How to derive the implied probability distribution from B-S volatilities?

• The general problem I have is visualization of the implied distribution of returns of a currency pair.

I usually use QQplots for historical returns, so for example versus the normal distribution:

Now I would like to see the same QQplot, but for implied returns given a set of implied BS volatilities, for example here are the surfaces:

USDZAR  1month  3month  6month  12month 2year
10dPut  15.82   14.59   14.51   14.50   15.25
25dPut  16.36   15.33   15.27   15.17   15.66
ATMoney 17.78   17.01   16.94   16.85   17.36
25dCall 20.34   20.06   20.24   20.38   20.88
10dCall 22.53   22.65   23.39   24.23   24.84

EURPLN  1month  3month  6month  12month 2year
10dPut  9.10    9.06    9.10    9.43    9.53
25dPut  9.74    9.54    9.51    9.68    9.77
ATMoney 10.89   10.75   10.78   10.92   11.09
25dCall 12.83   12.92   13.22   13.55   13.68
10dCall 14.44   15.08   15.57   16.16   16.34

EURUSD  1month  3month  6month  12month 2year
10dPut  19.13   19.43   19.61   19.59   18.90
25dPut  16.82   16.71   16.67   16.49   15.90
AtMoney 14.77   14.52   14.44   14.22   13.87
25dCall 13.56   13.30   13.23   13.04   12.85
10dCall 12.85   12.85   12.90   12.89   12.78


Anybody know how I could go about doing this? Any R packages or hints where to start on this? It doesn't necessarily have to be a qqplot, it could just be a plot of the density function; that would help me too. Thanks.

10 years ago

You can directly imply a probability distribution from a volatility skew.

Note that, for any terminal probability distribution $p(S)$ at tenor $T$, we have the model-free formula for the call price $C(K)$ as a function of strike $K$

$$C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS$$

Therefore we can write

$$e^{rT} \frac{\partial C}{\partial K}=\int_K^\infty (-1) \cdot p(S) dS$$ and by the fundamental theorem of calculus $$e^{rT} \frac{\partial^2 C}{\partial K^2} = p(K)$$

Therefore, all you need, in order to find the value of $p(x)$ for any $x$, is the second derivative of call prices at strike $x$.

Usually, one uses a fitted skew (such as a polynomial fit) to the available volatility values at the given tenor. In your case, with just 5 points, I would recommend fitting a parabola in log strike space. Once you have a continuous skew $\sigma(K)$ then you just need to find

$${\left. \frac{\partial^2 }{\partial x^2}\right|} BS_{\text{Call}}(S_0, x, \sigma(x), r, T, q)$$

evaluated at $x=K$ which can be done either with a bunch of symbol-jiggling, or by simply finite differencing. In your case I recommend the latter.

Once you have probability distribution values, of course, the process of generating the qq plots is one you have already mastered.

Edit: Sign error correction, per @Robino

Would this actually work? Once again, I'm not an expert in options pricing here, but aren't you essentially assuming the answer when you assume the skew is polynomial in log-strike space? Wouldn't you potentially get widely different answers (especially with so few data points) for different assumptions on the order of the polynomial?

It does work, for the conforming definitions of the word "work". The plots you get will certainly depend on the skew curve you fit, which (as you clearly realize) need not necessarily be a parabola in log(S) space. Cubic splines are another common choice. Try both and compare! If you want to be completely model-free, you can form finite-difference approximations to the second derivative using the 5 vol points you have. This of course gives you only 5 sample points on your distribution, and that's assuming you use 1-sided approximations on the endpoints.

The first derivative also gives the CDF at 5 points, right? what are the advantages of using second derivative?

Why is the first formula model-free? If the model leads to an incomplete market, which is often the case, then there may not be a single price for a call option - and the formula just gives one of the prices (for a particular choice of a martingale measure $p$).

In derivatives pricing the term "model-free" is shorthand for "does not arise from any particular SDE".

You have a mistake. The second equation should be negative of what it is. So $e^{rT} \frac{\partial C}{\partial K}=- \int_K^\infty 1 \cdot p(S) dS$. This is because the call price should DECREASE as the strike increases. Probably dropped a minus sign in the integration by parts...

@BrianB No problem. Don't forget to upvote any comments which you find helpful. ;)

May I ask what does 'symbol-jiggling' mean?