ECOM044 Advanced Asset Pricing and Modelling 2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
May Examination Period 2022
ECOM044 Advanced Asset Pricing and Modelling
Question 1 [25 marks]
In this question you are asked to prove some no arbitrage restrictions. The proofs should be by contradiction, that is, by assuming that an inequality does not hold and showing how you can then construct an arbitrage. Explain your steps and provide intuition.
(a) Consider a European call option and a European put option, both with strike price K, written
on a non-dividend-paying stock. Prove that c < S and p < PV (K). [10 marks]
(b) Assume now that the stock pays dividends. The present value of dividends is denoted I . Prove
the put-call parity, c = p + S − I − PV (K). [15 marks]
Question 2 [40 marks]
A stock price evolves in a standard binomial tree. Each period it can either go up to u = 1.1 times its previous price or down to d = 0.9 times its previous price. Consider a two-period model (t = 0, 1, and 2), as depicted below. The risk-free (net) return between t = 0 and t = 1 is r1 = 5%, while between t = 1 and t = 2 it is r2 = 6%. The initial stock price is S = 120 and the stock pays no dividends.
S
uS
%
&
dS
u2 S
%
&
udS
%
&
d2 S
t=0 t=1 t=2
(a) Price a European call option on the stock with strike price K = 110 and maturity at t = 2
using risk-neutral pricing. [6 marks]
(b) How would you replicate the European call option at the u node at time 1, cu . Compute ∆d
and ∆0 as well. Comment on the three values of ∆ . [8 marks]
(c) Explain in words why the price of the American call option with K = 110 and maturity at t = 2 is the same as the price of the European call option in part (a). [6 marks]
(d) Price a European put option and an American put option on the stock with strike price K = 119 and maturity at t = 2 using risk-neutral pricing. [10 marks]
(e) An average price Asian put option with maturity at t = 2 gives the payo↵ max(K − Savg , 0) to the buyer, where Savg = 
, and no early exercise is possible. Assume that K = 120. Price this option using risk-neutral pricing. [10 marks]
Question 3 [35 marks]
Consider the risk-neutral dynamics of the price process of a non-dividend-paying stock, dSt = rSt dt + σSt dWt ,
where r is the constant instantaneous risk-free rate, σ is a positive constant, and W is a standard Brownian motion under the risk-neutral probability measure Q.
(a) Assume that St = g(t,Wt ) = S0 exp ⇣⇣r − 
⌘ t + σWt ⌘ . Use Itˆo’s Lemma to show that dSt
is the above stochastic di↵erential equation. [7 marks]
(b) Let Xt = St(2) + t. Use Itˆo’s Lemma to find dXt . [8 marks]
(c) Consider a derivative Yt written on the stock which only pays at a prespecified maturity T and has the following payo↵:
8 2, if ST ≥ K2 ,
YT = < 1, if K2 > ST ≥ K1 ,
: 0, if K1 > ST ,
where 0 < K1 < K2 are constants agreed upon at time 0.
Draw the payo↵ diagram of this derivative, and express the derivative as a portfolio of dig- ital options (i.e., options that pay £1 if the value of the underlying asset exceeds some predetermined value, and zero otherwise). Find Y0 , the price of this derivative at time 0. xxxxxxxxxxxxxxx [10 marks]
(d) Let S0 = 1. Consider a derivative Zt written on the stock which only pays at a prespecified maturity T and has the following payo↵:
ZT = 
where K > 0 is a constant agreed upon at time 0.
Draw the payo↵ diagram of this derivative. Find Z0 , the price of this derivative at time 0. xxx [10 marks]
2022-07-29