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Fixed Income Modeling Review 4

Tao 发表于 02/07/2011 | 分类 技术 | 标签 , | 现有0条评论

After reviewing Ito lemma and SDE, it is time to have a look at the standard market model which are forward contracts, future contracts, options on bond, interest rate caps, floors, collars and swaptions.

Forward contracts and future contracts are similar but there are also some crucial differences: forward contracts are over the counter and future contracts are traded on exchanges; future contracts are settled daily which reduces the credit risks, and after daily settlement future contracts is worth zero.

Forward price and future price are prices make the contracts worth zero at the beginning or the time after each settlement. Usually they are different but if the interest rates are deterministic, or more generally, the underlying asset price process is independent of the rate process then thery are the same.

Fwrd(t)=\frac{S(t)}{P(t,T)}=\frac{E^B_t[D(t,T)S(T)]}{E^B_t[D(t,T)]}=E^B_t[S(T)]=Fut(t)

One more thing is that if the underlying asset price is positively correlated with the interest rates then the futures price will be higher than the forward price and this is why the rate implied from Eurodollar futures price is higher than the corresponding forward rate.

For options on futures we have Black's model which is completely consistent with the Black-Scholes formula for stock options. Black's model is so popular that the implied volatility from the option prices has become the way most popular options are quoted, and it has also become a popular pricing and quoting tool on fixed income markets thought it assumes deterministic interest rates.

There are two basic assumptions for Black's model:
(1) The value V(T) of the option is lognormally distributed with the standard deviation of \ln(V(T)) equal to \sigma(T-t)^{1/2}.
(2) The expected value of V(T) is F(t) under T-forward measure.

And so the Black's formula becomes

Call(t)=P(t,T)[F(t)\cdot N(d_1)-K\cdot N(d_2)]
Put(t)=P(t,T)[K\cdot N(-d_2)-F(t)\cdot N(-d_1)]

Bond option is an option to buy or sell a particular bond by a certain date for a particular price. In addition to trading in the OTC market, bond options are often embedded in bonds when they are issued like callable bonds or puttable bonds. By assuming the bond price S(T) at the maturity of the option T is lognormally distributed with standard deviation of \ln(S(T)) equal to \sigma(T-t)^{1/2} we can apply the Black's model to compute the price of European call and put options.

Interest rate cap is an OTC floating rate contract which is a series of call options with the underlying being the market rate and the strikes equal to a fixed rate. Each of the call options is called a caplet. For floors and floorlets, they are similar, only with call options changed to put options.

Every caplet can also be seen as a bond option:

V_i(t_{i-1})=P(t_{i-1},t_i)L\tau(t_{i-1},t_i)\max[R(t_{i-1},t_i)-R_K),0]
=\max[L-L(1+\tau(t_{i-1},t_i)R_K)P(t_{i-1},t_i),0]

Thus under the assumptions of Black's model we have the pricing formulas for caplets and floorlets:

Caplet_i(t)=L\tau(t_{i-1},t_i)P(t,t_i)[F_t(t)\cdot N(d_1)-R_k\cdot N(d_2)] Floorlet_i(t)=L\tau(t_{i-1},t_i)P(t,t_i)[R_K\cdot N(-d_2)-F_i(t)\cdot N(-d_1)]

In the formulas above we use different volatilities for each caplet and there is another approach which is to use the same volatility for one maturity. Volatility in the first approach is called spot volatilities and in the latter one it is called flat volatilities. In real market we usually use flat volatility for quoting.

European swaption is another OTC contract. It gives its holder the right to enter into a swap contract at a certain time in the future called exercise time. We could take a fixed-for-float swap to examine the nature of the contract. Since at the start of the contract the value of the floating leg is equal to the notional principle of the swap, the swaption could be seen as an option on a fixed rate bond with the strike being equal to the bond priciple.

Payoff(T_{Exe})=\max[R_K-R_{swap}(T_{Exe},T_{Stl},T_{Mat}),0]\cdot A(t,T_{Stl},T_{Mat})\cdot L

We can easily calculate the swap rate by

R_{Swap}(t,T_{Stl},T_{Mat})=\frac{P(t,T_{Stl})-P(t,T_{Mat})}{A(t,T_{Stl},T_{Mat})}

then a pricing formula for the swaption is

ReceiverSwaption(t)=L\cdot A(t,T_{Stl},T_{Mat})[R_K\cdot N(-d_2)-R_{Swap}(t)\cdot N(-d_1)]

Sometimes we need adjustments to price options on bond and they are convexity adjustment and timing adjustment. If the bond price is given in terms of the yield, then we could not simply assume that the forward price F(t) is equal to expected bond value and this is when convexity adjustments take place. If payment date is delayed until a later time T^*>T, then in order to compute expectation of a variable at time T with respect to T^* forward measure, we need timing adjustments.

