14. Options on Indices and Currencies#

14.1. General Framework#

14.1.1. 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 \(q\). The asset \(S\) then pays every instant \(t\) a dividend of \(q \Delta t\) shares and therefore \(q S_{t} \Delta t\) dollars. 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 \(S_{t + \Delta t} + q S_{t} \Delta t\).

Figure made with TikZ

You will notice that the payment of the dividend at time \(t + \Delta t\) is known at time \(t\). Indeed, this payment depends on the value of the stock at time \(t\) and not \(t + \Delta t\). This is not a mistake or a convenience, but reflects the fact that we want to incorporate in the total return of investing in the stock from time \(t\) to \(t + \Delta t\) both capital gains and dividends:

(14.1)#\[\begin{equation} \underbrace{\frac{S_{t + \Delta t} + q S_{t} \Delta t - S_{t}}{S_{t}}}_{\text{Total Return}} = \underbrace{\frac{\Delta S_{t}}{S_{t}}}_{\text{Capital Gains}} + \underbrace{\vphantom{\frac{\Delta S_{t}}{S_{t}}} q \Delta t}_{\text{Dividends}} \label{total_return_with_dividends} \end{equation}\]

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. We must note that specially for American type options, modelling lump-sum dividends as a continuous yield might induce errors in computing the optimal early-exercise policy. It could also lead to the wrong risk-neutral adjustment if dividends are paid, say, twice per year, and we want to risk-adjust the underlying asset process for the next three months just after a dividend has been paid.

14.1.2. Replicating A Derivative#

As we did in the previous chapter where there were no-dividends, to price a call or put option we take the point of view of a trading desk that makes the market for such contracts. Their sales team just sold a European option \(H\) written on a stock \(S\) with maturity \(T\) to a client. The stock in this case, though, pays a continuous dividend yield \(q\).

The traders of the desk will replicate the option by buying (or selling) \(N_{S}\) units of the stock and \(N_{B}\) units of a bond with face value \(K\) and maturity \(T\), respectively. The difference in this case compared to the no-dividend stock is that the number of shares bought will grow, so that the trader will have to buy a little less in order to hedge a call option, for example.

If we call \(V\) the value of such replicating portfolio, we have that:

(14.2)#\[\begin{equation} V_{t} = N_{S}(t) S_{t} + N_{B}(t) B_{t} \label{self_financing_condition} \end{equation}\]

At time \(t+\Delta t\), and because the underlying asset pays dividends, the value of the replicating portfolio is:

\[\begin{equation*} V_{t + \Delta t} = N_{S}(t) (S_{t + \Delta t} + q S_{t} \Delta t) + N_{B}(t) B_{t + \Delta t}, \end{equation*}\]

which implies that:

\[\begin{equation*} \Delta V_{t} = N_{S}(t) (\Delta S_{t} + q S_{t} \Delta t) + N_{B}(t) \Delta B_{t}. \end{equation*}\]

As \(\Delta t \rightarrow 0\), we have that:

(14.3)#\[\begin{align} dV & = N_{S} (dS + q S dt) + N_{B} dB \notag \\ & = N_{S} (dS + q S dt) + r (N_{B} B) r dt \notag \\ & = N_{S} (dS + q S dt) + r (V - N_{S} S) dt \notag \\ & = \left(r V - (r - q) N_{S} S\right) dt + N_{S} dS \label{dynamics_long_position} \end{align}\]

where in the second line we used the fact that \(dB = r B dt,\) and in the third line we applied the self-financing condition \(\eqref{self_financing_condition}\) re-written as \(N_{B} B = V - N_{S} S.\)

As in the previous chapter, equation \(\eqref{dynamics_long_position}\) captures the dynamics of the replicating portfolio needed to hedge the short position. For the hedge to be successful, the dynamics of the long position must match the dynamics of the short position that are given by:

(14.4)#\[\begin{equation} dV = \frac{\partial V}{\partial S} dS + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} dt + \frac{\partial V}{\partial t} dt \label{dynamics_short_position} \end{equation}\]

Equating \(\eqref{dynamics_long_position}\) and \(\eqref{dynamics_short_position}\) shows that:

(14.5)#\[\begin{equation} \left(\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + \frac{\partial V}{\partial t}\right) dt + \frac{\partial V}{\partial S} dS = \left(r V - (r - q) N_{S} S\right) dt + N_{S} dS \label{replicating_condition} \end{equation}\]

Equation \(\eqref{replicating_condition}\) reveals that, even in the presence of a dividend yield, the number of shares required to hedge the option must equal the partial derivative with respect to the stock price. Therefore, choosing \(\alpha = \frac{\partial V}{\partial S}\) implies that:

(14.6)#\[\begin{equation} \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + (r - q) S \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} - r V = 0 \label{black_scholes_pde} \end{equation}\]

with boundary condition \(V_{T} = F(S_{T}).\)

Equation \(\eqref{black_scholes_pde}\) is the Black-Scholes partial differential equation (PDE) that must satisfy all derivatives written on an asset that pays a dividend yield.1The Black-Scholes PDE given by \(\eqref{black_scholes_pde}\) is only valid for derivative instruments (or assets) that do not pay dividends themselves, even though they are written on a dividend paying asset. The stock, for example, does not satisfy equation \(\eqref{black_scholes_pde}\) since it pays a dividend. A generalized PDE that allows for the derivative to also pay a dividend yield can be found in the appendix.

