Options on Assets Paying a Dividend Yield

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

General Framework

Pricing Formulas

  • Consider an asset S that pays a continuous yield \textcolor{red}{q} and that follows a GBM under the risk-neutral measure: dS = (r-\textcolor{red}{\delta}) S dt + \sigma S dW
  • The price of European call and put options with strike price K and time-to-maturity T are given by: \begin{aligned} C & = S \textcolor{red}{e^{-\delta T}} \mathop{\Phi}(d_{1}) - K e^{-r T} \mathop{\Phi}(d_{2}) \\ P & = K e^{-r T} \mathop{\Phi}(-d_{2}) - S \textcolor{red}{e^{-\delta T}} \mathop{\Phi}(-d_{1}) \end{aligned} where d_{1} = \frac{\ln(S/K) + (r - \textcolor{red}{\delta} + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \text{ and } d_{2} = d_{1} - \sigma \sqrt{T}

Modeling Dividends

  • It is usually convenient to model dividends as a percentage yield paid over time.
  • We will denote the continuously-compounded dividend yield by \delta.
  • The asset S then pays every instant t a dividend of \delta S_{t} \Delta t.
  • Therefore, if you purchase one unit of the asset at time t for S_{t}, the value of the portfolio at time t + \Delta t will be S_{t + \Delta t} + \delta S_{t} \Delta t.
  • 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.

Replicating A Derivative

  • Consider a derivative H with maturity T written on an asset S that pays a continuous dividend yield \delta.
  • The derivative can be replicated by buying N_{S} units of the stock and N_{B} units of a bond with face value K and maturity T, respectively.
  • If we call V the value of such replicating portfolio, we have that: V_{t} = N_{S, t} S_{t} + N_{B, t} B_{t}.
  • At time t+\Delta t, the value of the replicating portfolio is: V_{t + \Delta t} = N_{S, t} (S_{t + \Delta t} + \delta S_{t} \Delta t) + N_{B, t} B_{t + \Delta t}, which implies that: \Delta V_{t} = N_{S, t} (\Delta S_{t} + \delta S_{t} \Delta t) + N_{B, t} \Delta B_{t}.

Replication in Continuous-Time

  • As \Delta t \rightarrow 0, we have that: \begin{aligned} dV & = N_{S} (dS + \delta S dt) + N_{B} dB \\ & = N_{S} (dS + \delta S dt) + r (N_{B} B) r dt \\ & = N_{S} (dS + \delta S dt) + r (V - N_{S} S) dt \\ & = \left(r V - (r - \delta) N_{S} S\right) dt + N_{S} dS \end{aligned}
  • Also, Ito’s Lemma implies that: dV = \left(\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + \frac{\partial V}{\partial t}\right) dt + \textcolor{blue}{\frac{\partial V}{\partial S} dS} = \left(r V - (r - \delta) N_{S} S\right) dt + \textcolor{blue}{N_{S} dS}

The Risk-Neutral Process for the Underlying Asset

  • Again, choosing N_{S} = \frac{\partial V}{\partial S} implies that: \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + (r - \delta) S \frac{\partial V}{\partial S} - r V = 0 with boundary condition V_{T} = H_{T}.
  • Using the same logic as before, we conclude that S follows a GBM under the risk-neutral measure given by: dS = (r - \delta) S dt + \sigma S dW

Pricing a Forward Contract

  • We can use the risk-neutral approach to price a long forward contract with maturity T and forward price F. \begin{aligned} V_{t} & = e^{-r(T - t)} \operatorname{E}^{*}_{t}(S_{T} - F) \\ & = e^{-r (T - t)} (S_{t} e^{(r - \delta)(T - t)} - F) \\ & = S_{t} e^{-\delta (T - t)} - F e^{-r (T - t)} \end{aligned}
  • The forward price F is determined such that at inception the value of the contract is zero: V = S e^{-\delta T} - F e^{-r T} = 0 \Rightarrow F = S e^{(r - \delta) T}
  • The value of the long position, in general, will change over time.

Pricing a European Call Option

  • As before, we can price a European call option written on the asset with maturity T and strike price K: \begin{aligned} C & = e^{-r T} \operatorname{E}^{*}\left((S_{T} - K) \large\mathbb{1}_{\{S_{T} > K\}} \right) \\ & = e^{-r T} \operatorname{E}^{*}\left(S_{T} \large\mathbb{1}_{\{S_{T} > K\}} \right) - e^{-r T} \operatorname{E}^{*}\left(K \large\mathbb{1}_{\{S_{T} > K\}} \right) \\ & = S e^{-\delta T} \mathop{\Phi}(d_{1}) - K e^{-r T} \mathop{\Phi}(d_{2}) \end{aligned} where \begin{aligned} d_{1} & = \frac{\ln(S/K) + (r - \delta + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T} \end{aligned}

Call Premium vs. Spot Price

The figure displays the Black-Scholes call premium C(S) where r = 0.05, \delta = 0.08, \sigma = 0.45, T = 1 and K = 100. It also shows the call option payoff given by \max(S - K, 0) and the lower bound for a European call given by \max(S e^{-\delta T} - K e^{-r T}, 0).

