108156.pdf

50 Mathematics for Finance Example 3.3 In the situation considered in Example 3.2 the returns are random variables taking the fo ...

Risky Assets 51 Example 3.4 Suppose thatS(0) = 100 dollars. Consider a scenario in whichS(1) = 110 andS(2) = 100 dollars. In ...

52 Mathematics for Finance Exercise 3.6 In each of the following three scenarios find the one-step returns, assum- ing thatK(1) ...

Risky Assets 53 Remark 3.3 If the stock pays a dividend of div(n) at timenand this is reflected in the price S(n), then the fo ...

54 Mathematics for Finance scenarios are−6%, 4%, 30%, respectively, then the expected annual return is −6%× 1 4 +4%× 1 2 + 30%× ...

Risky Assets 55 Exercise 3.11 Suppose that the time step is taken to be three months,τ=1/4, and the quarterly returnsK(1),K(2) ...

56 Mathematics for Finance Condition 3.1 The one-step returnsK(n) on stock are identically distributed independent random variab ...

Risky Assets 57 d)n−iat timen.Thereare (n i ) such scenarios, the probability of each equal to pi(1−p)n−i. As a result, S(n)=S ...

58 Mathematics for Finance Figure 3.4 Three-step binomial tree of stock prices Exercise 3.13 FinddanduifS(1) can take two values ...

Risky Assets 59 later chapters. To begin with, we shall work out the dynamics of expected stock prices E(S(n)).Forn=1 E(S(1)) ...

60 Mathematics for Finance in Chapter 5, where a precise definition of risk will be developed. The border case of a market in wh ...

Risky Assets 61 Figure 3.5 Geometric interpretation of risk-neutral probabilityp∗ Figure 3.6 Barycentric interpretation of ris ...

62 Mathematics for Finance Figure 3.7 Subtree such thatS(1) = 120 dollars 2 3 ×144 + 1 3 ×108 = 132 dollars, which is equal to 1 ...

Risky Assets 63 Proposition 3.5 Given that the stock priceS(n) has become known at timen, the risk-neutral conditional expecta ...

64 Mathematics for Finance 3.3.1 Trinomial Tree Model............................... A natural generalisation of the binomial tr ...

Risky Assets 65 The triple (p∗,q∗, 1 −p∗−q∗) regarded as a vector inR^3 is orthogonal to the vector with coordinates (u−r, n−r ...

66 Mathematics for Finance 3.3.2 Continuous-Time Limit Discrete-time and discrete-price models have apparent disadvantages. They ...

Risky Assets 67 Introducing a sequence of independent random variablesξ(n), each with two values ξ(n)= { + √ τ with probabilit ...

68 Mathematics for Finance In order to pass to the continuous-time limit we use the approximation ex≈1+x+ 1 2 x^2 , accurate for ...

Risky Assets 69 We shall use the Central Limit Theorem^2 to obtain the limit asN→∞of the random walkwN(t). To this end we put ...