since a long time I'm struggling with a particular question regarding the parametric estimation of the risk-neutral density (or implied probability) from option prices.

I want to pursue the parametric approach of minimizing the squared deviations from theoretical and observed prices as it is described e.g. in Bahra (1997, p. 22 and the following) (http://www.bankofengland.co.uk/archive/Documents/historicpubs/workingpapers/1997/wp66.pdf).

As we know and as stated in equation (10) and (11) of Bahra (1997), the theoretical call and put prices are given by the discounted expected pay-offs, i.e.

\begin{equation} c(X, \tau) = e^{-r\tau}\int_{S_T = K}^{\infty} q(S_{T})(S_{T} - X)dS_{T} \\ p(X, \tau) = e^{-r\tau}\int_{S_T = K}^{\infty} q(S_{T})(X - S_{T} )dS_{T}, \end{equation}

with $X$ as the strike price, $\tau$ the time to maturity, $r$ the risk-free interest rate, $S_{T}$ the underlying asset price at maturity and $q(S_{T})$ the risk-neutral density with some specific functional form (log-normal, mixture of log-normals etc.).

The parameters $\theta$ of the risk-neutral density are obviously unknown, and that's why we minimize the squared distance between the theoretical and the real-world call/put prices, $\hat{c}_{i},\hat{p}_{i}$ w.r.t. $\theta$, i.e.

\begin{equation} min_{\theta} ( \sum_{i=1}^{n} (c(X, \tau) - \hat{c}_{i})^2 + \sum_{i}^{m}(p(X, \tau) - \hat{p}_{i})^2 ) \end{equation}

This is a simplified version of eq. (17) in Bahra.

Say we assume $q(S_{T})$ to be lognormal with expectation $\alpha$ and variance $\beta$. Personally, I would now use some sort of optimizer to obtain these parameters. However, in the literature of estimating the risk-neutral density, one usually further assumes that $\alpha = ln S_{t} + (\mu - 0.5\sigma^2)\tau$ and $\beta = \sigma\sqrt\tau$, where $\mu, \sigma$ are parameters of the Black-Scholes model.

My question now is whether we actually need to make this assumption or if we can simply optimize over two parameters that are free of any assumption?

I have this concern, because in my specific case, my underlying of the option is not an asset, but interest rates, i.e. an index (this type of options is called caps and floors). Otherwise, the approach is the same, but I'm wondering whether it would make sense to assume $\alpha = ln S_{t} + (\mu - 0.5\sigma^2)\tau$ and $\beta = \sigma\sqrt\tau$ (for instance, interest rates, i.e. $S_{t}$, could be negative, which would cause a problem)

I would appreciate any remarks.

  • 2
    $\begingroup$ Quick side comment. Your option price formulae are incorrect as the call payoff should be $(S_T-K)^+$, i.e. the max function and likewise for the put. $\endgroup$
    – Dom
    Mar 8, 2017 at 10:14
  • $\begingroup$ Also, why not solve for the implied volatility for each call and put option you see and then fit that to some simple polynomial to capture the smile or skew. Then use the continuum of option volatilities as a function of strike with the Breeden-Litzenberger formula to calculate the market-implied risk neutral probability density function. In that way you do not need to make any distributional assumptions. Your main assumption is the shape of the polynomial used to fit the volatility smile/skew. $\endgroup$
    – Dom
    Mar 8, 2017 at 10:19
  • 1
    $\begingroup$ @Dom this is indeed an alternative approach adopted by many practitioners. But it necessitates detecting and correcting smile parameters leading to (static) arbitrage opportunities, which you do not have to do when directly parameterising the density. $\endgroup$
    – Quantuple
    Mar 8, 2017 at 10:30
  • $\begingroup$ True. It can lead to negative densities if there are second order discontinuities in the option price as a function of strike. However it is a simple first approach that could even be used to guide a parametrisation of the density. $\endgroup$
    – Dom
    Mar 9, 2017 at 11:07

1 Answer 1

  1. If you assume that, under $\Bbb{Q}$, you have the following parameteric distribution for asset prices $$ S_T \sim logN(\alpha,\beta) $$ you can indeed directly optimise for $\alpha$ and $\beta$. However, since your objective function involves call/put prices and not the density itself, you should express these prices as functions of $\alpha$ and $\beta$(*). Given the lognormal assumption, you fall-back onto the famous Black-Scholes modelling framework, hence the established link between $(\alpha,\beta)$ and $(\mu=r-q, \sigma)$.

  2. The lognormal assumption does not make sense if you are to model negative interest rates indeed. You can postulate any distribution whose support is $\Bbb{R}$ instead of $\Bbb{R}^+$ e.g. $$ S_T \sim N(\alpha, \beta) $$ and optimise for $\alpha$,$\beta$. Again, the only thing left to be done is to find the expression call/put prices under the postulated modelling assumptions (here normal distribution, hence Bachelier model).

(*) Actually, you could simply express the prices using the integral forms you mention in your original post, but this is less computationally efficient (requires a numerical integration) than if you use a closed-form formula, obviously.

  • $\begingroup$ Thanks for the answer, it clarified a lot. I assume that with "closed-form formula" you mean the Black-Scholes equation, right? But how can I integrate in there the assumption of a log-mixture? By numerical integration I could simply replace $q(ST)$ by a mixture, but how do I put that in the B-S formula? $\endgroup$
    – Walter
    Mar 9, 2017 at 13:50
  • $\begingroup$ You cannot simply "put that in the BS formula". The BS formula gives the price of a European option assuming $q(S_T)$ is lognormal. Similarly the Bachelier formula gives the price of a European option assuming $q(S_T)$ is normal. If you want $q(S_T)$ to be a log-mixture, you need to derive the formula yourself. $\endgroup$
    – Quantuple
    Mar 9, 2017 at 14:31

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