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 given for the best FIVE answers.
Electronic calculator
(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 that 1^= Aa= 0 and V^ = 0.
On the bottom branch we have
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 equation
dXt= (~Y^
Let (Yt)t>o be a process such that ^i =
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) gcfw_x).
(a) Find a stochastic representation for vcfw_t,
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 = (S\ - K)+ with strike price K = 100.
(a) For which va
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 lower bound,
and that we are interested in pricing and h
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 and maturity T > 0. The value
of this option is
V(t, s)
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 Brownian motion.
(a) The price of a zero coupon Bond with mat
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 )t0 is a one
jump process which takes the values
0 t<
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 cfw_St)t>o and (Pt)t>o denote the
solutions to the SDE