MAT00069M Computational Finance 2018/9
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MAT00069M
BA, BSc and MMath Examinations 2018/9
Computational Finance
1 (of 3). Consider a risky asset S (t) that follows the Geometric Brownian motion (GBM)
below under the risk neutral measure Q:
dS (t) = rS (t)dt +ìS (t)dW(t): (1)
where r is the risk free rate, the constant ì is the volatility and W(t) is a standard Brownian motion under Q.
To approximate this continuous process, a binomial tree has been built on the equidistant times below, for ti = i△t and △t = T/M,
0 = t0 \ t1 \ t2 \ . . . \ tM = T.
In each of the above periods, the stock goes up by a factor u or down by a factor d,
i.e.
S (ti ) = (2)
where d \ er△t \ u.
(a) In the above binomial tree, find the risk neutral Q-transition probability q of going up such that
EQ ┌ ' Fti- 1 , = :
where B(t) is the bank account which moves as B(t) = B(0)ert and Fti- 1 is the
filtration including all price history up to time ti- 1 . [5 Marks]
(b) Compute the conditional second moment of the risky asset at time ti EQ (S (ti )2 |Fti- 1 )
for both the GBM in Eq. 1 and the binomial tree in Eq. 2. [15 Marks]
(c) In the Jarrow-Rudd model, the Q-transitional probability is set to q = 1/2. Derive the up and down risky asset movement factors u and d for the Jarrow- Rudd model assuming the conditional second moments of the GBM and Jarrow-Rudd binomial tree are the same. [20 Marks]
[Total: 40 Marks]
2 (of 3). Consider the second order partial diferential equation (PDE)
= c2 - :
where c ! 0 is a real constant and u is subject to the the initial conditions u(x: 0) = f (x) and boundary conditions u(0: t) = u(L: t) = 0.
A grid in the portion of the (x: t) plane satisfying 0 s x s L and 0 s t s T is spaced out using points xj = j△x and tk = k△t, where △t,△x ! 0 are small and j and k are integers. Let uj:k denote the approximation to u(x: t) at the grid points such that
uj:k s u(xj : tk )?
(a) Using the Forward diference approximation to discretize and the central diference approximations to discretize both space derivatives, write down the iterative scheme in matrix form which would allow you to solve for u(x: t) at the grid points. Give the lower diagonal, diagonal and upper diagonal elements of the matrix. [10 Marks]
(b) Using the Backward diference approximation to discretize and the central diference approximations to discretize both space derivatives, write down the iterative scheme in matrix form which would allow you to solve for u(x: t) at the grid points. Give the lower diagonal, diagonal and upper diagonal elements of the matrix. [10 Marks]
(c) Using the lemma on the first page, determine the condition on △t and △x
which would ensure that the iterative scheme is stable. (Hint: Use the ap-
proximation A1 - x2 s 1 - x for small x) [20 Marks]
[Total: 40 Marks]
Suppose that X satisfies the stochastic diferential equation
dXt = rXtdt +ìXt(2)dWt :
where both r and ì are constants.
(a) Find a function F such that Yt = F(Xt ) is a difusion process with constant difusion coe伍cient 1. That is, Y satisfies a stochastic diferential equation of
the form
dYt = b(t: Yt )dt +dWt .
(b) For your choice of F, compute the drift function b(t: Yt ). [5 Marks]
(c) Write down a discretization scheme for X with no discretization error. [10 Marks]
[Total: 20 Marks]
2022-08-12