Breeden–Litzenberger Formula¶

Introduction¶

The Breeden-Litzenberger formula (1978) establishes a fundamental model-free relationship between European option prices and the risk-neutral probability density of the underlying asset. This remarkable result shows that the entire risk-neutral distribution can be recovered from observable option prices without specifying any particular model for asset dynamics.

The Fundamental Result¶

1. Statement of the Theorem¶

Theorem 4.2.1 (Breeden-Litzenberger Formula)
Let $C(K, T)$ denote the price of a European call option with strike $K$ and maturity $T$. The risk-neutral probability density function $q(S_T)$ of the terminal asset price $S_T$ is given by:

\[ q(K) = e^{rT} \frac{\partial^2 C}{\partial K^2}\bigg|_{K} \]

Equivalently, for the cumulative distribution function:

\[ Q(K) = \mathbb{P}^{\mathbb{Q}}(S_T \leq K) = e^{rT} \left(1 + \frac{\partial C}{\partial K}\bigg|_{K}\right) \]

2. Interpretation¶

The second derivative of the call price with respect to strike extracts the discounted risk-neutral density:

\[ \frac{\partial^2 C}{\partial K^2} = e^{-rT} q(K) \]

This is a model-free result: it holds regardless of whether the underlying follows geometric Brownian motion, jump-diffusion, local volatility, or any other process—it relies only on the absence of arbitrage.

Derivation via Risk-Neutral Pricing¶

1. Starting Point: Call Price Formula¶

Under the risk-neutral measure $\mathbb{Q}$, the call price is:

\[ C(K, T) = e^{-rT} \mathbb{E}^{\mathbb{Q}}[\max(S_T - K, 0)] \]

Expanding the expectation:

\[ C(K, T) = e^{-rT} \int_0^\infty \max(S - K, 0) q(S) dS = e^{-rT} \int_K^\infty (S - K) q(S) dS \]

2. First Derivative: Delta with Respect to Strike¶

Differentiating under the integral sign:

\[ \frac{\partial C}{\partial K} = e^{-rT} \frac{\partial}{\partial K} \int_K^\infty (S - K) q(S) dS \]

By Leibniz integral rule:

\[ \frac{\partial C}{\partial K} = e^{-rT} \left[ -\int_K^\infty q(S) dS + (K - K) q(K) \right] = -e^{-rT} \int_K^\infty q(S) dS \]

Recognizing the tail probability:

\[ \frac{\partial C}{\partial K} = -e^{-rT} \mathbb{P}^{\mathbb{Q}}(S_T > K) = -e^{-rT}(1 - Q(K)) \]

Rearranging:

\[ Q(K) = 1 + e^{rT} \frac{\partial C}{\partial K} \]

3. Second Derivative: Extracting the Density¶

Differentiating once more:

\[ \frac{\partial^2 C}{\partial K^2} = -e^{-rT} \frac{d}{dK} \int_K^\infty q(S) dS = -e^{-rT} \cdot (-q(K)) = e^{-rT} q(K) \]

Therefore:

\[ q(K) = e^{rT} \frac{\partial^2 C}{\partial K^2} \]

This completes the derivation. □

Regularity Conditions¶

1. Assumptions for Validity¶

The Breeden-Litzenberger formula requires:

Twice differentiability: $C(K, T)$ must be twice continuously differentiable in $K$
No-arbitrage: Call prices satisfy monotonicity and convexity constraints
Integrability: The risk-neutral density $q(K)$ must integrate to 1

2. Smoothness of Option Prices¶

In reality, option prices are observed on a discrete grid and may contain noise. For the formula to apply:

Proposition 4.2.1 (Regularity of Call Price Surface)
If the underlying asset price has a continuous risk-neutral density $q(S)$, then the call price function $C(K, T)$ is: - $C^0$ continuous everywhere - $C^1$ differentiable everywhere - $C^2$ twice differentiable almost everywhere

Proof sketch. The integral representation:

\[ C(K) = e^{-rT} \int_K^\infty (S - K) q(S) dS \]

is continuous in $K$ by dominated convergence. First derivative exists by fundamental theorem of calculus. Second derivative exists where $q$ is continuous. □

3. Violation of Smoothness¶

In practice, two issues arise:

Atoms in the distribution: If $\mathbb{P}^{\mathbb{Q}}(S_T = K_0) > 0$ for some $K_0$, then $q$ contains a Dirac delta:

$$ q(S) = q_c(S) + p_0 \delta(S - K_0) $$

The call price has a kink at $K_0$ (non-differentiable first derivative).

