Reference no: EM132264045

Question 1. Assume that individual preferences can be characterized by a CRRA utility function:

u(W) = -W^-1

Assuming that the individual's initial wealth is 10,000, the individual faces a risk Z ~ can be expressed as:

       100   1/4probability
Z ~ =   0     1/2probability
     - 100   1/4probability

Try to calculate how much insurance the individual is willing to pay to avoid such a risk.

Try the Arrow-Pratt approximation to calculate, compare and analyze the results?

Question 2. Assume that individual preferences can be characterized by a CRRA utility function:

u(W) = -W-1

Assuming that the individual's initial wealth is 100, the individual faces two risks Z ~1 and Z ~2, which can be expressed as:

(1) Try to calculate how much insurance the individual is willing to pay to avoid such two risks, which risk the individual prefers

(2) Try the Arrow-Pratt approximation to calculate, andcompare with (1) and analyze the results?

Question 3. Consider the US put option with the underlying asset as a stock, the maturity period is 3 months, the strike price of the option is $10, the current price of the stock is $10, the volatility of the stock is 20%, and the risk-free interest rate is 5% (continuous compound interest ).

Calculate the theoretical price of the American put option by using the three-stage binary tree method If the underlying asset of the American put option is a future, assuming that the futures volatility is 30% and the current futures price is $11, Use the three-stage binary tree method to calculate the theoretical price of the American futures option.

Question 4. Consider the US put option with the underlying asset as a stock, the maturity period is 2 months, the strike price of the option is $10, the current price of the stock is $10, the volatility of the stock is 20%, and the risk-free rate is 5% (continuous compound interest )

Try the two-stage binary tree method to calculate the theoretical price of the American put option. If the underlying asset of the American put option is a future, assuming that the futures volatility is 30% and the current price of the futures is $11, Use the two-stage binary tree method to calculate the theoretical price of the American futures option.

Question 5. Note f is the exchange rate between the RMB and the Hong Kong dollar (in terms of how many RMB/1 Hong Kong dollar), and assumed to obey the following random process:

df/f = μ_fdt + σ_fdW_f

Assume that the price of an A-share traded in the Shanghai market is subject to the following stochastic process:

dA/A = μ_Adt + σ_AdW_A

The corresponding H-share prices traded in Hong Kong are subject to the following random process:

dH/H = μ_Hdt + σ_HdW_H

whendW_f,dW_A,dW_His Brownian motion

Applying Itô tempering: What kind of random process is the difference between the price of Hong Kong stock denominated in RMB and the price of A shares?

Question 6. Suppose there are three types of long-lived securities in the economy, j = 1, 2, 3, they pay dividends only at t = 2, the price after deducting dividends at t = 0, 1 and the dividend payments at t = 2 as shown in the diagram.

(1) Try to calculate the equivalent martingale measures and risk-free interest rates in the economy;
(2) Try to calculate the price of the European call option with t = 0, the target on asset 1, maturity is 2, and execution price is 2.
(3a) Try to calculate the American put price of the target at asset 2, the maturity is 2, and the strike price is 3.
(3b) Try to calculate the price of the European put option with the target price of 3, the mature period is 2, and the execution price is 3 when calculating t = 0.
(4a) If the price of the three assets becomes (1, 1, 1) when the event (ω₁, ω₂, ω₃) occurs at t = 1, try to calculate whether there is an equivalent martingale measures in the economy; if not, construct a Arbitrage opportunities.
(4b)If the price of the three assets becomes (1, 1, 3) when the event (ω₁, ω₂, ω₃) occurs at t = 1, try to calculate whether there is an equivalent martingale measures in the economy; if not, construct a Arbitrage opportunity

Question 7. Assume that there are four equal-probability natural states ω₁, ω₂, ω₃ and ω₄ in the economy, assuming that the immediate return rater ~_1 of asset 1 in the economy and the stochastic return rate r ~_m of the market combination m are given by the following table, risk-free interest rate r ~_fis equal to 1.5. Does the CAPM theorem hold in this economy?

natural states return rate	ω1	ω2	ω3	ω4
r1(ω)	3	2	0	5
rm(ω)	2	-1	4	3

Question 8. It is assumed that there are four natural states ω₁, ω₂, ω₃ and ω₄ in the economy. It is assumed that there are two risk assets in the economy, and the values of random returns r ~₁?r ~₂andr ~₃ are given by the following table.

natural states return rate	ω1	ω2	ω3	ω4
r~1(ω)	2	2	0	0
r~2(ω)	1	3	1	3
r~3(ω)	4	2	2	4

1) Try to calculate the expected rate of return, variance and covariance of these three assets;

2) Try to calculate the forward porfolio of these two assets;

3) Suppose there is another portfolio in the economy whose stochastic rate of return obeys: r ~_p (ω₁ )= 3,r ~_p (ω₂ )= 2,r ~_p (ω₃ )= 1,r ~_p (ω₄ )= 2 try to calculate the expected rate of return, variance of the asset, and decompose the asset into a sum of a forward portfolio and a non-systematic risk.