# Tag Info

6

Since $dW_A$ and $dW_B$ are already correlated as per the way you construct it, your portfolio being the sum of the two is already correlated. If you want it very explicitity written out, then you could rewrite $dW_B = \rho dW_A + \sqrt{1-\rho^2}dW_Z$ where $dW_Z$ is independent of $dW_A$. More generally (higher dimensions) you can use Cholesky. Now with ...

3

Given efficient markets, asset prices should be unpredictable in the sense that any upcoming returns are uncorrelated with current or past returns. Hence for traded assets the price should follow something more similar to a GBM than an O-U process. However, many financial metrics are not prices; for example interest rates or volatility. O-U processes may ...

3

I would say Take log of first equation to get rid of dependence on $x_t$ Apply Kalman filter equations to estimate parameters I believe Conrad and Kaul (1988) J of Business do exactly what you describe.

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This is a very broad question and a large number of issues have been discussed in the literature. As such, it's hard to give specific advice except that it is better to model returns instead of prices directly. What I would do if I were you: Read some of the available literature to get a good overview. This is an interesting paper but many more exist. ...

2

Note that \begin{align*} E\bigg(\frac{S_{i+1}}{S_i}\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) &=zE\bigg(\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) \\ &=zP\bigg(\frac{S_{i+1}}{S_i}<z\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)^+\bigg). \end{align*} Then you ...

2

If you have a linear/gaussian state space model and you're using a Kalman Filter, you can use maximum likelihood estimation or the EM algorithm. I personally prefer the former since you don't need to know anything about smoothing. If you use the EM, you do. If your observations are $z_1, \ldots, z_n$, then you can write down an innovations likelihood ...

2

Most practitioners think of option prices in terms of implied volatility. It is easier to interpret and to model. One can consider the implied volatility surface as a random field : $\Sigma : \Omega \times \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ and apply PCA. The first 3 eigenmodes correspond to absolute level (ATM vol), slope in the strike ...

2

It turns out that GBM with constant drift and constant volatility is not really used in real life. It is well known that volatility as well as drift may vary over time. Hence, if you want to use a model with time-varying parameters, you need to come up with a model to define $\mu_t$ and $\sigma_t$. There are classic models that use some mean-reverting ...

2

Yes of course, credit rates depend on interest rates (i.e. https://en.wikipedia.org/wiki/Libor), which are set by some group of banks in almost every country Going further bankers analyze the market situation and also national interest rates, which are set by central bankers in every country which has a central bank ...

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Normal distribution makes most sense these days for ratesthat are very low, or even negative, like euribor, chf libor Normal distribution is what is assumed by option brokers impliedvolatility quotes for these currencies

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I would recommend you the following econometrics textbook Basics Econometrics, with a particular focus on multinomial logit / probit models. I guess the challenging part in your case will consist of specifying the exogenous variables, collecting data, before doing the computations. The latter being quick to perform. As far as I am concerned it's better to ...

1

Model them individualy and as a group. When you model them as a group you are essentially building a stock index that you can compare the performance of individual stocks to and can then calculate a subgroup beta for each stock. You can also calculate a beta coefficient for the group as a whole to the wider market. Since I assume that you are modeling them ...

1

Quant finance is about finding prices of illiquid assets in terms of more liquid assets. So if you have the the data for liquid small house prices you should be able to come up with a reasonable guess for less liquid larger houses, for example. That's basically what's been done all the time - replication of complicated derivatives wrt more liquid assets. ...

1

1) People usually consider an instantaneous covariance while you are considering a integrated covariance. In a model $$dS_t = S_t \cdot (\alpha(t)dt + A(t)dW_t)$$ the integrated covariance of log-returns is simply the integral over time of the instantaneous covariance: $$Cov(R_i(t),R_j(t)) = (i,j)\textrm{-coeff. of} \int_0^t \underbrace ... 1 Here is a related previous StackExchange question: Modelling with negative interest rates Also, it seems that Black-Scholes option pricing breaks down. 1 You need to see the deals on these options and/or have deep knowledge of how these prices are marked to be able to have a better model. First thing first, I believe that the prices that you see are usually either "marked" (set) by one or several treaders, or they are the prices on last transaction before the close of the market/first transaction of the day ... 1 Continuous time has a so-called elegance, but it is rarely correct. Most Q-measure people rarely care about correctness anyway, since they usually don't root their models in statistics. With no goodness of fit measures, continuous time models are elegant theory. In general, we also see that most ex-ante hedges are rarely good, ex-post. They have large ... 1 What you are looking for is the partial expectation of \frac{S_{i+1}}{S_i}. Since \frac{S_{i+1}}{S_i} is lognormally distributed, you can use the following result: For a lognormal random variable X \sim LND(m,v^2),$$ E(X | X < z) = E[X] \Phi\left( \frac{\log(z)-m-v^2}{v} \right)  In your case, $m = (r-\frac{1}{2}\sigma^2) (t_{i+1}-t_{i})$, \$v^2 ...

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Thank you guys. Sorry for the late reply, I just solved it in matlab using maximum likelihood estimation. Turns out that all we need to do is to specify a state space model, then estimate the coefficient using MLE. The linearity and normality here makes things pretty simple.

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It depends on the use of your model as pointed out in the comments. If a discretized version is sufficient then state space models could be a solution. You can check out the free online textbook by Athana­sopou­los and Hyndman. State space model describe time series in terms of level/trend (and seasonality) on an additive or multiplicative way. There are ...

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I think Matt Wolf had the best answer by far, but the only point I would add is that the normal distribution can actually be a bit of a dangerous assumption at times, I actually believe this is the reason that more emphasis has been placed on risk management (especially recently) as opposed to pricing models. The main reason for GBM is that it creates ...

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Number one, the central limit theorem means a lot of things that may not be normal end up looking normal when lots of little 'experiments' or impacts are added up. Number 2, when dealing with finance you need a model that seems plausible. An arithmetic Brownian motion could go negative, but stock prices can't. On the other hand, it seems quite plausible ...

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