Instructions 介紹

 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.
所有的學生都要單獨和獨立的完成這份案例，提交相似的dissertation將會被認為是剽竊，將會被按照常規的方法來處理。
 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.提供了豐富的細節和解釋在解決方案中，對商標部分來進行詳細的解釋。

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.
這種情況下，研究的主題利率債券的估值是有一定的浮動，確定下來的股票的指數將會被確定。
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.隨著初等概率的應用。另一種風險性的中性定價也會得到應用。