Dividend jumps: Ex-dividend jumps create discontinuities in $S_T$ distribution.

Connection to Arrow-Debreu Securities¶

1. Digital Options as Building Blocks¶

An Arrow-Debreu security (digital option) pays $1 if $S_T \in [K, K + dK]$ and 0 otherwise. Its price is:

\[ \text{Digital}(K, dK) = e^{-rT} \mathbb{P}^{\mathbb{Q}}(S_T \in [K, K + dK]) = e^{-rT} q(K) dK \]

2. Static Replication via Butterflies¶

A digital option can be replicated using a butterfly spread: - Long 1 call at $K - \Delta K$ - Short 2 calls at $K$ - Long 1 call at $K + \Delta K$

Cost of butterfly:

\[ B(K, \Delta K) = C(K - \Delta K) - 2C(K) + C(K + \Delta K) \]

As $\Delta K \to 0$:

\[ \lim_{\Delta K \to 0} \frac{B(K, \Delta K)}{(\Delta K)^2} = \frac{\partial^2 C}{\partial K^2} = e^{-rT} q(K) \]

Therefore:

\[ \text{Digital}(K, dK) \approx B(K, \Delta K) \quad \text{for small } \Delta K \]

Interpretation: The risk-neutral density can be extracted by observing the prices of butterfly spreads across all strikes.

Discrete Strike Grid: Finite Differences¶

1. Practical Implementation¶

In practice, options trade at discrete strikes $K_1 < K_2 < \cdots < K_n$. The density is approximated using finite differences:

\[ q(K_i) \approx e^{rT} \frac{C(K_{i-1}) - 2C(K_i) + C(K_{i+1})}{(\Delta K)^2} \]

where $\Delta K = K_{i+1} - K_i$ (assuming equal spacing for simplicity).

2. Second-Order Accurate Formula¶

For non-uniform grids, use:

\[ \frac{\partial^2 C}{\partial K^2}\bigg|_{K_i} \approx \frac{2}{K_{i+1} - K_{i-1}} \left( \frac{C(K_{i+1}) - C(K_i)}{K_{i+1} - K_i} - \frac{C(K_i) - C(K_{i-1})}{K_i - K_{i-1}} \right) \]

This is exact for quadratic functions and $O((\Delta K)^2)$ for smooth $C$.

3. Interpolation and Smoothing¶

To obtain density at arbitrary strikes:

Interpolate call prices: Use cubic splines or other $C^2$ interpolants
Differentiate analytically: Compute second derivative of interpolant
Apply Breeden-Litzenberger: Multiply by $e^{rT}$

Caution: Interpolation can introduce spurious oscillations, leading to negative densities (arbitrage). Use monotonicity-preserving and convexity-preserving schemes.

Extension to Put Options¶

1. Put-Call Symmetry¶

By put-call parity:

\[ P(K, T) = C(K, T) - S_0 e^{-qT} + K e^{-rT} \]

Differentiating twice:

\[ \frac{\partial^2 P}{\partial K^2} = \frac{\partial^2 C}{\partial K^2} \]

Therefore, the Breeden-Litzenberger formula applies equally to puts:

\[ q(K) = e^{rT} \frac{\partial^2 P}{\partial K^2} \]

2. Practical Advantage of Puts¶

For low strikes ($K \ll S_0$), put prices have higher liquidity and tighter spreads than calls. Using puts for the left wing and calls for the right wing provides more accurate density estimation.

Consistency Conditions¶

1. Non-Negativity Constraint¶

Since $q(K) \geq 0$ is a probability density, we require:

\[ \frac{\partial^2 C}{\partial K^2} \geq 0 \quad \text{for all } K \]

This is the no-butterfly-arbitrage condition. Violation indicates: - Mispriced options - Bid-ask spread effects - Model-free arbitrage opportunity

2. Normalization: Probability Sums to One¶

The density must integrate to unity:

\[ \int_0^\infty q(S) dS = 1 \]

Equivalently:

\[ e^{rT} \int_0^\infty \frac{\partial^2 C}{\partial K^2} dK = 1 \]

