金融学dissertation：利率债券的估值浮动案例分析

Instructions 介绍

 This case study must be completed individually and independently by all students.Any part of your submission that is too similar to that of another student will bedeemed to be plagiarism, and will be dealt with accordingly.
所有的学生都要单独和独立的完成这份案例，提交相似的dissertation将会被认为是剽窃，将会被按照常规的方法来处理。
 Your submission must be received by 18h00 on Friday, 28 October 2011. Latesubmissions will attract a penalty of 20% per calendar day that they are late.  Make sure you have attached a cover sheet to your submission.  Prepare your submission using a word-processor or typesetting software. Do notsubmit hand-written work.  Provide plenty of details and explanation in your solutions—marks are awardedfor explanation.提供了丰富的细节和解释在解决方案中，对商标部分来进行详细的解释。

Background背景

The topic for this case study is the valuation of a certain floating-rate bond, whosecoupon payments are determined by the performance of an equity index.
这种情况下，研究的主题利率债券的估值是有一定的浮动，确定下来的股票的指数将会被确定。
In particular, ifthe index performs well over a certain coupon period, then the investor receives a largercoupon. On the other hand, the investor is guaranteed to receive a certain minimalcoupon payment even if the index performs poorly. So, in a sense, the investor gets thebest of both worlds—the guaranteed return of a fixed-income security, and exposure tothe equity market when conditions are good. We shall refer to this instrument as anequity performance bond (EPB).To get a more detailed idea of how the EPB works, let S0be the initial value of anequity index that follows a geometric Brownian motion, with volatility σ and dividendyield q. We shall write Stto denote the value of the index at any subsequent time t. Nowlet 0 = t0< t1< . . . < tN 1< tN= T be a sequence of equally-spaced times, and define t := ti  ti 1to be the interval between any two successive times. The current time isobviously t0= 0, and the maturity of the EPB is tN= T . The cash-flows of the EPB arethen specified as follows:http://ukthesis.org/jr/

  On each of the dates ti, for i = 1, . . . , N, the EPB pays a couponCi:= max{Cmin, Cmin(1 + Ri)},where Cminis the guaranteed minimal coupon payment and Ri:= Sti/Sti 1  1 isthe nominal return of the index over the period from ti 1to ti.    EPB:

(1) Demonstrate that the coupons of the EPB can be written as follows:Ci= a + f(Sti 1) max{Sti  Sti 1, 0}, (1)for each i ∈ {1, . . . , N}, where a is a constant and f(Sti 1) is a function of theindex value at time ti 1. In particular, you should find expressions for a andf(Sti 1).(2) Fix i ∈ {1, . . . , N}, and let ci(t) denote the value at time t ∈ [0, ti] of the couponpayed at time ti. In particular, it therefore follows that ci(ti) = Ci. In order toprice the EPB, you must first figure out how to determine the initial value ci(0) ofthis coupon. To begin with, note that the risk-neutral valuation principle yieldsci(0) = e rti 1E ci(ti 1) . (2)Another application of risk-neutral valuation, together with some elementaryprobability theory.随着初等概率的应用。另一种风险性的中性定价也会得到应用。