HW 4, My solution

Welcome Gasperini

Tao 发表于 25/06/2011 | 分类 体育 | 标签 , , | 现有10条评论

莱昂那多还是离开了国米，不过滑稽的是，以他在球队的表现，主动提出分手的竟然不是国米而是他，anyway，终于不用再担心会被强队血洗了。

后来选帅也是提心吊胆，因为传来莫拉蒂喜欢米哈的消息，看到这样的新闻我已经无语到极点，这不等于送走一个莱昂又找来另一个吗？为什么莫拉蒂就是喜欢这种球员出身，执教经验几乎没有，意味着极大风险的“教练”呢。这些年，除了瓜迪奥拉以外，克林斯曼，多纳多尼，马拉多纳，莱昂那多哪一个不是输的灰头土脸？即使是瓜迪奥拉，也是巴塞罗那的特殊情况。看来，莫拉蒂还是只能稳定稳定大局，具体事务尽量不要插手为好，上赛季若不是他的阻拦，我们也不会高薪养着米利托一个伤号以及麦孔这样出工不出力的球员。

幸好，加斯佩里尼最终成为国际米兰的教练，西蒙尼之后国米聘请的第一个意大利联赛培养起来的教练。应该说，我还是比较相信意大利教练的能力，现在好奇的是， 这些教练往往能把小球队带的非常难缠，如果给了他们充足的资源和一流的球星，他们又能把大球队带成什么样呢？另外很重要的一点就是，莫拉蒂能不能给加斯佩里尼足够的支持，去年贝尼特斯的改革虽然冒进，但球队成绩不好的另一重大原因无疑是主席对教练的信任不够，貌似从开始到最后就一直没给教练完全的支持，我始终相信，如果下半赛季的教练是贝尼特斯，我们绝对不可能被沙尔克04这样的球队淘汰。

莫拉蒂，请忘记瓜迪奥拉，全力支持加斯佩里尼吧。

Box-Muller transform

Tao 发表于 30/05/2011 | 分类 技术 | 标签 | 现有5条评论

蒙特卡罗方法在金融数学中有很重要的作用, 比如可以方便的处理与路径相关的金 融衍生品定价, 它的基本原理是生成随机数, 进而生成多个随机路径, 假设这些路 径具有相同的概率, 于是根据

V(0)=E[e^{-rT}V(T)]

就可以通过对所有的路径取平均而得到 V(0), 即衍生品的价格. 它的一个缺点是计 算出来的 V(0) 取决于生成的随机数, 因为每次的随机数必不相同(几乎是几乎处处), 所以 V(0) 也互不相同.

可见如何生成随机数是蒙特卡罗方法的关键, 并且蒙特卡罗方法一般都是通过计算 机编程实现(很难想象进行手算蒙特卡罗的应用), 于是如何找到快速有效低耗的生 成随机数的方法就相当重要. 通常的步骤是: 生成伪随机数 --> 生成一致分布的随 机数 --> 生成正态分布的随机数. 所谓伪随机数就是一列整数 I_n\in[0,RAND\_MAX], 可以通过 u_n=I_n/RAND\_MAX 转化为[0,1]一致分布 的随机数. 各个编程语言都提供了生成伪随机数的算法, 比如 Microsoft C++ 就 有一套令 RAND_MAX=32,767 的标准算法, 但这对于金融应用并不够, 比如要进行 一个360月的债券的定价, 在第91个路径之后随机数便开始重复了. Numerical Recipes in C 这本书里据说讨论了一些非常好的随机数生成算法, 但是我还没看.

当获得了一致分布的随机数之后, 要将它们转化成正态分布 N(0,1), 这里有几个 方法:

(1) 最基本的是生成 u_i\sim U(0,1),i=1,2,...,12, 然后令 \epsilon=\sum^{12}_{i=1}u_i-6. 这种方法简单而且快, 但缺点是不够准确.

(2) 一种非常常用的方法叫 Box-Muller 方法. 它的原理是假设有两个独立同分布 的 U_1,U_2\sim U(0,1), 那么根据 Inverse transform sampling 定 理, R^2=-2\ln U_1 是指数分布, 同时也是 \mathcal{X}^2 分布, 这也就是说它是一 个二维正态分布的范数的平方, 然后再通过极坐标投影,

Z_0=R\cos(\theta), Z_1=R\sin(\theta)

是两个独立标准正态分布, 其中 \theta=2\pi U_2. 这种方法还能进行转化, 令 u=R\cos\theta,v=R\sin\theta, 那么

Z_0=\sqrt{-2\ln U_1}\cos(2\pi U_2)=u\cdot\sqrt{\frac{-2\ln s}{s}}

Z_1=\sqrt{-2\ln U_1}\sin(2\pi U_2)=v\cdot\sqrt{\frac{-2\ln s}{s}}

这种转化的好处是避免了在计算机中进行正弦, 余弦函数的运算. 最后是一段简单 的在 C++ 中通过 Box-Muller 方法生成正态分布随机数的代码

double GetOneGaussianByBoxMuller()
{
double result;
double x;
double y;
double sizeSquared;
do
{
x = 2.0*rand()/static_cast(RAND_MAX)-1;
y = 2.0*rand()/static_cast(RAND_MAX)-1;
sizeSquared = x*x + y*y;
}
while
( sizeSquared >= 1.0);
result = x*sqrt(-2*log(sizeSquared)/sizeSquared);
return result;
}

Fixed Income Modeling Review 3

Tao 发表于 26/05/2011 | 分类 技术 | 标签 , | 现有0条评论

During the whole semester, topic 3 was probably the easiest. Because I have known the material of Ito integral and stochastic differential equation(SDE) in advance of the lecture. But it was still good as a complement to my knowledge.

