The whole white model is a short rate model, and we're going to discuss partly what that is to begin with, but then also look at the details of this modeling framework.

整個白色模型是一個短速率模型，我們將首先部分討論什麼是短速率模型，然後再看看這個模型框架的細節。

So let's get started.

那麼，讓我們開始吧。

First of all, we want to describe what a short rate is and an introduction to short rate models in general.

首先，我們想介紹一下什麼是短利率，以及一般短利率模型的介紹。

We can think of short rates as the continuously compounded and annualized interest rate at which an entity can borrow money for an infinitesimally short period of time.

我們可以把短期利率看作是一個實體在極短的時間內可以借到錢的連續複利年化利率。

Now that's a bit abstract, so we have that illustrated here by means of an equation.

現在看來有點抽象，所以我們在這裡用一個等式來說明。

We have some price of a risk-free asset at the time t plus delta t, and that can be expressed as the price at the initial time point, Pt, times this factor here.

我們有一些無風險資產在 t 時間點的價格加上 delta t，可以表示為初始時間點的價格 Pt 乘以這裡的係數。

And this is an exponential of the short rate, Rt, times the time interval delta t.

這是短速率 Rt 乘以時間間隔 delta t 的指數。

So the short rate tells us how much this value gain is going to be over some short time interval if we invest in a risk-free asset.

是以，短期利率告訴我們，如果我們投資於無風險資產，在某個較短的時間間隔內，價值收益會是多少。

And it is what's called continuously compounded, since these interest rate payments happen all the time.

這就是所謂的連續複利，因為這些利率支付一直在進行。

It is not like in some other models where you have interest rate payments at some sort of frequency, but this compounds continuously.

它不像其他一些模式那樣以某種頻率支付利息，而是連續複利。

And it is annualized in the sense that we need to express our interest rate in terms of some time interval.

它是按年計算的，因為我們需要用某個時間間隔來表示我們的利率。

We would say you get 10% per year, for example, or 10% per week, or something like that.

例如，我們會說你每年能得到 10%，或每週 10%，或類似的數字。

It's annualized, so we always talk about years when we talk about these interest rates.

這是按年計算的，所以我們在談論這些利率時總是以年為組織、部門。

So a short rate model then describes the future evolution of these short rates by generating a term structure.

是以，短期利率模型通過生成一個期限結構來描述這些短期利率的未來演變。

So the short rates would vary over time.

是以，短期利率會隨著時間的推移而變化。

You can see that Rt here has a time index to it, so the rates are not constant in the market.

您可以看到，這裡的 Rt 有一個時間指數，是以市場上的利率並不是恆定的。

And the short rate models describe how this Rt varies over time.

而短速率模型則描述了 Rt 如何隨時間變化。

So with that, let's focus on the whole wide model now.

是以，現在讓我們來關注整個寬幅模型。

This model implies a mean reverting and normally distributed assumption on the interest rates.

這一模型意味著利率的均值回覆和正態分佈假設。

And this gives us lognormal bond prices as a consequence.

這樣一來，對數正態的債券價格就出來了。

We can see the stochastic differential equation that summarizes the whole wide model in this slide.

我們可以在這張幻燈片中看到總結整個廣義模型的隨機微分方程。

And we can see that Drt, so the difference in the rate, is given by this drift term here plus some Brownian motion.

我們可以看到，Drt，也就是速率的差異，是由這裡的漂移項加上一些布朗運動給出的。

And the drift term has two components.

而漂移項有兩個組成部分。

First we have a theta of t here, and then we have a constant alpha times the rate itself.

首先，我們這裡有一個 t 的 Theta，然後我們有一個常數 alpha 乘以速率本身。

Now this theta is, as you can see, a function of time.

現在，正如你所看到的，θ 是時間的函數。

And this is chosen explicitly so that the model fits with the term structure that is being observed in the market right now.

明確選擇這一點，是為了使模型符合目前市場上觀察到的期限結構。

And that way you can calibrate this model to be accurate to the kind of term structures that are existing currently.

這樣，你就可以對這一模型進行校準，使其準確地適應目前存在的期限結構。

And then what the model gives you is a way to extrapolate, based on the current term structure, what they are going to be in the future.

然後，該模型給你提供了一種方法，根據當前的期限結構來推斷未來的期限結構。

The second part of this drift term is alpha Rt.

漂移項的第二部分是 alpha Rt。

And alpha is a mean reversion parameter.

而阿爾法是一個均值迴歸參數。

So it tells us how fast we are going to revert to the mean value of this process.

