security markets stochastic models - darrell duffie

[...]... portfolio x with non-zero market value, let R,(i) = x/(pi x) Finally, assuming the , for any time T Noting that B2 = (A-lpPA)(A-lpPA) = A - l p 2 P 2 ~and similarly that Bt = A-lptPt A for any t 2 1, we see that BT converges to the zero matrix as T goes to infinity, leaving INTRODUCTION where II = CE"=,t a series calculation, II = A-'(I - pP )-' A - I By Equivalently, or the current value of a security is the... let Atk = tk - t k - I AXk = X(tk) -X(tk-I), and ADk = B(tk)-B(tk-I), for k 1 A stochastic difference equation for the motion of X might be: a , > It then follows that lim T-0 E In other words, Ito's Lemma tells us that the expected rate of change of j at any point x is V f(x) 18 INTRODUCTION J We apply Ito's Lemma to the following portfolio control problem We assume that a risky security has a... commodity market equilibrium ~ ' 3 ~ C Financial security markets are an effective alternative to contingent commodity markets We take the ( S states, C commodities) contingent consumption setting just described Security markets can be characterized by an S x N dividend matrix d, where N is the number of securities The n-th security is defined by the n-th column of d, with d,, representing the number... let the security dividend matrix d be the identity matrix, meaning that the n-th security pays one dollar in state n and zero otherwise Let the security price vector be q = (1,1, , I ) E RN, and take the spot price matrix 11, = r For each agent i and state s, let Qj = Qs (x: - w;), equating the number of units of the s-th security held with the spot market cost of the net consumption choice x - wf... matrix-x E L is a consumption choice and 6 E R~ is a Figure 1 State-Contingent Consumption security portfolio The dollar payoff of portfolio 0 in state s is 6 d, Given security and spot prices (q, $J),a plan (8, x) is budget feasible for agent i if q - e5 0 (1) and A budget feasible plan (8,x) is optimal for agent i if there is no budget feasible plan (p,y) such that ui(y) > ui(x) A security- spot... expanded in Section 21 The continuous-time portfolio-consumption control solution of Paragraph J is due to Merton (1971); more general results are presented in Section 24 Chapter I STATIC MARKETS This chapter outlines a basic theory of agent choice and competitive equilibrium in static linear markets, providing a foundation for the stochastic theory of security markets found in the following three... is no budget feasible plan (p,y) such that ui(y) > ui(x) A security- spot market equilibrium is a collection ((O1,xl), - - ,(e1,x'), (9,TJJ)) with the property: for each agent i, the plan (8',xi) is optimal given the security- spot price pair (q, $J); and markets clear: I I Cei=O Csi-ui=0 i=l and i=l INTRODUCTION 4 To see the effectiveness of financial securities in this context, suppose ( x l , ,... full rank S x N dividend ' matrix d, whose n-th column dn E R is the vector of dividends of security n in the S states Drawing from Paragraph C, we can convert the contingent commodity market equilibrium into a security- spot market equilibrium in which the spot price $J, for consumption is one for each s E (0, ,S}, and in which the market value of the n-th security is INTRODUCTION 7 of the bond whose... maximize E [: e-pt u [c(x~ ) dt ~-PTv(xT)] for any time J I T By our calculations (and some technical arguments) this is equivalent to the problem: + max A(w),C(w) + + VV(w) - pV(w) + u [C(w)] The first order necessary conditions for (24) are where p(x) A(x)x(m 6 - T) TX - C(x), a ( x ) = A(x)xv, and w > 0 is the given initial wealth The indirect utility for wealth w is and (m + 6 - r)wV1(w) + ~... stock per day Details are found in Section 22 The Black-Scholes Formula (9) was originally deduced by much different methods, however, using a continuous-time model F One of the principal applications of security market theory is the explanation of security prices We will look at a simple static model of security prices and follow this with a multi-period model The static Capital Asset Pricing Model, . Continuous-Time Pricing 0 A-K 1 A-G 2 A-G 3 A-F 7 A-E 4 A-E 6 A-G 11 A-E 12 A-G 9 A-D 14 A-F 18 A-E 19 A-H 20 A-C 15 A-D 21 A E 22 A-I 23 A 24 A-E 25 A-M - *Background. Atk = tk -tk-I. AXk = X(tk) -X(tk-I), and ADk = B(tk)-B(tk-I), for k > 1. A stochastic difference equation for the motion of X might be: It then follows that lim E T-0 In other. security- spot market equi- librium is a collection ((O1,xl), . - -, (e1,x'), (9,TJJ)) with the property: for each agent i, the plan (8',xi) is optimal given the security- spot