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Fixed Income Modeling Review 6

Tao 发表于 03/02/2012 | 分类技术 | 标签 Fixed Income, Mathematical Finance | 现有0条评论

To apply the theoretical models, let us first see how to use interest rate trees to price and calibrate. As mentioned in Review 5, markovian property of short rate models is needed to implement recombining lattice.

Usually, a short rate model can be written as

$r(t)=F(u(t)+\varphi(t))$
$du(t)=-au(t)dt+\sigma dW(t).$

Hence, we first discretize the process $u(t)$ . Typically, people use binomial tree and trinomial tree but for mean reverting model, binomial tree cannot be used. The time step $\Delta t$ is arbitrary, the state step $\Delta u$ and branch probabilities $p_u$ , $p_m$ and $p_d$ should be chosen in such a way that the discrete dynamics has the first few moments matched. Since we could solve

$u(t+\Delta t)=e^{-a\Delta t}u(t)+\sigma\int^{t+\Delta t}_te^{-a\Delta t}dW(s)$

then by moment-matching, we can set

$\Delta u=\sqrt{3V}$ , $V=\sigma^2\frac{1-\exp(-2a\Delta t)}{2a}$

and we select the middle branching node at next step to be the closest one to the mean of the continuous process:

$k=\int[je^{-a\Delta t}+\frac{1}{2}$

Branching probabilities are determined as

$p_u=\frac{1}{6}+\frac{1}{2}(\beta^2+\beta)$
$p_m=\frac{2}{3}-\beta^2$
$p_d=\frac{1}{6}+\frac{1}{2}(\beta^2-\beta)$
$beta=je^{-a\Delta t}-k$

Once tree for the stochastic driver $u(t)$ is built, we need to convert it to the short rate tree according to the functions:

normal: $r(i,j)=r_0(i)+u(i,j)$
lognormal: $r(i,j)=r_0(i)\cdot\exp(u(i,j))$

where mean level vector $r_0(i)$ is calibrated to the term structure of zero coupon bond prices.

Since trees represent discretization of the rate dynamics, then within each time step the evolution of short rate is not described by tree and thus needs to be specified. Usually we assume short rate is constant within each period. To price derivatives, we do backwards

$S(t)=E^B_t[\exp(-\int_t^{t+\Delta t}r(s)ds)(S(t+\Delta t)+CF(t+\Delta t))]$

where $CF(t+\Delta t)$ is the cash flow at time $t+\Delta t$ . And now the remaining problem is how to calibrate to get mean level vector $r_0(i)$ . Denote prices of zero coupon bonds with maturities $t_i=i\Delta t$ by $P(i)$ , then apply the following iterative search process forward:

(1) Set $r_0(0)=-\ln(P(1))/\Delta t$
(2) Search for the value of $r_0(1)$ such that lattice price of the zero coupon bond maturing at $t_2$ equals $P(2)$
(3) ...
(4) Assuming $r_0(0),\ldots,r_0(i-1)$ are done, search for the value of $r_0(i)$ to fit $P(i+1)$
(5) ...

We need to keep in mind that we always need to do numerical calibration for $r_0(i)$ even when there are analytic formulas for the no-arbitrage drift because the use of trees alters the continuous dynamics. To be more specific about step (2) and step (4), we introduce Arrow-Debreu Security which pays 1$ once the node (i,j) is reached and pays nothing otherwise. Thus in trees we have

$AD(i+1,j)=\sum_k AD(i,k)\exp(-r(i,k)\Delta t)P_{i,k\rightarrow i+1,j}$

One very useful property of Arrow-Debreu prices is that

$P(0,t_i)=\sum_j AD(i,j)$

Recombining interest rate trees have a serious shortcoming that the rate history is totally lost and thus it is incompatible with any path dependent instrument. To overcome this problem, one can use the tower law and record the cash flows at the lattice nodes not when they are paid but rather when they are certain and express them with appropriate discounting.

