The Impact of Dividends

Options and Futures

Lorenzo Naranjo

Spring 2024

Assets Paying Cash Dividends

Put-Call Parity with Dividends

For European options written on dividend paying stocks, the put-call parity is modified as follows: \[ C - P = S - D - K e^{-r T} \] where $D$ is the present value of dividends paid during the life of the option.
In order to derive this expression we will proceed as before by trying to build a covered call in two different ways.

Building a Covered Call

Supose that we did the same as in a previous lecture.
Strategy A: Long stock and short call \[ \begin{align*} \text{Cost} & = S - C \\ \text{Payoff} & = \begin{cases} S_{T} + \textit{FV}(D) & \text{if $S_{T} \leq K$} \\ K + \textit{FV}(D) & \text{if $S_{T} > K$} \end{cases} \end{align*} \]
Strategy B: Long bond and short put \[ \begin{align*} \text{Cost} & = K e^{-r T} - P \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{align*} \]
Both strategies no longer have the same payoff at maturity because the stock pays dividends.

Adjusting Strategy A

Strategy A: Long stock, borrow $D$ and short call \[\begin{align*} \text{Cost} & = S - D - C \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{align*}\]
Strategy B: Long bond and short put \[\begin{align*} \text{Cost} & = K e^{-r T} - P \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{align*}\]
Note that in A we use the dividends, reinvested at $r$, to repay the loan and generate the same payoff as in B.

Example 1

Suppose that $S = 110$, $r = 5\%$, $K = 110$, $T = 9$ months, and $C = 13.30$.
The stock is expected to pay dividends of $2 in 6 months, and $2.5 in 1 year.
What should be the no-arbitrage price of a European put with the same strike and maturity as the European call?
The present value of the relevant dividends is: \[ D = 2 e^{-0.05 \times 6/12} = 1.95 \]
Then, according to put-call parity we should have that: \[ P = 13.30 - 110 + 1.95 + 110 e^{-0.05 \times 9/12} = 11.20 \]

Example 2

What if in the previous example everything stays the same, but you find that the put trades for $11?
- Then we have an arbitrage opportunity since the put is relatively cheap compared to what it should trade.
- Hence, we should buy the put and sell the synthetic put. Note that the stock will pay a dividend of $2 in six months that can be used to pay the loan at that time.

Example 2 (cont’d)

Lower Bound on European Options with Cash Dividends

With dividends, we modify the lower bounds on European call and put options as follows: \[ \begin{align*} C & \geq \max(S - D - K e^{-r T}, 0) \\ P & \geq \max(K e^{-r T} - S + D, 0) \end{align*} \]
As for the case with no dividends, these results are a consequence of put-call parity and the fact the option premium is never negative.

Example 3

Suppose that you have $S = 110$, $r = 5\%$, $K = 110$, and $T = 9$ months.
The stock is expected to pay dividends of $2 and $2.5, in six and twelve months, respectively.
The previous result implies that: \[ C \geq \max(110 - 2 e^{-0.05 \times 6/12} - 110 e^{-0.05 \times 0.75}, 0) = 2.10 \]
Note that we only include the dividend paid in 6 months since the maturity of the option is 9 months.
Also note that the bound is lower than the bound for an otherwise equivalent option written on a non-dividend paying stock.
- For European options you can only purchase the stock at maturity and therefore you miss the dividend paid in 6 months.

Upper Bound on European Options with Cash Dividends

Since the payoffs of strategies A and B described above are positive, the cost of both strategies must be positive.
Therefore, if the asset pays cash dividends, the upper bounds on European call and put options are as follows: \[ \begin{align*} C & \leq S - D \\ P & \leq K e^{-r T} \end{align*} \]

Example 4

Suppose that you have $S = 110$, $r = 5\%$, $K = 110$, and $T = 9$ months.
The stock is expected to pay dividends of $2 and $2.5, in six and twelve months, respectively.
The previous result implies that: \[ C \leq 110 - 2 e^{-0.05 \times 6/12} = 108.05 \]
Hence, no matter how high the volatility is on this European call option with strike $110 and maturity 9 months, its premium must be less than $108.05.

Assets Paying a Dividend Yield

The Dividend Yield

There are many assets that pay dividends continuously, like a currency, or that can be modeled as such, like a stock index.
In these cases it is convenient to model dividends as a percentage yield paid over time.
We will denote the continuously-compounded dividend yield by $q$.
The asset $S$ then pays every instant $t$ a dividend of $q S_{t} \Delta t$.
Therefore, the dividend yield can be seen as the units of the asset growing over time at the rate $q$.
In practice, this is the approach used to model options on stock indices and currencies, although some practitioners also use it to model individual stocks as well.

