Extracting Risk-Neutral Density From a Volatility Smile
You are trading options on a commodity forward $F$ under Black's model. The market quotes implied volatilities $\sigma(K)$ across strikes, forming a volatility smile.
- Starting from the smile $\sigma(K)$, explain how to obtain call prices $C(K)$ as a function of strike. Then show how to extract the risk-neutral probability density $f(x)$ of the terminal forward $F_T$ from $C(K)$.
- State the conditions on $\sigma(K)$ (or equivalently on the total implied variance $w(K) = \sigma(K)^2 T$) that guarantee no butterfly arbitrage exists.
- Describe one practical numerical approach to fit a smooth volatility smile to market quotes while enforcing that $C(K)$ remains convex in $K$.
Hints
- Think about what a butterfly spread pays off and how its price relates to the probability of $F_T$ landing near a given strike.
- The Breeden-Litzenberger result connects $\partial^2 C / \partial K^2$ to the risk-neutral density -- convexity of $C(K)$ is equivalent to a non-negative density.
- For the no-arbitrage conditions, work in terms of total implied variance $w(k) = \sigma^2 T$ as a function of log-moneyness $k = \ln(K/F)$ and look up the Gatheral-Jacquier density condition.
Worked Solution
How to Think About It: This is one of the foundational results in derivatives pricing -- the connection between option prices across strikes and the market's implied probability distribution. The key economic insight is simple: a butterfly spread (long calls at $K - \delta$, short two at $K$, long at $K + \delta$) pays off only if $F_T$ lands near $K$. Its price is therefore proportional to the probability density at $K$. Making that precise leads to the Breeden-Litzenberger result: the risk-neutral density is the second derivative of call prices with respect to strike. Everything about no-arbitrage smile conditions follows from requiring that density to be non-negative.
Quick Sanity Checks: - Call prices $C(K)$ must be decreasing in $K$ (otherwise sell the low-strike call, buy the high-strike -- free money). - $C(K)$ must be convex in $K$ (otherwise a butterfly spread has negative price, i.e., arbitrage). - The second derivative $\partial^2 C / \partial K^2$ must be non-negative everywhere, since it is proportional to a probability density.
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Part (a): From Smile to Density
Given the implied volatility smile $\sigma(K)$, you obtain call prices by plugging each $(K, \sigma(K))$ into the Black formula:
$C(K) = e^{-rT}\bigl[F\,\Phi(d_1) - K\,\Phi(d_2)\bigr]$
where $d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma(K)^2 T}{\sigma(K)\sqrt{T}}$ and $d_2 = d_1 - \sigma(K)\sqrt{T}$.
Note that $\sigma$ depends on $K$, so $C(K)$ is not the standard Black formula with a flat vol -- each strike gets its own implied volatility.
The Breeden-Litzenberger result then gives the risk-neutral density:
$f(x) = e^{rT}\frac{\partial^2 C}{\partial K^2}\bigg|_{K=x}$
The intuition: consider a butterfly centered at $K$ with width $\delta$. Its payoff is a tent function peaking at $K$, and its price is $C(K-\delta) - 2C(K) + C(K+\delta)$. Dividing by $\delta^2$ and sending $\delta \to 0$ gives $\partial^2 C / \partial K^2$, which is the discounted density. This is why convexity of $C(K)$ in $K$ is equivalent to non-negative density -- i.e., no butterfly arbitrage.
In practice, you rarely differentiate $C(K)$ directly (noisy). Instead, you can express $\partial^2 C / \partial K^2$ in terms of $\sigma(K)$ and its derivatives using the chain rule through Black's formula.
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Part (b): No-Butterfly-Arbitrage Conditions
Define total implied variance $w(K) = \sigma(K)^2 T$. In the log-strike variable $k = \ln(K/F)$, the no-butterfly-arbitrage conditions are:
- Non-negative density: $\frac{\partial^2 C}{\partial K^2} \geq 0$ for all $K > 0$. Equivalently, $C(K)$ is convex in $K$.