是以，它告訴我們，我們恢復到這一過程平均值的速度有多快。

It is typically left as a user input and can, for example, be estimated from historical data.

它通常由用戶輸入，例如可以根據歷史數據進行估算。

The last part of this equation is sigma that you can see here, which is the volatility parameter.

等式的最後一部分是西格瑪，也就是波動參數。

And it is determined by a calibration to a set of caplets and swaptions that are tradable in the market.

它是通過校準一組可在市場上交易的小數點和掉期來確定的。

So that is also something that would be derived from market data.

是以，這也是根據市場數據得出的結果。

And, as we said in the beginning, theta is something that is being explicitly chosen so that this model matches the term structures that are existing in the market right now.

正如我們在一開始所說的，θ 是明確選擇的，以便該模型與目前市場上存在的期限結構相匹配。

And that gives us essentially this expression, can be derived for what theta should be.

這樣，我們就可以得出θ 應該是多少的表達式。

And so plugging this into theta would give you essentially a model that fits the current term structure.

是以，將其插入 Theta，基本上就能得到一個符合當前期限結構的模型。

And it uses the instantaneous forward rate at time 0 for the maturity t here as a component of this value.

它使用 0 時到期日 t 的瞬時遠期利率作為該值的組成部分。

So that is where the market values themselves actually make it into the model.

是以，這就是市場價值本身實際進入模型的地方。

So that is fair enough.

所以這很公平。

We can see here an example of a realized process of the Holowite model.

在這裡，我們可以看到霍洛維特模型實現過程的一個實例。

So we can see here the short rate, which is what the model ultimately predicts over time.

是以，我們可以在這裡看到短期利率，也就是模型最終預測的長期利率。

And we can see a time in years on the x-axis here.

我們可以在 x 軸上看到以年為組織、部門的時間。

And as we extrapolate, something happens to this rate and it follows some sort of behavior that is stochastic in nature.

隨著我們的推斷，這個比率會發生一些變化，並遵循某種隨機性質的行為。

We can see the parameters that have been used for this evaluation here.

我們可以在這裡看到評估所使用的參數。

And this particular trajectory that we are analyzing now is going to be used for some of the comparisons coming up in the preceding slides.

我們現在分析的這一特定軌跡將用於前面幻燈片中的一些比較。

Nevertheless, what you need in terms of parameterization are the parameters alpha and sigma that we discussed before.

不過，在參數化方面，你需要的是我們之前討論過的參數 alpha 和 sigma。

You also need how long you want to predict.

您還需要預測多長時間。

So in this case, big T is 2 and the number of points that you want to simulate as well.

是以，在這種情況下，大 T 是 2，也是您要模擬的點數。

In addition to that, you need a yield curve or some sort of future rates or forward rates to actually plug into the model to calibrate the theta.

除此之外，你還需要一條收益率曲線或某種未來利率或遠期利率，以實際插入模型來校準 theta。

And this is based on a zero or a zero yield curve.

而這是以零或零收益率曲線為基礎的。

So that is an introduction to the Holowite model.

以上就是對霍洛維特模型的介紹。

And if we look at some of the mathematical properties of the stochastic differential equation that we saw in the last slide, we can note the following.

如果我們看看上一張幻燈片中隨機微分方程的一些數學特性，就會發現以下幾點。

So firstly, this can be integrated.

是以，首先可以將其整合起來。

It can be integrated and solved for, which means that R of T can be explicitly derived to be the following value here.

可以對其進行積分和求解，這意味著 T 的 R 可以明確地推導出如下值。

So this is R of T expressed in terms of an earlier time s.

是以，這是用更早的時間 s 表示的 T 的 R。

And this time s doesn't necessarily have to be, I mean, we can pick this to be zero if we want.

這個時間 s 不一定是，我是說，如果我們願意，也可以把它設為零。

But also if we have another time, sort of halfway through, we want to extrapolate based on that, we can do so as well.

但是，如果我們有另一個時間，比如在中途，我們想在此基礎上進行推斷，我們也可以這樣做。

And we introduced a new term beta in this expression.

在這個表達式中，我們引入了一個新詞貝塔。

And that is explained here what that expression is.

這裡解釋了這種表達方式是什麼。

And with this derivation, which we won't go over in detail, you essentially arrive at a normal distribution.

有了這個推導（我們就不細說了），基本上就得出了正態分佈。

So that's something we commented on in the last slide, that the Holowite framework gives normal distributions for the short rates.

這就是我們在上一張幻燈片中提到的，霍洛維特框架給出了短期利率的正態分佈。

And we can see the mean and variance of this normal distribution outlined here.