Integrating by parts:

\[ e^{rT} \left[ \frac{\partial C}{\partial K}\bigg|_0^\infty - \int_0^\infty \delta(K - K') \frac{\partial C}{\partial K} dK' \right] \]

Using boundary conditions: - $\frac{\partial C}{\partial K}\big|_{K \to 0} = 0$ (deep ITM call has delta 1) - $\frac{\partial C}{\partial K}\big|_{K \to \infty} = -e^{-rT}$ (OTM call worthless)

This recovers:

\[ e^{rT}(0 - (-e^{-rT})) = 1 \quad \checkmark \]

Applications¶

1. Extracting Risk-Neutral Moments¶

Once the density is known, compute moments:

Mean (first moment):

\[ \mathbb{E}^{\mathbb{Q}}[S_T] = \int_0^\infty S \cdot q(S) dS = S_0 e^{(r - q)T} \]

(This is the forward price under no-arbitrage.)

Variance (second central moment):

\[ \text{Var}^{\mathbb{Q}}(S_T) = \int_0^\infty S^2 q(S) dS - (\mathbb{E}^{\mathbb{Q}}[S_T])^2 \]

Skewness and kurtosis: Higher moments characterize tail behavior.

2. Model-Free Implied Volatility¶

The Breeden-Litzenberger density can be used to define a model-free implied variance:

\[ \sigma_{\text{MF}}^2 = \frac{2e^{rT}}{T} \int_0^\infty \frac{C(K) - \max(S_0 e^{-qT} - K e^{-rT}, 0)}{K^2} dK \]

This integral of option prices across all strikes provides a volatility estimate independent of Black-Scholes.

3. Testing Model Assumptions¶

Compare the empirical density $q_{\text{market}}(K)$ extracted from option prices to model-implied densities: - Black-Scholes: $q_{\text{BS}}$ is lognormal - Local volatility: $q_{\text{LV}}$ from Dupire equation - Stochastic volatility: $q_{\text{SV}}$ from Heston, SABR, etc.

Deviations indicate model misspecification.

Relationship to Characteristic Function¶

1. Fourier Inversion¶

The risk-neutral density can also be expressed via the characteristic function:

\[ q(K) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{-i\omega \ln K} \phi(\omega) d\omega \]

where $\phi(\omega) = \mathbb{E}^{\mathbb{Q}}[e^{i\omega \ln S_T}]$ is the characteristic function.

2. Carr-Madan Formula¶

Carr and Madan (1999) show that call prices can be recovered from the characteristic function:

\[ C(K) = \frac{e^{-rT}}{\pi} \int_0^\infty \text{Re}\left[ \frac{e^{-i\omega \ln K} \phi(\omega - i)}{(\omega^2 + 1)} \right] d\omega \]

Combining with Breeden-Litzenberger provides an alternative path: $\phi \to C \to q$.

Numerical Examples¶

1. Example 1: Black-Scholes Density¶

For $S_T \sim \text{Lognormal}(\mu, \sigma^2 T)$ with $\mu = (r - q - \sigma^2/2)T$:

\[ q_{\text{BS}}(K) = \frac{1}{K \sigma \sqrt{2\pi T}} \exp\left( -\frac{(\ln K - \ln S_0 - \mu)^2}{2\sigma^2 T} \right) \]

Compute $C(K)$ via Black-Scholes formula, then verify:

\[ e^{rT} \frac{\partial^2 C_{\text{BS}}}{\partial K^2} = q_{\text{BS}}(K) \]

Verification:

\[ \frac{\partial C_{\text{BS}}}{\partial K} = -e^{-rT} \Phi(d_2) \]

\[ \frac{\partial^2 C_{\text{BS}}}{\partial K^2} = -e^{-rT} \phi(d_2) \frac{\partial d_2}{\partial K} = e^{-rT} \frac{\phi(d_2)}{K \sigma \sqrt{T}} \]

Since $\phi(d_2) = \frac{1}{\sqrt{2\pi}} e^{-d_2^2/2}$ and $d_2 = \frac{\ln(S_0/K) + (r - q - \sigma^2/2)T}{\sigma\sqrt{T}}$:

\[ e^{rT} \frac{\partial^2 C_{\text{BS}}}{\partial K^2} = \frac{1}{K \sigma \sqrt{2\pi T}} e^{-d_2^2/2} = q_{\text{BS}}(K) \quad \checkmark \]

