关于期权定价的SV模型的几点疑问

最近在做股票指数期权定价的论文，没有基础，很多不懂的地方想请教大家我需要用SV模型，问题如下：
1.这个期权定价的SV模型和时间序列分析中的SV 模型有什么关系？我感觉应该是相互联系的，可是从书上查阅到的公式差别很大，不明白
2.我是不是应该先利用已有的股指数据 建立起SV模型，然后再通过数学推导求出微分方程，求解这个微分方程就能得到期权的价格啊？
3.操作层面上，怎么实现第二个问题？？
求指点，问题有点多
在人大经济论坛也是新手，实在不富裕，所以悬赏的论坛币不多。。见谅见谅

卡拉巫 发表于 2013-7-21 13:40
谢谢您的解答。
您的意思是说用SV模型 是不能求解的么？用蒙特卡洛模拟也无法求出么？
我现在要对一股票 ...

No, I didn't mean that. I just mean no closed form solution, but we can have other solutions such as approximation or numerical ones.

If you are not familiar with SV model. You can take a look at my article:
https://bbs.pinggu.org/thread-2449559-1-1.html

There are many SV models, and many ways to estimate them, so it depends on which model you use. What I recommend is Heston (1993) model which is the most successful one. You can check papers on how people use Heston model to price options. The main issue is 1) calibrate (estimate) the model with real data, 2) pricing numerically.

You do not need to create some methods, just follow other peoples results.

Best,

1. They have some relationship. If you discretize the continuous model in option pricing, you will get a time series model.

2. For the second question, what I recommend is you directly build the SV in the model rather than first time series then pricing. You can take a reference to Heston (1993) model. Stochastic volatility model is usually hard to solve, and usually no close form solution in time domain.

best,

Chemist_MZ 发表于 2013-7-21 13:02
1. They have some relationship. If you discretize the continuous model in option pricing, you will g ...

谢谢您的解答。
您的意思是说用SV模型 是不能求解的么？用蒙特卡洛模拟也无法求出么？
我现在要对一股票指数期权定价，您有没有什么建议给我ne
再次感谢

FOr Heston (1993) model, we've a formula in the form of numerical integration. But how to calibrate it efficiently is not so easy.

1.个人觉得两个SV模型只是形式上有关联，实质上并不同。表面上看，期权定价中的SV是“连续时间模型”，其实质目的是通过将波动率随机化修正经典模型中的“常波动率假设”，也就是解释“波动率微笑”现象，最终目的是更好地定价；时间序列中的SV是“离散时间模型”，实质目的是将波动率随机化（不过不是完全的随机化，需要服从一个时间序列模型）来解释波动率序列存在的自相关性，也就是解释“波动率聚集”现象，最终目的是预测。PS：如果波动率聚集和波动率微笑有联系，以上说法要打折扣。
初看起来连续时间模型比离散时间模型复杂，其实不然。在离散框架下，通常难以建立非线性的模型，所以时间序列分析中的模型绝大部分是线性的；但是在连续框架下，借助Ito公式，一些复杂的非线性的元素可以转化成PDE中的线性元素（比如Heston模型，3/2模型，具体见wiki），微分方程驾驭复杂问题的能力要强于差分方程。

2.以我个人的经验，虽然期权定价中的SV差别很大，但是实证得到的数值结果很相似，所以我觉得可以先指定一个SV模型，然后通过统计计量的方法估计模型参数，再用MC或PDE的方法求解。得到定价的结果不是最终目的，还要看对“波动率微笑”的解释力度和拟合程度。

3.解PDE的话通常用有限差分法（Heston模型有闭式解），一个SV模型的PDE维度至少是3，解的难度比较大。如果用MC的话，因为有两个随机源，也不太容易。

xuruilong100 发表于 2013-7-22 10:20
1.个人觉得两个SV模型只是形式上有关联，实质上并不同。表面上看，期权定价中的SV是“连续时间模型”，其实 ...

1. Agree, the time series model can be better viewed as "time-varying volatility" model. They have different roles. SV provides a distribution for pricing, while TS model is used to predict the volatility. But actually they have some relationship as I mentioned. See John Hull (Chapter 22 page 503 8ed) he gives a very simple example showing that GARCH(1,1) model is actually equivalent a stochastic process. Something like a lognormal Vasicek model.

2. Agree. But MC is not good for those models with non-normal density, such as CIR process.

3. As I know, Heston has a "closed" form solution in frequency domain, not in time domain. At last, still need numerical integral. What people usually do for SV model now is to use numerical method to find the joint density of the price and volatility. Then you can either use MLE to estimate the parameters or use numerical integral to do the pricing.

best,

Chemist_MZ 发表于 2013-7-22 10:38
1. Agree, the time series model can be better viewed as "time-varying volatility" model. They have ...

我想到了一个有意思的问题：
给我股票数据，在常波动率假设下通过计量统计技术可以算出“波动率”；如果在随机波动率假设下，我希望能“逆向计算”出波动率的轨迹，或者退一步计算出和波动率有关的一条轨迹。如果波动率只是价格的函数，上述想法是可行的。如果波动率被另外的随机源驱动，得到的轨迹也许是某种“平滑”的结果。

xuruilong100 发表于 2013-7-22 17:50
我想到了一个有意思的问题：
给我股票数据，在常波动率假设下通过计量统计技术可以算出“波动率”；如果 ...

Good thinking. This is probably another way to link the stochastic process with time series model.