model is driven by a two-dimensional Brownian motion. Therefore it is
incomplete according to the rule of thumb. As a way out we consider a
liquid European call option as an additional hedging instrument. We
denote its price process by C(t) = c(t, S(t), (
mesh size = M/N. The quality of this approximation depends on the
smoothness of g and in particular of t . This is determined by the
existence of high moments of X(T) or, put differently, by the thickness of
the tails of the law of X(T). If we set m := 1
W(ti1)dt = Z ti ti1 (ti t)dW(t) N(0,(ti ti1) 3 /3) = N(0,(t) 3 /3),
I01 = Z ti ti1 Z t ti1 dsdW(t) = Z ti ti1 (t ti1)dW(t) N(0,(ti ti1) 3
/3) = N(0,(t) 3 /3), I11 = Z ti ti1 Z t ti1 dW(s)dW(t) = Z ti ti1 (W(t)
W(ti1)dW(t) = 1 2 (W(ti) W(ti1)2 (ti ti1) 2
European call with z(s) = E e rT s exp (r 2 /2)T + T X K + = E e
rT exp log s + (r 2 /2)T + T X K + = Es e rT exp (r 2 /2)T +
T X K + , where X is N(0, 1)-distributed under P and N(log(s)/
2T , 1)-distributed under Ps. In the above notation (but with s
variance t, we have found a simple method to simulate an
approximation of X on the grid. The algorithm of this Euler method in
pseudo code reads as follows. Initialise t = T/m, t0 = 0, Y0 = X(0), Loop
For i = 1, 2, . . . , m compute ti = ti1 + t W = Z t w
random vector Z = (Z1, . . . , Zd) is Gaussian with mean vector 0 and
covariance matrix 1d. Determine a matrix A R dd with AA> = , e.g.
the Cholesky decomposition is a lower triangular matrix. Now set X = AZ
+ , X is Gaussian because it is an affine trans
function of the present stock price S(t) and time t. In other 28 CHAPTER
3. ITO CALULUS words, it does not depend on the past stock prices
which do not play a role for the expected payoff. We denote this
function also as v, i.e. we write the discounted op
because the option is worthless otherwise. It makes sense to assume
that the option price V (t) is a function of time t, the present stock price
S(t) and the running minimum M(t) = minst S(s). Of course, V (t) = 0 if
M(t) H because the expected payoff van
that this PDE is solved by v(t, x) = x log( x K ) + (r + 2 2 )(T t) T
t ! H x 1+ 2r 2 log( H2 Kx ) + (r + 2 2 )(T t) T t ! e r(T
t)K log( x K ) + (r 2 2 )(T t) T t ! H x 2r 2 1 log( H2
Kx ) + (r 2 2 )(T t) T t ! . For the replicating strategy = (0,
1), t
The dM(t)-term does not contribute anything unless the stock price S(T)
happens to coincide with its running minimum M(t). The discounted
stock price S(t) is a Q-martingale. In order for the sum V (t) to be a Qmartingale as well, the drift part must vanis
(3.2). We will see in the next section how it is computed in more
specific situation. At this point we only note that it vanishes if both
either X or Y is continuous and either of them is of finite variation.
Moreover, [X, Y ] is itself of finite variatio
fe(Riu) = fe(R + iu) and (uiR) = (u iR). 40 CHAPTER 4. INTEGRAL
TRANSFORMS This yields V (0) = e rT 2 Z 0 fe(R + iu)(u iR) + fe(R
iu)(u iR) du = e rT 2 Z 0 fe(R + iu)(u iR) + fe(R + iu)(u iR)
du = e rT Z 0 Re fe(R + iu)(u iR) du. (4.4) Similarly, we obta
constant growth. But the fundamental idea of analysis is the
observation that they often do so on a local scale. In other words, a
large class of functions are of the form dX(t) = (t)dt, i.e. in a
neighbourhood of t they resemble a linear function with gr
to evaluate an investor's margin position based on changes in asset
values. B. Options 1. The candidate will be able to define and recognize
the definitions of the following terms: Materials for Study, 2010 Exam 2
E2-4 a. Call option, Put option b. Expira
tv(t, x) rv(t, x) + rx xv(t, x) + 1 2 2x 2 2 x2 v(t, x) (v(t, x) g(t,
x) = 0, v(T, x) g(T, x) = 0. (3.36) This linear complementarity problem,
variational inequality, or free boundary value problem does not
generally allow for a closed form solution. We w
quite different nature. Why do we not stick to the one method which
proves superior in all circumstances? It simply does not exist. Some
methods are faster, others easier to implement, even others more
generally applicable. In this course, we repeatedly c
mathematics and by Albert Einstein in his 1905 paper on thermal
molecular motion. Brownian motion as the only continuous process
with stationary and independent increments has a natural extension to
the multivariate case. The law of such a continuous R d
with some constant z. For the sequel, we even need to take complexvalued z into account, i.e. we consider payoffs of the form f(x) = e zx = e
Re(z)x (cos(Im(z)x) + isin(Im(z)x), where z C and i = 1. Of course,
complex-valued options do not make sense from
subtle technicalities, a market does not allow for arbitrage
opportunities if and only if there exists an equivalent martingale
measure (EMM), i.e. a probability measure Q P such that the
discounted price processes S i are Q-martingales for i = 0, . . . ,
On the other hand, it seems to jump to 0 at the stopping time = infcfw_t
0 : M(t) H = infcfw_t 0 : S(t) = H, i.e. at the first time that the stock
price touches the barrier. This apparent contradiction is resolved by
imposing the the boundary condition v
tv(t, x, ) = rv(t, x, ) rx xv(t, x, ) ( ) v(t, x, ) 1 2
x2 2 x2 v(t, x, ) 1 2 e 2 2 2 v(t, x, ), (3.41) in undiscounted
terms and x -notation. For the final value we have v(T, x, ) = f(x)
(3.42) 36 CHAPTER 3. ITO CALULUS as in the Black-Scholes case. The
American options in the Black-Scholes model. Suppose that the exercise
process is of the form X(t) = g(t, S(t) for some function g of time and
stock price. For standard call and put options, the payoff function g(t, x)
= (x K) + resp. g(t, x) = (K x) + do
Closed-form expressions for option prices are available only in rare
cases. Often, the probability densities of S(T) or log S(T) are unknown so
that computation by numerical integration of the payoff is not obvious
either. We consider here an approach to
the local diffusion coefficient (t) now depend on time t and possibly
also on randomness. (3.6) does not make sense in ordinary calculus but
it can be interpreted as an integral equation X(t) = X(0) + Z t 0 (s)ds + Z
t 0 (s)dW(s) (3.7) in the sense of the
process of Brownian motion X = (X1, . . . , Xd) in R d with covariance
matrix c is given by [Xi , Xj ](t) = cij t and in particular [Wi , Wj ](t) = t if i
= j, 0 otherwise for standard Brownian motion W in R d . 3.2 Ito
processes We do not need the theory