Stand Alone Complex - 不存在的原创产生原创的拷贝

Fixed Income Modeling Review 2

Tao 发表于 19/05/2011 | 分类技术 | 标签 Fixed Income, Mathematical Finance | 现有0条评论

This review will concentrate on bond mathematics and yield curves. I have come up with typical assumptions in fixed income modeling last time now we first need to examine more detailed assmptions on zero-coupon bonds, which illustrate the differences between theoretical formulas and market reality.

-- Most of the bonds pay regular coupons
-- Available bond maturities do not span the whole time axis
-- Different bonds trade at different spreads to benchmark securities because of different credit quality and liquidity level
-- Most corporate bonds are callable

After addressing the theoretical assumptions, we now turn to a very important concept in fixed-income: yield. Bond yield or yield to maturity can be defined as a constant discount rate through the whole life of a coupon bond. So we can see that the bond price is decreasing with respect to the yield. Furthermore, the bond price equals the principle if and only if yield equals the coupon rate. For short term bonds paying no coupon we define the discount yield which is a simply compounded rate. For government bonds we often have another quoting method called Bond Equivalent Yield (BEY) which is defined as a discounted rate semi-annually.

In my opinion, fixed-income market is a market of interest rate derivatives, so we definitely need to examine one derivative's interest rate risk. PVBP is one indicator to show a security's sensitivity to the change of interest rate. It is defined as

$P=100[\sum^N_{i=1}\frac{c/2}{(1+y/2)^i}+\frac{1}{(1+y/2)^N}]$
$P^+=100[\sum^N_{i=1}\frac{c/2}{(1+(y+0.0001)/2)^i}+\frac{1}{(1+(y+0.0001)/2)^N}]$
$PVBP=P-P^+$ .

PVBP can also be approximately computed by $PVBP=-\frac{\partial P}{\partial y}0.0001$ .

Duration is a similar measure to PVBP but rather in a relative way: $D=-\frac{\partial P}{\partial y}\frac{1+y/k}{P}$ .

When $k\rightarrow\infty$ , we have $D=-\frac{\partial P}{\partial y}\frac{1}{P}$ which is also defined as modified duration. One important property of duration is that duration can be expressed as a weighted average of cash flow times.

$D=-\frac{\partial P}{\partial y}\frac{1+y/k}{P}=\sum^N_{i=1}\frac{1}{P}\frac{CF(t_i)t_i}{(1+y/k)^{kt_i}}=w_i\cdot t_i$ .

PVBP and duration measure the first order sensitivity to the change of rate while convexity measures the second order sensitivity.

$Cx=\frac{1}{2}\frac{\partial^2P}{\partial y^2}\frac{1}{P}$

Hence we can hedge against small parallel shifts in yield curve by making portfolio's PVBP zero and hedge against large parallel shifts by making convexity zero.

We have seen the importance of zero coupon bond curve. We can and we have to use them to calculate the present value of a cash flow. In practical world, we often use a way called bootstraping-interpolation to strip the zero-coupon bond price out of market data. Basically, it is to solve linear equations of cash flows to get the zero bond price at every node. By using this way, we could not only calculate zero bond price but also continuous forward rate, continuous spot rate and discretely compounded yields.

The last thing for this review is how to interpolate zero coupon bonds. From my point of view, the most popular way and its special cases can be called polynomial spline, which briefly speaking is to assume the zero bond price between two different time nodes could be expressed as polynomials.

HW 2, My Solution

Fixed Income Modeling Review 1

Tao 发表于 11/05/2011 | 分类技术, 科学人 | 标签 Fixed Income, Mathematical Finance | 现有0条评论

Fixed income modeling is one of the major areas of mathematical finance, which is usually more complicated than stock market modeling. Not surprisingly, "Theory and Practice of Fixed Income Modeling" is among the core courses of our program. The lecturer is Andrei Lyashenko who used to be a mathematics professor at UIC and now he is working at QRM as a quant. We were taught the theoretical part in class and our homework is done in excel as practitioners do in industry. I just finish this semester and if I let it go I would probably forget the details in 3 months. So it is better for me to have a whole review of this class.

The first thing is to justify all types of risks in fixed income market: credit risk, interest rate risk, liquidity risk, cash flow timing risk, cash flow amount risk, foreign exchange risk, inflation risk and taxation risk. Different products are exposed to different risks. For example, since the US government is unlikely to get default, the credit risk does not present for treasure bonds. However, for debt issued by corporations such as citigroup, credit risk does present. Callability, fixed or floating rate, a lot of aspects need to be taken into account as you consider the potential risk.

The very basic contract considered in fixed income modeling is zero-coupon bond and the first tricky question is does $P(t,T)\leq 1$ for any $t<T$ . While the first thought comes to most people is: if the market does not allow negative rates, then it is true. However, this question actually has not much to do with positive or negative rates. Since you can always hold 1 dollar instead of invest it by buying zero-coupon bond, then if the price of zero-coupon bond is higher than 1 such contract would not exist any more. So what are the assumptions when we think about fixed income modeling?

-- Zero bonds of all maturities are traded

-- No transaction cost

-- Securities are perfectly divisible

-- Short selling is permitted

-- No arbitrage

Because of no arbitrage, discount factor $D(t,T)$ should equal the zero bond price $P(t,T)$ .