Pricing a European Put Option

  • Consider now a European put option with the same characteristics as the previous call.
  • According to put-call parity, it must be the case that: C - P = S e^{-\delta T} - K e^{-r T}
  • Hence, \begin{aligned} P & = C - (S e^{-\delta T} - K e^{-rT}) \\ & = S e^{-\delta T} \mathop{\Phi}(d_{1}) - K e^{-r T} \mathop{\Phi}(d_{2}) - (S e^{-\delta T} - K e^{-rT}) \\ & = K e^{-r T} (1 - \mathop{\Phi}(d_{2})) - S e^{-\delta T} (1 - \mathop{\Phi}(d_{1})) \\ & = K e^{-r T} \mathop{\Phi}(-d_{2}) - S e^{-\delta T} \mathop{\Phi}(-d_{1}) \end{aligned}

Put Premium vs. Spot Price

The figure displays the Black-Scholes put premium P(S) where r = 0.05, \delta = 0.08, \sigma = 0.45, T = 1 and K = 100. It also shows the put option payoff given by \max(K - S, 0) and the lower bound for a European put given by \max(K e^{-r T} - S e^{-\delta T}, 0).

Example 1 A stock that pays a continuous dividend yield of 8% currently trades for $100. The instantaneous volatility of returns is 30% per year and the risk-free rate is 5% per year, continuously compounded and constant for all maturities. Consider ATM call and put options written on the stock with maturity 10 months. Then, \begin{aligned} d_{1} & = \frac{\ln(100/100) + (0.05 - 0.08 + 0.5(0.30)^{2})(10/12)}{0.30\sqrt{10/12}} = 0.0456 \\ d_{2} & = 0.0456 - 0.30\sqrt{10/12} = -0.2282 \end{aligned} Therefore, \mathop{\Phi}(d_{1}) = 0.5182 and \mathop{\Phi}(d_{2}) = 0.4097, which implies that: \begin{aligned} C & = 100e^{-0.08(10/12)}(0.5182) - 100e^{-0.05(10/12)}(0.4097) = \$9.18 \\ P & = 100e^{-0.05(10/12)}(1 - 0.4097) - 100e^{-0.08(10/12)}(1 - 0.5182) = \$11.54 \end{aligned}

Delta of European Call and Put Options

  • For an asset that a pays a continuous dividend yield q, we have that for a European call option: \frac{\partial C}{\partial S} = e^{-\delta T} \Phi(d_{1})
  • We can see that if q > 0, the number of shares required to hedge the call is lower than in the case of a non-dividend paying asset.
    • The shares that you buy to hedge the call grow over time at the rate \delta, which means that you need to buy less.
  • Similarly, for a European put option we have that: \frac{\partial P}{\partial S} = - e^{-\delta T} \Phi(-d_{1})

Example 2 In the previous example, we found that \mathop{\Phi}(d_{1}) = 0.5182 and \mathop{\Phi}(d_{2}) = 0.4097. Hence, \begin{aligned} \frac{\partial C}{\partial S} & = e^{-0.08(10/12)} (0.5182) = 0.4848 \\ \frac{\partial P}{\partial S} & = - e^{-0.08(10/12)} (1 - 0.5182) = -0.4507 \end{aligned} This means that an OTC dealer who sells a call option needs to buy 0.4848 units of the asset while borrowing 100 e^{-0.05 (10/12)} (0.4097) = \$39.30 at the risk-free rate.

To hedge a put option, the dealer needs to short-sell 0.4507 units of the asset and invest 100 e^{-0.05 (10/12)} (1 - 0.4097) = \$56.62 in the money-market account.

Options on Indices

Options on Stock Indices

  • Most stock indices such as the S&P 500 (SPX) do not reinvest their dividends.
  • Hence, to replicate an option written on the index we can use a portfolio of stocks that mimics the value of the index and that will pay a dividend yield over time.
  • We will assume that the replicating portfolio exactly matches the composition of the index at any point in time so that S_{t} represents both the value of the index and of the tracking portfolio.

SPX Options

  • One of the most liquid option contracts in the world.
  • Characteristics:
    • European style exercise
    • Cash settled
    • Each contract is written on 100 times the value of the index
  • There are also mini-SPX index options written over XSP which is an index 10 times smaller than SPX.
  • More information can be found at https://cdn.cboe.com/resources/spx/spx-fact-sheet.pdf

Example 3 The SPX index is currently at 4,251, has a dividend yield of 1.33% per year and an instantaneous volatility of 17% per year. The risk-free rate is 3% per year, continuously compounded and constant for all maturities. Say we want to compute the price of an SPX call option contract with maturity 3 months and strike 4,300. Then, \begin{aligned} d_{1} & = \frac{\ln(4251/4300) + (0.03 - 0.0133 + 0.5(0.17)^{2})(3/12)}{0.17\sqrt{3/12}} = -0.0432 \\ d_{2} & = -0.0432 - 0.17\sqrt{10/12} = -0.1282 \end{aligned} Hence, \mathop{\Phi}(d_{1}) = 0.4828 and \mathop{\Phi}(d_{2}) = 0.4490, which implies that: \begin{aligned} C = 4,251 e^{-0.0133(3/12)}(0.4828) - 4,300 e^{-0.03(3/12)}(0.4490) = \$129.193 \end{aligned} Therefore, a standard SPX call option contract should cost $12,919.30, whereas a mini-SPX call option contract should trade for $1,291.93.