Semester One Final Examinations, 2017
MATH7039 Financial Mathematics
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School of Mathematics & Physics EXAMINATION
Semester One Final Examinations, 2017
MATH7039 Financial Mathematics
This paper is for St Lucia Campus students.
120 minutes 10 minutes
Examination Duration:
Reading Time:
Exam Conditions:
This is a Central Examination
This is a Closed Book Examination – specified materials permitted During reading time – write only on the rough paper provided
This examination paper will be released to the Library
Materials Permitted In The Exam Venue:
(No electronic aids are permitted e.g. laptops, phones) Calculators – Casio FX82 series or UQ approved (labelled) Materials To Be Supplied To Students:
1 x 14 Page Answer Booklet
Instructions To Students:
This exam paper contains five questions, each carries the number of marks shown.
There are a total of 50 marks.
Write your working and solution to each question in the answer booklet provided. Additional exam materials (eg. answer booklets, rough paper) will be provided upon request.
For Examiner Use Only
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Page 1 of 6
Total ________
Semester One Final Examinations, 2017 MATH7039 Financial Mathematics
Q 1. ANZ Bank enters into a credit default swap as the protection buyer to insure the bank against default on a large fixed interest loan it has made with BHP. The loan ANZ Bank has with BHP is for $100 million over 3 years at a fixed rate of 7% with annual loan payments. ANZ is insured for 75% of the loss given default. The yearly probability of default is constant at 15% and the yearly recovery rate is constant at 80%. The notional principal is the loan amount.
Assume that the following yield spot curve is observed: y0,1 = 3%, y0,2 = 3.297%, and y0,3 = 3.630%, and the swap rate is 1%. Find the credit default swap value. [8 marks]
Q 2. In each of the following parts, you will be given two quantities to compare. Let a and b denote the left-hand-side and the right-hand-side, respectively. There are exactly seven possible answers. Under the assumptions given in each part,
• Answer: “=” if a = b must hold.
• Answer: “<” if a < b must hold.
• Answer: “>” if a > b must hold.
•Answer: “≤”ifa
impossible.
•Answer: “≥”ifa>banda=bareeachpossible;buta
Pick your answer and provide a brief justification.
Note: As usual, our standing assumptions of frictionless markets and no arbitrage apply to all parts. Aside from these general assumptions, each part stands on its own (even if they share the same notation), unless otherwise instructed.
Let P denote the physical probability measure, and let E denote expec- tation with respect to P.
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Semester One Final Examinations, 2017 MATH7039 Financial Mathematics
Let P denote the risk-neutral probability measure, and let E denote ex- pectation with respect to P.
There exists a bank account Bt = ert, where r > 0 is constant.
There exists a stock S such that St > 0, ∀t > 0, with probability 1.
Let T, K, K1, and K2 denote positive constants.
(a) Let a = (K − S0)+. Let b be the time-0 price of a T-expiry, K-strike European put option on S. Compare a and b. [2.5 marks] Solution: Remember to draw diagrams.
Note that (K −ST)+ ≥ K −ST and (K −ST)+ ≥ 0, we can conclude thatb≥(Ke−rT −S0)+. Also,(K−ST)+ ≤K,sob≤Ke−rT.
In summary (Ke−rT − S0)+ ≤ b ≤ Ke−rT . As such, a <=> b.
(b) Let a and b respectively be the time-0 price of a T-expiry, K1-strike and K2-strike European call option on the underlying asset S. Com- pareaandbwhenK1
(c) Asset S satisfies P(ST = 100) = 0.7 and P(ST = 0) = 0.3. Compare S0 and 50. [2.5 marks] Solution: S0 <=> 50.
If the student used S0 = e−rT E[ST ] = 70e−rT , deduct 1 mark, since P is not risk-neutral.
(d) Assets A and B are worth 100 and 110 today, respectively. Asset A will be worth 110 tomorrow with a physical probability of 0.1 and 90 otherwise. Asset B will be worth 120 tomorrow with physical prob- ability 0.9 and 100 otherwise. Let a be today’s price of a European call on A with strike K1 = 100. Let b be today’s price of a European call on B with strike K2 = 110. Compare a and b. [3 marks] Solution: Since for both options, the payoff are the same (10 for the un-move and zero for the down-move), just need to compare the risk-
neutral probability for the up-move.
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Semester One Final Examinations, 2017 MATH7039 Financial Mathematics
For the option on A, pAu = 5erT − 4.5.
FortheoptiononB,pBu =5.5erT −5. Sincer>0,pBu >pAu,sob>a.
Q 3. Consider a one-period market with time points 0 and T, and three out- comes Ω = {ωu, ωm, ωd} where each outcome has nonzero physical proba- bility. The market has three assets: bank account B, stock S, and option C, where
B0 = 1, BT(ωu) = BT(ωm) = BT(ωd) = 1.2,
S0 = 90, ST(ωu) = 180, ST(ωm) = 120, ST(ωd) = 60,
C0 = 35, CT(ωu) = 0, CT(ωm) = 60, CT(ωd) = 60.
(a) State our basic definition of an equivalent martingale measure.
Solution: It is a probability measure, equivalent to the physical proba- bility measure, such that each asset price divided by B is a martingale.