2. Example 2: Discrete Grid Recovery¶

Given market call prices $\{(K_i, C_i)\}_{i=1}^n$:

Fit smooth interpolant (e.g., cubic spline with no-arbitrage constraints)
Compute $C''(K_i)$ from interpolant
Multiply by $e^{rT}$ to get $q(K_i)$
Verify $\int q(K) dK \approx 1$ using trapezoidal rule

Extensions and Generalizations¶

1. Time-Dependent Interest Rates¶

For stochastic interest rates, the discount factor $e^{-rT}$ is replaced by the price of a zero-coupon bond $B(0, T)$:

\[ q(K) = B(0, T)^{-1} \frac{\partial^2 C}{\partial K^2} \]

2. Dividends and Jumps¶

Discrete dividends create point masses in the density. Continuous dividend yield $q$ modifies the forward price but preserves the Breeden-Litzenberger structure.

3. Multi-Asset Options¶

For basket options, the joint risk-neutral density can be extracted using higher-dimensional derivatives:

\[ q(K_1, K_2) = e^{rT} \frac{\partial^2 C}{\partial K_1 \partial K_2} \]

This requires options on the basket at a two-dimensional grid of strikes.

Limitations and Practical Considerations¶

1. Sparse Strike Grid¶

Real markets have limited strike coverage, especially in the wings. Extrapolation is required, introducing model dependence.

2. Bid-Ask Spreads¶

Using mid-prices can lead to inconsistencies. Best practice: - Use bid prices for sold options (butterfly short legs) - Use ask prices for bought options (butterfly long legs) - Result: Conservative density estimate

3. Negative Probabilities¶

If $\partial^2 C / \partial K^2 < 0$ at some strike, the extracted "density" is negative—indicating arbitrage or measurement error. Solutions: - Regularization (smooth to enforce convexity) - Projection onto arbitrage-free space - Discard suspect data points

4. Tail Behavior¶

Breeden-Litzenberger requires knowledge of $C(K)$ for all $K \in [0, \infty)$. In practice: - Lower tail: $K \to 0$ may have no traded options - Upper tail: $K \to \infty$ requires extrapolation

Standard approach: assume parametric tail (e.g., power law, exponential decay).

Summary¶

The Breeden-Litzenberger formula:

\[ q(K) = e^{rT} \frac{\partial^2 C}{\partial K^2} \]

provides a model-free method to extract the risk-neutral probability density from option prices. Key properties:

No model assumption: Holds for any arbitrage-free market
Direct observable: Second derivative of call prices
Consistency check: Non-negativity equivalent to no-butterfly-arbitrage
Complete information: Full distribution recovered from option surface

Applications include: - Risk-neutral moment calculation - Model validation and comparison - Density forecasting and tail risk assessment - Static replication of exotic payoffs

The formula forms the foundation for model-free results in option pricing, establishing that the option market is the probability distribution under the risk-neutral measure.

Exercises¶

Exercise 1. Starting from the risk-neutral pricing formula $C(K, T) = e^{-rT} \int_K^\infty (S - K) q(S) \, dS$, derive the Breeden-Litzenberger formula $q(K) = e^{rT} \frac{\partial^2 C}{\partial K^2}$ by differentiating twice with respect to $K$. State clearly which differentiation rule you use at each step.

Solution to Exercise 1

Starting from the risk-neutral pricing formula:

\[ C(K, T) = e^{-rT} \int_K^\infty (S - K) q(S) \, dS \]

First derivative. Apply the Leibniz integral rule to differentiate with respect to the lower limit $K$ and the integrand:

\[ \frac{\partial C}{\partial K} = e^{-rT} \left[ \underbrace{-(K - K) q(K)}_{\text{boundary term } = 0} + \int_K^\infty \frac{\partial}{\partial K}(S - K) q(S) \, dS \right] \]

The boundary term vanishes because the integrand $(S - K)$ evaluates to zero at $S = K$. The partial derivative of the integrand gives $-1$, so:

\[ \frac{\partial C}{\partial K} = -e^{-rT} \int_K^\infty q(S) \, dS = -e^{-rT} \mathbb{P}^{\mathbb{Q}}(S_T > K) \]

Second derivative. Differentiate again with respect to $K$ using the fundamental theorem of calculus:

\[ \frac{\partial^2 C}{\partial K^2} = -e^{-rT} \frac{d}{dK} \int_K^\infty q(S) \, dS = -e^{-rT} \cdot (-q(K)) = e^{-rT} q(K) \]

Solving for the density:

\[ q(K) = e^{rT} \frac{\partial^2 C}{\partial K^2} \]

This is the Breeden-Litzenberger formula. The first step used the Leibniz rule for differentiating an integral with a variable limit, and the second step used the fundamental theorem of calculus.

