– Assignment 3 –
MATH3090/7039: Financial mathematics
Assignment 3
Semester I 2021
Due 5pm Thursday May 27 Weight 10 %
MATH3090/7039 total marks 35 marks
Submission: Hardcopy to the submission box by 5pm Friday May 31. Submit a scanned copy of
your assignment on Blackboard.
Assignment questions - all students
1. (15 marks) Based on the material presented in the guest lecture, answer the following ques-
tions.
a. (5 marks) Explain the efficient market hypothesis.
. (5 marks) Provide 2 examples of an inefficient market (Hint: use a company stock price
or index).
c. (5 marks) Explain how social amplification contributes to the efficient market hypothe-
sis.
2. (10 marks) (Replication and bounds)
Assume that discount bonds maturing at T have time-0 price 1.5 per unit. Assume that
T -expiry standard European calls on a [real-valued] random variable ST are available at the
following strikes K and time-0 prices C0(K). Exactly six basic assets are available: the bond
and these five calls, nothing else. As defined in class, these standard European calls pay
(ST −K)+. The following are some derivatives:
K C0(K)
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
For K1 < K2, a T -expiry (K1,K2)-strikes call spread on ST is a portfolio of longing the
T -expiry K1-strike call on ST and shorting the T -expiry K2-strike call on ST .
A T -expiry K-strike binary call on ST pays at time T either 1 if ST ≥ K, or 0 if ST < K.
A T -expiry K-strike binary put on ST pays at time T either 1 if ST < K, or 0 if ST ≥ K.
Answer the following questions.
a. (1 marks) Find the time-0 price of a T -expiry (17.5, 20.0)-strikes call spread on ST and
a T -expiry (20.0, 22.5)-strikes call spread on ST .
MATH 3090/7039 – 1 – Roxane Foulser-Piggott
– Assignment 3 –
. (3 marks) Use part (a) to find the upper and lower bounds on the time-0 price of a T -
expiry 20-strike binary call on ST . You should find a static “supe
eplicating” portfolio
to obtain an upper bound, and a static “su
eplicating” portfolio to obtain a lowe
ound.
c. (3 marks) Find upper and lower bounds on the time-0 price of a T -expiry 20-strike binary
put on ST .
d. (3 marks) Find the time-0 price of a security which pays (5− |ST − 20|)+ at time T .
Hints and Tips for Question 2
In general, when you are given a set of basic assets and asked to price or value a “derivative”,
this means to assign a price to the derivative in such a way that a frictionless “extended
market” consisting of the basic assets together with the derivative [at your proposed price]
would admit no a
itrage
If the derivative can be perfectly replicated by the basic assets, then we showed in lecture
that the unique no-a
itrage price of the derivative is the value of the replicating portfolio.
For example, this was the case in L6.17, where the derivative was a forward contract, and the
asic assets were the underlying S and the bond Z.
If the derivative cannot be perfectly replicated by the basic assets, then there is a range of
prices that could be assigned to the derivative, all of which would be consistent with no-
a
itrage. For example, this was the case in L6.22, where the derivative was a call, and
the basic assets were the underlying non-dividend-paying stock S and the bond Z, and we
considered only static “buy-and-hold”) portfolios. We could not exactly replicate the call
payoff and find a unique price. Instead, what we did, as shown in the diagram, was to super-
eplicate the call payoff using ST , and to su
eplicate the call payoff using ST −K (and also
using the zero payoff). By no-a
itrage, we concluded that the time-0 value of the call is
ounded above by the supe
eplicating portfolio’s time-0 value, and bounded below by the
su
eplicating portfolios’ time-0 values. So there was not a unique price, but there was a
ange of no-a
itrage prices that had upper and lower bounds, expressible in terms of the
prices of the basic assets.
In Question 2, the basic assets are the five (standard) calls and the bond. The derivative in
question depends on what part of the problem you are doing. For the derivative in Question
2(a)-(d), perfect replication is possible. For the derivatives in Question 3(b, c), perfect repli-
cation is impossible (unless you make further assumptions such as assuming dynamics for ST
ut I am not allowing you to make such extra assumptions).
Suggestions: Draw a payoff diagram, like the ones we did in class. Include both the derivative’s
payoff and your [supe
sub]-replicating portfolio’s payoff (as we did in class), to make sure
that your proposed portfolio really does [supe
sub]-replicate. In parts (b)-(c), you do not
need to show that your bounds are tight. However, bounds which are not tight will lose some
credit. Tight means that they cannot be improved (lowered in the case of an upper bound,
aised in the case of a lower bound) without making further assumptions.
3. (10 marks) (Black-Scholes) Fill in the details of L9.24. Specifically,
MATH 3090/7039 – 2 – Roxane Foulser-Piggott
– Assignment 3 –
a. (3 marks) Assume that C(S, t) satisfies the first PDE of L9.24 (Black-Scholes). Show
that U(z, τ) satisfies the second PDE of L9.24 (the heat/diffusion equation).
