TRINOMIAL TREE APPROXIMATION TO BROWNIAN MOTION
MARK H.A. DAVIS, 25.05.11
1. Random walk model. This is equivalent to a random walk model in which Brownian motion
W (t) with variance parameter is approximated for t = , 2, . . . by
k
W (k ) Xk =
Zj ,
j =1
Finite Difference Methods
The Cubic Spline Formulas
The data points are (xj , yj ), j = 0, . . . , n. On the interval [xj , xj +1 ] we write (following Numerical Recipes )
S (x) = A(x)yj + B (x)yj +1 + P (x)
(1)
where the first two terms linearly interpol
MSc Course in Mathematics and Finance
Imperial College London, 2010-11
Finite Difference Methods
Mark Davis
Department of Mathematics
Imperial College London
www.ma.ic.ac.uk/mdavis
Part II
3. Numerical solution of PDEs
1
4. Numerical Solution of Partial D
MSc Course in Mathematics and Finance
Numerical Methods in Finance
Solutions to January 2009 Examination
1. (i) We have the random variable
n
ai eYi .
X=
i=1
Thus
1
ai EeYi =
m1 = EX =
2
ai e 2 i ,
i
while
m2 = EX 2 = E
ai aj eYi +Yj
i,j
=
i,j
12
2
ai aj
THE BLACK SCHOLES FORMULA
MARK H.A. DAVIS
If options are correctly priced in the market, it should not be possible to make sure profits by
creating portfolios of long and short positions in options and their underlying stocks. Using this
principle, a theo
THE DUPIRE FORMULA
MARK H.A. DAVIS
1. Introduction. The Dupire formula enables us to deduce the volatility function in a local volatility
model from quoted put and call options in the market 1 . In a local volatility model the asset price model
under a ri
Finite Difference Methods
Mark Davis, Summer Term 2011
(Based on notes by R. Nrnberg and H. Zheng)
u
These notes summarize information about numerical methods, covering many of the topics included
in the course and much more besides. You may find it a use
AMERICAN OPTIONS IN THE BINOMIAL MODEL (REVISED)
MARK H.A. DAVIS
1. The binomial tree. The tree has N time steps corresponding to times k = 0, 1, . . . , N , and
models an asset price Sk . The price is normalized to S0 = 1, and at each branch in the tree
COORDINATE TRANSFORMATION IN THE BLACK-SCHOLES PDE
MARK H.A. DAVIS, 25.05.11
The standard Black-Scholes PDE, in the usual notation, for the call option price function C (t, S ) is
(0.1)
C
C
1
2C
+ S
+ 2 S 2 2 rC = 0, (t, S ) [0, T [R+ ,
t
S
2
S
C (T, S )
BUILDING A YIELD CURVE GENERATOR
MARK H.A. DAVIS
1. Libor and Swap Rates. Libor rates are quoted every day for standard maturities 1 month,
3 months, . They are quoted in the form of an annualized rate L, and an accrual basis which is
actual/360 except in