Exercise 2. Three call options with the same maturity $T = 0.5$ and $r = 3\%$ have prices $C(95) = 10.20$, $C(100) = 7.50$, $C(105) = 5.30$. Using the finite-difference approximation $q(K) \approx e^{rT} \frac{C(K - \Delta K) - 2C(K) + C(K + \Delta K)}{(\Delta K)^2}$ with $\Delta K = 5$, estimate the risk-neutral density $q(100)$. Verify that $q(100) > 0$.

Solution to Exercise 2

Given $T = 0.5$, $r = 0.03$, $C(95) = 10.20$, $C(100) = 7.50$, $C(105) = 5.30$, and $\Delta K = 5$, apply the finite-difference approximation:

\[ q(100) \approx e^{rT} \frac{C(95) - 2C(100) + C(105)}{(\Delta K)^2} \]

Compute the numerator of the finite difference:

\[ C(95) - 2C(100) + C(105) = 10.20 - 2(7.50) + 5.30 = 10.20 - 15.00 + 5.30 = 0.50 \]

Compute the denominator:

\[ (\Delta K)^2 = 25 \]

Compute the discount factor:

\[ e^{rT} = e^{0.03 \times 0.5} = e^{0.015} \approx 1.01511 \]

Therefore:

\[ q(100) \approx 1.01511 \times \frac{0.50}{25} = 1.01511 \times 0.02 \approx 0.02030 \]

Since $q(100) \approx 0.0203 > 0$, the non-negativity condition is satisfied, confirming no butterfly arbitrage at this strike.

Exercise 3. The cumulative distribution function is given by $Q(K) = e^{rT}(1 + \frac{\partial C}{\partial K})$. If the call price function is $C(K) = e^{-rT}(F - K + \sigma_0 \sqrt{T} \phi(d))$ for some approximation near ATM, show that $Q(F) \approx 0.5$ (the probability of finishing in the money is near 50% for an ATM option).

Solution to Exercise 3

The CDF is given by $Q(K) = e^{rT}(1 + \frac{\partial C}{\partial K})$. We need to show that $Q(F) \approx 0.5$ for an ATM option (where $K = F$, the forward price).

From the first derivative result in the Breeden-Litzenberger derivation:

\[ \frac{\partial C}{\partial K} = -e^{-rT} \mathbb{P}^{\mathbb{Q}}(S_T > K) \]

At $K = F$ (at-the-money forward):

\[ Q(F) = e^{rT}\left(1 + \frac{\partial C}{\partial K}\bigg|_{K=F}\right) = e^{rT}\left(1 - e^{-rT}\mathbb{P}^{\mathbb{Q}}(S_T > F)\right) = e^{rT} - \mathbb{P}^{\mathbb{Q}}(S_T > F) \]

Alternatively, directly:

\[ Q(F) = \mathbb{P}^{\mathbb{Q}}(S_T \leq F) \]

For a distribution that is approximately symmetric around the forward (in particular, the lognormal distribution with small $\sigma\sqrt{T}$), we have $\mathbb{P}^{\mathbb{Q}}(S_T \leq F) \approx 0.5$. More precisely, in the Black-Scholes model:

\[ Q(F) = \Phi(d_2)\big|_{K=F} = \Phi\left(\frac{\sigma\sqrt{T}}{2}\right) \]

Wait — correcting: $d_2 = \frac{\ln(F/K) + (-\sigma^2/2)T}{\sigma\sqrt{T}}$. At $K = F$, $\ln(F/F) = 0$, so $d_2 = -\frac{\sigma\sqrt{T}}{2}$. Thus:

\[ Q(F) = \Phi\left(-\frac{\sigma_0\sqrt{T}}{2}\right) \]

For small $\sigma_0\sqrt{T}$, $\Phi(-\sigma_0\sqrt{T}/2) \approx 0.5 - \frac{\sigma_0\sqrt{T}}{2\sqrt{2\pi}} \approx 0.5$. The deviation from $0.5$ is of order $\sigma_0\sqrt{T}$, which is small for short maturities or moderate volatility. Hence $Q(F) \approx 0.5$.