. (5 marks) Using L9.23, solve for U .
Write your final answer using the standard normal CDF N .
Hint: Replace the lower limit of integration (the −∞) with a co
ectly chosen quantity
that lets you get rid of the “positive-part function” in the integrand, thus replacing
(· · · )+ with (· · · ). Then split the integral into two integrals. Introduce a new variable
of integration x into each integral. In the first integral, let x := (ζ − z− σ2τ)/(σ
√
τ); in
the second integral, let x := (ζ − z)/(σ
√
τ).
Also have a look at Question 3 on tutorial week 9.
c. (2 marks) Transform the solution U(z, τ) back to find C(S, t).
MATH 3090/7039 – 3 – Roxane Foulser-Piggott
L9.1
MATH3090/7039: Financial mathematics
Lecture 9
Roxane Foulser-Piggott
L9.2
A
itrage in continuous time
Black-Scholes: PDE and formula
Fundamental theorem in continuous time
Solution to the BS PDE
Outline
A
itrage in continuous time
Black-Scholes: PDE and formula
Fundamental theorem in continuous time
Solution to the BS PDE
L9.4
A
itrage
Consider an Ft-adapted vector price Itô process Xt = (X1t , . . . , XNt ).
A portfolio/trading strategy is an Ft-adapted vector process
Θt = (θ
1
t , . . . , θ
N
t ) of quantities held in each asset 1, . . . , N .
Say that the trading strategy is self-financing if its value Vt = Θt ·Xt
satisfies (with probability 1) for all t
dVt = Θt · dXt, equivalently Vt = V0 +
∫ t
0
Θu · dXu.
A
itrage is a self-financing trading strategy Θt such that
V0 = 0, and both:
P(VT ≥ 0) = 1
P(VT > 0) > 0
o
V0 = 0 and P(VT ≥ 0) = 1
L9.5
Replication
• Definition: a trading strategy Θ replicates a time-T payoff YT if
it is self-financing, and its value VT = YT (with probability 1).
• Law of one price: At any time t < T , the no-a
itrage price of an
asset paying YT must be the value of the replicating portfolio.
• To hedge could mean to [try to] replicate a payoff (or the portion
of a payoff attributable to some particular source of risk), but
usually it means to [try to] replicate the negative of the payoff.
For example, to hedge a position that is short one option usually
means to [try to] replicate a position that is long the option.
I say “try to” because “hedge” can mean an approximation to
eplication such as super-replication, or
oadly speaking, any
strategy to reduce some notion of risk.
Outline
A
itrage in continuous time
Black-Scholes: PDE and formula
Fundamental theorem in continuous time
Solution to the BS PDE
L9.7
Motivation for GBM to model a stock price
BM is a natural starting point for model-building. But some problems
with Wt or αt+ βWt as a model for a stock price
• BM can go negative, and so can scaled BM with drift αt+ βWt.
• If dSt = αdt+ βdWt then each St+1 − St is independent of Ft.
A 10+ dollar move is equally likely, whether St is at 20 or 200.
For a GBM S, the drift and diffusion are proportional to S
• S stays positive
• Each log return log(St+1/St) is independent of Ft.
A 10+ percent move is equally likely, whether St is at 20 or 200.
L9.8
Black-Scholes model
In continuous time, consider two basic assets:
• Money-market or bank account: each unit has price Bt = ert.
Equivalently, it has dynamics
dBt = rBtdt, B0 = 1
• Non-dividend-paying stock: share price S has GBM dynamics
dSt = µStdt+ σStdWt, S0 > 0
where volatility σ > 0 and W is BM, under physical probabilities.
Find: price Ct of call which pays CT = (ST −K)+ at time T (K > 0).
We first do an intuitive, but flawed, derivation, then do a careful
proof.
L9.9
Plan of intuitive derivation: Replicate B using C and S
• We will price options using replication.
The other approach is to use the martingale
isk-neutral pricing
approach: Apply Fundamental Thm, and take E of discounted
payoff. We do not cover this in this course.
• Construct risk-free portfolio of (C, S).
Risk-free means zero dW term
• If self-financing, then it must grow at the risk-free rate r, else
there is a
itrage of portfolio vs B.
• On the other hand, if Ct = C(St, t) for some smooth function C,
then Itô rule says that the portfolio value’s drift can be expressed
in terms of C’s partials.
• Therefore C(S, t) satisfies a PDE.
• Solve this PDE to obtain formula for C.
L9.10
Construct a risk-free portfolio
Use (1 option, -at share), choosing at to cancel the option risk
Portfolio value is
Vt = Ct − atSt.
So some authors assert that
dVt = dCt − atdSt.
But the Itô’s rule says
d(atSt) = atdSt + Stdat + (dat)(dSt),
so it’s not true that d(atSt) = atdSt. Ignoring this point for now . . .
L9.11
Construct a risk-free portfolio (cont)
Assume Ct = C(St, t) where C