Breeden-Litzenberger-formula-for-risk-neutral-densities

The Breeden-Litzenberger formula, proposed by Douglas T. Breeden and Robert H. Litzenberger in 1978, is a method used to extract the implied risk-neutral probability density function from observed option prices The Breeden-Litzenberger formula, proposed by Douglas T. Breeden and Robert H. Litzenberger in 1978, is a method used to extract the implied risk-neutral probability density function from observed option prices.

The formula is based on the principle that the second derivative of the price of a call option with respect to the strike price is equal to the risk-neutral density. In mathematical terms, this is expressed as:

$$\frac{\partial^2 C(K, T)}{\partial K^2} = e^{rT}f_Q(K)$$

where:

• $C(K, T)$ is the price of a call option with strike price $K$ and time to maturity $T$,
• $r$ is the risk-free interest rate,
• $f_Q(K)$ is the risk-neutral density function.

This formula allows us to derive the market's implied probability distribution for the future price of the underlying asset. It's important to note that the formula assumes that the market is complete and arbitrage-free.

Real-Life Application of the Formula

The Breeden-Litzenberger formula can be used in practice in several ways:

1. Risk Management: The risk-neutral density (RND) extracted from option prices can be used to assess market expectations of future price movements, which can be useful for risk management.

2. Option Pricing: The formula can be used to price exotic options or other derivatives that depend on the future distribution of the underlying asset.

3. Arbitrage Opportunities: By comparing the RND with an investor's own forecast of the future price distribution, potential arbitrage opportunities can be identified.

4. Trading Strategies: The formula can show the theoretical relevance of some popular option strategies used in practice.

Here is a practical example of how to implement the Breeden-Litzenberger formula to compute the market-implied risk-neutral densities for the S&P 500 for some quoting dates. The steps are as follows:

• Step 1: Extract the call strikes c_strikes for a given maturity T and the corresponding market prices css.
• Step 2: Compute the implied volatilities via a function like ImpliedVolatilities.m.
• Step 3: Interpolate the implied volatility curve using a command like interp1 in Matlab. The vector xq is the strikes grid and the vector vq is the corresponding interpolated implied volatilities.
• Step 4: Compute the market implied density: f(K) = erT ∂2C(K, T) ∂K2 ≈ erT C(K +ΔK, T) − 2C(K, T) + C(K −ΔK, T) (ΔK)2 where ΔK = 0.2 is the strike grid size.

Please note that this is a simplified example and actual implementation may require additional steps and considerations, such as handling extreme market conditions and ensuring the robustness of the numerical methods.

Other methods for extracting the risk-neutral density:

There are several methods that have been proposed to extract risk-neutral densities from option prices. These methods can be grouped into different categories:

1. Parametric methods: These methods assume a specific functional form for the risk-neutral density. The parameters of the function are then estimated to best fit the observed option prices.

2. Nonparametric methods: These methods do not assume a specific functional form for the risk-neutral density. Examples include kernel regression methods and maximum entropy methods.

3. Implied volatility spline methods: These methods fit a spline to the implied volatility smile, and then derive the risk-neutral density from the fitted spline.

4. Price dynamics-related methods: These methods are based on specific assumptions about the dynamics of the underlying asset price.

5. Implied binomial/trinomial tree methods: These methods construct a binomial or trinomial tree that is consistent with the observed option prices.

Each method has its own strengths and weaknesses, and the choice of method depends on the specific application and available data.