我們可以在這裡看到正態分佈的均值和方差。

Now what this means is if we're standing at the starting point, let's say, and we want to estimate what the short rate is going to be in one year, that's going to be normally distributed across or around some mean.

這意味著，如果我們站在起點上，比方說，我們想估算出一年後的短期利率是多少，這將在某個平均值上或周圍呈正態分佈。

We don't have the mean plotted here, but let's say the mean is up here, and then you would have some variance outside of that.

我們沒有在這裡繪製平均值，但假設平均值在這裡，那麼在平均值之外會有一些方差。

And so you could estimate a given point in this yield curve or in this short rate curve, if you wanted to.

是以，如果你願意，可以估算出這條收益率曲線或這條短期利率曲線上的某個點。

In order to make this a bit more tangible, we're going to assume that theta is a constant.

為了讓這個問題更加具體，我們假設 Theta 是一個常數。

So I said before that you would calibrate theta based on market parameters, and that would ultimately be a function of time.

是以，我之前說過，你將根據市場參數來校準 theta，而這最終將是時間的函數。

But to make some of these properties a bit more analytical, we're going to set it to be constant for a second.

不過，為了讓這些屬性更具分析性，我們要把它設置為常數。

We can see that if it's constant, the expression for r of t becomes a bit more simple.

我們可以看到，如果它是常數，那麼 t 的 r 的表達式就會變得簡單一些。

It becomes the following.

現在變成了

And just like before, this would be normally distributed.

和之前一樣，這將是正態分佈。

And if we compare or if we now generate the expected value of the rate at a given time, and also the variance of this, we would get these two expressions.

如果我們比較一下，或者說如果我們現在生成特定時間內利率的預期值以及它的方差，我們就會得到這兩個表達式。

Now these are a bit more approachable than the ones we saw on the last slide here.

現在，與上一張幻燈片相比，這些更容易接近了。

Also, I made the change that rather than extrapolating from another time s, I simply set them to be zero in this example.

此外，我還做了一個改動，在這個例子中，我沒有從另一個時間 s 開始推斷，而是直接將它們設為零。

So we can see the variance predicated on, well, the filtration at zero.

是以，我們可以看到以過濾為零為前提的方差。

If you don't know what that is, never mind, but nevertheless, we don't extrapolate based on another time point.

如果你不知道那是什麼，沒關係，但無論如何，我們不能根據另一個時間點進行推斷。

We just take the starting point as our baseline.

我們只是把起點作為基線。

And we can see these two analytical expressions here, and I've plotted the corresponding curves in these two figures.

我們可以看到這兩個分析表達式，我在這兩幅圖中繪製了相應的曲線。

So if we begin by analyzing the expected value, you can see that down here, essentially, is the starting value.

是以，如果我們從分析預期值開始，你可以看到，這裡基本上就是起始值。

And unsurprisingly, as we let t approach zero, r of t converges to r of zero.

毫不奇怪，當我們讓 t 接近於零時，t 的 r 趨近於零值的 r。

I mean, that's the starting point value we use for this model, so we would expect that to converge.

我的意思是，這是我們在這個模型中使用的起點值，所以我們希望它能收斂。

In this case, r of zero is a small value here, so that's why we're close to zero.

在這種情況下，0 的 r 是一個很小的值，所以我們才接近於 0。

And as we then let t tend to infinity, if we allow theta to be constant or set it to be constant, what's going to happen with the convergence is that the expected value here is going to approach sigma over alpha.

當我們讓 t 趨於無窮大時，如果我們讓 theta 保持恆定或將其設置為恆定，那麼收斂的結果就是這裡的期望值將接近 sigma 而不是 alpha。

That's going to be the long-term mean of this process.

這將是這一進程的長期意義所在。

And now depending on what these two parameters are, so depending on how volatile your model is calibrated to be, and then also depending on the mean inversion parameter here, you're going to get some sort of stable equilibrium for the mean of this process.

現在，取決於這兩個參數是什麼，所以取決於你的模型校準的波動性有多大，然後也取決於這裡的均值反演參數，你會得到這個過程均值的某種穩定平衡。

Now, if we look at the variance, there are two points that I want to raise.

現在，如果我們看一下差異，我想提出兩點。

First of all, we have a limit similar to what we have for the expectation value, and that is in this case sigma squared over two alpha.

首先，我們有一個類似於期望值的極限，在這種情況下就是兩個 alpha 的西格瑪平方。

What this means is that as t goes to infinity, so as we look infinitely far into the future, we don't get more and more uncertainty beyond a certain point.

