20
20.1
MATH3733
American options: denition
Denition 1 An American option is a contract similar to European option with a corresponding payo (call (St K)+ , put (K St )+ , general option gt (St ) 0), but which may be exercised any time to expiry.
20.2
Bin
11
11.1
MATH3733
Random Walk
Wiener process or Brownian motion (BM) is a continuous time analogue of a random walk with very small steps. There is a strict theorem like CLT behind this. Random walk Wn (t) with steps = 1/n is defined as Wn (0) = 0; Wn (k +
7
7.1
MATH3733
Information
Information is crucial for the market. Hence, a mathematical nancial theory uses it under the name ltration; other names: information, sigma-eld, sigma-algebra). Denition 1 Let market consist of one stock which is modelled by a
12
12.1
1.
MATH3733
Wiener process properties
EW (t) = 0, Exercise 1 Show this. EW (t)2 = t.
2. Paths of W (t) are continuous. (Just believe that it may be proved.) 3. Paths of W (t) are not differentiable. [Hence, f (s)dW (s) cannot be defined as f (s)W
6
6.1
MATH3733
One step CRR model: Implied probabilities
+ S K r + S0 + e S S + S K + , + S S S
Remind the formulae from the previous lecture for call C0 = S0 + B0 =
+ (S K)
and for any synthetic derivative with payo f (S ), (, are dierent now) + f (S
14
14.1
MATH3733
Its formula, chain rule
o
The following result includes the previous theorem and usually is called Its formula.
o
Theorem 1 (Its formula as a chain rule) Let process Xt have a stochastic difo
ferential,
dXt = bt dt + t dWt .
Then for any
7 Lecture: Central Limit Theorem Another Proof
1. Theorem Let X1 , X2 , . . . are IIDRV's with EX1 = 0, var(X1 ) = 2 (0, ), Sn = n Xk . Then, k=1
Sn / n 2 = N (0, 1)
2. Proof (not compulsory): Hint-1. It suces to show convergence
(1)
Eg(Sn / n 2 ) Eg(Z)
3
3.1
MATH3733
European Call option: mathematical notations
Assume that the standard interest rate r = 0. Denote the price of our underlying stock and our call option at time t by St and Ct (from Call) correspondingly. The price St is given to us by the m
5
5.1
MATH3733
Cox-Ross-Rubinstein model: One step analysis
This is a discrete time model with time t = 0 and t = > 0. On our market there is a bank account Bt (a riskless bond in [BR, p. 10]) and a stock. The interest rate is r 0, i.e. B = B0 exp(r). The
8
8.1
MATH3733
Conditional distributions for random vectors
Denition 1 1. Let a n-dimensional random vector (X1 , . . . Xn ) have a joint (ndimensional) density p(x1 , . . . , xn ). Denote Y = (X1 , . . . , Xn1 ). We call a conditional density of Xn given
9
9.1
MATH3733
More about conditional expectations
The following formula can be shown by induction, similarly to the proof of lemma 1 from lecture 6. X X X E(E(Xn |Fn-1 )|Fn-2 ) = E(Xn |Fn-2 ). More generally, for any 0 < k n, by induction
X X X E(E(Xn |
16
16.1
MATH3733
Cameron-Martin-Girsanov transformation
In the CRR model, we have seen how to calculate the implied probabilities at each step and for each branch of the binomial tree. We can say that we transform the real probabilities p and 1 p for each
17
17.1
MATH3733
Black-Scholes formula via Cameron-Martin-Girsanov
This is another way to nd the value of call option which is better in the sense that you need not know how to solve partial dierential equations; all you need is to know Gaussian integrals
13
13.1
MATH3733
Stochastic dierential
dXt = bt dt + t dWt
Denition 1 We say that the process Xt has a stochastic dierential i Xt = X0 +
0
t
bs ds +
t 0
s dWs
where the rst integral is a usual Riemann one while the second one is a stochastic one. In parti
10
10.1
MATH3733
Drift and volatility in the binomial model
We consider a binomial (CRR) model with several steps and probabilities up and down 1/2 & 1/2; assume that at any node S the values S are dened as S + = S exp( + ), S = S exp( ). The mean value o