大概一个月前, 带我实习的法国 boss 开始表达希望我暑假能留下来继续实习的想法, 并且很热心的要帮忙解决工作许可的问题, 于是我一段曲折起伏的心路历程开始了.

在他和我谈话之前, 我的想法是暑假专心做研究, 可是法国人偏偏提出了一个让我很难拒绝的提议: 就是暑期工作时可以用50%的时间做数学. 之所以诱人, 是因为这样的安排已经是一个普通 quant 的时间分配了(我顿时体会到 PhD 三个字母在对方心目中的分量), 可另一方面, 实在也担心老板会反对我暑假继续工作. 于是带着犹豫的心理, 回学校找老板谈话. 老板当然是不用听我说完就能明白我的心思, 跟我分析了好久目光长短的问题, 然后表态说可以在不超过某一限度的范围内继续实习. 我像吃了定心丸一样, 开始着手要解决工作许可的问题. 可没想到过两天, 系里一年一度 Menger Day 大会上, 颁奖嘉宾突然叫道我的名字让我上台去领暑期研究的奖金, 之前老板说因为我还没开始做, 所以这次奖金的申请是不指望的, 结果自然是让我大感意外, 同时是真想不通为什么最近好事多的让我无法抉择. 说实话, 钱对现在的我来说是很有用的, 但更重要的是, 数学系没有选择那么些已经在读的博士, 偏偏把奖金给了我, 真的让人很有压力, 再一想导师对我整个博士期间的规划, 千里之行, 始于足下, 为了开一个好头, 我还是放弃了暑假实习的想法.

现在回顾这一段实习经历还有点过早, 因为有很多东西在忙碌的实习期内来不及消化, 特别是数学上的. 非常幸运, 在读博前可以有机会了解真实的市场, 的确学到不少学校内学不到的东西, 而且能加入 Clearing 部门参与 OTC Clearing 的项目, 我记得 Quant 组的 Director 在第一次跟我谈话时就说, OTC Clearing 现在是金融业的一件大事, 不光如此, 未来 OTC 交易转 Exchange 肯定也会成为趋势, 所以, 能有机会参与到最前沿的工作当中, 心里很高兴. 这两个多月里, 我发现大部分 quant 们都是跟普通人不太一样的 math geek, 自己也受到一些影响, 感觉如此过一生, 大概就是最适合自己也感觉最舒服的一条路.

冬天和同学去看 Feist 的时候, 在一个破破烂烂的剧场里, 我们坐在二层, 当时我还说坐着听听 Feist 可以, 但要是看摇滚乐队演出就比较奇怪了, 结果没想到这次 The Black Keys 和 Arctic Monkeys 的演出, 我们便是在联合中心球馆的 300 Level 坐着看完的. 之前因为在附近的意大利餐馆吃饭耽误了时间, 到场的时候已经迟了, Arctic Monkeys 开始唱 I Bet You Look Good on the Dance Floor 的时候我们还在上楼梯找座位, 懊悔不已.

想到去年 Lollapalooza 上看 Muse 的经历, 结合网上一些 Placebo 在美国演出的视频, 再加上这次看 Arctic Monkeys, 真的发现英国乐队在美国的不易, Muse 已经是我看到的在美国最成功的英国乐队了. 就拿这次 Arctic Monkeys 来说, 我错过的部分不算, 全场也就 Fluorescent Adolescent 响起时欢呼声稍微大一点, 其它时间观众太安静了, 这还是在猴子们唱了所有他们最流行的歌曲的情况下. 我只能很欣慰的讲, 中场出去抽烟时, 听到有人说他们是来看北极猴的.

The Black Keys 受到的待遇明显不同, 从头至尾高分贝欢呼不断, 尽管他们很多首歌听起来很重复, 但可以看出他们绝对是一支踏实认真的乐队, 再加上风格特别合适现场演出, 所以, 我觉得他们还是配得上观众的热情. 之所以认为 The Black Keys 朴实努力, 是因为 1. 主唱的话很少, 不像一些乐队中途要说很多, 他就一个劲的唱; 2. 唱的歌很多, 让观众过足了瘾, 而且返场更炫, 能让人不由自主的起立跟着他们摆动. 同学说, 感觉这次 The Black Keys 把所有的歌都唱了一遍.

