. Now assume that V t can be
written as V (S t , t) where V (, ) is a certain k
set A F defined by 1A() = 1 if A, 0 if 6 A. A simple
function is a linear combination of finitely many characteristic functions.
For a simple function Pn i=1 ci1Ai , its integral is defined as Z Xn i=1
ci1Ai () ! P(d) := Xn i=1 ciP(Ai). The integral of a
+ R]T M(0), M(t) = 1 [1 + R] tT 1 [1 + R]T M(0). In our example, R
= 0.005, T = 180, M(0) = 100, 000 so the monthly payment is P = 100,
000 0.005/[1 (1.005)180] = $843.85 (iii) The present value of the
mortgage borrowers cash flow, under constant interest
> 0) distributed, or with mean and standard deviation (i.e.
variance 2 ) if it has the probability density (x) := 1 22 exp (x
) 2 2 2 . When = 0, a N(, 0) random variable X becomes a
deterministic constant function X() for (almost) all . In the
sequel,
satisfy the Black-Scholes equation. 2. Now let V be the solution to the
Black-Scholes equation with initial condition V (s, T) = f(s) for all s > 0.
Let ns and nb be defined as in (3.9). Consider the portfolio consisting of
ns(S t , t) shares of stock and
the portfolio exactly pays the claim. Thus there is no future payoff and
we have a profit at time t. Similarly, if V (S t , t) < V (S t , t) one can do
other way around. Since such arbitrage is excluded from the
mathematical perfection, we conclude that t
Exercise 3.21. Assume risk-free rate is r and volatility of a stock is > 0.
Both r and are constants. Find the price for European put and call
options, with duration time T and strick price K. 104 CHAPTER 3. ASSET
DYNAMICS Chapter 4 Optimal Portfolio Grow
information on short term risk-free interest rates needed in a state
model is a complicated matter. We omit detailed study here. 3. Pricing a
Security with Dividend Suppose we are to price a derivative security
which has a cash flow cfw_(t, d(t, )tT where
beauty and deep theoretical analysis, we would like to take the limit as
t 0 to obtain continuous models, for which, calculus will be very
useful. 2. Assets. In our consideration there are two assets: a0 : a riskfree asset with constant continuous compoun
individuals age. Financial planers can obtain such function by asking
certain questions and based on answers to have values on certain
parameters in a general formula, typically linear combinations of
exponential functions. Exercise 4.1. (certainty equiva
(market expected) short term interest rates, for the first half and second
half year respectively. Construct portfolios that support your conclusion.
72 CHAPTER 2. FINITE STATE MODELS Exercise 2.22. Suppose a stock,
currently $100 per share, pays a fixed
probability is needed to build upon a subset of the space Map(T; R) of
functions. Exercise 3.5. Assume that f : R R is a simple function. Prove
the law of unconscious statistician. Exercise 3.6. Let = cfw_1, 2, 3, 4. Let F
be the smallest -algebra that co
than V (S t , t) at some time t < T and some spot stock price S t , then
there is an arbitrage. Exercise 3.19. Note that in (3.10) can be
expresses as = S t+dt[ns(S t+dt, t + dt) ns(S t , t)] + B t+dt[nb(S t+dt, t
+ dt) n t b (S t , t)] = S t+dtdns + B t+
stochastic process cfw_Wtt>0 defined earlier. Here the probability space
for the process is determined by the probability space associated with
the random variables cfw_X1 i=1. A standard probability space (, F, P)
for a sequence cfw_Xi i=1 of binary rand
on (, F, P) and B is a Borel set of R, then X1 (B) := cfw_ | X() B
is a measurable set. In the sequel, we use notation, for every Borel set B
in R, Prob(X B) := P(X1 (B) = P(cfw_ | X() B). Note that the
mapping: PX1 : B B P(X1 (B) defines a probability me
important to know that plays no rule here. This is one of the Black
Scholes most significant contribution towards the investment science.
That is irrelevant is due to the fact that only risk-neutral probability
play roles here. 3.8 The BlackScholes Equati
presented by Sharpe in [28] and later developed by Cox, Ross,
Robinstein [3] and also Rendleman and Bartter [23]. 73 74 CHAPTER 3.
ASSET DYNAMICS 3.1 Binomial Tree Model To define a binomial tree
model, a basic period length of time is established (such a
invertor prefers a particular utility function U and has a total capital V 0
> 0 to invest among m assets a1, , am. For i = 1, , m, the asset ai
has an initial unit share price S 0 i > 0 and a final (end of period) price
Si , a non-negative bounded random
binary lattice cfw_(tk, uid ki ) | k = 0, 1, , i = 0, , k of size Pn i=0(i +
1) = (n + 1)(n + 2)/2. From each node, there are two outgoing arrows,
one for up and one for down. For a tree, each node (except root) has
only one incoming arrow; for lattice, e
the optimal portfolio. Exercise 4.3. Show that (i) the absolute risk
aversion coefficient is a constant for exponential utility functions, and
(ii) the relative risk aversion coefficient is constant for logarithmic and
power utilities. Exercise 4.4. Suppo
length, both with probability 1/2. Such digitized Brownian motion
seems very special, nevertheless, in its limit, it can form most of the
known stochastic processes. This phenomena has its root from the
central limit theorem which asserts that almost all
is u kd nk fold of its initial price. While there are C k n many ways to
reach this price, if only the stock prices are relevant to the problem,
then we can combine all those nodes which give the same price. In this
ways, the 2n binary states can be repla
reality, the model parameter (u, d, q) has to satisfy the matching
condition qu + (1 q)d = 1 + E(R), q(1 q)(u d) 2 = Var(R). There are
three parameters for two equations, so one of the parameter is free.
One can show that if the single period is sufficien
portfolio that matches exactly the required payment for contingent
claim, regardless of which state the stock price lands on. Here is the
same situation. The randomness is get rid of by matching the
coefficients of two dV 0 s. The former from the actual b