債券定價(jià)原理的證明

1、.Theorem 1: Bond prices move inversely to the yield to maturity of bonds.P對(duì)r示導(dǎo)得：式子第二項(xiàng)小于0，因此要證明第一項(xiàng)小于0即可，即證明分子部分小于0：所以有：，即債券價(jià)格與到期收益率反向變動(dòng)。Theorem 2. For a given change in yield to maturity from the nominal yield, the change in bond prices are greater, the longer is the term to maturity.當(dāng)時(shí)，當(dāng)發(fā)生變動(dòng)時(shí)，(1)如變小

2、，由原理1可知債券價(jià)格會(huì)上升，此時(shí)債券價(jià)格變動(dòng)數(shù)值為：的大小與的關(guān)系體現(xiàn)在，令對(duì)其求導(dǎo)得：上式的符號(hào)由決定，前面假設(shè)變小，所以，因此。即越大，增加得越多。(2) 如變大，由原理1可知債券價(jià)格會(huì)下降，此時(shí)債券價(jià)格變動(dòng)數(shù)值為：的大小與的關(guān)系體現(xiàn)在，同樣令對(duì)其求導(dǎo)得：由于前面假設(shè)變大，所以，因此。即越大，增加得越多。Theorem 3: The percentage change described in theorem 2 increases at a diminishing rate as the time to maturity increases.隨著時(shí)間的增加，的數(shù)值越來(lái)越大，但增加的部分

3、越來(lái)越少。即求其二階導(dǎo)。(1)如變小， 此時(shí)0，(2) 如變大，此時(shí)因此原理3成立。Theorem 4: Price movements resulting from equal absolute (or, what is the same, from equal proportionate) increases and decreases in the yield to maturity are asymmetric; that is, a decrease in yield to maturity raises bond prices more than the same increase

4、 in yield to maturity lowers prices.收益率相同的增加或減少時(shí)，債券價(jià)格的變動(dòng)是不對(duì)稱的。收益率的減少導(dǎo)致的債券價(jià)格增加超過(guò)收益率的增加導(dǎo)致的債券價(jià)格的減少。由原理1已知，所以我們要證明上式的符號(hào)由分子決定，即決定。這個(gè)式子中，最后一項(xiàng)大于0，因此只要部分大于0即可，該部分為：因此有：Theorem 5: The higher is the coupon carried by the bond, the smaller will be the percentage price fluctuations for a given change in the yield to maturity except for one year securities and consols.息票率越高，債券價(jià)格變化的百分比越小。設(shè)故有：前面已經(jīng)證明，故因此，息票率越高，債券價(jià)格變化的幅度越小。設(shè)