ECO00040M Theory of Finance 2021–22
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECO00040M
MSc Degree Examinations 2021–22
Theory of Finance
SECTION A. Answer at least ONE question
1.
(a) The Vasicek model assumes that the short term (spot) interest rate yt is normally distributed. What type of probability distribution do yields obey in this model? When is this assumption likely to be unrealistic? Is it a good model in the current economic environment?
(b) Why would it be more realistic to use the model:
where yt is the short rate and µ > 0?
(c) Assume that the short rate follows the process in Eq. (1) and that the zero-coupon
bond prices are exponentially affine in the short rate, i.e. Dm,t = exp(dm,t ) =
exp( −am + bmyt ). Find the expected value of the one-period ahead price Dm −1,t+1
(d) Assume that the log stochastic discount factor in the economy is given by
− mt+1 = yt + λξt+1 , (3)
where yt is the same shock as in Eq. (1). Recall that the conditional log risk premium adjusted for convexity is given by
Et [rn,t+1] − yt + Vart [rn,t] = −Covt [rn,t+1,mt+1],
(4)
where rn,t+1 is the log return on the n−period zero coupon bond, rn,t+1 = dn −1,t+1−
2. Suppose that we have N risky assets with random returns ri , means E[ri] = µi , vari-
ances Var[ri] = σii , (i = 1, ...,N) and covariances Cov[ri ,rj ] = σij (i,j = 1, ...,N; i
j). Let xi be the share held in the i − th asset. Define the N × 1 vectors ι = [1, ..., 1]\ ,
wealth in the risky portfolio. The N × N covariance matrix for the vector of risky returns
is represented by Σ = E[(r − µ)(r − µ)\]. Suppose that all investors have the same
Fj = E[rp] − γj Var[rp], (5)
where γj is a risk aversion coefficient for the j − th investor. The portfolio return rp is:
rp = y + x(r − ιy),
where y is the risk-free rate on a safe asset.
(a) Prove that if they optimize a linear function of the mean and variance of the per- centage return on the portfolio over the same fixed investment horizon using the basic assets of the system, they can do just as well by holding the safe asset and a specially-constructed composite portfolio or mutual fund.
(b) Briefly outline the implications of result (a) for the fund management industry.
(c) The expected return of j − th investor is
E[r ] = ρjj E(rm ) + (1 − ρj )y, (6)
where rm is the return on the mutual fund derived in (a) (the market portfolio) and ρj is the proportion of wealth invested in rm . Derive the demand for the risky portfolio ρj for the j − th investor in terms of µ , Σ and ρj .
(d) If all investors had the same degree of relative risk aversion, γj = γ, what would be the demand for the safe asset if it is provided in zero net supply?
SECTION B. Answer at least ONE question
3.
(a) Derive the put-call parity relationship using an arbitrage argument for European options and show it on a graph.
(b) Use the put-call parity to establish the lower bound on the European call option.
How could you make a profit if this bound was violated?
(c) Show that a reverse straddle consisting of a short put and short call with the same strike price K and expiry date T is equivalent to:
• being short one T − maturing forward struck at the forward price F = K and
(d) You are given the following information on the price of 3 and 6 month European call and put options written on an equity price. The equity is currently trading at 1, 055. Your client believes that the volatility implicit in the March option prices is
Calls Puts
Strike price: Mar Jun Mar Jun
1, 040
1, 050
1, 060
43.6 37.6 33.9
5 2
47.7
23.2
27.1
33.3
3 8
−
too high. Explain how a strangle could be set up that would allow her to benefit if
prices stays in the range 1, 040 − 1, 060. Construct and price a butterfly spread for
4. Assume that at time t an oil company share price is St = 500. A major legal judgement concerning an oil spill is due to be announced at time T = 1 year. If this is favourable, its share price will move up to ST,u = 600, otherwise it will move down to ST,d = 300. The risk-free rate is a constant y = 0.10 simply compounded. Consider a riskless portfolio with value Vt at time t and VT at time T that holds θ shares and one put option that costs pt , has a strike price X such that ST,d < X < ST,u and expires at time T.
(a) Derive the value of θ that makes the portfolio riskless. Find the value of the port- folio at t and T, Vt and VT .
(b) Find the price of the put option at times t and T, pt and pT .
(c) Find the prices of contingent claims. Interpret the sum of all contingent claims Check that the contingent claims price all securities, i.e. the stock, the safe asset and the put option.
(d) Find the risk-neutral probabilities held at time t for the two states at time T. Do they sum up to one? Check that the risk-neutral probabilities price all three assets.
SECTION C. Answer at least ONE question
5. Suppose that a security price P follows a geometric Brownian motion (GBM) under the
risk-neutral measure:
dP(t) P(t)
= ydt + σdzN .
(7)
(where the risk-neutral expectation EN [dzN ] = 0) and also follows a GBM under the real-world probability measure P:
dP(t) P(t)
= νdt + σdzP .
(8)
(where the expectation under the real-world measure P is EP [dzP ] = 0) y , ν and σ are
constant, P(0) = 1 and:
dzP = dzN − σdt.
Also define:
Q(t) = exp (−t + σzP (t)) .
(9)
(10)
Evaluate:
• zP (T) − zN (T);
• EP [zN (T)].
(b) Use Ito’s lemma to show that Q(t) is a martingale under the real-world measure.
(c) Using the fact that the probability density under the real-world probability measure P is related to the probability density under the risk-neutral measure N by the
relationship:
Q(T)fP (zP (T)) = fN (zN (T))
where:
fP (zP (T)) = g(zP (T)) =
find an expression for fQ (zQ (T)).
(11)
(12)
(d) Denote by xt a total payoff on a stock, which includes all compounded dividends and price at time t. The risk-neutral pricing formula says that
P0 = e −yt E0(N) [xt], (13)
where P0 is the price of the stock at time t = 0. The result from (c) implies that
E0(N) [xt] = E0(P) [xt Q(t)]
for some random variable xt . Define the stochastic discount factor as Mt = e−yt Q(t),
where y is a (constant) risk-free rate. Show that
e −yt E0(N) [xt] = E0(P) [xt Mt].
What is the interpretation of this relation when x is a payoff for a security?
6.
(a) Derive the Black-Scholes formula for the price c0 at time 0 of a European call option on a zero dividend stock with price S0 using put-call parity and the formula for the price of a European put option with the same strike price X and expiry date
T > 0 :
p0 = e−yT XΦ (−d1 + σ ^T) − S0 Φ(−d1 )
where:
d1 = log(S0 /X) + (y + σ2 /2)T
σ ^T .
(b) Show that the delta of this put is [Φ(d1 ) − 1].
(c) Show that the value of a long position in a forward on a this stock with contract price F = X and expiry T is V0 = [S0 − Xe−yT ].
(d) Show that this forward has the same value as the long call c0 and the short put (−)p0 . What is the delta of this forward?
(e) Use the results at (b) and (d) to find the delta of the call option with price c0 .
2022-08-03