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kiann
Topic Author
Posts: 18
Joined: April 16th, 2008, 6:39 pm

### Conversion of realized volatilities pdf into probability density function in Black-Scholes

Hi experts, I am looking for advise and help in some volatility mapping I am trying to do.

Ignoring the asset-class (FX, interest rate etc), I have a time-series of extremely short-dated (seconds) market-stream. These are converted into a series of rolling (instantaneous) volatility (e.g. stdev[A1:(A1, 30,0)].
I now have a set of X number of (instantaneous) volatilities, and I bucket these into a kind of 'histogram' of probabilities (y-axis) versus volatilities; where probability = (no. of counts of vols in that bucket)/(total no. of counts).

Now, is there a way to convert this pdf (prob versus vol) into the pdf of the underlying variable S under the risk-neutral black-scholes framework? More precisely, my understanding is that the standard pdf of a variable S can be estimated using the 2nd derivative of the call-price

pdf(K=strike) = d^2C/dK^2 where C = call-Price(S, K, rf, T-expiry, vol[K])
and we assume pdf(K = strike) = exp^1/2 * [S - mean/vol(K)]

how can I generate a function of vol[K] to reflect pdf[K=Strke] from the histogram/curve of pdf[vol]

Kind regards
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outrun
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

Suppose you historical vol distribution has just two buckets: vol=0.20 30% of the time and vol= 0.50 70% of time, then the option price is  simply
Price = 0.30*CallPrice(vol=0.2) + 0.70*Call(vol=0.5)

Once you have the price you can go back to implied vol if you like.

Hoever,. I see a lot of things that might go wrong depending on your application.

Black Scholes assumes that the returns are Markov but at the seconds time scale they are highly mean reverting, bouncing between bid-ask all the time. If you see prices like 4,5,4,4,5,4,4,5,5,4 you might say it changes on average with 1 point per second, and B&S will assume it will have moved +/- 3 after 10 seconds and price options accordingly,.. whereas in reality if will still likely be bouncing between 4-5 10 seconds from now.

An easy way to check this is to compare the standard deviation of 1 second returns with that of 2,4,8 second returns. You'll see it won't scale sqrt(dt). Looking at this scaling is very important if you want to extrapolate to the n-step distribution from 1 step historical vol estimate.

Another issue is that you are looking at the"unconditional vol" but vol is highly persistent. If the vol in the last 30 seconds was 10x higher than average then the next 10 second will also be higher than average for sure. Your vol distribution doesn't capture that.

kiann
Topic Author
Posts: 18
Joined: April 16th, 2008, 6:39 pm

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

thanks outrun, that does sound like a good idea. There is this observation in the markets (most asset types) that vols tend to be mean-reverting, or that short-dated vols are higher than longer-dated vols and volatility profiles tend to have a hump-shape.

I would try that idea you just proposed. I was also thinking whether there are any 'analytical' based form, to convert the historical vol distribution to the variable pdf though.

outrun
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

Yes maybe it doable to do it analytically

if regular B&S is viewed as expected payoff:
C(K) = int Pn(S,vol) F(S,K) dS

- with Pn() the stock distribution at expiration having distribution parameter vol. The n stands for the fact that the returns are assumed to be normal distributed.
-  F() the payoff of the option

when you have a vol *distribution* you get
C(K) = int int Pv(vol=v) Pn(S,v) F(S,K) dS dv

which can be written as (which is what I tried to suggest)
C = int Pv(vol=v) [int Pn(S,v) F(S,K) dS] dv
with the bit between [] the regular B&S call integral, and the Pv(vol=v) acting as a weight.

I think your goal is to find a v(K) smile such that
C(K) = int int Pv(vol=v) Pn(S,v) F(S,K) dS dv
C(K) = int int Pv(vol=v) Pn(S,v) dv F(S,K) dS

C(K) = int Pn(S,v(K)) F(S,K) dS

?

bearish
Posts: 5158
Joined: February 3rd, 2011, 2:19 pm

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

Somewhere here is lurking an unstated assumption that the vol is independent of (or at least uncorrelated with) the stock price level. That won’t matter much for near the money options, but would affect the tails of the distribution.