Exercise 4. Using the Breeden-Litzenberger formula, explain why the butterfly spread $C(K - \Delta K) - 2C(K) + C(K + \Delta K) \geq 0$ is equivalent to the non-negativity of the risk-neutral density. What economic interpretation does a negative density have, and why is it inconsistent with no-arbitrage?

Solution to Exercise 4

The butterfly spread cost is:

\[ B(K, \Delta K) = C(K - \Delta K) - 2C(K) + C(K + \Delta K) \]

As $\Delta K \to 0$, this approaches:

\[ B(K, \Delta K) \approx \frac{\partial^2 C}{\partial K^2} (\Delta K)^2 \]

By the Breeden-Litzenberger formula, $\frac{\partial^2 C}{\partial K^2} = e^{-rT} q(K)$, so:

\[ B(K, \Delta K) \approx e^{-rT} q(K) (\Delta K)^2 \]

Equivalence: The butterfly spread price is non-negative ($B \geq 0$) if and only if $\frac{\partial^2 C}{\partial K^2} \geq 0$, which is equivalent to $q(K) \geq 0$.

Economic interpretation of a negative density: If $q(K) < 0$ at some strike $K$, then $B(K, \Delta K) < 0$, meaning the butterfly spread has a negative cost. Since the butterfly payoff is always non-negative (it pays $(K - |S_T - K|)^+ / \Delta K \geq 0$ approximately), buying it for a negative price yields a guaranteed non-negative payoff for a positive cash inflow. This is a pure arbitrage: risk-free profit with no initial investment. Such a situation is inconsistent with the no-arbitrage assumption, which requires that any claim with non-negative payoff must have a non-negative price.

Exercise 5. Suppose the risk-neutral density extracted from SPX options is bimodal (two peaks). What does this indicate about the market's view of future price distribution? Provide a market scenario (such as a pending binary event) that would produce a bimodal risk-neutral density.

Solution to Exercise 5

A bimodal risk-neutral density has two distinct peaks, meaning the market assigns elevated probability to two separate price regions while assigning lower probability to intermediate prices.

Interpretation: The market expects the underlying to end up near one of two distinct levels, with relatively low probability of finishing in between. This indicates the market is pricing a binary outcome — an event whose resolution will push the price decisively in one of two directions.

Market scenario: Consider a pharmaceutical company awaiting FDA approval for a key drug:

Approval (positive outcome): The stock jumps to approximately $120 (first peak)
Rejection (negative outcome): The stock drops to approximately $60 (second peak)
Current price: $90

The risk-neutral density would show peaks near $60 and $120 with a trough near $90. Other examples include pending merger announcements (deal closes vs. falls apart), election outcomes affecting regulated industries, and central bank rate decisions with two likely scenarios.

In the implied volatility space, a bimodal density corresponds to an unusually pronounced smile with very high wing volatilities and a relatively low ATM volatility, creating a deep U-shape.

Exercise 6. Derive the relationship $\frac{\partial C}{\partial K} = -e^{-rT} \mathbb{P}^{\mathbb{Q}}(S_T > K)$ from the risk-neutral pricing formula. Use this to show that the call delta with respect to strike gives the (discounted) complement of the CDF. Verify that $\frac{\partial C}{\partial K} \to 0$ as $K \to 0$ and $\frac{\partial C}{\partial K} \to -e^{-rT}$ as $K \to \infty$.

Solution to Exercise 6

Starting from the risk-neutral call pricing formula:

\[ C(K, T) = e^{-rT} \int_K^\infty (S - K) q(S) \, dS \]

Differentiate with respect to $K$ using the Leibniz rule:

\[ \frac{\partial C}{\partial K} = e^{-rT} \left[-(K - K)q(K) + \int_K^\infty (-1) q(S) \, dS\right] = -e^{-rT} \int_K^\infty q(S) \, dS \]

Since $\int_K^\infty q(S) \, dS = \mathbb{P}^{\mathbb{Q}}(S_T > K)$:

\[ \frac{\partial C}{\partial K} = -e^{-rT} \mathbb{P}^{\mathbb{Q}}(S_T > K) \]

This shows the call delta with respect to strike equals the discounted complement of the CDF. Equivalently, $-e^{rT}\frac{\partial C}{\partial K} = 1 - Q(K)$, the survival function.

