MATH49102
Three hours
UNIVERSITY OF MANCHESTER
STOCHASTIC MODELLING IN FINANCE
24 May 2013
9:45 12:45
Answer FIVE of the SEVEN questions. If more than FIVE questions
are attempted, then credit will be
(b) Find the replicating strategy for C. The next figure shows that payoff of the downand-in put option
c=o
C=2.9
Figure 2: Price process diagram
We can conclude that when Si = 11
so we may conclude t
Stochastic Modelling in Finance - Solutions to sheet 4
4.1 Let (Wt)t>o be a Brownian motion,
(a) Show that the process (Xt)t>Q defined by
Xt = (l-t) [ ~^
Jo 1 -
solves the stochastic differential equa
Stochastic Modelling in Finance - Solutions to sheet 5
5.1 Suppose that a > 0 and aGR. Let v(t, x) denote the solution to PDE
v(t,x) + (a- x)-q^v(^x) + 2a2
subject to the terminal condition v(T, x) gc
S0(l+al^\
S0(l+bJ
Figure 2: l.b. independent increments
2.2 We consider a single period model with r > 0, So = 100 and
P(Sl = So + 20) = P(Si = So - 10) = 1/2.
Let C be a European call option, i.e. C
Stochastic Modelling in Finance - Solutions 9
9.1 Suppose that the stock price (St )t0 is modeled using
dSt = (St ) dt + (St ) dWt
where the functions and are bounded and additionally has a positive l
Stochastic Modelling in Finance - Solutions 7
7.1 Consider the Black & Scholes Model with volatility a > 0, drift /i, interest rate r and
initial stock price Sq. Take a put-option with strike /C > 0 a
Stochastic Modelling in Finance - Solutions 10
10.1 Suppose that the short-rate is modeled under the pricing measure P as the solution
to the SDE
dr(t) = (t) dt + r(t) dWt
where (Wt )0tT is a P Browni
Stochastic Modelling in Finance - Solutions to sheet 8
8.1 The price of a defaultable asset can be modeled as
dSt
= dt + dWt dNt
St
where , are constants, (Wt )t0 is a standard Brownian motion and (Nt
Stochastic Modelling in Finance - Solutions to sheet 6
6.1 Suppose that (Wt)t>o and (Bt)t>o are correlated Brownian motions with correlation
coefficient p [-1,1] on a probability space (fi,^7, P). Let