The next important thing is to clarify the kinds of rates:

-- Spot rate: is the constant rate that makes the investment of $P(t,T)$ at current time become 1 unit of currency at maturity

-- Forward rate: is a generalization of spot rate. Only assume that the investment happens at a future time and the maturity is some time further. 3-months or 1/4-year forward rates are usually called Libor Rates. By examine the contract of FRA we can prove that $P(t,T)\cdot P(t,T,S)=P(t,S)$ . There are also instantaneous forward rates defined as $F(t,T)=\lim_{T\rightarrow S}F(t,T,S)=-\frac{\partial\ln(P(t,T))}{\partial T}$ , and it has nothing to do with day counts and compounding. Remember forward and short rate ( $r(t)=F(t,t)$ ) are purely theoretical and are not observed on the market.

-- Par rate: is the bond coupon rate that makes its price equal to its principle. It can be explained as the weighted average of Libor rates.

At the same time, spot rate is an average of instantaneous forward rates. Actually, $F(t,T)=-\frac{\partial\ln(P(t,T))}{\partial T}$ implies that $\int^T_t F(t,s)ds=-\ln(P(t,T))$ , and therefore, $R(t,T)=\frac{1}{T-t}\int^T_t F(t,s)ds$ .

Price of fixed income derivatives are often quoted by yield, and yield curve is a plot of rates against their maturities.

Last for review 1 is to introduce the fixed income derivatives:

-- Bonds

-- FRA: OTC, a rate decided ahead, lending and borrowing starting and ending in the future

-- Interest rate swap

-- Swaption: is an OTC contract giving its holder the right but not the obligation to enter into a swap contract

-- Caps and Floors: cash flows at one time is called caplet or floorlet and are defined as

$CF(T_i)=L\tau(T_{i-1},T_i)\max[0,R(T_{i-1},T_i)-R_K]$

$CF(T_i)=L\tau(T_{i-1},T_i)\max[0,R_K-R(T_{i-1},T_i)]$ .

HW 1, My Solution

Assassin's Creed

Tao 发表于 10/05/2011 | 分类极客 | 标签游戏 | 现有1条评论

期末考完后花了两三天时间把《刺客信条》1和2打通关，再加上之前闲暇时通关的《兄弟会》，这个游戏目前的三个 PC 版本都玩过了。之所以愿意花时间还是因为喜欢游戏主角的暗杀风格，同时整个剧情的编写也是引人入胜。

这段视频是《刺客信条2》的开场，描写的是主人公在威尼斯狂欢节上进行刺杀，囊括了人物的作战风格：首先是出其不意的用袖剑暗杀胖子；接下来飞扑向一个卫兵同样用袖剑刺杀；然后通过反击杀死拿刀的卫兵；再夺下卫兵的长矛；一连串的飞檐走壁后用火枪杀死第二个头目。我觉得算是整个系列游戏最精彩的一段动画。

故事的核心围绕着“伊甸园的金苹果”——可以产生幻象控制人类思想的神器展开，“圣殿骑士”妄图夺取金苹果并控制世界，而代表正义与自由的刺客们则与对手进行战斗。在不断的战斗中，主人公发现在金苹果的背后其实隐藏着更深的秘密，神灵出现，讲述了人和神的历史，并告诉主人公他就是先知，并且肩负着重大的使命。

有趣的是，游戏制作方还在剧情中结合了很多历史人物与事件，比如《君主论》的作者马基雅维利在游戏里就是刺客团体的一员，达芬奇也是刺客们的好友，他的研究和发明为盟友提供了重要的帮助。每一部游戏都在结尾处留下一个疑团，让人期待下一部的作品，听说《刺客信条3》年内就会发布。

我纯粹是听歌的

Tao 发表于 21/03/2011 | 分类影音 | 标签 Placebo | 现有4条评论

好歌就是用来翻唱的, 等到快四分钟终于有人声了. 以前不喜欢军鼓, 不过最近听 Explosions in the Sky, 因而印象大有改观.

Bayern Munich 2-3 Internazionale

Tao 发表于 16/03/2011 | 分类体育 | 标签国际米兰, 拜仁慕尼黑, 欧洲冠军联赛 | 现有9条评论

整整大半个赛季没写足球, 其实是因为整整大半个赛季没看到一场让人激动的比赛, 今天的欧冠够惊险, 上半场若不是拜仁领先后缺少了一点点坚定, 恐怕比赛的悬念早就结束. 不过话说回来, 罗本今年真是塞萨尔的克星, 两轮比赛两次让塞萨尔脱手, 主场那球就不提了, 今天的失误更是业余, 这之后的国米几乎崩溃, 后防乱不成军, 在那一刻我还真有一点想念穆里尼奥.

不过莱昂纳多也是奇人, 至今的三个欧冠胜场全部在客场, 国米成了历史上第四支首轮主场告负还能晋级的球队. 这场比赛又让我对欧冠有了新的认识, 不管之前被打的多么千疮百孔, 只要苟延残喘保住了那一点希望, 一切往往就在一瞬间发生改变. 看看米兰对热刺时消失的伊布, 或者想想曾经那支年年十六郎的球队, 不得不说, 现在就像换了一支球队, 不管胜利对我们来说是否更艰难, 至少我们的性格不再软弱而又神经质了.

疯狂的国米, Finlay 在校内上说的好: 不要低估一个冠军的心.

第 5 页，共 182 页« 最新 «...456...»最旧 »

Stand Alone Complex ← 不存在的原创产生原创的拷贝

Fixed Income Modeling Review 2

Fixed Income Modeling Review 1

Assassin's Creed

我纯粹是听歌的

Bayern Munich 2-3 Internazionale