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Mathematical Finance | 金融数学 – 这道题目金融数学的代写任务, 涵盖了金融数学等方面
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BSc and MSci Examination
6CCM388A Mathematical Finance I: Discrete Time
Summer 2021
Time Allowed: Two Hours
All questions carry equal marks.
Full marks will be awarded for complete answers to FOUR questions.
If more than four questions are attempted, then only the best FOUR will count.
You may consult lecture notes and use a calculator.
Kings College London
- Part I.In the single-period binomial asset pricing model consider two assets: a stock and a bank account. The price of one share of stock isS 0 at timet= 0 and moves up toS 1 (H) with probabilityp >0 and down toS 1 (T) with probability q= 1p. One unit in the bank account at timet= 0 becomes 1 +rat time t= 1. Recall thatHrepresents the head,Tthe tail andrthe interest rate. A dealer contracts at timet= 0 the obligation to pay an investor the amount S^21 at timet= 1, whereS 1 is eitherS 1 (H) orS 1 (T), whichever turns out. The dealer hedges this derivative by trading in the two assets. Assume that there is no arbitrage and the stock pays no dividends.
i. Find the replicating portfolio of this derivative. That is:
a. Show the number of shares of stock must be:
=S 1 (H) +S 1 (T).
[10%]
b. Show the dealer must invest:
S 1 (H)S 1 (T)
1 +r
in the bank account.
[10%]
ii. Show that the price of this derivative must be:
V 0 =S 0 [S 1 (H) +S 1 (T)]
S 1 (H)S 1 (T)
1 +r
.
[10%]
iii. Define pand qin terms ofS 0 ,S 1 (H),S 1 (T) andrsuch that the valueV 0
can be expressed in the form of a discounted expectation:
V 0 =
1
1 +r
E [S^21 ],
whereE is the expectation with respect to the probability measure for
whichS 1 =S 1 (H) with probability pandS 1 =S 1 (T) with probability q.
[10%]
iv. Show that if the obligation of the dealer is to payS 1 instead ofS 12 , then
V 0 =S 0. What is the value of and the money invested in the bank
account in this case?
[10%]
Part II.Consider anN-period binomial asset pricing model and two assets: a stock and a bank account. The price of one share of stock isS 0 at time t= 0 and moves up toS 1 (H) with probabilityp >0 and down toS 1 (T) with probabilityq = 1p. One unit in the bank account at timet= 0 becomes 1 +rat timet= 1. Consider an European call option with strike priceKand maturityN and the corresponding European put option (with same maturity and strike). Let us denote byVnE,call (resp. VnE,put) the price process of the European call option (resp. European put option). Show that the following formula holds:
VnE,callVnE,put=SnK
1
(1 +r)Nn
, n= 0,...,N.
[50%]
- i. LetNNbe a natural number and Nthe set of all sequences of length N of coin tosses with possible outcomes in{T,H}. Define a probability measurePon N that assigns equal probability to every sequence ofN tosses. Find the probability of the event the last (i.e.N-th) toss is a head (i.e.H). [10%]
ii. Let (N,P) be the probability space defined in a. DefineXj() = 1 if
thej-th toss ofNresults in a head andXj() =1 if thej-th toss
results in a tail. Define further the sequence of random variablesM 0 = 0
andMn=
n
j=1Xjfor anyn{^1 ,^2 ,...,N}.
a. Prove that the sequence{M =Mj; j= 0,...,N}is a martingale.
State carefully all the properties of conditional expectations you need
to apply. You may use these properties without proof. [10%]
b. DefineYn:=Mn^2 n, forn 0 ...N. Prove that the sequence{Y =
Yj; j= 0,...,N}is a martingale. State carefully all the properties of
conditional expectations you need to apply. [20%]
iii. A gambler wins or loses one pound in each round of betting, with equal
chances and independently of the past events. She starts betting with
the firm determination that she will stop gambling when either she won
npounds or she lostmpounds.
a. What is the probability that she is winning when she stops playing
further. [30%]
b. What is the expected number of her betting rounds before she will
stop playing further. [30%]
- Consider anN-period binomial model. AnAsian optionhas a payoff based on the average price, i.e.
VN=f(
1
N+ 1
N
n=
Sn),
where the functionfis determined by the contractual details of the option.
a. DefineYn=
n
k=0Sk and use the independence lemma to show that the
two-dimensional process (Sn,Yn),n= 0,...,Nis Markov.
[30%]
b. The price of the Asian option at timenis some functionvnofSnandYn
i.e.
Vn=vn(Sn,Yn), n= 0, 1 ,...,N.
Give a formula forvn(s,y) and provide an Algorithm for computingvnin
terms ofvn+1.
[30%]
c. ForN = 2 andf(x) = 3xassume thatS 0 = 8,u= 2,d= 1/2,r= 1/ 4
and risk-neutral probabilities are p= q= 1/2. Using partbor a different
method computev 0 (8,8), which is the time 0 price of an Asian option
expiring at time 2.
[40%]
- i. Define the following terms:
a. replicating portfolio for an option.
b. Markov property.
c. American put option.
d. Martingaleprocess,submartingaleprocess andsupermartingalepro-
cess.
[15%]
ii. Consider a two-period binomial model withS 0 = 4, up factoru= 2, down
factord= 0.5 and interest rater= 0. Assume the probability that the
stock price moves up isp= 2/3.
a. Compute the actual probabilities:
P(HH),P(HT),P(TH),P(TT) and the
risk-neutral probabilities:
P (HH),P (HT),P (TH),P (TT). Recall that with each headHthe
stock price moves up and with each tailTit moves down.
[15%]
b. Compute the Radon-Nikodym derivative random variableZ.
[10%]
c. Compute the Radon-Nikodym derivative processZn, n= 0, 1 ,2, dis-
play it in a tree diagram and show that it is a martingale under the
market probabilityP.
[10%]
d. Give the formula and compute the state price densities:
2 (),{HH,HT,TH,TT}.
[10%]
e. Using the numbers computed in partd.find the time-zero price of
an Asian option whose payoff at time 2 is (^13 Y 2 4)+, whereYn =
n
k=0Skis the sum of stock prices between time 0 and timen.
[20%]
iii. Suppose the initial stock price isS 0 = 80, and with each head the stock
price increases by 10, and with each tail the stock price decreases by 10.
In other words,S 1 (H) = 90,S 1 (T) = 70,S 2 (HH) = 100, etc. Assume
the interest rate is always zero.
Consider a European call option with strike price 80, expiring at time three. What is the price of this call at time zero? [20%]
- a. i. What assumptions are used when modelling the interest rate as a stochastic process? Define a zero-coupon bond and an interest rate swap. [30%]
ii. Letrnbe the stochastic interest rate in anN-period model. Prove
that the value at timenof the fixed income derivative paying out
rN 1 at timeNis equal toBn,N 1 Bn,N, whereBn,N 1 (resp.Bn,N)
is the timenprice of the zero-coupon bond with maturityN1 (resp.
N). [30%]
iii. Prove that the price of the fixed income derivative inii. is greater
than or equal to zero so long asrN 1 0. [10%]
b. In the Black-Dermann-Toy model, the interest rate at time 0nN is
given by
rn( 1 ...n) =anb#nH{^1 ...n}
where the power ofbn is the number of heads in the scenario 1 ...n.
Suppose thatan= 10 ..^052 n andbn= 1.44. The risk-neutral probabilities are
p = q= 1/2.
There are two zero coupons, one with maturity 1 and the other with
maturity 2. Compute the price of these zero-coupon bonds at timesn=
0 ,1. [30%]
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