期貨理論與實務英文版課件：Chapter 4 Discrete Time Market Models

1、Chapter 4Discrete Time Market Modelsmain content4.1 Stock and Money Market Models4.1.1 Investment Strategies4.1.2 The Principle of No Arbitrage4.1.3 Application to the Binomial Tree Model4.1.4 Fundamental Theorem of Asset Pricing4.2 Extended Models4.1 Stock and Money Market ModelsThe risky positions

2、 in assets number 1, . . . , m will be denoted by x1, . . . , xm,respectively, and the risk-free position by y. The wealth of an investor holding such positions at time n will benegative y(n),xm(n) means what?Assumptions1.Randomness：The future stock prices S1(n), . . . , Sm(n) are random variables f

3、or any n =1, 2, . . . . The future prices A(n) of the risk-free security for any n = 1, 2, . . . are known numbers.2.Positivity of Prices：S(n) 0 and A(n) 0 for n = 0, 1, 2, . . . .3.Divisibility, Liquidity and Short Selling：x1, . . . , xm, y R.4.Solvency：V (n) 0 for n = 0, 1, 2, . . . .5.Discrete Un

4、it Prices：share prices S1(n), . . . , Sm(n) are random variables taking only finitely many values.4.1.1 Investment StrategiesA process that the investor alters the position of risky and risk-free assets.A portfolio is a vector (x1(n), . . . , xm(n), y(n) indicating the number of shares and bonds hel

5、d by an investor between times n 1 and n. A sequence of portfolios indexed by n = 1, 2, . . . is called an investment strategy. The wealth of an investor or the value of the strategy at time n 1 isAt time n = 0 the initial wealth is given bySelf-FinancingAssumption: no consumption or injection of fu

6、nds takes place. (In real life cash can be taken out of the portfolio for consumption or injected from other sources.)Definition：if the portfolio constructed at time n 1 to be held over the next time step n + 1 is financed entirely by the current wealth V (n), that is,ExampleV(0) = 3000, x1(1) = 20,

7、 x2(1) = 65, y(1) = 5V(1) = 2065 + 6515 + 5110 = 2,825; At that time the number of assets can be altered by buying or selling some of them, as long as the total value remains $2,825.x1(2) = 15, x2(2) = 94 , y(2) = 4 V(1) = 1565 + 9415 + 4110 = 2,825; Short Selling in RealityIn practice some security

8、 measures to control short selling may be implemented by stock exchanges. Typically, investors are required to pay a certain percentage of the short sale as a security deposit to cover possible losses. If their losses exceed the deposit, the position must be closed. The deposit creates a burden on t

9、he portfolio, particularly if it earns no interest for the investor.However, restrictions of this kind may not concern dealers who work for major financial institutions holding large numbers of shares deposited by smaller investors. These shares may be borrowed internally in lieu of short selling. (

10、Why?)PredictableAssumption: An investor has no knowledge of future stock prices. In particular, no insider dealing is allowed. Definition: An investment strategy is called predictable if for each n=0,1,2,.the portfolio (x1(n+1),.,xm(n+1),y(n+ 1) constructed at time n depends only on the nodes of the

11、 tree of market scenarios reached up to and including time n.PropositionGiven the initial wealth V (0) and a predictable sequence (x1(n), . . . , xm(n), n = 1, 2, . . . of positions in risky assets, it is always possible to find a sequence y(n) of risk-free positions such that (x1(n), . . . , xm(n),

12、 y(n) is a predictable self-financing investment strategy.Admissibleif it is self-financing, predictable, and for each n = 0, 1, 2, . . . V (n) 0 with probability 1.ExerciseConsider a market consisting of one risk-free asset with A(0) = 10 and A(1) = 11 dollars, and one risky asset such that S(0) =

13、10 and S(1) = 13 or 9 dollars. On the x, y plane draw the set of all portfolios (x, y) such that the one-step strategy involving risky position x and risk-free position y is admissible.4.1.2 The Principle of No ArbitrageDefinition:There is no admissible strategy such that V (0) = 0 and V (n) 0 with

14、positive probability for some n = 1, 2, . . . .ExerciseShow that the No-Arbitrage Principle would be violated if there was a self-financing predictable strategy with initial value V(0) = 0 and final value 0V(2)0, such that V(1)0.ExerciseConsider a market with a risk-free asset such that A(0) = 100,

15、A(1) =110, A(2) = 121 dollars and a risky asset, the price of which can follow three possible scenarios,Scenario S(0) S(1) S(2) 1 100 120 144 2 100 120 96 3 100 90 96 Is there an arbitrage opportunity if a) there are no restrictions on short selling, and b) no short selling of the risky asset is all

16、owed?4.1.3 Application to the Binomial Tree Modelwhy the binomial tree model admits no arbitrage if and only if d r u?proof:In Chapter 1, we have proofed no arbitrage d r u.Suppose that dr0 (a cash loan invested in stock). Then V(1) =a(dr)0 if the price of stock goes down.3)a0 (a long position in bo

17、nds financed by shorting stock). In this case V(1) =a(ur)0 if stock goes up.Arbitrage is clearly impossible when dru.Several steps. Let d r 0 at one or more of these nodes. By the one-step case this is impossible if d r 0 for each scenario and the discounted stock price satisfy for any j = 1, . . .

18、, m and n = 0, 1, 2, . . . , where denotes the conditional expectation with respect to probability P* computed once the stock price S(n) becomes known at time n. ExampleLet A(0) = 100, A(1) = 110, A(2) = 121 and suppose that stock prices can follow four possible scenarios: Scenario S(0) S(1) S(2) 1

19、90 100 112 2 90 100 106 3 90 80 90 4 90 80 80 The tree of stock prices is shown in Figure 4.2. The risk-neutral probability P*is represented by the branching probabilities p*, q*, r* at each node.4.2 Extended ModelsPrimary Securities:traded independently of other assets (such as stock).Derivative Se

20、curities:contingent on the prices of other securities (such as options or forwards).Assumptions:1.Randomness:The asset prices S1(n), . . . , Sm(n),A(n),D1(n), . . . , Dk(n) are random variables for any n = 1, 2, . . . .2.Positivity of Prices:S1(n), . . . , Sm(n),A(n) 0 for n = 0, 1, 2, . . . .3.Divisibility, Liquidity and Short Selling:x1, . . . , xm, y, z1, . . . , zk R.4.Solvency:V (n) 0 for n = 0, 1, 2, . . . .5.Discrete Unit Prices:For each n = 0, 1, 2, . . . the prices S1(n), . . . , Sm(n),A(n),D1(n), . . . , Dk(n) are