Practice Exam for the rst Midterm (Time: 90 minutes)
1. Evaluate the following integrals: (a) (b)
8
x3 +
0
1 x
dx
e2x + 3 cos (2x) dx
(Remark: your nal answer may not contain any sin or cos-func
derivative asset (see example sheet 1). We will mostly explore the first
perspective in this chapter, but will return to the second perspective in
our study of HJM models. 2. Bank accounts to bond pri
equation, 81 law of iterated expectations, 34 Lebesgue measure, 103
local martingale, 37 local volatility, 86 long interest rate, 95 market price
of risk, 75 Markovs inequality, 107 martingale, 34 mar
subsets of such that (1) if A F then Ac F, (2) if A1, A2, . . . F
then S i=1 Ai F. The terms sigma-field and sigma-algebra are
interchangeable. The Borel sigma-field B on R is the smallest sigma-field
complex derivative securities. He lied simply to appear respectable!
There you have it. Its not fashionable to be a financial mathematician
these days. On the plus side, these quants or financial math
formula for the moment generating function is hard to call beautiful, it
is very explicit. In particular, given the set of model parameters (v0, , v,
c ), the function can be evaluated very quickly on
seventies, the market of financial derivatives has grown in notional
amount to about $600 trillion in 2007. This compared to the world GDP
of order $45 trillion. Amongst financial derivatives, as of 2
The market has no arbitrage and is incomplete. 6. The Black
Scholes model and formula We will consider the simplest possible
model of the type studied introduced above. Consider the case of a
market
T)t[0,T] and ( (i) (t, T)t[0,T] . Let the short rate be given by rt = f(t,
t) and the bank account dynamics by dBt = Bt rt dt. Finally, let the bond
prices be given by P(t, T) = e R T t f(t,s) ds . Th
have been around for some time (we do not discuss futures in this
chapter; they are very similar to the forward contracts discussed
below). The oldest and the largest futures and options exchange, The
variable X pX the mass function of a discrete random variable X fX the
density function of an absolutely continuous random variable X X the
characteristic function of X E(X) the expected value of the
continuously differentiable in the x variable. Then df(t, Xt) = f t (t,
Xt)dt + Xn i=1 f xi (t, Xt) dX(i) t + 1 2 Xn i=1 Xn j=1 2 f xixj (t, Xt)
dhX (i) , X(j) it 4. Girsanovs theorem As we have seen
t if i = j 0 if i 6= j. Then (Xt)t0 is a standard m-dimensional Brownian
motion. Proof. Fix a constant vector R m and let i = 1. Consider
Mt = e iXt+| 2 t/2 . By Itos formula, dMt = Mt i dXt + | 2 2 d
Xtn1 ) 2 Z t 0 f 0 (Xs)dXs + Z t 0 1 2 f 00(Xs)dhXis. 3.2. The multidimensional version. We now introduce the vector version of Itos
formula. It is basically the same as before, but with worse notatio
Z T 0 e tr0 + (1 e t)r 2 2 2 (1 e t) 2 dt so that f(0, T) = e
tr0 + (1 e t)r 2 2 2 (1 e t) 2 By the time-homogeneity of
the Vasicek model, we can actually deduce the formula f(t, t + x) = rte
x + r(1
r p 1 then Xn X in Lr Xn X in L p . Definition. Let A1, A2, . . .
be events. The term eventually is defined by cfw_An eventually = [ NN \
nN An and infinitely often by cfw_An infinitely often = \ NN [
stochastic integration theory. Consider the stochastic process (Zt)t0
given by Zt = e 1 2 R t 0 |s| 2ds+ R t 0 sdWs where (Wt)t0 is a mdimensional Brownian motion and (t)t0 is a m-dimensional
predicta
Vasicek model. In 1977, Vasicek proposed the following model for the
short rate: drt = (r rt)dt + dW t for a parameter r > 0 interpreted
as a mean short rate, a mean-reversion parameter > 0, and a vol
price of a two liter ketchup bottle should be twice the price of a one
liter ketchup bottle, otherwise by following the sacred mantra of buy
low and sell high one can create an arbitrage, that is, ins
and a Brownian motion (Wt)tR+ for P. Note that while in a complete
stock market model there was only one equivalent martingale measure,
no such choice is possible since the short rate is not traded. H
admissible strategy H such that Ht Pt = 1 Yt E P (YT T |Ft). In particular,
the strategy replicates the payout T . Remark. That is to say, the
quantity E(YT T )/Y0 is the minimal amount of money neede
e (ts) )ds + Z t 0 0e (ts) dW s = f(0, t) + 2 0 2 2 (1 e t) 2
+ Z t 0 0e (ts) dW s The short rate dynamics are given by drt = f 0
0 (t) + 2 0 e t(1 e t) dt + 0dW t Z t 0 0e (ts) dW s dt
= f 0 0 (t) +
insurance or hedge against associated risks. This chapter briefly
discusses some such popular derivatives including those that played a
substantial role in the economic crisis of 2008. Our primary foc
1. More generally, if X is a random vector valued in R n then X : R n
C defined by X(t) = E(e itX) is the characteristic function of X. Theorem
(Uniqueness of characteristic functions). Let X and Y b
a m-dimensional Brownian motion W = (Wt)t0 is defined, and let the
filtration (Ft)t0 be the filtration generated by W. Let X = (Xt)t0 be a
continuous local martingale. Then there exists a unique predi
model P = (B, S) has dynamics dBt = Btrtdt dSi t = S i t i tdt + Xm j=1 ij
t dWj t ! for i = 1, . . . , d as before, or in vector notation, these
equations can be written as dSt = diag(St)(tdt + tdWt)
r) = 1 Suppose P(t, T) = V (t, rt). Then the discounted price process e R
t 0 rsdsP(t, T) is a Q-local martingale. Proof. Itos formula implies d e
R t 0 rsdsV (t, rt) = rte R t 0 rsdsV (t, rt)dt +e R