Put-Call Parity with a Dividend Yield

For European options written on assets paying a dividend yield, the put-call parity is modified as follows: \[ C - P = S e^{-q T} - K e^{-r T} \] where $q$ is the dividend yield paid continuously by the asset during the life of the option.
In order to derive this expression we will proceed as before by trying to build a covered call in two different ways.

Two Strategies with the Same Payoff

Strategy A: Long $e^{-q T}$ units of the asset and short call \[ \begin{align*} \text{Cost} & = S e^{-q T} - C \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{align*} \]
Strategy B: Long bond and short put \[ \begin{align*} \text{Cost} & = K e^{-r T} - P \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{align*} \]
Note that in A the asset grows at the rate $q$, so the total number of “units” of the asset at maturity is $e^{-q T} e^{q T} = 1$.

Example 5

Suppose that $S = 110$, $r = 5\%$, $q = 3\%$, $K = 110$, $T = 9$ months, and $C = 13.30$.
What should be the no-arbitrage price of a European put with the same strike and maturity as the European call?
According to put-call parity we should have that a European put with the same strike and maturity as the call should cost: \[ P = 13.30 - 110 e^{-0.03 \times 9/12} + 110 e^{-0.05 \times 9/12} = 11.70. \]

Example 6

What if in the previous example everything stays the same, but you find that the put trades for $11?
Then we have an arbitrage opportunity since the put is relatively cheap compared to what it should trade.
Hence, we should buy the put and sell the synthetic put.

Lower Bound on European Options with Dividend Yields

If the asset pays a dividend yield, we modify the lower bounds on European call and put options as follows: \[ \begin{align*} C & \geq \max(S e^{-q T} - K e^{-r T}, 0) \\ P & \geq \max(K e^{-r T} - S e^{-q T}, 0) \end{align*} \]
As for the case with no dividends, these results are a consequence of put-call parity and the fact the option premium is never negative.

Example 7

Suppose that you have $S = 110$, $r = 5\%$, $q = 3\%$, $K = 110$, and $T = 9$ months.
The previous result implies that: \[ C \geq \max(110 e^{-0.03 \times 9/12} - 110 e^{-0.05 \times 9/12}, 0) = 1.60 \]
Hence, no matter how low the volatility is on this European call option with strike $110 and maturity 9 months, its premium must be higher than $1.60.

Upper Bound on European Options with Dividend Yields

Since the payoffs of strategies A and B described above are positive, the cost of both strategies must be positive.
Therefore, if the asset pays a dividend yield, the upper bounds on European call and put options are as follows: \[ \begin{align*} C & \leq S e^{-q T} \\ P & \leq K e^{-r T} \end{align*} \]

Example 8

Suppose that you have $S = 110$, $r = 5\%$, $q = 3\%$, $K = 110$, and $T = 9$ months.
The previous result implies that: \[ C \leq 110 e^{-0.03 \times 9/12} = 107.55 \]
Hence, no matter how high the volatility is on this European call option with strike $110 and maturity 9 months, its premium must be less than $107.55.

Feasible Prices for European Call Options

The graph below describes the region of feasible prices for European call options written on an asset that pays a positive dividend yield such that $q > r$.

Feasible Prices for European Put Options

The graph below describes the region of feasible prices for European put options written on an asset that pays a positive dividend yield such that $q > r$.

Binomial Pricing

Pricing options when the asset pays a dividend yield requires to adjust the risk-neutral probabilities accordingly.
Say that over the next period $\Delta t$ the asset price can go up to $S^{u} = S u$, or down to $S^{d} = S d$, and that we want to price a derivative $X$ that pays either $X^{u}$ or $X^{d}$ in each state, respectively.

Risk-Neutral Pricing

The risk-neutral probability of an up-move in this case is given by: \[ p = \frac{e^{(r - q) \Delta t} - d}{u - d} \]
The price of the derivative is then: \[ X = (p X^{u} + (1 - p) X^{d}) e^{-r \Delta t} \]
Note that we can make this model consistent with the Black-Scholes model by choosing $u = e^{\sigma \sqrt{\Delta t}}$ and $d = 1/u$, where $\sigma$ represents the annualized volatility of the asset returns.

Example 9

Suppose that $S = 110$, $r = 5\%$, $q = 3\%$, $\sigma = 30\%$, $K = 110$, $T = 9$ months.
Using a one-period binomial tree, let’s compute the no-arbitrage price of a European call option.
The binomial trees for the asset and the call are as follows:

Example 9 (cont’d)

The risk-neutral probability of an up-move is \[ p = \frac{110 e^{(0.05 - 0.03) \times 9/12} - 84.83}{142.63 - 84.83} = 0.4642 \\ \]
The price of the call is then given by \[ C = (32.63 p + 0 (1 - p)) e^{-0.05 \times 9/12} = 14.59 \]