我会嘲笑美国人那么粉 Muse, 可我又非常赞同他们对于 The Black Keys 的好感, 事实上, 我还很希望能在今年的 Lolla 上再看他们一次.

不过, 今天正式确认下来, 拿到了数学系的 offer, 接着的三至四年将进行我的 PhD 生活, 这意味着从小学算起到30岁左右毕业时, 我将整整在校园里待25年, 很恐怖的一个数字.

为什么读 PhD, 因为心里有着做 pure quant 的梦想, 一方面此类工作对学历的要求都是博士, 另一方面我也深感自己还不够 quantitative, 所以希望能够得到更多的数学训练. 在美国待了快两年, 发现了和国内不同的地方. 出国前, 听到最多的声音是, 继续读书再不出来工作就和社会脱节了; 提起学数学, 很多人第一反应是"你要当老师啊", 总之我认为国内弥漫着"读书无用论"的风气. 在美国的这段时间里, 参加过一些招聘, 听过一些业内人士的讲座, 在实习的时候和同事有一些交流, 大家总把"PhD"挂在嘴边:"你们不是 PhD, 所以你们只能怎样怎样", "这个问题以前也有 PhD 来问, 尚且花了很多时间才搞懂", 大概意思就是 PhD 在人们的眼中是一群 super smart guys, 这个行业的顶尖人才都是 PhD 出身. 导师在跟我谈的时候说的没错, 有了博士的头衔, 别人对待你的方式就不同了, 比如某某某, 现在在公司只有他能和公司的一号人物在一起研究论文. 美国好吗? 我觉得其实一般, 这里有无形的歧视, 有难吃的食物, 人们的品味普遍出现问题, 但我能确信一点, 美国是最发达的国家, 因为这个国家真正的尊重知识和追求创新. 想好好干一翻工作, 还是这个国家最合适.

还有选择学校的问题, 事实上, 我觉得除了整体排名低一点, 每个月生活费比一些学校少一点以外, 其它各方面都算非常理想了. 我想做的是金融数学, 数遍美国各个学校的数学系, 基本上在这个方向上都不超过两位教授, 最厉害的是卡耐基梅隆, 有两个大牛, 接下来可能是普林斯顿, 哥伦比亚等等, 选择面其实很窄, 除了现在在的学校以外就只剩下顶级牛校. 在这种情况下, 本校也可以接受了, 有两个金融数学方向的老师, 两个人都算我的导师, 真正官方的导师是和 Brigo 这样的大牛齐名的人物, 同时从主观角度来说, 当这样的大牛能够从你的身上发现别人看不到的优点, 能够交给你很难的题目并且说 You have the brain for it 的时候, 我觉得无法离开. 古人说, 知遇之恩当涌泉相报, 我想就是这个道理.

Anyway, 接下来的三至四年会比较辛苦, 因为这一组是号称数学系最苦逼的一组, 导师对我发火是迟早的一件事情, 但是不付出努力, 别的只能免谈了, 在国内的时候荒废了那么多年, 那就接下来补回来好了.

其实, 想到每年都还有 Pitchfork, Lollapalooza 和数不清的演唱会, 还可以在这里等着 Placebo 来, 真是一件非常开心的事情.

三月份开始要在 CME Group(芝加哥商业交易所集团) 实习, 职位是 Quantitative Risk Research Intern, 所在小组负责利率衍生品场外交易清算的模型建立, 开发和测试. 本来给自己定的目标是通过前五, 六次面试进行锻炼, 没想到第三次就成功了.

说实话, 自己在找实习方面运气还是不错的: 第二学期投的简历相对多一点, 得到了美国一家顶尖的自营交易公司的电面, 面试内容是两位数乘两位数的心算和概率题的心算, 结果由于不熟悉面试的流程没发挥好; 上学期只投了一份简历, 得到了面试机会, 但因为没什么统计和大规模数据处理的经验, 再加上紧张, 又失败了, 决定毕业读 PhD 以后, 去 NYU 全美金融工程专业招聘会上打了一圈酱油, 把普华永道的 representative 搞的 pissed off, 我也被他搞的 pissed off; 这学期要不是老师帮忙把简历递到 CME, 估计是一份都不会投的.

