the so-called martingale transform of X by B. The second, and more
difficult, step is show that it is possible to approximate an arbitrary g
L 2 ad by a sequence of step processes gn L 2 ad such that
the payoff. The rebate may be paid as soon as the barrier is triggered or
not until expiry. 5.2 Asian options Asian options give the holder a payoff
that depends on the average price of the underlying
value of the underlying stock, namely St , is known. The expiry date T is
time T t away. Thus, we simply re-scale our original Black-Scholes
solution so that t is the new time 0, the new initial price
slight modification of Example 4.9 shows that cfw_In, n = 0, 1, 2, . . . is a
martingale with respect to the filtration cfw_Fn, n = 0, 1, . . .. Note that the
requirement that Yj1 be previsible is exa
risk-free interest rate. Theta measures sensitivity to the passage of
time. Sometimes the financial definition of is V T . With this
definition, if you are long an option, then you are short theta. Ve
than that provided by interest from a bank deposit. Thus, to find the
fair value of the option V (t, St), 0 t T, we will set up a replicating
portfolio of assets and bonds that has precisely the same
equation (PDE) given by equation (8.15) on page 79. We now mention
two important points. (i) The drift parameter in the asset model does
NOT appear in the Black-Scholes PDE. (ii) Actually, we have not
and let It(g) = Z t 0 g(s) dBs and It(h) = Z t 0 h(s) dBs. (a) If , R are
constants, then It(g + h) = It(g) + It(h). (b) It(g) is a random variable
with I0(g) = 0, E(It(g) = 0 and Var(It(g) = E[I 2 t
n = 0, 1, 2, . . .. Solution. We find E(Xn+1|Fn) = E(Xn Yn+1|Fn) =
XnE(Yn+1|Fn) (by taking out what is known) = XnE(Yn+1) (since Yn+1 is
independent of Fn) = Xn 1 = Xn and so cfw_Xn, n = 0, 1, 2, . .
p0 < 1). Correspondingly, the payoff from the option will become either
Vu (for up-movement in the underlying asset price) or Vd (for downmovement). The following argument is similar to that of contin
T 0 E[g 2 (t)] dt < for every T > 0. Our goal is to now define It(g) = Z t 0
g(s) dBs for g L 2 ad. This is accomplished in a more technical manner
than the construction of the Wiener integral, and th
that dt term V (t, St) + 2 2 S 2 t V 00(t, St) rD(t, St) + St V 0 (t, St)
A(t, St) reduces to V (t, St) + 2 2 S 2 t V 00(t, St) rD(t, St) since we
already need A(t, St) = V 0 (t, St) for the dBt piec
observed is the volatility of the underlying asset price which is a
measure of our uncertainty about the returns provided by the
underlying asset. Typically values of the volatility of an underlying a
r are all known, and consider V (0, S0). Since we are assuming that only
is unknown, we will emphasis this by writing V (). 80 Implied
Volatility Thus, if we have a quoted value of the option price,
is easy to implement by the binomial tree method. A question: if there
is no dividend payment during the life of the option, whether is there a
chance to exercise a Bermudan call option prior to the e
relation. But, the following put-call symmetry relation holds for both
European and American options: C(S, t; X, r, q) = P(X, t; S, q, r) (3.13)
where the underlying price and the strike price in the
T , and d2 T = d1 T 2 T = log(S/E) + (r + 1 2 2 )T
2T3/2 2 T = log(S/E) + (r 1 2 2 )T 2T3/2 . (a) Delta. Since V =
S (d1) EerT (d2), we find = V S = (d1) + S (d1) S
EerT (d2) S = (d1) + S 0 (d1) d1
slightly different than the one in Higham [12]. We are differentiating V
with respect to the expiry date T as opposed to an arbitrary time t with
0 t T. This accounts for the discrepancy in the minus
random walk. If S0 = 0 and Sn = Y1 + + Yn where Pcfw_Y1 = 1 = 1 Pcfw_Y1
= 1 = p, 0 < p < 1/2, then E(Sn) = (2p 1)n. Hence, Sn does NOT have
stable expectation so that cfw_Sn, n = 0, 1, 2, . . . cannot
setting I0 = 0 and It = Z t 0 g(s) dBs for t > 0, then (a) the process cfw_It , t
0 is a continuous-time martingale with respect to the Brownian
filtration cfw_Ft , t 0, and (b) the trajectory t 7 It
and so the increment Sj Sj1 is exactly the outcome of the jth round of
this fair game. 27 28 Brownian Motion as a Model of a Fair Game
Suppose that we bet on the outcome of the jth round of this game
T ! gives c = 1.875. In general, it is not possible to invert the formula so
that is expressed as a function of S0, X, r, T, and c. But, it is not hard to
use Matlab to find a numerical solution of be
2S 2 2V S2 + rS V S rV = 0 At time t = T1, the option values are
known as V (S, T1) = Sf(T1). So we will solve the Black-Scholes equation
in (S, t) (0,) [0, T1) with the above final condition. It is n
motion with B0 = 0, and suppose that for any fixed 0 t < 1 we define
the random variable It by It = Z t 0 B1 dBs. Since B1 is constant (for a
given realization), we might expect that It = Z t 0 B1 dBs
the option at the expiry time T, and (ii) V (0, S0) denotes the value of
option at time 0. Example 16.2. Assuming that the function V C 1
([0,) C 2 (R), use Itos formula on V (t, St) to compute dV (t,
a long forward contract, and digital options (binary options). 2.1.
CONTINUOUS-TIME MODEL: BLACK-SCHOLES MODEL 11 2.1.3 RiskNeutral Pricing and Theoretical Basis of MonteCarlo Simulation The
expected
what happens at time 0; therefore, we will simply define X0 = 0.
Notice that the sequence cfw_Xj , j = 1, 2, . . . tracks the outcomes of the
individual games. We would also like to track our net numb
representation is consistent with the terminal payoff. Then the pricing
model is formulated by min V t LV, V (S, A) = 0 V (S, A, T) =
(S, A) where L = ( SA t A + 1 2 2S 2 2 S2 + (r q)S S r, for
arithm
Sn1) (5.1) is a martingale. As we will soon see, there are natural
Brownian motion analogues of each of these martingales, particularly
the stochastic integral (5.1). 6 Brownian Motion as a Model of a
00(t) 2! t + f 000(t) 3! (t) 2 + . At this point we see that if t 0,
then lim t0 f(t + t) f(t) t = lim t0 f 0 (t) + f 00(t) 2! t + f
000(t) 3! (t) 2 + = f 0 (t) 55 56 Itos Formula (Part I) which is
ex
Accounting policies are the companys way to adhere to the accounting
principles. These are the specific rules and procedures which are used by the
company while maintaining and preparing their financi
58.
Which of the following is not true of the
terms debit and credit?
a. They can be abbreviated as Dr. and Cr.
b. They can be interpreted to mean
increase and decrease.
c. They can be used to describ
E3-7
The Ledger of Passehl Rental Agency on March 31 of the current year includes the selected accounts, shown below, before
adjusting entries have been prepared.
Debit
Prepaid Insurance
$ 3,600
Suppl
22/6/2014
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