Measuring volatility for option prices

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Mobius Striptease

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maratikus Wrote:
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>
> What do you mean by estimating volatility under
> risk-neutral measure?

i mean something like this:
http://pages.stern.nyu.edu/~rengle/B303352/Duan.pdf

more specifically theorem 3.1, the GARCH process is different under real-world vs. the risk-neutral measure used for option pricing, that lambda which shows up there. not sure how to calibrate it in practice
 
Mobius Striptease Wrote:
-------------------------------------------------------
> maratikus Wrote:
> --------------------------------------------------
> -----
>
> >
> > What do you mean by estimating volatility under
> > risk-neutral measure?
>
> i mean something like this:
> http://pages.stern.nyu.edu/~rengle/B303352/Duan.pd
> f
>
> more specifically theorem 3.1, the GARCH process
> is different under real-world vs. the risk-neutral
> measure used for option pricing, that lambda which
> shows up there. not sure how to calibrate it in
> practice

That's interesting, Mobius. I will check it out.
 
Mobius Striptease Wrote:
-------------------------------------------------------
> maratikus Wrote:
> --------------------------------------------------
> -----
>
> >
> > What do you mean by estimating volatility under
> > risk-neutral measure?
>
> i mean something like this:
> http://pages.stern.nyu.edu/~rengle/B303352/Duan.pd
> f
>
> more specifically theorem 3.1, the GARCH process
> is different under real-world vs. the risk-neutral
> measure used for option pricing, that lambda which
> shows up there. not sure how to calibrate it in
> practice

If I just straight up estimate the GARCH equation, run a monte carlo simulation, and apply 2.5, you're saying that I would not be taking into account risk-neutral pricing (all the talk about risk neutral measures makes my head hurt) and I would have some bad mojo?
 
im not saying anything, mr Duan is, and seeing that his paper is posted on the website of nobel-prize winner Engle, the mac-daddy of all things ARCH, it seems quite legit.

In all seriousness, I haven't read the paper in detail to fully understand all derivations and nuances, but it seems very reasonable that if the volatility is specified by a GARCH process, and hence it is dependent on lagged returns, and returns differ under real and risk-neutral measures, the GARCH equation for the volatility will take a different form under the risk-neutral measure if you require the price process to be a Q-martingale
 
Perhaps you could do this - run risk-neutral MC with standard GARCH estimate for the volatility process, and discount the expected stock price at T by the risk-free rate (instead of an option payoff). Then see if you recover the starting stock price. If I'm interpreting this right, unless you adjust the GARCH by the lambda factor, the stock price process won't be a martingale under the risk-neutral measure so you won't recover the starting price?
 
jmh530 Wrote:
> If I just straight up estimate the GARCH equation,
> run a monte carlo simulation, and apply 2.5,
> you're saying that I would not be taking into
> account risk-neutral pricing (all the talk about
> risk neutral measures makes my head hurt) and I
> would have some bad mojo?

When you estimate GARCH, you do that using a probability measure. To simplify things we can just look at one step binomial model and compare volatility under physical and risk-neutral measures. Let's say

x = a with prob p and b with prob 1- p, risk-free rate is r. risk neutral probability q is such that EQ(x)=r -> q = (r-b)/(a-b) //has nothing to do with physical measure p.

Variance of x under P is VaRP(x)=(a-b)^2*p*(1-p), Variance of x under Q is (a-b)^2*q*(1-q). Clearly, those variances will most likely be different because it's highly unlikely that physical and risk-neutral measures will coincide (that would only happen if EP(x)=r.

I hope this helps.
 
I think what you said makes sense, but the whole process seems a little perplexing to actually set up properly. I wonder how this result compares to volatility skew/smile adjusted black-scholes and whether just plugging in GARCH vol (or expected GARCH vol) into that equation makes an improvement.
 
bchadwick Wrote:
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> I'm curious... what's the normal practice on using
> standard deviation of returns vs standard
> deviation of log returns here?

I think the standard practice is using log returns... like, in VarSwaps, volatility is calculated using log returns. In option pricing models too.

But it depends, if you are generating efficient forntier, then volatility of absolute returns makes more sense as you are concerned with absolute change in prices.

Rationale for log return for calculating probability is the assumption of normal distribution of returns I think, which is packaged with assumption of log normal distribution of price, taking log of returns expands the range from -inf to +inf, making it more 'normal'.
 
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