這意味著，當 t 變為無窮大時，也就是當我們無限遙望未來時，我們不會在某一點之外獲得越來越多的不確定性。

So let's say we're comparing, I mean, 25 years here to let's say 50 years in the future.

是以，讓我們來比較一下，我是說，這裡的 25 年和未來的 50 年。

There would not be a significant difference in how much uncertainty we have in the rate over those time periods.

在這些時間段內，我們在費率方面的不確定性不會有很大差別。

So after a while, due to the convergence behavior of this model, the rates don't just explode in variability.

是以，在一段時間後，由於該模型的收斂行為，速率的可變性不會突然爆炸。

So that's a good thing.

所以這是件好事。

We can see also that in the beginning here, we have some form of slope of this curve in variance.

我們還可以看到，在一開始，這條曲線在方差上有某種形式的斜率。

So as we're very close to zero, the variance is zero of the rate.

是以，當我們非常接近零時，速率的方差就是零。

We can sort of see that intuitively as well, because if we are at time zero, we know exactly what the rate is going to be because we can measure it, and it's directly implied from the market, so we don't have any uncertainty, but uncertainty grows with time.

我們也可以直觀地看到這一點，因為如果我們在零時，我們確切地知道利率會是多少，因為我們可以測量它，而且它直接來自市場的暗示，所以我們沒有任何不確定性，但不確定性會隨著時間的推移而增加。

And how fast it grows with time is given by the tangent here, and we can see that if we go through time at the starting point, then that will come out to be sigma squared.

它隨時間增長的速度由這裡的正切給出，我們可以看到，如果我們從起點開始穿越時間，那麼它將是西格瑪的平方。

So that's how fast the uncertainty grows in the beginning.

這就是一開始不確定性增長的速度。

Okay, so now we looked at some analytical properties of this.

好了，現在我們來看看它的一些分析特性。

Let's actually see what term structures we get as a consequence.

讓我們來看看會產生什麼樣的術語結構。

So we have a graph here of different term structures, and by taking the weighted average of interest rates that prevailed over any one period, we can obtain the effective interest rate for that specific period.

是以，我們這裡有一張不同期限結構的圖表，通過對任何一個時期的利率進行加權平均，我們可以得出該特定時期的實際利率。

And if we plot these over time, that's how we obtain a yield curve.

如果我們將這些數據隨時間推移繪製成曲線，就能得到收益率曲線。

So we can see a number of different yield curves here, and they look a bit different depending on when we start to analyze these, because they are ultimately dependent on the short rate, and because the short rate varies over time, so will these yield curves.

是以，我們可以在這裡看到許多不同的收益率曲線，它們看起來有些不同，這取決於我們何時開始分析這些曲線，因為它們最終取決於短期利率，而由於短期利率隨時間而變化，這些收益率曲線也會隨之變化。

And with the trajectory of the whole white model that we saw in the beginning, these are essentially the different yield curves you would get for different points in time.

根據我們一開始看到的整個白色模型的軌跡，這些基本上是不同時間點的不同收益率曲線。

So we have the initial yield curve here, then we have the yield curve after 6 months, and we have the yield curve after 12 months, and so on.

是以，這裡有最初的收益率曲線，然後是 6 個月後的收益率曲線，12 個月後的收益率曲線，以此類推。

And depending on which instantiation, which time point you start looking at the yield curve, it's going to look slightly different.

根據不同的實例、不同的時間點，收益率曲線會略有不同。

So that's the first point, I would say.

這就是我要說的第一點。

And we can see that there is some variability between the curves as well, which seems to be at around 10% of the value or so.

我們可以看到，曲線之間也存在一些變化，似乎在數值的 10%左右。

And one more thing to notice, as the time to maturity here grows very large, these yield curves tend to converge.

還有一點需要注意的是，隨著到期時間的增長，這些收益率曲線趨於收斂。

And this is expected if we think about it, because we have mean reverting behavior exhibited by the whole white model, and so we would expect these rates to converge over long time periods to essentially the same value.

如果我們仔細想想，這也是意料之中的，因為整個白色模型都表現出了均值回覆行為，是以我們預計這些比率會在很長一段時間內收斂到基本相同的值。

And as such, the yield of a zero-coupon bond in that case would come out to be the same value as well.

是以，在這種情況下，零息債券的收益率也會得出相同的值。

And it is the yield that we are plotting on the y-axis here.

這就是我們在 Y 軸上繪製的收益率。

Okay, so I spoke about the variance between these curves, and let's look at the actual expression for this variance as well.

