STAC70 Statistics & Finance I
Task 1
Due date: Feb four, 2021 (by 11:59pm). Late submissions might be penalized.
Be aware: For some issues it’s extra handy (and even essential) to make use of the pc to
carry out the computations. For those who do, carry out all computations in R. Embrace your codes and
outputs in your submission.
1. Think about the only interval binomial mannequin with d < 1 + r < u. Show from first rules
(i.e., with out utilizing the elemental theorems of asset pricing) that there’s a distinctive
equal martingale measure.
2. Think about the only interval binomial mannequin. Suppose r = zero.02, u = 1.05, d = zero.9 and
S0 = 50. Think about a put possibility on the inventory with strike value Okay = 49.
(a) Compute the equal martingale measure (qu, qd).
(b) Discover the no-arbitrage value of the declare at time zero.
(c) Discover the replicating portfolio.
three. Think about a single interval binomial mannequin which satisfies the no-arbitrage situation d <
1 + r < u. Think about two contingent claims C and C
zero the place
(cu, cd) = (1, 2) and (c
zero
u
, c0
d
) = (1, four).
You’re provided that the no-arbitrage value of C is 1.5, and the no-arbitrage value of C
zero
is 2.
(a) Discover the rate of interest r. Trace: Use the 2 given claims to generate different cashflows.
(b) Discover the risk-neutral chances qu and qd.
four. Write down two totally different units of parameters (u, d, r, S0) of the only interval binomial
mannequin which give the identical equal martingale chances. Should they provide the identical
costs for all contingent claims? Clarify your reasoning.
5. Think about the trinomial mannequin in Instance 2.1 of the lecture notes: r = zero, S0 = 1, u = three,
m = 2 and d = half of. Characterize all contingent claims that may be replicated. That’s,
give a normal expression of the payoffs cu, cm, cd that may be replicated.
6. Think about a single-period mannequin with four states and three dangerous belongings (other than the risk-free
asset). Write Ω = . Assume that
(S
j
zero
)zero≤j≤three =
1 1.7 2 2
and
(S
j
1
(ωi))1≤i≤four,zero≤j≤three =
1 1 three 1
1 2 three three
1 2 1 1
1 1 1 three
.
1
(Right here the column index j runs from zero to three. So the risk-free rate of interest is zero.)
(a) Present that the mannequin is arbitrage-free and full.
(b) Think about the contingent declare
(C(ωi))1≤i≤four =
1
zero
zero
1
.
Discover the no-arbitrage value of C at time zero and discover a replicating portfolio. Is the
replicating portfolio distinctive?
7. Think about the playing technique in Instance three.2 of the lecture notes, the place the preliminary
wealth is zero. Discover the precise distribution of the wealth M10 at time t. (This implies discovering
the likelihood mass operate P(M10 = x) for the attainable values of x.) Confirm numerically
that E[M10] = zero.
2
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Task 1 for STAC70 Statistics & Finance I
The deadline is February four, 2021. (by 11:59pm). Late submissions will incur penalties.
Nota bene: For some conditions, it’s extra handy (or maybe required) to make the most of a pc to unravel them.
perform the calculations For those who do, ensure to do your whole calculations in R. Embrace your whole codes and
in your submission’s outputs
1. Check out the only interval binomial mannequin with d 1 + r u. Show it from the bottom up.
(i.e., with out utilizing the elemental theorems of asset pricing) that there’s a distinctive
equal martingale measure.
2. Think about the only interval binomial mannequin. Suppose r = zero.02, u = 1.05, d = zero.9 and
S0 = 50. Think about a put possibility on the inventory with strike value Okay = 49.
(a) Compute the equal martingale measure (qu, qd).
(b) Discover