Alan
Posts: 9865
Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

@kiann,
Well, the "risk-neutral black-scholes framework", unfortunately, means different things to different people. If you mean the framework discussed in Black and Scholes' seminal paper, then the pdf associated to that is a log-normal density. True volatility in that framework is a constant parameter and your realized volatility pdf represents the true volatility +  sampling noise (under that framework).

kiann
Topic Author
Posts: 18
Joined: April 16th, 2008, 6:39 pm

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

@kiann,
Well, the "risk-neutral black-scholes framework", unfortunately, means different things to different people. If you mean the framework discussed in Black and Scholes' seminal paper, then the pdf associated to that is a log-normal density. True volatility in that framework is a constant parameter and your realized volatility pdf represents the true volatility +  sampling noise (under that framework).
Hi Alan, true. Black-scholes assumes a constant (and log-normal) volatility. However, the industry has worked around this assumption, mainly either by generating the volatility itself as a stochastic variable, or creating a parametric form for the vol such that there is a one-to-one mapping between implied vol versus observed market (European Vanilla Price).
What I am merely trying to do (and feel for), is that if
a) I take the realized volatility pdf as the assumed distribution of the volatility
b) is there a way to map this assumed distribution of the volatility into the underlying variable pdf
Kind regards

Alan
Posts: 9865
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

So, now apparently "Black-Scholes framework" just means the existence of a BS-implied volatility (IV)? If so, then if you have a set of option prices, you should get your pdf via Breeden-Litzenberger (B-L), which I see from your post you are aware of. In turn, that's done by making a smooth IV-fit to the option prices and differentiating.

I suspect you are actually trying to infer what option prices would be for a traded asset with *no* currently existent options. If that's the case, you should have stated that as your question.  AFAIK, there's no model-independent way to answer the latter question by your realized vol route.

However, taking your question as you stated it, for a model-dependent answer:

- Take the assumption that the volatility is an uncorrelated diffusion.
- Modify what you are doing to get the distribution of the (integrated) variance U(T) over the time period T of the option. [This is trivial re-scaling].
- Get option prices from the distribution of U(T) by mixing. [For details, there is a chapter in my first book].
- Use B-L to get the pdf you want from the mixing formula.  Equivalently, that pdf is now conditionally a lognormal -- then mixed over the distribution of U(T)! [If your true question was inferring option prices, however, then actually getting this pdf is not necessary, as you already have them from mixing].

The main problem with this answer is that the associated inferred smiles/pdf will be symmetric w.r.t. $x_K \equiv \log K/S_0$ and this may be a dubious result. In any event, the specifics of the underlying/asset class will matter here in assessing how grossly wrong all these many assumptions may be.  A rather subtle hidden assumption, I believe, is that the procedure assumes that the P and Q distributions of U(T) are identical.

Amin
Posts: 2157
Joined: July 14th, 2002, 3:00 am

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

@kiann,
Well, the "risk-neutral black-scholes framework", unfortunately, means different things to different people. If you mean the framework discussed in Black and Scholes' seminal paper, then the pdf associated to that is a log-normal density. True volatility in that framework is a constant parameter and your realized volatility pdf represents the true volatility +  sampling noise (under that framework).
Hi Alan, true. Black-scholes assumes a constant (and log-normal) volatility. However, the industry has worked around this assumption, mainly either by generating the volatility itself as a stochastic variable, or creating a parametric form for the vol such that there is a one-to-one mapping between implied vol versus observed market (European Vanilla Price).
What I am merely trying to do (and feel for), is that if
a) I take the realized volatility pdf as the assumed distribution of the volatility
b) is there a way to map this assumed distribution of the volatility into the underlying variable pdf
Kind regards
Kiann, yes of course you can do that. You will have to know what is the assumption you made about the dynamics of the asset price when you calculated the realized probability density function of the volatility. I do not know what was the underlying evolution SDE you assumed but one possibility could be that you use lognormal SDE (the one we use for Black model) for the underlying and then use filtering or some other method to infer the volatility distribution. In such a case, you can find the underlying pdf by using the formula below

$p(x)=\int_0^{\infty} \frac{1}{x \sigma \sqrt{2 \pi t} } \exp(- {\frac{(\log(x)-\log(x_0)-.5 {\sigma}^2 t)^2}{{2 \sigma}^2 t}}) f(\sigma) d \sigma$

Here $f(\sigma)$ is the distribution of volatility that you have already calculated. You could also have some other SDE for the underlying model when you calculated your volatility and then you would change the density of the underlying in the above equation appropriately. If you let us know about your underlying model SDE assumptions, may be I can help you with a formula more relevant to your case.

kiann
Topic Author
Posts: 18
Joined: April 16th, 2008, 6:39 pm

### Re: Conversion of realized volatilities pdf into probability density function in Black-Scholes

thanks all, for now (before using the mathematical interpretation), I have tried to use a brute force monte-carlo run technique.
Basically, I just do X number of trials, with uniform prob. distribution against the volatilities. Then for each trial (Xi), I generate the normal gaussian * vol(given the random sample).

Below are the graph I obtained (where prob = normal gaussian, and probV = using the vols I have).
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