因为心里还有别的小九九, 所以去 CME 面试时抱的是无所谓的态度, 心态也前所未有的轻松. 虽然经历了四个人轮番轰炸, 但还算应付自如. 第一个面试官本应该是未来的 boss, 结果他当天有事, 临时换成一个刚从纽约调来的女孩儿, 此人麻省理工数学系代数方向毕业, 我有两个问题没答好, 一是短期和长期 yield curve 中间的 gap 可以通过哪种衍生品价格补全, 我想了半天估摸着说可能是掉期期权, 她说我把问题搞复杂了, 掉期期货就可以; 二是掉期期权的 Black-Scholes 定价公式, 我这种从来不记公式的人当场就傻了, 又是搞了半天在黑板上从头到尾把公式推了一遍, 然后她建议我有空还是要把 John Hull 的那本"圣经"好好看看... 第二位是个有些诡异的美国人, 见面也不和我打招呼, 然后一开始的气氛就有些尴尬, 我们在我的简历上花了好多时间, 后来我发现跟他说话还是挺轻松的, 于是面试就有往聊天方向发展的趋势, 他问我对 Black-Scholes model 的看法, 我扯了一通然后往自己比较擅长的 Heston model 上靠, 末了不忘加一句, "其实 Heston model 还有完善的余地", 他又问我为什么从数学转到金融, 我说我想把数学运用在实际当中, 他说那还有别的方向可以用数学, 我说"你指工程吗, 我感觉也就用用傅里叶变换, 挺无趣的", 他想了半天居然说了一句"我觉得还要用偏微分方程"... 快结束时他问我知不知道 Garch model, 我就完全放肆了, 心想 Heston 我都知道, Garch 不是反而还差些吗, 于是说仅仅听到过这个模型, 还反问他"这个模型好不好呀"... 第三第四个面试官都比较随和, 我答的也不错, 问题大概就是 Heston model 好在哪里, 为什么要用蒙特卡洛方法, 怎么用蒙特卡洛方法, 还有几个期权方面的基本数学问题. 走出办公楼的时候我比较轻松, 心想反正过不了我就继续留在学校看美女. 等了一个星期, 中间听说班上另一个去面试的同学录取了, 我则是收到邮件说周四再去一次, 这次接我上楼的人上次面过我, 对我说不要担心, 今天跟我谈的是我 boss, 随便聊一聊. 我一开始都没反应过来, 后来面着面着发现他开始跟我谈工作时间, 工作前景的问题, 我想这下有戏了, 也的确是果不其然.

走出办公楼, 给家里打了一个电话通报一声, 兴奋了大概一个小时. 然后就是纠结实习和讨论班的冲突问题, 还有半个月, 希望船到桥头自然直.

有实习经验的话, 后面工作也会比较好找, 比如利率衍生品的方向, QRM 这样的公司应该是很有希望的. 不过, 因为心里面还抱着做100%时间用来推公式的 pure quant 的梦想, 所以下定决心要读 PhD, 希望再经历一个三到四年的训练, 能够达到自己期望的标准. 美国的同学曾经问, 什么样的工作可以让我放弃读博. 我说, 至少起薪10万刀吧.

说的有些过分了, "堤高于岸, 浪必催之", 低调, 慎行.

and if is defined as

then satisfies

, .

The theorem can be stated the other way around. Thus, Feynman-Kac formula could be applied to interest rate models, for example, under Hull-White model, zero coupon bond price satisfies

,

we can deduce the analytic solution for by using the theorem described above.

Another application of Feynman-Kac formula in finance is discretization: explicit schemes and implicit schemes. For explicit schemes, we use such discretization method:

,
,

substitute these into the PDE then the equation could be written as

This kind of scheme has equivalence to trinomial tree valuation

and setting the size of rate step leads to the most efficient approximation, under which the scheme becomes 4th order w.r.t. to .

While applying explicit finite difference methods, boundary conditions have very little effect. We need to have

hold cause otherwise small numerical errors can grow uncontrollably.

