Breeden-Litzenberger Formula

Worked Solution

How to Think About It: The Breeden-Litzenberger formula is one of the most elegant results in derivatives theory -- it says that option markets implicitly contain the full risk-neutral distribution of the underlying. Before writing any math, think about it this way: a call option's value depends on the entire right tail of the distribution above the strike. If you have call prices for every possible strike, you have encoded every tail probability. Taking derivatives just peels off the layers. The first derivative gives you the survival function (tail probability), and the second derivative gives you the density itself. This is not just a theoretical curiosity -- traders use this daily to extract implied distributions from option chains.

Quick Sanity Checks: - The density $f_{\mathbb{Q}}(K)$ must be non-negative everywhere. Since the market is arbitrage-free, call prices are convex in strike ($\partial^2 C / \partial K^2 \geq 0$), so this checks out. - As $K \to \infty$, both $C(K) \to 0$ and $f_{\mathbb{Q}}(K) \to 0$. Consistent. - As $K \to 0$, the call approaches its intrinsic value $S_0 e^{-qT} - K e^{-rT}$ (for a non-dividend-paying stock, $C \to S_0 - K e^{-rT}$), so $\partial C / \partial K \to -e^{-rT}$ and $\partial^2 C / \partial K^2 \to 0$, meaning the density vanishes at zero. That makes sense for a lognormal-like distribution. - The factor $e^{rT}$ is just undoing the discounting baked into call prices.

Derivation:

Start from the risk-neutral pricing formula for a European call:

$C(K) = e^{-rT} \int_K^{\infty} (S_T - K) \, f_{\mathbb{Q}}(S_T) \, dS_T$

Split the integrand:

$C(K) = e^{-rT} \left[ \int_K^{\infty} S_T \, f_{\mathbb{Q}}(S_T) \, dS_T - K \int_K^{\infty} f_{\mathbb{Q}}(S_T) \, dS_T \right]$

Differentiate with respect to $K$. Using Leibniz's rule (the boundary terms from the first integral cancel since $S_T = K$ at the boundary gives $(K - K) f_{\mathbb{Q}}(K) = 0$):

$\frac{\partial C}{\partial K} = e^{-rT} \left[ -\int_K^{\infty} f_{\mathbb{Q}}(S_T) \, dS_T \right] = -e^{-rT} \left[1 - F_{\mathbb{Q}}(K)\right]$

where $F_{\mathbb{Q}}(K) = \int_0^K f_{\mathbb{Q}}(S_T) \, dS_T$ is the risk-neutral CDF. This already tells us something useful: the slope of the call price curve with respect to strike gives us the (discounted) risk-neutral survival probability.

Differentiate again:

$\frac{\partial^2 C}{\partial K^2} = e^{-rT} f_{\mathbb{Q}}(K)$

Rearranging:

$\boxed{f_{\mathbb{Q}}(K) = e^{rT} \frac{\partial^2 C}{\partial K^2}}$

Practical Estimation (Butterfly Spread):

In practice, you do not have a continuous call price function -- you have discrete strikes. Approximate the second derivative with the standard finite-difference formula:

$f_{\mathbb{Q}}(K) \approx e^{rT} \frac{C(K + \Delta K) - 2C(K) + C(K - \Delta K)}{(\Delta K)^2}$

Notice that the numerator $C(K + \Delta K) - 2C(K) + C(K - \Delta K)$ is exactly the cost of a butterfly spread centered at $K$ with wing width $\Delta K$. So the economic interpretation is:

• A butterfly spread is a bet on the stock landing near $K$ at expiry.
• Its price, properly scaled, gives you the risk-neutral probability of $S_T$ being near $K$.
• As $\Delta K \to 0$, this converges to the density.

In practice, you take your option chain, compute butterfly prices at each available strike, normalize by $(\Delta K)^2$, and multiply by $e^{rT}$. This gives you a discrete approximation to the implied risk-neutral distribution. Smoothing (e.g., fitting a spline to implied vols first) helps avoid noisy density estimates from bid-ask spreads.

Practical Interpretation: Traders use extracted risk-neutral densities constantly. Comparing the implied density to a lognormal benchmark tells you where the market is pricing extra probability mass -- fat tails, skewness, or bimodality around events. Before earnings, you often see bimodal implied densities reflecting the market's view of a big move in either direction. Risk managers use the implied density to stress-test portfolios under the market's own probability measure rather than relying on historical distributions.

Answer: The Breeden-Litzenberger formula states $f_{\mathbb{Q}}(K) = e^{rT} \, \partial^2 C / \partial K^2$. The risk-neutral density is recovered from the convexity of call prices in strike. Practically, this is estimated via butterfly spread prices: $f_{\mathbb{Q}}(K) \approx e^{rT} [C(K+\Delta K) - 2C(K) + C(K-\Delta K)] / (\Delta K)^2$. A butterfly spread centered at $K$ is a pure bet on the underlying landing near $K$, making its price a direct readout of the implied probability density at that point.