Estimating Risk-Neutral Density from Unequal-Strike Options
Summary
The document explains how to approximate the curvature of call prices around an at-the-money strike when option strikes are unevenly spaced. Since the risk-neutral density is related to the second derivative of call price with respect to strike, the method uses a generalized finite-difference formula derived from Taylor expansions at three points with unequal distances from the center.
For an iron butterfly with an out-of-the-money put, an at-the-money put and call, and an out-of-the-money call, put-call parity is used to express the put price as an in-the-money call price. The resulting in-the-money, at-the-money, and out-of-the-money call prices and strikes provide the inputs to the unequal-step approximation. The source gives the derivation and notes that equal spacing reduces to the familiar centered second-difference formula. It does not provide empirical validation or discuss market frictions, quote noise, or the sensitivity of density estimates to strike selection; the approximation is local and depends on reliable option prices.
Key ideas
- The risk-neutral density can be estimated from the second derivative of call price with respect to strike.
- Unequal strike spacing calls for a generalized finite-difference approximation.
- Put-call parity converts the out-of-the-money put into an equivalent in-the-money call input.
- With symmetric strike spacing, the generalized formula becomes the standard centered second difference.
- The estimate depends on local option prices and offers no treatment of quote noise or empirical accuracy.
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# Implied Risk Neutral Probability Density from Iron Butterfly
# Implied Risk Neutral Probability Density from Iron Butterfly
It's known that we can get the implied RND from a butterfly as (DF is discount factor)
```
P(S(T) = K) = DF * [C(K + X)+ c(K - X) - 2C(K)]/X^2
```
But how to get the same density from 4 strikes : `K(ATM PUT) , K(OTM PUT) , K(ATM CALL) , K(OTM CALL)`(i.e Iron Butterfly). The 4 strikes are chosen such that the net delta of the structure is zero.
## Answer by Kermittfrog (score 3, accepted)
https://quant.stackexchange.com/a/77709
### Background
If I understand you correctly, you want to know how to approximate a second derivative to a function i.e. $\frac{\partial^2f}{\partial x^2}$, using a generalized finite difference approximation with unequal step sizes.
This task is usually approached by solving a linear equation system based on Taylor series expansions of the function $f$ around some point $x_0$, here up to the second order:
$$ \begin{align} f(x_0+a)&\approx f(x_0)+af'(x_0)+\frac{1}{2}a^2f''(x_0)\\ f(x_0)&=f(x_0)\\ f(x_0-b)&\approx f(x_0)-bf'(x_0)+\frac{1}{2}b^2f''(x_0)\\ \\ \Rightarrow \begin{pmatrix}f(x_0+a)\\f(x_0)\\f(x_0-b)\end{pmatrix}&=\begin{pmatrix}1&a&\frac{1}{2}a^2\\ 1&0&0\\ 1&-b&\frac{1}{2}b^2\end{pmatrix}\begin{pmatrix}f(x_0)\\f'(x_0)\\f''(x_0)\end{pmatrix} \end{align} $$ This 3x3 linear equation system is solved via (I used some online matrix inversion tool):
$$ \begin{align} \begin{pmatrix}f(x_0)\\f'(x_0)\\f''(x_0)\end{pmatrix}&=\begin{pmatrix}1&a&\frac{1}{2}a^2\\ 1&0&0\\ 1&-b&\frac{1}{2}b^2\end{pmatrix}^{-1}\begin{pmatrix}f(x_0+a)\\f(x_0)\\f(x_0-b)\end{pmatrix}\\ &=\begin{pmatrix}0&1&0\\ \frac{b}{a(a+b)}&\frac{1}{b}-\frac{1}{a}&-\frac{a}{b(a+b)}\\ \frac{2}{a(a+b)}&-\frac{2}{ab}&\frac{2}{b(a+b)} \end{pmatrix}\begin{pmatrix}f(x_0+a)\\f(x_0)\\f(x_0-b)\end{pmatrix} \end{align} $$
And we find the generalized approximation as:
$$ f''(x_0)\approx \frac{2}{a(a+b)}f(x_0+a)-\frac{2}{ab}f(x_0)+\frac{2}{b(a+b)}f(x_0-b) $$
Note that if $a=b=h$, the usual approximation formula results, i.e.
$$ f''(x_0)=\frac{1}{h^2}\left(f(x_0+a)-2f(x_0)+f(x_0-b)\right) $$
### Application
The first step would be to transform the OTM PUT option price into an ITM CALL option price using put call parity. Then we'd have all required ingredients for our equation system, i.e. $K_{ITM}, C(K_{ITM})$, $K_{ATM}, C(K_{ATM})$ and $K_{OTM}, C(K_{OTM})$.
$$ \begin{align} C(K_{OTM})&\approx C(K_{ATM})+C'(K_{ATM})(K_{OTM}-K_{ATM})+\frac{1}{2}C''(K_{ATM})(K_{OTM}-K_{ATM})^2\\ C(K_{ATM})&= C(K_{ATM})\\ C(K_{ITM})&\approx C(K_{ATM})-C'(K_{ATM})(K_{ITM}-K_{ATM})+\frac{1}{2}C''(K_{ATM})(K_{ITM}-K_{ATM})^2 \end{align} $$
and we can now reuse the solution from above with $a=K_{OTM}-K_{ATM}$ and $b=K_{ITM}-K_{ATM}$.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.