Extracting Implied Forward and Implied Dividend Curves from the Options Market: Theory and Practice

1. Implied Forward for a Single Strike

Basic Principle: Put-Call Parity

For European options with the same expiration date $ T $ and the same strike price $ K $, the no-arbitrage condition requires:

$$C(K) - P(K) = DF(0,T) \times [F(0,T) - K]$$

Where:

• $ C(K) $ and $ P(K) $ are the prices of the call and put options, respectively.
• $ DF(0,T) $ is the discount factor for maturity $ T $, e.g., $ e^{-rT} $.
• $ F(0,T) $ is the theoretical forward price for maturity $ T $.

Implied Forward Formula

Solving the parity relationship yields the forward price implied by that strike:

$$F_K(0,T) = K + \frac{C(K) - P(K)}{DF(0,T)}$$

Practical Handling Suggestions

Use mid-prices to reduce the impact of bid-ask spreads:

$$C = \frac{C_{\text{bid}} + C_{\text{ask}}}{2}, \quad P = \frac{P_{\text{bid}} + P_{\text{ask}}}{2}$$

Signal Verification: Check if $ |C - P| $ is reasonable and exclude strikes with abnormal quotes.

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2. Synthesizing Forward Price from Multiple Strikes: Weighted Averaging Method

Problem Context

The $ F_K $ values derived from different strike prices exhibit slight variations, primarily due to:

• Bid-ask spreads in option quotes
• Market microstructure noise
• Illiquidity of deep out-of-the-money options

Weighted Averaging Method

For multiple strikes $ K_i $ with the same expiration date:

$$F(0,T) = \frac{\sum_i w_i F_{K_i}}{\sum_i w_i}$$

Weight Design: Based on Spread Uncertainty

Method 1: Simple Sum of Spreads

$$s_i = \text{spread}(C_i) + \text{spread}(P_i)$$

where $ \text{spread}(X) = X_{\text{ask}} - X_{\text{bid}} $.

Method 2: Conservative Sum of Squares (Recommended)

$$s_i = \sqrt{ \text{spread}(C_i)^2 + \text{spread}(P_i)^2 }$$

Weight Allocation: Use inverse-variance weighting.

$$w_i = \frac{1}{s_i^2}$$

Smaller spreads (better liquidity) receive higher weights.

Strike Screening Criteria

1. Liquidity Priority: Use only strikes near at-the-money (typically Delta in the 0.2-0.8 range).
2. Outlier Removal:
   • Strikes with bid prices of 0 or abnormally low.
   • Spreads exceeding a specified threshold (e.g., 20% of the mid-price).
   • Strikes that clearly violate no-arbitrage put-call parity.
3. Data Validation: Check the sensitivity of the weighted $ F(0,T) $ to weight choices.

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3. Constructing the Implied Dividend Curve

Theoretical Basis

The implied forward curve $ F(0,T_j) $ embeds the market's expectations for future dividends. Assuming the following are known:

• Current spot price $ S_0 $
• Discount factors $ DF(0,T) $ for each maturity

Method A: Cumulative Dividend Present Value Curve (Recommended)

Discrete Dividend Model Relationship:

$$F(0,T) = \frac{S_0 - D^{PV}(T)}{DF(0,T)}$$

where $ D^{PV}(T) $ is the present value of all dividends paid before time $ T $.

Inversion Formula:

$$D^{PV}(T) = S_0 - F(0,T) \times DF(0,T)$$

Curve Construction Steps:

1. For each expiration date $ T_j $, calculate $ D^{PV}(T_j) $.
2. Check monotonicity: $ D^{PV}(T) $ should be non-decreasing with $ T $.
3. Use monotonic interpolation methods (e.g., piecewise linear or monotonic splines) to build the complete curve.

Advantage: Directly corresponds to actual dividend payments, facilitating alignment with seasonal dividend patterns.

Method B: Equivalent Continuous Dividend Yield

Continuous Model Relationship:

$$F(0,T) = S_0 e^{(r - q)T}$$

where $ q $ is the continuous dividend yield.

Calculation Formula:

$$q^{\text{avg}}(0,T) = r^{\text{avg}}(0,T) - \frac{1}{T} \ln\left( \frac{F(0,T)}{S_0} \right)$$

where $ r^{\text{avg}}(0,T) $ is the average continuous interest rate for maturity $ T $.

Application Scenario: Suitable for pricing models requiring a continuous dividend yield (e.g., Black-Scholes).

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4. Practical Operation Essentials

Data Preparation

1. Option Data: Ensure synchronized call/put quotes are used.
2. Interest Rate Curve: Construct an accurate zero-coupon curve or forward rate curve.
3. Spot Price: Use the same underlying definition as at option expiration.

Quality Control

1. Liquidity Filtering: Based on trading volume, open interest, and bid-ask spreads.
2. Arbitrage Checks: Verify that put-call parity is not significantly violated.
3. Curve Smoothing: Perform reasonableness checks on the generated forward and dividend curves.

Seasonal Handling

For the dividend curve:

• Identify known ex-dividend date patterns.
• Compare implied dividends with the company's announced dividend schedule.
• Consider differences between special and regular dividends.

Sources of Error

1. Bid-Ask Spreads: The largest source of noise.
2. Borrowing Costs: Stock loan rates not captured in standard models.
3. Early Exercise: Impact of American-style options (significant for high-dividend stocks).
4. Taxes and Transaction Costs: Frictions present in actual arbitrage.