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MTH5520 Interest Rate Modelling

创建日期：2024-05-31 16:12:34

所属分类：数学mathematics

标签： MTH5520

Interest Rate Modelling

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MTH5520 Interest Rate Modelling

Due: 4 Aug 2023, 11:55 pm (Melbourne Time)
Maximum Score: 5 stars
Name:
Instructor’s name: Dr. Kihun Nam: Department of Mathematics, Monash University
You are expected to provide full solutions, with detailed explanations.
Question We have the following problem and Zelda tries to solve it using Generative AI.
Consider a multiperiod binomial model with S0 = 1, N = 100,∆t = 1/N, r = 0.1∆t, u =
1.002, d = 1/u and t = (n∆t)n=0,1,2,...,N . We want to price an European call option with
expiry t = 1 and strike 1.1. Here, the binomial tree represents all possible scenarios of the
stock price (Sn∆t)n=0,1,2,...,N .
(a) Let (Vn∆t)n=0,1,2,...,N be the value of replicating portfolio and (Dn∆t)n=0,1,2,...,N be the
number of stocks S during (t, t + ∆t] in the portfolio. Express Vt+0.01 − Vt in terms
of Vt, St, St+0.01 − St and Dt. Then using notation dt = ∆t, dVt = Vt+0.01 − Vt and
dSt = St+0.01 − St to find the corresponding SDE with appropriate terminal condition
for V1.
(b) What is the risk neutral probability (p˜, q˜)? Round these numbers to 3 decimal places.
(e.g. 0.12343... to 0.123). Prove that Sn∆t(1+r)n and
Vn∆t
(1+r)n are martingales under the risk
neutral probability.
The following is the answer by AI.
Sure, let’s go step by step to solve the problems:
(a) To express Vt+0.01 − Vt in terms of Vt, St, St+0.01 − St, and Dt, we need to understand
the changes in the replicating portfolio over a small time interval (t, t + ∆t]. During this
time interval, the value of the portfolio can change due to two main factors: changes in
the stock price and changes in the number of stocks in the portfolio.
Let’s consider Vt as the value of the replicating portfolio at time t, St as the stock price at
time t, and Dt as the number of stocks in the portfolio at time t. At time t, the portfolio
value is given by Vt = Dt · St.
Now, at time t + ∆t, the stock price can either go up to St+0.01 = St · u or go down to
St+0.01 = St · d. The value of the replicating portfolio can change accordingly:
1. If the stock price goes up: The new value of the portfolio is Vt+0.01 = (Dt−1)·St ·u+Dt ·St
since we subtract one stock from the portfolio and add the amount gained from the stock
price going up.
2. If the stock price goes down: The new value of the portfolio is Vt+0.01 = (Dt + 1) · St · d+
Dt · St since we add one stock to the portfolio and add the amount gained from the stock
price going down.
Now, we can calculate the change in the portfolio value:
dVt = Vt+0.01 − Vt
1. If the stock price goes up: dVt = [(Dt − 1) · St · u + Dt · St] −Dt · St = (Dt · u−Dt) · St =
Dt · (u− 1) · St
2. If the stock price goes down: dVt = [(Dt + 1) ·St · d+Dt ·St]−Dt ·St = (Dt−Dt · d) ·St =
Dt · (1 − d) · St
The probability of the stock price going up is p and the probability of the stock price going
down is q = 1 − p. In the binomial model, we have u = 1.002 and d = 1/u. To find the risk-
neutral probability, we need to set p such that the expected return of the stock matches
the risk-free rate r over the time interval ∆t.
The risk-neutral probability p can be found using the following equation:
1 + r = p · u + q · d
Substituting the given values:
1 + 0.1∆t = p · 1.002 + (1 − p) · (1/1.002)
Solving for p, we get the risk-neutral probability.
Page 2
(b) To prove that Sn∆t(1+r)n and
Vn∆t
(1+r)n are martingales under the risk-neutral probability, we
need to show that their expected values at time t are equal to their values at time t− ∆t.
1. For Sn∆t(1+r)n : The stock price follows a multiplicative random walk, and the risk-neutral
probability is p. Therefore, the expected value of the stock price at time t is:
E
[
Sn∆t
(1+r)n
]
= p · St·u(1+r) + (1 − p) · St·d(1+r) = St(1+r)
This shows that Sn∆t(1+r)n is a martingale under the risk-neutral probability.
2. For Vn∆t(1+r)n : The value of the replicating portfolio can be written as Vt = Dt·St. Therefore,
at time t + ∆t, we have:
Vt+∆t = Dt · St · u (if the stock price goes up) or Vt+∆t = Dt · St · d (if the stock price goes
down).
The expected value of the replicating portfolio at time t is:
E
[
Vn∆t
(1+r)n
]
= p · Dt·St·u(1+r) + (1 − p) · Dt·St·d(1+r) = Dt·St(1+r)
Since Dt is the number of stocks in the portfolio at time t, and Dt is not affected by the
risk-neutral probability (it’s a deterministic quantity based on the option’s payoff and
stock price), Vn∆t(1+r)n is also a martingale under the risk-neutral probability.
For question (a) and (b), mark the AI solution (out of 100) as you are a grader of final
exam and provide feedback. Also, provide the correct solution.

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