Recovering an FX Risk-Neutral Density from Option Prices

# How would a FX price probability distibution function look?


# How would a FX price probability distibution function look?












I would like to see how the currency price levels are distributed in a probability function. But I don't even know if there is such a thing or if perhaps its just common knowledge and readily available.

Possibly related questions:

- How do you synthesize a probability density function (pdf) from equally weighted price data?

Q: How can I plot the price probability function for a currency pair, for a certain period, $x$?

(Are there any readily available tools for doing this?)

UPDATE:

As a clarification to some of the comments, I would like to be more specific, and mention that I have nearly 4 months of EUR/USD M1 OHLCV data (5x ~112K points) that I would like to turn into a PDF. The answers provided so far has not gotten me any closer to this task.

## Answer by roz (score 3, accepted)

https://quant.stackexchange.com/a/50450

You can extract the risk neutral density implied by option prices and have a look at that. The implied probabilities are given by the prices of butterfly spreads in the market. This is common knowledge. Page 241 of this book explains how you could go about doing it in Excel: https://gaussiandotblog.files.wordpress.com/2018/02/wiley-trading-giles-peter-jewitt-fx-derivatives-trader-school-2015-wiley.pdf

## Answer by user39119 (score 4)

https://quant.stackexchange.com/a/50451

Assume we knew the density function $f$ of the FX price that we observe in the market. Then the market price of a call option $C(K)$ with strike $K$ would be \begin{align*} C(K)&=e^{-rT} \int_0^{\infty}(s-K)^+ f(s)ds \\ &=e^{-rT} \left( \int_K^{\infty} s f(s)ds - K \int_K^{\infty}f(s)ds \right) \tag*{(1)}. \end{align*}

$C(K)$ is a market price and we want to recover $f$ from Eq(1). So we just have to differentiate twice with respect to K. We differentiate once to get \begin{align*} \frac{d}{dK}C(K)e^{rT} &= -K f(K) - \left( \int_K^{\infty} f(s)ds - K f(K) \right) \\ &= - \int_K^{\infty} f(s)ds \end{align*} Differentiating again $$\frac{d^2}{dK^2}C(K)e^{rT} = f(K). \tag*{(2)}$$ This shows that the "real" density can be obtained from the call prices. As an easy approximation to Eq(2), we can use the following $$f(K) \cong \left[ \frac{ C(K+h)+C(K-h)-2C(K)}{h^2} \right]e^{rT}. \tag*{(3)}$$ How do we use this formula? We can collect a set of market prices of calls, for all the strikes corresponding to some maturity, and interpolate the volatilities. Suppose we have 6 market prices. Then we get the implied volatilities from the options prices and interpolate the vols to have a " more dense" set of vols. For example if we originally have market implied vols 85%, 90%, 95%, 100% 105% 110%, we interpolate to obtain 70%, 71%, 72%, ... , 140%. Then we use Black-Scholes formula to obtain $C(K)$ and plug it into Eq(3) in order to get the desired implied density $f$. This can be done in Excel quite easily.

Of course there are many ways to interpolate volatilities and also to extrapolate them in both ends.