Fixed Income Modeling Review 5

Tao 发表于 03/02/2012 | 分类技术, 科学人 | 标签 Fixed Income, Mathematical Finance | 现有0条评论

This topic is mainly about modeling of short rate $r(t)$ which plays a central role in the theory and practice because zero coupon bonds can be priced as

$P(t,T)=E_t[\exp(-\int^T_tr(s)ds)]$

and so could discount factor.

Assume there is only one source of uncertainty represented by a Brownian motion $W(t)$ under the risk-neutral measure $Q$ . We want the stochastic process to be markovian which is referred to as path independence and hence allows for the use of recombining lattice. We don't want to use non-recombining lattice cause the number of nodes growing exponentially w.r.t. the number of time steps and thus is not practical. There are several important things we need to think about when choosing a model:

-- Does the dynamics guarantee positive rates?
-- What distribution does the dynamics imply for the short rate?
-- Can the model be calibrated to fit today's structure of zero coupon bond prices, how easy, and does the calibration stable?
-- Are the zero bond prices computable by means of analytic formulas?
-- Are there analytic pricing formulas available for call/put bond option?
-- Is the model mean-reverting?
-- How do the volatility structures implied by the model look like?
-- How suited is the model for building recombining lattices?

If the parameters of a model are constant like $dr=a(b-r(t))dt+\sigma dW$ (Vasicek) then it is called equilibrium model. It's simple but could not be calibrated to fit the current term structure of rates. If we make the parameter of the drift term time dependent then we can overcome this problem and make the model no-arbitrage.

Based on empirical observation, people find that interest rates (unlike stock prices) tend to be pulled back to some mean level. So we want our model to be mean reverting; we also want to have the analytical formulas for zero coupon bond, call/put bond options, etc., and hence we require the model be affine which means the drift term is linear to short rate and square of the volatility is also linear to short rate. Regarding the volatility, typically we have two kinds of model for volatility: normal and lognormal. It is very hard to say which one is better, normal assumption results in analytic tractability and lognormal assumption leads to positive rates. Also, normal dynamics suit low rate environments and for lognormal, the other way around.

Several important models:

Ho-Lee: $dr=\theta(t)dt+\sigma dW$
Hull-White: $dr=a(\theta(t)-r)dt+\sigma dW$
CIR model: $dr=a(\theta(t)-r)dt+\sigma\sqrt{r}dW$
BDT model: $d\ln(r)=(\theta(t)+(\sigma'(t)/\sigma(t))\ln(r))dt+\sigma(t)dW(t)$

Among these models, Hull-White has a positive probability to generate negative rates but we can cut that and it does not affect pricing very much. CIR model is the only model with full analytic tractability which also has positive rates. While doing calibration, $\theta(t)$ term in CIR must be solved for numerically. Under lognormal assumption, the total volatility is proportional to the rate level and thus it grows/declines with the rates, which produces rate path distribution skewed upward. Theoretically, we could deduce such equation under lognormal model:

$E_0[B(\Delta t)]=\infty$

therefore we can make infinite money in an arbitrary small time from $1 initial investment, however in numerical implementations of a short rate model on a tree this problem does not appear as the number of states is finite.

In order to make our models better, we could add a deterministic shift:

$r(t)=r_0(t)+\varphi(t)$
$dr_0(t)=a(\theta(t)-r_0)dt+\sigma\sqrt{r_0}dW(t)$ .

This kind of extension could preserve the tractability but not the positiveness of rates. Shifted CIR model could perfectly fit the market price of zero coupon bond and according to this:

when rates are low: $\sigma(t)(r(t)+\alpha)\approx\sigma(t)\alpha$
when rates are high: $\sigma(t)(r(t)+\alpha)\approx\sigma(t)r(t)$

we know that shifted lognormal models capture the phenomenon that rates follow close-to-normal dynamics in low rate environments and close-to-lognormal one in high rate environments.

The Best of 2011 - Uzak

Tao 发表于 31/12/2011 | 分类电影评论 | 标签 Nuri Ceylan | 现有0条评论

首先, 我觉得中文译名不够好, 对于本片, 从涵盖主题的角度来说, "远方"并不如英文译名"Distant"来的精确.

