期貨理論與實務英文版課件：Chapter 3 Risky Assets

1、Chapter 3 Risky Assetsmain content3.1 Dynamics of Stock Prices3.1.1 Return3.1.2 Expected Return3.2 Binomial Tree Model3.2.1 Risk-Neutral Probability3.2.2 Martingale Property3.3 Other Models3.3.1Trinomial Tree Model3.3.2 Continuous-Time Limit3.1 Dynamics of Stock PricesThe price of stock at time t wi

2、ll be denoted by S(t).S(t) can be represented as a positive random variable on a probability space , that is,S(t) : (0,).The probability space consists of all feasible price movement scenarios . We shall write S(t, ) to denote the price at time t if the market follows scenario .Example：Suppose that

3、there are three possible market scenarios, = 1, 2, 3, the stock prices taking the following values over two time steps:Scenario S(0) S(1) S(2) 1 55 58 60 2 55 58 52 3 55 52 53These price movements can be represented as a tree, see Figure 3.1.1 ReturnIf the stock pays no dividends:The rate of return,

4、 or briefly the return K(n,m) over a time interval n,m (infact m, n ), is defined to be the random variableIf the stock pays a dividend of div(m) at time m:Exercise: 除權、復息公式3.1.1 Returnsome conclusions1 + K(n,m) = (1+K(n + 1)(1 + K(n + 2) (1 + K(m)S(m) = S(n)(1 + K(n + 1)(1 + K(n + 2) (1 + K(m).loga

5、rithmic returnThe logarithmic return over a time interval n,m (more precisely, n, m) is a random variable k(n,m) defined byk(n,m) = k(n + 1) + k(n + 2) + + k(m), if there are no dividends.The relationship between the K(m,n) and k(m,n)3.1.2 Expected ReturnDefinition:Suppose that the probability distr

6、ibution of the return K over a certain time period is known. Then we can compute the mathematical expectation E(K),called the expected return. It is important to note, however, that the expected return is usually based on historical data and is not guaranteed.ExerciseSuppose that the stock prices in

7、 the following three scenarios are Scenario S(0) S(1) S(2) 1 100 110 120 2 100 105 100 3 100 90 100with probabilities 1/4, 1/4, 1/2, respectively. Find the expected returns E(K(1), E(K(2) and E(K(0, 2). Compare 1 + E(K(0, 2) with (1 +E(K(1)(1 + E(K(2).some conclusionsIf the one-step returns K(n + 1)

8、, . . . , K(m) are independent, then:1 + E(K(n,m) = (1 + E(K(n + 1)(1 + E(K(n + 2) (1 + E(K(m)In the case of logarithmic returns additively extends to expected returns, even if the one-step returns are not independent. NamelyE(k(n,m) = E(k(n + 1)+E(k(n + 2)+ + E(k(m)3.2 Binomial Tree ModelThe model

9、is defined by the following conditions:The one-step returns K(n) on stock are identically distributed independent random variables such thatat each time step n, where 1 d u and 0 p 1.In an n-step tree of stock prices each scenario (or path through the tree) with exactly i upward and ni downward pric

10、e movements produces the same stock priceAs a result,with probabilityexample:The stock price S(n) at time n is a discrete random variable with n+1 different values. The distribution of S(n) is shown in Figure 3.2 for n = 10, p = 0.5, S(0) = 1, u = 0.1 and d = -0.1.An example of a two-step binomial t

11、ree of stock prices is shown in Figure 3.3 and a three-step tree in Figure 3.4.In both figures S(0) = 1 for simplicity.ExerciseSuppose that stock prices follow a binomial tree, the possible values of S(2) being $121, $110 and $100. Find u and d when S(0) = 100 dollars.Do the same when S(0) = 104 dol

12、lars.3.2.1 Risk-Neutral ProbabilityWhile the future value of stock can never be known with certainty, it is possible to work out expected stock prices within the binomial tree model. It is then natural to compare these expected prices and risk-free investments.To begin with, we shall work out the dy