Boundary verification:

As $K \to 0$: Every positive terminal value exceeds $K = 0$, so $\mathbb{P}^{\mathbb{Q}}(S_T > 0) = 1$ (assuming $S_T > 0$ a.s.). But the call at $K = 0$ is just the forward itself, $C(0) = e^{-rT}F$, and $\frac{\partial C}{\partial K}\big|_{K=0} = -e^{-rT}(1) = -e^{-rT}$.

Correction: Re-examining, we note from the formula $\frac{\partial C}{\partial K} = -e^{-rT}\mathbb{P}^{\mathbb{Q}}(S_T > K)$. As $K \to 0^+$, $\mathbb{P}^{\mathbb{Q}}(S_T > K) \to 1$, so $\frac{\partial C}{\partial K} \to -e^{-rT}$.

As $K \to \infty$: $\mathbb{P}^{\mathbb{Q}}(S_T > K) \to 0$, so $\frac{\partial C}{\partial K} \to 0$.

These match the economic intuition: a deep ITM call ($K \to 0$) loses exactly $e^{-rT}$ per unit increase in strike (it behaves like a short bond minus strike), while a deep OTM call ($K \to \infty$) is insensitive to strike changes.

Exercise 7. The Breeden-Litzenberger formula requires the second derivative $\frac{\partial^2 C}{\partial K^2}$, but market data provides prices at discrete strikes. Compare three numerical methods for estimating this derivative: (a) centered finite differences, (b) fitting a smooth parametric curve (such as SVI) and differentiating analytically, (c) kernel smoothing. Discuss the tradeoff between accuracy and robustness to noise for each method.

Solution to Exercise 7

(a) Centered finite differences:

\[ \frac{\partial^2 C}{\partial K^2}\bigg|_{K_i} \approx \frac{C(K_{i-1}) - 2C(K_i) + C(K_{i+1})}{(\Delta K)^2} \]

Accuracy: Second-order accurate $O((\Delta K)^2)$ for smooth functions. Uses only three adjacent data points.
Robustness: Very sensitive to noise. Numerical differentiation amplifies errors: if option prices have noise of magnitude $\epsilon$, the second derivative estimate has noise of order $\epsilon / (\Delta K)^2$. With $\Delta K = 5$ and $\epsilon = \$0.05$, the noise in the density estimate is $0.05/25 = 0.002$, which can be comparable to the density value itself.
Tradeoff: Simple and unbiased for smooth data, but unreliable with noisy market quotes.

(b) Parametric curve fitting (e.g., SVI):

Fit a smooth parametric model such as the SVI (Stochastic Volatility Inspired) parametrization to the implied volatility data:

\[ \sigma_{\text{IV}}^2(k) = a + b\left[\rho(k - m) + \sqrt{(k - m)^2 + s^2}\right] \]

where $k = \ln(K/F)$. Convert to call prices and differentiate analytically.

Accuracy: Provides smooth analytical derivatives. The accuracy depends on how well the parametric form matches the true smile shape. If the model is correct, accuracy is excellent.
Robustness: Highly robust to noise because the fit averages over many data points. However, model misspecification introduces systematic bias — the extracted density inherits the shape constraints of the parametric family.
Tradeoff: Most robust to noise but introduces model dependence. The density can only take shapes permitted by the parametric family.

(c) Kernel smoothing:

Smooth the call prices using a kernel regression:

\[ \hat{C}(K) = \frac{\sum_i K_h(K - K_i) C(K_i)}{\sum_i K_h(K - K_i)} \]

where $K_h$ is a kernel function with bandwidth $h$, then differentiate the smoothed function.

Accuracy: Non-parametric, so no model bias. Accuracy depends on bandwidth selection: too small preserves noise, too large over-smooths and flattens the density.
Robustness: Intermediate between (a) and (b). More robust than raw finite differences but less constrained than parametric fits. Does not automatically enforce no-arbitrage conditions (may produce negative densities).
Tradeoff: Flexible and fairly robust, but requires careful bandwidth tuning. The bias-variance tradeoff is controlled by bandwidth: larger $h$ reduces variance but increases bias.

Summary: Method (a) is best for dense, clean data. Method (b) is best for sparse, noisy data where the parametric form is trusted. Method (c) offers a middle ground with no parametric assumptions but requires bandwidth selection.