好了，我說了這些曲線之間的方差，我們也來看看這個方差的實際表達式。

So we can see here the volatility term structure, and that is given by the expression here.

是以，我們可以在這裡看到波動率的期限結構，它是由這裡的表達式給出的。

Now there is one thing that is very noticeable about this, and it is that the graphs that we are plotting here, these plots are essentially of this volatility term structure for the instantiations of the model that we discussed before.

現在有一件事非常值得注意，那就是我們在這裡繪製的圖表，這些圖表基本上就是我們之前討論過的模型實例的波動率期限結構。

We can see that these are the same, regardless of what time period we are considering, or rather regardless of what starting point we are considering.

我們可以看到，無論我們考慮的是哪個時期，或者說無論我們考慮的是哪個起點，這些都是一樣的。

And the reason for that is that this expression here of the volatility term structure only depends on big T minus small t.

原因在於，這裡的波動率期限結構表達式只取決於大 T 減去小 t。

It doesn't depend on small t explicitly, and so there is no dependence on when we actually start to analyze this.

它並不明確依賴於小 t，是以也不依賴於我們何時開始分析。

And so we could see that the volatilities of this term structure sort of narrow in on itself, as explained by the graph that we can see here, where the spot rate volatility goes to, well, becomes smaller over time.

是以，我們可以看到，這種期限結構的波動率會自我收窄，正如我們在這裡看到的圖表所解釋的那樣，即期利率的波動率會隨著時間的推移而變小。

Okay, we have some limits here as well for the volatility term structure, and we can see that as we get to the starting point, so as big T minus small t goes to zero, this volatility term structure goes to sigma.

好的，對於波動率的期限結構，我們也有一些限制，我們可以看到，當我們到達起點， 即大 T 減小 T 為零時，這個波動率的期限結構會變為西格瑪。

And the sigma value that we have in these parameters is 0.01, and you can see also that the spot rate volatility here goes to 0.01 in this limit.

這些參數中的西格瑪值為 0.01，你也可以看到，在這個限制條件下，即期匯率的波動率為 0.01。

And as we let t go to infinity, the expression for this goes to zero, and that's also observed by this sort of tapering off or exponential decay of the volatility term structure.

當我們讓 t 變為無窮大時，這個表達式就會變為零，這也可以從波動率期限結構的這種逐漸減弱或指數衰減中觀察到。

Okay, cool.

好吧

So then I also wanted to speak quickly about the bond pricing under the whole-white model, specifically for a zero-coupon bond.

是以，我還想快速談談全白模型下的債券定價，特別是零息債券的定價。

So after a lot of analytical calculations, you can arrive at the following price for a zero-coupon bond under the whole-white model.

是以，經過大量的分析計算，可以得出以下全白模型下的零息債券價格。

And we're not going to go over the calculations, but what I want to highlight is that the distribution of this is going to be log-normal.

我們不打算重複計算，但我想強調的是，它的分佈將是對數正態分佈。

So the rates in the whole-white model are normal, and as a consequence, the bond prices are going to be log-normal.

是以，全白模型中的利率是正態的，是以債券價格也將是對數正態的。

And we can see in the graph down here, the zero price for a zero-coupon bond with nominal $1 over, well, as a function of the time to maturity.

我們可以在下面的圖表中看到，名義價值為 1 美元的零息債券的零價格是到期時間的函數。

So naturally, as the time to maturity increases, the prices of these are going to decrease, because we're getting paid $1 at some point in the future, and the longer away that is, the more we're going to discount it.

是以，隨著到期時間的延長，這些產品的價格自然會下降，因為我們要在未來的某個時間點獲得 1 美元的報酬，時間越長，我們的折扣就越大。

And we can see the different instantiations here that we saw before as well, but now reflected in terms of their effect on the zero price for a nominal of $1.

我們可以在這裡看到之前看到的不同實例，但現在反映的是它們對 1 美元名義零價格的影響。

And we have some variability between these, not too much though.

它們之間有一些差異，但不會太大。

So one thing that's important to note is if you calibrate a model like this to, well, market parameters, the level of sophistication that you need is quite dependent on the application.

是以，需要注意的一點是，如果你根據市場參數來校準這樣一個模型，你所需要的複雜程度就取決於應用。

For a lot of applications, a quite simplistic model might do, because ultimately, the variance in terms of bond prices is not going to be very big under normal market conditions.

對於很多應用來說，一個相當簡單的模型就可以了，因為在正常的市場條件下，債券價格的差異最終不會很大。

Nevertheless, if you need that extra granularity, a sophisticated model like the whole-white model might be, well, appropriate.