- In terms of total variance $w(k)$, the Gatheral-Jacquier condition states that the smile is free of butterfly arbitrage if and only if:
$g(k) = \left(1 - \frac{k\,w'(k)}{2w(k)}\right)^2 - \frac{w'(k)^2}{4}\left(\frac{1}{w(k)} + \frac{1}{4}\right) + \frac{w''(k)}{2} \geq 0$
for all $k$, where primes denote derivatives with respect to $k$.
- Slope bounds: $|w'(k)|$ must not be too large. Specifically, the effective local volatility (which involves - k w'/(2w)$) must remain positive.|k|$ as $|k| \to \infty$ (the Lee moment formula bound).
- Boundary behavior: As $K \to 0^+$, we need $C(K) \to e^{-rT}F$ (the forward value), and as $K \to \infty$, $C(K) \to 0$. In terms of total variance, $w(k)$ must not grow faster than
A practical sufficient condition that is often checked: the total variance slice $w(k)$ should be smooth, increasing in the wings at a sub-linear rate in $|k|$, and satisfy $g(k) \geq 0$ everywhere.
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Part (c): Numerical Fitting With Convexity Enforcement
One standard approach is SVI (Stochastic Volatility Inspired) parameterization with constrained optimization:
1. Parameterize the total variance as a function of log-moneyness $k$: $w(k) = a + b\bigl[\rho(k - m) + \sqrt{(k - m)^2 + s^2}\bigr]$ where $(a, b, \rho, m, s)$ are five parameters. This form is flexible enough to capture typical smile shapes.
- Fit the parameters to market quotes by minimizing the sum of squared implied vol errors (or price errors) across observed strikes.
- Enforce convexity by imposing the constraint $g(k) \geq 0$ at a dense grid of strikes during the optimization. This is typically done via a constrained nonlinear solver (e.g., SQP or interior-point method). Alternatively, check the constraint after fitting and adjust parameters if violated.
- Verify that the resulting $C(K)$ is decreasing and convex by computing the density $f(x)$ and checking it is non-negative everywhere.
An alternative approach uses natural cubic splines on the total variance $w(k)$ with knots at observed strikes. Convexity of $C(K)$ is enforced by adding linear inequality constraints on the spline coefficients at a set of grid points. The fit is then a quadratic program (QP), which is fast and globally solvable. The downside is that spline extrapolation in the wings requires care -- often the SVI form is used for wing behavior and the spline for the body.
Answer:
- (a) Plug the smile $\sigma(K)$ into Black's formula to get $C(K)$. The risk-neutral density is $f(x) = e^{rT}\,\partial^2 C / \partial K^2|_{K=x}$ (Breeden-Litzenberger).
- (b) No butterfly arbitrage requires $C(K)$ to be convex in $K$, which translates to $g(k) \geq 0$ in the Gatheral-Jacquier condition on total variance, plus appropriate slope and boundary constraints.
- (c) Fit an SVI parameterization (or constrained spline) to market quotes, with the $g(k) \geq 0$ constraint enforced during optimization to guarantee convexity of call prices.
Intuition
The deep idea here is that a complete set of European option prices across all strikes encodes the entire risk-neutral distribution of the underlying. This is not a theoretical curiosity -- it is the foundation of volatility surface construction on every derivatives desk. When you interpolate a vol smile, you are implicitly choosing a probability distribution, and if that distribution has negative mass anywhere (i.e., $C(K)$ is locally concave), you have created a butterfly arbitrage that any competent trader would exploit.
In practice, this is why smile fitting is so much harder than it looks. You cannot just spline through market quotes and call it a day. The fitted smile must produce a valid density everywhere, including in the wings where you have no data. The SVI parameterization became popular precisely because it has the right asymptotic behavior (linear in log-moneyness, matching the Lee moment formula) and is flexible enough for most equity and commodity smiles. The constraint $g(k) \geq 0$ is the single most important check in any smile construction pipeline -- if it fails, your local volatility surface will blow up and your exotic pricing will be garbage.