(b) Find all equivalent martingale measures. (A probability measure on a finite space can be specified by giving the vector of probabilities of individual outcomes.) [3 marks] Solution: P is an equivalent martingale measure iff (pu, pm, pd) = (P(ωu), P(ωm), P(ωd)) satisfies
1.2 180 0 19035=1.2pupmpd 1.212060
andpu,pm,pd >0. The solution is
pu = 0.3,pm = 0.2,pd = 0.5. Page 4 of 7
Semester One Final Examinations, 2017 MATH7039 Financial Mathematics
There is only one equivalent martingale measure. So the market is complete.
(c) Suppose you want to replicate the payoff XT where XT(ωu) = 90, XT(ωm) = 30, XT(ωd) = 0.
Show that this can be done using a portfolio of B,S,C. Solution: Solving
we find that θB = 0, θS = 0.5 and θC = −0.5 units of respectively B, S, C together replicate X.
(d) Find the no-arbitrage time-0 price of payoff XT in two ways: use the pricing probabilities from part (b), and use the replicating portfolio from part (c). [3 marks] Solution:
Using the replicating portfolio gives the price 0.5×90−0.5×35 = 27.5. Using the risk-neutral probabilities gives the price (1/1.2) × (0.3 × 90 + 0.2 × 30 + 0.5 × 0 = 27.5.
Q 4. Recall the Black–Scholes model discussed in class. Specifically, adopt the frictionless assumption and consider two basic assets, namely a bank account with dynamics
dBt = rBtdt, B0 = 1, and a non-dividend-paying stock with dynamics
(1) dSt =μStdt+σStdWt, S0 >0.
Here, r > 0 is the constant risk-free rate, σ > 0 is the instantaneous volatility, μ is the drift, and W is Brownian motion under physical proba- bilities. Let C(S,t) be the price at time t ∈ [0,T] of a T-expiry, K-strike European call option written on S.
1.21800θB 90 S
1.2 120 60θ =30 1.2 60 60 θC 0
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Semester One Final Examinations, 2017 MATH7039 Financial Mathematics
(a) Explain why the dynamics (1) are more suitable for modelling stock prices than the dynamics
dSt = μdt + σdWt.
Two reasons:
1. If dSt = αdt + βdWt, then St could become negative.
2. If dSt = αdt+βdWt then each St+1 −St is independent of Ft. Thus a 10+ dollar move is equally likely, whether St is at 20 or 200, which is very unreasonable.
(1.5 marks total)
For a GBM, the drift and diffusion are proportional to S so S stays positive, and each log return log(St+1/St) is independent of Ft. Thus, a 10+ percent move is equally likely, whether St is at 20 or 200, which is much more reasonable.
(1.5 marks total)
(b) State the Black–Scholes Partial Differential Equation (PDE) that gov- erns C(S,t). Indicate the terminal condition and the domains for S and t. [3 marks]
∂C+rS∂C+1∂2Cσ2S2=rC, (S,t)∈(0,∞)×[0,T), ∂t ∂S 2∂S2
C (S, T ) = (S − K )+
1.5 marks for the PDE, and 0.75 marks for the terminal condition and
0.75 marks for the domain.
(c) Let CBS(S,t,K,T,r,σ) be the Black–Scholes formula for a T-expiry, K-strike European call option written at time t ∈ [0,T] on S having a positive constant instantaneous volatility σ and constant risk-free
rate r. Give the expression for CBS(S, t, K, T, r, σ). terms that you use.
For t < T,
CBS(S,t) = SN(d1) − Ke−r(T−t)N(d2), Page 6 of 7
Define all new [2 marks]
Semester One Final Examinations, 2017 MATH7039 Financial Mathematics
where N(x) = 1 x e−x2/2dx, the CDF of the standard normal, √2π −∞
log(Ser(T −t)/K) σ√T − t
d1,2 ≡ d+,− = σ√T − t ± andCBS(S,T)=(S−K)+ =limt→T CBS(S,t).
1.5 marks for the formula and 0.5 marks for the terminal condition.
(d) Verify that CBS(S, t, K, T, r, σ) that you give in part (c) satisfies the Black–Scholes PDE with the appropriate terminal condition.
It is straightforward to show that −SN′(d1)σ
∂C/∂t= 2√T−t −rKe ∂C/∂S = N(d1),
2 2 N′(d1) ∂C/∂S =Sσ√T−t.
Substitutions show results. (3 marks total) For the boundary condition.
IfS
(1 mark total)
Q 5. Suppose that investors care only about the mean μP and the variance σP2 of portfolio returns, and that they want to minimise σP2 subject to a target portfolio return μP . Suppose that there are N risky assets. Also let
ω = (ω1,…,ωN) ∈ RN be the risky asset portfolio weights vector,
μ = (μ1,…,μN) ∈ RN the risky asset mean returns vector,
1 = (1,…,1) ∈ RN Page 7 of 7
Semester One Final Examinations, 2017 MATH7039 Financial Mathematics an N-dimensional vector of ones, and
for the optimal weights.
σ11 … σ1N Σ= . .
. . σN1 … σNN
the N × N risky asset variance–covariance matrix.
(a) State the mean–variance optimisation problem for a portfolio com-
posed of only risky assets.
(b) Solve the mean–variance optimisation problem in part (a) using the method of Lagrange multipliers. In particular, derive an expression
(c) Express the optimal portfolio’s variance, denoted by σP , in terms of μP , and give a geometric interpretation of the relationship between
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