# Using Box Spread to Determine Implied Dividend and Risk-free Rate Simultaneously

Dividends are a fundamental component of many investment strategies and a crucial aspect of equity ownership. They represent a portion of a company’s profits that is distributed to its shareholders, typically on a regular basis, such as quarterly or annually. Dividends are an input in the pricing model for equity options. When valuing options, the timing and magnitude of expected dividend payments are essential considerations. These payments can affect the option’s value by altering the future cash flows associated with owning the underlying stock.

Dividends can be estimated using historical dividend payouts. However, for stocks with traded options, dividends can be calculated or implied from traded option prices. We can utilize the put-call parity to derive the implied dividend. However, using this approach requires prior knowledge of the risk-free interest rate.

Reference [1] introduced a method for simultaneously determining the implied dividend and risk-free interest rate based on the no-arbitrage principle. This approach relies on utilizing the box spread to establish an equation for calculating both the implied dividend and the risk-free rate. The authors pointed out,

In this article, a simple no-arbitrage methodology allowing for the simultaneous estimation of option-implied interest rates and dividend yields via a robust estimation method has been proposed. The estimators for the slope and intercept are derived from first principles via the no-arbitrage relationship, clearly demonstrating the link between the sloped asset position with the intercept and the box spread with the slope. Using market options data, robust forward-looking model-free estimates of the interest rate and dividend yield are obtained, with the robustness of the repeated median estimator being more pronounced even for heteroscedastic data.

This approach is useful for determining the implied dividend yield, offering valuable assistance to dividend investors in the effective management of their dividend-paying stock portfolios. We observe, however, that the derivation is applicable to European options only. It would be interesting to extend the formula to American options.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Muoria Kamau & Ivivi J. Mwaniki, On a robust estimation of option-implied interest rates and dividend yields, Cogent Economics & Finance, 11:2, 2260658, DOI: 10.1080/23322039.2023.2260658

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