Black-Scholes-Merton Model with Continuous Dividends

Introduction

Welcome to our next lesson. In our previous session, we mastered the application of the Black-Scholes-Merton (BSM) formula for pricing European options on non-dividend-paying stocks. We established a systematic process using the five key inputs and saw how put-call parity offers a valuable shortcut.

Today, we will build directly on that foundation by relaxing one of the key assumptions: that the underlying stock pays no dividends. We will learn how to adjust the BSM formula to correctly price options on stocks that pay a continuous dividend yield. This is a crucial extension, as many real-world assets provide some form of regular income. By the end of this lesson, you will be able to confidently apply the modified BSM formula to solve exam-style problems involving dividend-paying stocks.

The Impact of Dividends on Option Prices

The first question to ask is: why do dividends affect option prices? The answer lies in who receives them. The owner of a share of stock is entitled to any dividends paid. The owner of a call option, however, is not. A call option is the right to buy the stock in the future, not an ownership of the stock today.

Therefore, from the perspective of a call option holder, every dividend payment is a cash flow they miss out on. This cash outflow reduces the value of the stock, which in turn makes the call option on that stock less valuable. Conversely, for a put option holder, a dividend payment is beneficial. The stock price is expected to drop by the dividend amount, which increases the probability of the put finishing in-the-money.

This brief video segment reinforces the core idea.

How to interpret N(d1) and N(d2) in Black Scholes Merton (FRM T4-12)

The video "How to interpret N(d1) and N(d2) in Black Scholes Merton" by Bionic Turtle provides a concise explanation of this effect.

Please watch the segment from this point, where the speaker explains that a dividend has the effect of reducing the current stock price because the option holder forgoes these payments.

Adjusting the Black-Scholes Formula

To account for this, the original BSM model, developed by Fischer Black and Myron Scholes, was extended by Robert Merton to incorporate dividends. The adjustment is elegant and intuitive. We assume the stock pays a continuous dividend at a known, constant yield, denoted by $δ$ .

The most straightforward way to incorporate this is to replace the current stock price, $S_{0}$ , in the formula with a dividend-adjusted price. We use the present value of the stock price after accounting for the continuous "leakage" of the dividend yield over the option's life.

The adjusted stock price to be used in the BSM formula is $S_{0} e^{- δ T}$ .

Essentially, we are pricing an option on an asset worth $S_{0} e^{- δ T}$ today, which pays no dividends. This simple replacement allows us to use the structure of the original B-S model.

The following resource provides a clear example of this method.

[PDF] Chapter 12 - The Black-Scholes Formula - OSU Math

This document from Ohio State University's math department provides solved problems related to the Black-Scholes formula. We'll look at a question that demonstrates this dividend adjustment technique.

Please read the section for Question 12.7. Focus on part b), which shows how a new, adjusted stock price is calculated as S(new) = 100 * exp(−0.03 * 0.75). The explanation highlights that the dividend yield's only role is to discount the stock price.

The Modified BSM (Merton) Formulas

This adjustment leads to a modified set of BSM formulas. While you can always perform the manual adjustment of $S_{0}$ as shown above, it's common to see the formulas written in their complete, dividend-inclusive form.

Modified Intermediate Value $d_{1}$ :

Notice that the dividend yield $δ$ is introduced into the drift component of $d_{1}$ . You can think of it as reducing the effective risk-free rate to $(r - δ)$ .

d_{1} = \frac{ln ( S _{0} / K ) + ( r - δ + \frac{1}{2} σ ^{2} ) T}{σ T}

d_{2} = d_{1} - σ T

Modified Call and Put Formulas:

The stock price term, $S_{0}$ , is now explicitly discounted at the dividend yield rate $δ$ .

European Call Price:

C = S_{0} e^{- δ T} N (d_{1}) - K e^{- r T} N (d_{2})

European Put Price:

P = K e^{- r T} N (- d_{2}) - S_{0} e^{- δ T} N (- d_{1})

You can verify that using $S_{0}$ and the modified $d_{1}$ in these formulas is algebraically identical to substituting $S_{n e w} = S_{0} e^{- δ T}$ into the original BSM formulas from our last lesson. This gives you two valid ways to approach the same problem.

