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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 for any n=1,2,.The future
3、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 Unit Prices：share prices S1(n),.,Sm(n)are random variables taking only finitely ma
4、ny 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 held by an investor between times n 1 and n.A sequence of portfolios indexed by n=1,2,.is called an i
5、nvestment 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 funds takes place.(In real life cash can be taken out of the portfolio for consumption or injected from other sour
6、ces.)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,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 the
7、m,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 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
8、security deposit to cover possible losses.If their losses exceed the deposit,the position must be closed.The deposit creates a burden on the portfolio,particularly if it earns no interest for the investor.However,restrictions of this kind may not concern dealers who work for major financial institut
9、ions holding large numbers of shares deposited by smaller investors.These shares may be borrowed internally in lieu of short selling.(Why?)PredictableAssumption:An investor has no knowledge of future stock prices.In particular,no insider dealing is allowed.Definition:An investment strategy is called
10、 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 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 ass
11、ets,it is always possible to find a sequence y(n)of risk-free positions such that(x1(n),.,xm(n),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 a
12、sset with A(0)=10 and A(1)=11 dollars,and one risky asset such that S(0)=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:The
13、re is no admissible strategy such that V(0)=0 and V(n)0 with 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 mark
14、et with a risk-free asset such that A(0)=100,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
15、 selling of the risky asset is allowed?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 do
16、wn.3)a0(a long position in bonds 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 sati
17、sfy for any j=1,.,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 90 100 112
18、 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 Securities:contin
19、gent 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.S
20、olvency: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 random variables taking only finitely many values.A portfolio is a vector (x1(n),.,xm(n),y(n),z1(n),.,zk(n)indicating the number of primary and derivative securities held by an investor
21、 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 isself-financing：predictable：for each n=0,1,2,.the portfolio(x1(n+1),.,xm(n+1),y(n+1),z1(n+1),.,zk(n+1)constructed at time n depends only on the nodes of the tree of market scenarios reached up to and including time n.admissible：V(n)0Fundamental Theorem of Asset Pricing
期货理论与实务英文版课件：Chapter 4 Discrete Time Market Models