For implicit scheme, we discretize the PDE like this:

,
,

then Feynman-Kac formula becomes

It has a relationship to trinomial trees:

Since we use time value to calculate , then if we want to do pricing backward, we need to solve a system of linear equations. Also, we need to care about the boundary conditions where we usually set and which is large enough. Hence, implicit scheme is slightly more difficult to implement than the explicit one. However, it is always stable and convergent. In practice, one method that is used very often is called Crank-Nicolson method which combines both explicit scheme and implicit scheme:

,
,

Although Crank-Nicolson method is a little bit harder to implement than both the explicit and implicit scheme, it is as stable as the fully implicit method and is the second-order accurate w.r.t. time step which is better than explicit and implicit schemes.

-- Generating stochastic paths for interest rates, exchange rates, and stock prices;
-- Numerical valuation of derivative instruments;

consider the risk-neutral pricing equation for security S with payoff

we can generate a large number of equally probable sample paths then the security value at time can be approximated as

.

For rate process , path generation is straightforward:

where is standard normal random number. For stochastic driver like this

we sample first and set .

When compared to the lattice valuation method, Monte Carlo approach has two important advantages:

-- Monte Carlo method easily handles path dependent instruments
-- Monte Carlo method is well suited to be used with multi-factor models

however, it also has drawbacks:

-- It is ill-suited for pricing interest rate derivatives with embedded exercise rights
-- It converges to true value very slowly
-- Longer rates along short rate paths cannot be implied from the paths
-- It does not give the same result twice since its value is random
-- Generating sample paths for high dimensional problems has considerable practical difficulties, which is an important feature of interest rate derivatives
-- MC has the opposite problem of recombining lattice: the full future is inaccessible

To implement Monte Carlo method, first we generate uniform random numbers then use methods such as Box-Muller or inverse transform to get normal random numbers. Because of the perfect foresight problem, zero coupon bonds prices cannot be computed without the use of either analytic pricing formulas or interest rate trees. Although MC method will not be used to calibrate the model, MC paths should be calibrated on the top of model calibration to make sure they price zero coupon bonds correctly. Instead of the continuous sampling as described above, we could first discretize the underlying continuous short rate process by means of a short rate lattice and then sample the lattice instead, which is call discrete sampling. The advantage of discrete sampling is that longer yields are readily available at each MC node and we also have a complete view of the stochastic future at each node on a path.

We want to decease the error of MC method as much as possible. One way is to increase the number of paths, the other is variance reduction. There are also two ways to do variance reduction: control variates and improving the sampling quality. The latter's advantage is it does not depend on characteristics of each instrument as the first one does. To improve the sampling quality, there is no other way better than improving the quality of Brownian motion sampled. Since we generate uniform random numbers first, then we can apply an approach called stratified sampling to improve their uniformity by

Based on the fact that Brownian motion is perfectly symmetric about zero, we generate Wiener sample paths which are symmetric to each other w.r.t. zero:

and this is called antithetic sampling. After the sample paths are generated, one can make a simple adjustment called moment matching to ensure the correct mean and standard deviation:

where and are the actual mean and standard deviation of the paths at time .

Usually, a short rate model can be written as

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

then by moment-matching, we can set

,

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

Branching probabilities are determined as

Once tree for the stochastic driver is built, we need to convert it to the short rate tree according to the functions:

normal:
lognormal:

where mean level vector 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

where is the cash flow at time . And now the remaining problem is how to calibrate to get mean level vector . Denote prices of zero coupon bonds with maturities by , then apply the following iterative search process forward:

(1) Set
(2) Search for the value of such that lattice price of the zero coupon bond maturing at equals
(3) ...
(4) Assuming are done, search for the value of to fit
(5) ...

We need to keep in mind that we always need to do numerical calibration for 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

One very useful property of Arrow-Debreu prices is that

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.

and so could discount factor.

Assume there is only one source of uncertainty represented by a Brownian motion under the risk-neutral measure . 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 (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:
Hull-White:
CIR model:
BDT model:

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, 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:

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:

.

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:
when rates are high:

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.

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.

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.