之所以喜欢这部电影, 还是在于导演对"疏离"这个元素的表达和描述. 我一直相信, 一部真正的好电影, 一定是重述和探索人们日常生活中的真实的. 就像摄影, 我们都喜欢那些捕捉到有趣, 动人瞬间的片子. 不是说商业剧情片不好, 我认为它们也有趣, 但代表了另一种乐趣, 如同摄影师用 Photoshop 把羊头天衣无缝的接到牛身上, 一种表象的刺激, 我们看到都会笑, 但笑完之后, 几乎没留下任何回味的余地. 回到"疏离感"上来, 这是美国影评家非常喜欢的一个词, 尽管他们的导演们似乎并不太喜欢向观众表现它. 其实这是多么普遍的一种体会和感受呢, 它来自于工业化, 来自于社会的进步, 来自于人内心的骄傲等等一切可以设置障碍的有形或无形的变化. 比如说已故的安东尼奥尼, "疏离感"深入到他电影的骨髓里, 建筑, 色彩, 主角的行为, 心理活动, 一切外在或内在无不表现着这一矛盾.

当代欧洲有很多出色的导演, Almodóvar 关注女性; Nanni Moretti 关注男性的气质危机; Fatih Akin 关注不同文化间的冲击; 唯一深入研究超越种族, 文化的人类的本质性矛盾的, 就我目前了解, 好像只有锡兰 (Nuri Ceylan) 一个人. 在他的作品中, 他总是把这种矛盾投射到各个方面: 伊斯坦布尔与小镇; 男人与女人; "文化"与"文盲"; 努力工作与游手好闲, 在这些差别中他展现和探讨人与人, 人与生活的距离. 我们喜欢欧洲, 不仅因为他们有很长的历史, 很多的雕塑与建筑, 有多元的文化, 有摇滚乐与足球, 还因为他们关注人, 有人文精神. 锡兰正是这种人文关怀的表达者.

锡兰本身还是一位出色的摄影师, "诗意的镜头语言"是塔科夫斯基的标志, 锡兰的影片有同样的特点, 这也是人们称他为"土耳其的塔科夫斯基"的原因, 将电影拍成照片, 又将照片连接成电影, 所以他的影片总在诗意中透出那种"愉悦的忧伤", 这种忧伤符合他的身份: 一个扎根于土耳其的导演, 他的国家由弱到强, 又盛极而衰. 他的作品偶尔提及历史 (在极少的情况下), 而聚焦于当下, 历史变迁, 形成了当代土耳其人的气质, 锡兰在作品中将这一切回归于普通的土耳其人, 做了平淡而又饱满的表达.

The Best of 2011 (候选)

Tao 发表于 28/12/2011 | 分类影音 | 标签 Fatih Akin, Ferzan Özpetek, Nuri Ceylan | 现有4条评论

又到年末, 照例会挑出一部当年最喜欢的电影, "新浪潮"的老板说我每次都在碟子卖完后才去他那问, 不光买碟, 看电影也一样, 今年豆瓣上打5颗星的电影最近也是2010年的, 后知后觉到我自己都不能忍了.

这年的年初看过几部电影, 学期中途总是很忙无暇看片, 这个寒假算是过了瘾, 差不多是以每天不止一部的速度在看, 其间不停的感慨, 不停的流泪, 但观影速度太快不能静下心来思考, 明知是对好电影的糟蹋, 又忍不住想看下一部是怎样的故事.

