Estimating FX Risk-Neutral Densities from Vanilla Option Data
Summary
The document describes how to estimate a foreign-exchange risk-neutral density from vanilla option data, in the context of comparing the distribution for GBP/EUR around two historical dates. The required inputs include option prices across strikes and maturities, option type, implied volatility, and the relevant risk-free rate. One proposed route is to obtain volatility-surface data, convert quoted deltas to strikes or moneyness, and use implied volatilities with an option-pricing model to derive prices.
The density or cumulative distribution can then be recovered from the way vanilla option prices vary with strike; the response gives a relationship between the derivative of call prices and the risk-neutral cumulative probability, adjusted for discounting. The document also points to spline-based estimation from option prices. Its guidance is brief and identifies one possible data workflow rather than a definitive best method. Correct currency conventions, discounting, quote quality, and interpolation across strikes and maturities matter to the result.
Key ideas
- Risk-neutral distributions can be inferred from vanilla option prices across strikes.
- The required market inputs include strikes, option types, implied volatilities, and a risk-free rate.
- FX volatility-surface quotes can be converted into option prices using an option-pricing model.
- The strike derivative of call prices is related to the risk-neutral cumulative distribution after discounting.
- Spline-based estimation is one possible approach, and the suggested workflow is not presented as uniquely best.
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Full text
# Risk-neutral density from spot prices?
# Risk-neutral density from spot prices?
I am currently working on a university project and I hope someone can help me out with a rather silly question :-) I want to analyse the change in the shape of risk-neutral density functions of spot GBPEUR before Brexit. I've chosen as potential dates May 1, 2015 and May 1, 2016. What data should I gather from Bloomberg? As aware as I am, Bloomberg would give me the volatility skews but not the risk-neutral densities of course. So, by using MATLAB, how can I get the risk-neutral density? What data should I get?
I was thinking of using this MATLAB app: https://uk.mathworks.com/company/newsletters/articles/estimating-risk-neutral-density-from-option-prices-with-a-matlab-app.html , but I would like to receive some advice about the data that I need to use.
Many thanks!
## Answer by DomingoBrown (score 2)
https://quant.stackexchange.com/a/44478
Retrieving the Risk Neutral Density from option prices is nicely developped by Figlewski and Birru here . Very basically what you will need is the spot price of options, their respective strike/type/IV (for the cubic splines), and the risk free rate
## Answer by Antoine Conze (score 1)
https://quant.stackexchange.com/a/43915
Do a quick search on this site to see how the risk neutral cumulative distribution function is related to the derivative of vanilla option prices with respect to strike.
In your case you would have for the EURGBP risk neutral CDF $$ P(\text{EURGBP}_T \leq K) = 1 - \frac{d}{dK}\text{Call}_{\text{EURGBP}}(T, K) / \text{discount}_{\text{GBP}}(T) $$
## Answer by Sanjay (score 1)
https://quant.stackexchange.com/a/44656
As mentioned in the two answers you can use option prices to do so.
What data should I gather from Bloomberg?
- In the search function you can type GBPEUR Crncy.
- Type OVDV and type enter which takes you to the Option volatility surface. Read about it here http://www.fintute.com/2013/06/15/option-volatility-surface-bloomberg-training/#.XI_vfa0kqEI
- Adjust the date and then you have different Implied Volatilities given for different Time to maturity and strike. The strikes are given by Delta which you can compute to Moneyness. See my old question: Calculate strike from Black Scholes delta
- You can compute the option prices by the Black Scholes formula where the volatilities are the implied volatiles in your data.
When computing the options prices then be aware to use the correct interest rate. See: https://www.researchgate.net/publication/275905055_A_Guide_to_FX_Options_Quoting_Conventions
This one way to get option prices from Bloomberg. This is possibly not the only nor best way. That I don't know.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.