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MSIN0107 Advanced Quantitative Finance

创建日期：2024-04-06 17:19:36

所属分类：金融Finance

标签： MSIN0107

Advanced Quantitative Finance

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MSIN0107 Advanced Quantitative Finance
Mock examination paper
Examination length: TWENTYFOUR (24) hours
NOTE: Although the window for completion is TWENTY-FOUR (24) hours, this
exam paper is designed to be completed in TWO (2) hours.
There is ONE (1) section to the examination paper. The section consists of FOUR
(4) compulsory questions. It is worth ONE HUNDRED (100) marks.
Mock exam: MSIN0107
1. [25 points] Consider the following model for an asset price St,
logSt = 0:08 + logSt1 + "t;
where "t; t = 1; 2; 3; ::::, are i.i.d. with
P ("t = 0:5) =
1
2
; P ("t = 0:5) = 1
2
:
(a) [6 points] Suppose that today we are in period t = 2 and we have observed
logS1 = 4 and logS2 = 0:08 + logS1 + 0:5. What is the distribution of
next periods price, S3, conditional on current and past prices.
(b) [6 points] What is the expected value of S3 conditional on current and past
prices.
(c) [6 points] In period 1, you purchased an option that expires in period t = 3
at which point it pays o¤
V3 = max

max
t=1;2;3
St 100; 0

:
Suppose that the risk-free rate is r = 3% per period and is continuously
compounded. How many units of the underlying asset do you need to go
short today (t = 2) in order to remove any risk associated with the option
that you purchased?
(d) [7 points] What is todays no-arbitrage price of the option?
2. [25 marks] You wish to price a so-called European barrier option whose pay-o¤
at expiration T depends on whether the price of the underlying crosses a barrier
before expiration. Let St be the price of the underlying asset, B = 90 be the
barrier that the price has go above, and K = 100 be the exercise price. Then
the pay-o¤ of the barrier option takes the form
VT = max fST 100; 0g I

max
0tT
St > 90

,
where I fmax0tT St > 90g = 1 if the price of the underlying gets above 90
during the life time of the option, and zero otherwise.
Suppose that St satises
dSt = Stdt+ StdWt; (1)
where Wt is a Brownian motion. Here, = 0:05, = 0:1, the risk-free rate is
5% per annum and the expiry date of the barrier option is T = 1 year.
MSIN0107 2 TURN OVER
Mock exam: MSIN0107
(a) [6 marks] Suppose that todays price of the underlying is S0 = 80. Provide
the expression of a Monte Carlo estimator of the option price at time 0,
V0.
(b) [6 marks] Provide an expression of the standard errors of your Monte Carlo
estimator.
(c) [6 marks] You wish to measure the sensitivity of the barrier option to
changes in the volatility of the asset. Explain how you would estimate the
marginal e¤ect on the barrier option price to a change in the volatility.
(d) [7 marks] Suppose that two months have now passed since the barrier
option was written, and the current price of the underlying asset is now
92. What is current price of the barrier option?
3. Debt and Equity as Options. Consider a rm with non-dividend-paying equity
outstanding, a senior zero-coupon bond, and a junior zero-coupon bond. Both
the senior and the junior bonds mature in year T = 8 and have the same face
value D = $35 million. Represent the year t values of the rms assets, senior
and junior bonds, and equity as At, Bst , B
j
t , and Et, respectively. The volatility
of the rms asset return is 30%. The annualized continuously compounding
interest rate is 4%. Suppose the value of the rms asset in year 5 is A5 =
$100 million. Also suppose that the rms asset value evolves as a Geometric
Brownian motion (i.e., satises the assumptions of the Black-Scholes formula).
(a) [3 points] Use Black-Scholes formula to calculate E5, i.e., the equity value
at year 5.
(b) [5 points] What is Bs5, i.e., the value of the senior bond at year 5?
(c) [5 points] What is Bj5, i.e., the value of the junior bond at year 5?
(d) [4 points] What is the senior bonds yield to maturity at year 5?
(e) [4 points] What is the junior bonds yield to maturity at year 5?
(f) [4 points] Are the yields of the senior and junior bonds the same? If not,
explain why.
4. Dynamic Portfolio Choice and Consumption. This question will guide you to
solve the Merton problem with nite T , u (c; t) = 0 and F (W ) =W 1=3. There
are two assets in the economy. One is a non-dividend stock, the price of which
follows
dSt
St
= dt+ dZt ,
where Zt is a standard Brownian motion. The other is a riskfree asset, the price
of which follows dXt = rXtdt. An agent starts with wealth W0 and needs to
choose consumption fctg and allocate between the two assets to maximize his
expected utility.
(a) [2 points] What is the optimal consumption ct at time t 2 [0; T ]?
MSIN0107 3 TURN OVER
Mock exam: MSIN0107
(b) [3 points] Let t denote the fraction of the agents wealth invested in the
risky asset at t. Derive the stochastic di¤erential equation for his wealth
Wt.
(c) [4 points] Let J (W; t) denote the value of having wealth W at time t 2
[0; T ]. Write down the HJB equation for J .
(d) [3 points] Derive the rst order condition with respect to .
(e) [5 points] Conjecture that
J (W; t) = k (t)W 1=3. (2)
Derive an ODE for k ().
(f) [5 points] Solve for k () from the ODE and then J (W; t).
MSIN0107 4 TURN OVER

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