The following resource formally presents these formulas and provides the numerical examples we will work through.

Extensions to the Black-Scholes Equation

This PDF by J. Robert Buchanan provides a thorough academic treatment of BSM extensions. We will focus on the practical formulas and examples.

First, review the section titled "Black-Scholes Option Pricing Formulas". Note the formulas for the call (C^e_\delta) and put (P^e_\delta) on a stock paying a continuous dividend \delta. (Note: the variable w in the text is equivalent to our d_1). Next, carefully work through the Call Option example. Finally, work through the Put Option example.

Let's walk through the call option example from the document together to solidify the process.

Call Option Example:

$S_{0} = 62$
$K = 60$
$r = 0.10$
$σ = 0.20$
$T = 5/12$ years
$δ = 0.03$

Step 1: Calculate $d_{1}$ and $d_{2}$

d_{1} = \frac{ln ( 62/60 ) + ( 0.10 - 0.03 + \frac{1}{2} ( 0.20 ) ^{2} ) ( 5/12 )}{0.20 5/12} = \frac{0.03279 + ( 0.09 ) ( 0.41667 )}{0.20 ( 0.6455 )} \approx 0.5445

d_{2} = 0.5445 - 0.20 5/12 \approx 0.5445 - 0.1291 = 0.4154

Step 2: Find $N (d_{1})$ and $N (d_{2})$

Using a standard normal table or calculator:

$N (0.5445) \approx 0.7069$
$N (0.4154) \approx 0.6611$

Step 3: Calculate the Call Price $C$

C = (62) e^{- 0.03 (5/12)} (0.7069) - (60) e^{- 0.10 (5/12)} (0.6611)

C = (62) (0.9876) (0.7069) - (60) (0.9592) (0.6611)

C = 43.268 - 38.046 \approx $5.22

This matches the document's result of $5.24, with slight differences due to rounding of intermediate steps. The document's observation is key: without the dividend, the call would be worth $5.80. The dividend has indeed lowered the option's value.

Put-Call Parity with Continuous Dividends

Just as with the BSM formula itself, the put-call parity relationship requires a small adjustment. The stock price term must be discounted to reflect the forgone dividends. The new relationship is:

C - P = S_{0} e^{- δ T} - K e^{- r T}

Rearranging to solve for the put price gives you a powerful shortcut, just as before:

P = C + K e^{- r T} - S_{0} e^{- δ T}

If you have already calculated the call price, you can find the put price with this simple formula rather than recalculating with the full put formula. This is an essential tool for exam efficiency.

Conclusion

In this lesson, we have extended our pricing toolkit to one of the most common real-world scenarios: options on dividend-paying assets. We saw that the presence of a dividend yield reduces the value of a call option and increases the value of a put option.

Key Takeaways:

The Core Adjustment: The BSM model is modified for a continuous dividend yield $δ$ by replacing the stock price $S_{0}$ with its dividend-adjusted present value, $S_{0} e^{- δ T}$ .
Modified Formulas: This adjustment results in the Merton formulas for calls ( $C = S_{0} e^{- δ T} N (d_{1}) - K e^{- r T} N (d_{2})$ ) and puts ( $P = K e^{- r T} N (- d_{2}) - S_{0} e^{- δ T} N (- d_{1})$ ), where the $d_{1}$ term now includes $(r - δ)$ .
Practical Application: We have worked through numerical examples, applying these formulas step-by-step to find option prices.
Adjusted Put-Call Parity: The invaluable put-call parity shortcut is also adjusted to $C - P = S_{0} e^{- δ T} - K e^{- r T}$ .

So far, we have treated volatility, $σ$ , as a given input. But where does this number come from? In our next lesson, we will explore the concept of implied volatility, where we use the market price of an option to work backward and solve for the volatility the market is "implying". This is a fundamental concept in trading and risk management.

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