Everyone should know what probability space, filtration, stochastic processes, adapted processes, martingale and Brownian motion are, cause these concepts are the very basics of mathematical finance. One should also know that Ito integrals are martingales and if the integrand is non-random then Ito integral and Stratonovich integral are the same thing, which means that the product rule for Riemann integral applies to Ito integral as well. Why do we choose Ito integral instead of Stratonovich integral given that Stratonovich integral preserves the product rule? The reason is when we construct Ito integral, we choose the left end points for the integrand so we don't look into the future, and this is essential to financial application.

One of the most important theory we need to use is Ito's lemma. It basically expands the concept of derivative we already know in calculus. We know the Taylor expansion:

df(x)=f'(x)dx

and now for Ito integral we have

dg(t,W(t))=g_tdt+g_xdW(t)+\frac{1}{2}g_{xx}dt

so essentially we are using the fact that dW(t)dW(t)=dt.

In order to compare different securities and remove the price appreciation due to the time value of money effect, we introduce the concept of numeraire and prices divided by numeraire are called dicounted prices. For any numeraire there exists a corresponding martingale measure which is equivalent to the probability measure that discounted prices are martingales under such measure. Actually this is not precisely correct cause martingale measure does not always exist, so this sentence I am saying is more with respect to change of numeraire. That is if you change the numeraire then the martingale measure changes and vice versa.

The first fundamental theory of asset pricing says if there exists a martingale measure then there is no arbitrage in the market. The converse is not true and you need to add another condition called no free lunch with vanishing risk to ensure that martingale measure exists given there isn't any arbitrage opportunity. One thing new I learnt from this lecture is that if martingale measure exists for one numeraire then for another numeraire we could also find martingale measure. The most commonly used numeraire are banking account and zero coupon bond.

Change of measure is another important technique with respect to mathematical finance. You first need to know what is Radon-Nikodym derivative from real analysis. Then you should know the Girsanov's Theorem:

If W is a Brownian motion under P, then for

dP^*=\rho dP, \rho=\exp\{-\int^t_0\theta(s)dW(s)-\frac{1}{2}\int^t_0\theta^2(s)ds\}

dW^*(t)=dW(t)+\theta(t)dt is a Brownian motion under P^*.

Martingale measure according to banking account is called risk neutral measure which is named after the fact that the market price of risk is zero in this case. Under risk neutral measure, all the prices discounted by momey market account are martingales and in case of random interest rate model we often change the numeraire to forward measure because of simplicity of calculation(this is not the only reason!). Forward measure is the EMM with respect to zero coupon bond and one of the professors said to us that if someone knows what forward measure is then he/she could skip the course of Mathematical Finance I. Notably, forward price is a martingale under forward measure.

In fixed income modeling, one rule I didn't know is that conditional expectation of short rate is equal to the forward rate: E^T_t[r(T)]=f(t,T).

HW 3, My Solution

蛋疼的表现

Tao 发表于 23/05/2011 | 分类 日记 | 标签 , | 现有12条评论

记得曾经折腾过用 Emacs 当做离线 blog 客户端, 后来重装系统便丢掉了, 今天 不知怎么突发奇想, 又把它折腾回来, 还变本加厉装了 Total Commander 和 firefox 的插件 Pentadactyl, 大有要把鼠标扔掉的趋势.

前两天在书店看到一本介绍用 C++, Matlab, Excel 进行金融建模的书, 内容本是 很好的, 可被翻译成中文就感觉有些奇怪和不知所云, 况且像中国这样连期权都没 有的国家, 我真的怀疑有没有引进此类书的必要.

暑假开始后, 老师说给我几个题目思考, 前几天终于把文档发给我了, 看了看, 基本就是把现在那几个 PhD 和刚毕业的 PhD 的研究方向给我总结了一下, 另外又 提了一个新题目, 还说以后会给我一些 Credit Risk 的问题. 我在想, 这是让我 在里面选个题目做, 还是指望我把所有的都做了呢? 倒不是说自己没那个智力, 而是, 要在一年内完成这么多问题根本就是 physically impossible. 不由得感到, 美国的研究生教育和中国真是不同, 美国教授也做科研, 但是对待学生非常负责; 中国的很多人, 只顾自己的工作, 却忘了还有一个身份叫做老师, 学生在摸索阶段, 最需要的就是他们的指点, 可这些人偏偏有各种理由推脱自己身为导师的责任. 中 国在解放后没有出过大师, 如今看来, 不仅仅是大学没有自治的原因, 具体到每一 个细节, 貌似都有腐烂的痕迹.

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