TOC
1. AD Pricing: Equilibrium
1.1 Arrow-Debreu Economy
Assume there are two dates: and . There are N possible states of nature at date 1, which is index by with probability
There is one perishable consumption good and agents, with endowment
Assume agent โs preferences:
where is consumption and is the time discount factor.
AD securities (state-contingent claims): Securities pays one unit of consumption if state occurs and nothing otherwise.
Agentโs Problem: agent maximize the expected utility:
The inequality contraint will typically hold with equality in a world of nonsatiation.
Equilibrium is a set of prices such that
- All agents obtain optimal given the price
- All markets clear: aggregate consumption = aggregate endowment
In market
In : for each state , market
An example is shown as below:
Two periods: . At , two states with . and . The discount factor is 0.9.
Denote as the price of one unit of consumption in date 1 state 1 and 2. The price of period 0 consumption is normalized to 1. Agent aโs problem
Using L-method
F.O.C.:
Similarly for agent b:
With market-clearing conditions:
Solve them jointly
- AD price:
- Consumption:
Thus
1.2 AD Price and Utility
Substitute the budget constraint into the objective function
F.O.C.
The price of AD security equals the products of 3 components:
- is the pure time discount factor
- is the probability for state
- The last term reflects the collective (market) assessment of the scarcity of the consumption in the future relative to today
is the marginal utility if state is realized.
Risk Sharing and Utility Improving
Agent aโs expected utility before trade
Agent aโs expected utility after trade
Similarly for b, his utility changes from 4.157 to 4.210.
Case: No Aggregate Risk
Assume at , two states with
EU Before:
EU After:
If there is No aggregate risk, people are perfectly insured.
From the first order condition
For consumer a and b, the MRS between state 1 and state 2 consumption are the same. If people have same risk averse: Investors share risk proportionately. Specially, if one of the two agents is fully insured (MRS=1), the other must as well. If people are differentially risk averse, the less risk averse will provide some insurance services to the more risk averse. That is, the agent most tolerant of risk bears a disproportionate share of risk.
1.3 Market Failure
Pareto Optimal Properties: It is impossible to rearrange the allocation of consumption so that the utility of one agent is higher without diminishing the utility of the other agent.
In PO allocation, the ratio of two agentsโ MU with respect to each goods (time & states) should be identical. In practice, the desired allocation is through impersonal security markets or through deals involving financial intermediaries. The reason is that in an organized market, the contracts implied by the purchase or sale of a security are enforceable.
Example:
Assume there are two periods: . At , two states with . and . The discount factor is 1
Introduce a security which entails 1 unit state 1 consumption with price . Denote as its demand. Ther are two markets and the intertemporal value transfer is only by the security .
Financial market clear , therefore
Utility before trade is
Utility after trade is
Agent is issuer of , if the effective cost of setting up the market is larger than 0.074, market failure unless agent shares cost. Sometimes there will be no issuer even total utility will increase.
2. AD Pricing: No-arbitrage
2.1 Options
Option is a contract entitling the holder to buy or sell a security at particular price at a time in the future.
Call option:
- European option: option that can be exercised only at expiration of the contract.
- American option: option that can be exercised any time up to expiration.
Index options of China
Upper Bound for Call Option Price
An call option gives the holder the right to buy one share of a stock for a certain price. The option can never be worth more than the stock .
Lower Bound for Call Option Price
Consider the two portfolio:
- A: Buy one call option, buy a zero-coupon bond that pays K at time T
- B: Buy one share of the stock
At time 0, portfolio Aโs cashflow (cost) is ; portfolio Bโs cashflow (cost) is .
At time T, portfolio Bโs cashflow is , for portfolio A:
- if , cashflow is
- if , cashflow is , which is bigger than
So portfolio Bโs future cashflow portfolio Aโs future cashflow. Therefore, Aโs cost should no less than Bโs cost, that is , and thus . Note that too.