缘起 Fatih Akin 的 Soul Kitchen, 最近这段时间看了相当多的土耳其导演作品, 还有 Ferzan Ozpetek 和 Nuri Ceylan, 打了5颗星的有: Solino, Im Juli., Head-on, Uzak, Mayıs Sıkıntısı, Mine vaganti, Hamam. 他们的风格互不相同, 这些不同又是缘于个人出身和经历的差别. Fatih Akin 是出生在汉堡的土耳其后裔, 他的电影始终关注着德国的土耳其移民, 移民在异乡的位置以及他们与故乡的关系; Ferzan Ozpetek 本人是一位同性恋(如果我没有理解错 an openly gay director 这句话的话), 他的作品不时聚焦在同性恋的身上, 描写社会对同性恋的排斥与宽容, 与同性恋角色与生俱来的问题是他们对自身的探索, 种种原因导致 Ozpetek 的电影中总是隐藏着秘密, 带有悬疑的味道; Nuri Ceylan 是很多人追捧的一位导演, 因为他的镜头有着与阿巴斯或者塔可夫斯基类似的风格, 他本人同时是摄影家, 在他的掌控下, 影片或像一张张流动的胶片, 或者就是一张定格的摄影作品, 也由于如此, 他的电影极其擅长通过人物面部的特写去展现心理活动.

除了这些土耳其制作, 09年的法国电影非法入境 (Welcome) 讲述伊拉克难民试图偷渡到英国的故事, 开放性的结尾画龙点睛, 温暖动人; 法斯宾德的罗拉 (Lola) 是他"女性三部曲"中我唯一没看过的, 赞叹导演的天才: 通过一个柔弱的女性角色爆发出巨大的能量, 完成对社会的强烈反讽; 乐与路则是一个简单, 带有明显日本风格的主流爱情片, 偏偏人就容易被这样的电影催泪; Inside Job 是关于金融危机的纪录片, 我们不能指望艺术工作者能够完全客观的看待这次危机: 有些剪辑显得刻意, 有些评论过于激进, 但总体而言, 很清晰明了的解释了金融危机的来龙去脉; 靖大爷和他的老主顾们也是一部纪录片, 完成于02年, 片长不足一小时, 年过八旬的老理发匠四处奔走为年纪相仿的老主顾们理发, 他见证了政权, 城市和人生的变迁, 送走一位位主顾, 也不得不随着北京城的发展而告别一个个传统, 八十多岁, 知天命, 按他自己的话说到了该死的年龄, 每天乐观而平静的活着.

接下来的几天会好好回味一下这些片子, 跨年之前挑出年度最喜爱的作品.

We Come in Pieces

Tao 发表于 24/11/2011 | 分类影音 | 标签 Battle for the Sun, Placebo | 现有12条评论

Placebo

This is a Thanksgiving announcement, reminding every single one of you, good people, to breathe underwater.

We Come in Pieces

真正爱上 Placebo 是从看了他们03年巴黎现场的 DVD 开始, 应该绝大部分人到现在都依然坚持那是 Placebo 最出色的一次现场, 这张 We Come in Pieces 是 Battle for the Sun 全球巡演的收官演出, 歌单涵盖了迄今的所有专辑:

Placebo - Nancy Boy, Teenage Angst;
Without You I'm Nothing - Every You Every Me;
Black Market Music - Taste in Men;
Sleeping with Ghosts - Soulmates, Special Needs, The Bitter End;
Meds - Meds, Infra-Red, Post Blue, Song to Say Goodbye.

几乎包含了所有我最喜欢的歌, 不知道等我真正能现场看到他们的时候还是不是这么幸运. 喜欢 Placebo 大约7年的时间, 感觉生命就像一次次的轮回, 从 Meds 开始到结束有一个女朋友, 从 Battle for the Sun 录制到他们巡演结束有另一个女朋友, 不知道下一张专辑的时候又会是怎样. 期间他们来了一次中国, 换了一次鼓手, 在每张专辑中都尝试了新的风格, 拿了一次欧洲 MTV 最佳另类乐队奖, 生活总在不停的变化之中, 唯有 Soulmates Never Die.

Fans

Anyway, Placebo 明年又要录新专辑了.

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Stand Alone Complex ← 不存在的原创产生原创的拷贝

Fixed Income Modeling Review 6

Fixed Income Modeling Review 5

The Best of 2011 - Uzak

The Best of 2011 (候选)

We Come in Pieces