13、namics of expected stock prices E(S(n). For n = 1E(S(1) = pS(0)(1 + u) + (1 p)S(0)(1 + d) = S(0)(1 + E(K(1),whereE(K(1) = pu + (1 p)dis the expected one-step return. This extends to any n as follows.The expected stock prices for n = 0, 1, 2, . . . are given by E(S(n) = S(0)(1 + E(K(1)of:Since t

14、he one-step returns K(1),K(2), . . . are independent, so are the random variables 1 + K(1),1 + K(2), . . . . It follows that E(S(n) = E(S(0)(1 + K(1)(1 + K(2) (1 + K(n) = S(0)E(1 + K(1)E(1 + K(2) E(1 + K(n) = S(0)(1 + E(K(1)(1 + E(K(2) (1 + E(K(n).Because the K(n) are identically distributed, they a

15、ll have the same expectation, E(K(1) = E(K(2) = = E(K(n),which proves the formula for E(S(n).It proves convenient to introduce a special symbol p* for the probability as well as E* for the corresponding expectation satisfying the condition E*(K(1) = p*u + (1 p*)d = rfor risk-neutrality, which implie

16、s thatWe shall call p* the risk-neutral probability and E* the risk-neutral expectation. It is important to understand that p* is an abstract mathematical object, which may or may not be equal to the actual market probability p. Only in a risk-neutral market do we have p = p*.Ask no Risk Compensatio

17、n3.2.2 Martingale Propertythe expectation of S(n) with respect to the risk-neutral probability p* issince r = E*(K(1).Example:Consider a two-step binomial tree model such that S(0) = 100 dollars, u = 0.2, d = 0.1 and r = 0.1. Then p* = 2/3 is the risk-neutral probability, and the expected stock pric

18、e after two steps isAfter one time step, once it becomes known whether the stock price has gone up or down, we shall need to recompute the expectation of S(2).Suppose that the stock price has gone up to $120 after the first step.Given that S(1) = 120 dollars, the risk-neutral expectation of S(2) wil

19、l therefore be 2/3144+ 1/3108 = 132 dollars, which is equal to 120(1+r).E*(S(2)|S(1) = 120) = 120(1 + r).Similarly, if the stock price drops to $90 after one time step, the set of possible scenarios will reduce to those for which S(1) = 90 dollars,the risk-neutral expectation of S(2) will be 2/3108+

20、 1/381 = 99 dollars, which is equal to 90(1 + r). This can be written as E*(S(2)|S(1) = 90) = 90(1 + r).The last two formulae involving conditional expectation can be written as a single equality, properly understood: E*(S(2)|S(1) = S(1)(1 + r).3.3 Other Models:Trinomial Tree ModelThe one-step retur

21、ns K(n) are independent random variables of the formwhere d n u and 0 p, q, p +q 1.The one-step return r on a risk-free investment is the same at each time step and d r u.Since S(1)/S(0) = 1 + K(1),ExerciseLet u = 0.2, n = 0, d = 0.1, and r = 0. Find all risk-neutral probabilities.3.3.2 Continuous-T

22、ime LimitWe shall consider a sequence of binomial tree models with time step = 1/N ,letting N . For all binomial tree models in the approximating sequence it will be assumed that the probability of upward and downward price movements is 1/2 in each step.We denote by m the expectation and by the stan

23、dard deviation of the logarithmic return k(1)+k(2)+ +k(N) over the unit time interval from 0 to 1, consisting of Nsteps of .It follows that m = E (k(1) + k(2) + + k(N) = E(k(1) + E(k(2) + + E(k(N) = NE(k(n), 2 = Var(k(1) + k(2) + + k(N) = Var(k(1) + Var (k(2) + +Var (k(N) = NVar (k(n)This means that

24、 each k(n) has expectation m/N = mand standard deviation so the two possible values of each k(n) must beIntroducing a sequence of independent random variables (n), each with two valuesNext, we introduce an important sequence of random variables w(n), called a symmetric random walk, such thatw(n) = (1) + (2) + + (n), and w(0) = 0. Clearly, (n) = w(n) w(n 1).proofThe stock price at time t = n is given by S(t) = S(0) exp(mt + w(t).Stochastic Differential Equation of Stock PriceW(t) is a Wiener process or Brownian motion 1.W