Put-Call Parity
Consider two portfolios:
- A: Buy one call option, buy one zero-coupon bond
- C: Buy one put option, buy one share of the stock
Cashflows are shown as below
Since future cashflows of two portfolios are the same, the price of them should also be the same, therefore
2.2 Market Completeness
Financial markets are said to be complete if there exist a set of assets , such that any payoff is spanned by the payoffs of those assets.
First Welfare Theory: In a complete market, the competitve allocation is Pareto optimal.
AD Pricing
The spanning set is most conveniently using the AD assets. Markets are complete if, for each state of nature , there exsits a AD security.
Once AD price is available, it provides the answer to the key valuation question. As long as the AD securities are traded, their prices constitute the essential building blocks.
If markets are complete, any cash-flow stream can be replicated as a portfolio of AD securities
Denote as the price of the AD security in state , with no arbitrage condition
Value Additivity Theorem
Assume two assets have payoffs and , with equilibrium price and . Suppose a third asset , can be replicated by
Then at the date 0
The proof is quite straightforward,
denote the payoff of asset in state by , then
Recovering AD Securities
Example: There are two assets in the market:
To recover AD securities, we can obtain that
General Case
Assume Non-AD Assets, , space . Construct portfolio with weights
AD Price from Coupon bond
Assume there are two 5-year coupon
To find the price of the 5-year A-D security, we can use replication
The cashflows are shown as below
Therefore, we build a 5-year zero-coupon bond. The price of the 5-year A-D security is then 176.92/230.77=23/30.
Complete the Market
Assume stock โs price is
state | price |
1 | |
2 | |
3 |
Denote the call option value will strike price is , then
together with and thus can constitute a complete set of securities markets.
Incomplete Market
Stock price
together with and do not span the . Thus, market is not complete.
Proposition: The stock or portfolio must distinguish between states by its payoff structure for option to complete the market.
2.3 AD and Option
By CAPM, the state of nature can be identified by market portfolio. State assumes any value in a range, to identify the states of nature with ranges of possible value for the market portoflio.
Recovering AD Prices from Options
Assume is the price of the underlying portfolio. AD security that pays 1 when
Construct Portfolio:
- Buy one call with
- Sell one call with
- Sell one call with
- Buy one call with
The payoff structure is shown as below
The payoff table is below
Portfolio Price
Buy unit of the portfolio to get 1 unit payoff. Let to eliminate the area of the triangular part โ payoff is 1 in .
AD price at is
This gives a price of contigent claim that pays when .
Example
There is a call option on an underlying with current price . The call optionโs price with a range of strike prices are shown as below. Calculate the AD price associated with
AD Pricing for Discrete States
3. Multi-period AD Pricing
3.1 Stationarity Hypothesis
Pricing long-term assets, requires some form of stationarity: N states and AD prices are constant through time. The price of the one-period AD securities is
where is the AD price of receiving 1 unit tomorrow if state occurs given we are in state today.
Stationarity hypothesis is equivalent to: Q is invariant through time.
3.2 Multi-Periods Pricing
Pricing matrix between time and
- : value of $1 in 2 periods if state 1 occurs and the intermediate state is 1.
- : value of $1 in periods if state 1 occurs and the intermediate state is 2.
Pricing matrix between time and is
Note that AD prices include information of discount factors and MU ratios.
In general, to price the cash flow CF at time
Assume , then 1 period discount bond:
furthermore
For 2 period discount bond
furthermore
APT and Arrow-Debreu
Consistency:
- Every individual asset or portfolio can be viewed as a complex security, or a combination of primitive securities.
Difference:
- Definition of the premitive securities
- APT: risk factors
- AD: AD securities
- In theory, AD securities is satisfying while APT factors have many shortcomings. However, in reality, APT factors are easier to be constructed than risk factors, and APT pricing is more frequently used.
- Pricing of the primitive securities
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