Example 14.1

Consider a forward contract written on an asset that pays a dividend yield \(q\) expiring at time \(T\) with forward price \(K\). The value \(V\) of the forward contract at time \(0 \leq t \leq T\) is:

\[\begin{equation*} V = S e^{-q (T - t)} - K e^{-r (T - t)} \end{equation*}\]

Let us check that the value of the contract satisfies the Black-Scholes PDE equation \(\eqref{black_scholes_pde}.\) The relevant derivatives are:

\[\begin{align*} \frac{\partial V}{\partial S} & = e^{-q (T - t)} \\ \frac{\partial^{2} V}{\partial S^{2}} & = 0 \\ \frac{\partial V}{\partial t} & = q S e^{-q(T - t)} - r K e^{-r (T - t)} \\ \end{align*}\]

The left-hand side of equation \(\eqref{black_scholes_pde}\) then becomes:

\[\begin{equation*} (r - q) S e^{-q(T - t)} + q S e^{-q(T - t)} - r K e^{-r (T - t)} - r (S e^{-q (T - t)} - K e^{-r (T - t)}) = 0 \end{equation*}\]

which clearly satisfy the claim.

14.1.3. The Risk-Neutral Process for the Underlying Asset#

As in Section 13, the replication argument is indifferent of the dynamics of the stock. This implies that the same logic should work in a hypothetical world where everyone is risk-neutral. In such a world, the expected total return of all assets is the risk-free rate. Hence, equation \(\eqref{total_return_with_dividends}\) implies that:

\[\begin{equation*} \frac{S_{t + \Delta t} + q S_{t} \Delta t - S_{t}}{S_{t}} = \frac{\Delta S_{t}}{S_{t}} + q \Delta t = r \Delta t + \sigma \Delta W_{t} \end{equation*}\]

As \(\Delta t \rightarrow 0\) we obtain the continuous-time analog of risk-neutral process followed by the stock:

\[\begin{equation*} dS = (r - q) S dt + \sigma S dW \end{equation*}\]

We conclude that \(S\) follows a GBM under the risk-neutral measure with drift \(r - q\) and volatility \(\sigma.\)

Example 14.2 (Pricing a Forward Contract)

We can use the risk-neutral approach to compute the value \(V\) of a long forward contract with maturity \(T\) and forward price \(K\).

\[\begin{align*} V & = e^{-r T} \ev(S_{T} - K) \\ & = e^{-r T} (S e^{(r - q) T} - K) \\ & = S e^{-q T} - K e^{-r T} \end{align*}\]

The value of the long position, in general, will change over time. The forward price \(F\) is determined such that the value of the contract is zero when the contract is signed:

\[\begin{equation*} V = S e^{-q T} - F e^{-r T} = 0 \Rightarrow F = S e^{(r - q) T} \end{equation*}\]

This is consistent with what we would get using put-call parity:

\[\begin{equation*} C - P = S e^{-q T} - K e^{-r T} \end{equation*}\]

The right hand-side of this expression is the cost of a forward contract with forward price \(K\). The left hand-side says that a forward contract can be synthesized by buying a call and selling a put.

Figure made with TikZ

Remember that under the risk-neutral measure, the value of any asset is computed as its expected payoff discounted at the risk-free rate. Therefore, the price of a European call option with maturity \(T\) and strike price \(K\) written on the asset that pays a continuous dividend yield \(q\) is given by:

\[\begin{align*} C & = e^{-r T} \ev\left((S_{T} - K) \1{S_{T} > K} \right) \\ & = e^{-r T} \ev\left(S_{T} \1{S_{T} > K} \right) - e^{-r T} \ev\left(K \1{S_{T} > K} \right) \\ & = S e^{-q T} \cdf(d_{1}) - K e^{-r T} \cdf(d_{2}) \end{align*}\]

where

\[\begin{align*} d_{1} & = \frac{\ln(S/K) + (r - q + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T} \end{align*}\]

Figure made with TikZ

Fig. 14.1 The figure displays the Black-Scholes call premium \(C(S)\) where \(r=0.05\), \(q = 0.08\), \(\sigma=0.45\), \(T=1\) 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 \(\max(S e^{-q T} - K e^{-r T}, 0)\).

Note that, even though the risk-free rate is positive, the time-value of deep ITM call options is now negative due to the positive dividend yield. Indeed, the lower bound asymptote has a slope coefficient less than one, making the option price to cross the option’s intrinsic value.

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:

\[\begin{equation*} C - P = S e^{-q T} - K e^{-r T} \end{equation*}\]

Hence,

\[\begin{align*} P & = C - (S e^{-q T} - K e^{-rT}) \\ & = S e^{-q T} \cdf(d_{1}) - K e^{-r T} \cdf(d_{2}) - (S e^{-q T} - K e^{-rT}) \\ & = K e^{-r T} (1 - \cdf(d_{2})) - S e^{-q T} (1 - \cdf(d_{1})) \\ & = K e^{-r T} \cdf(-d_{2}) - S e^{-q T} \cdf(-d_{1}) \end{align*}\]

The figure below displays the Black-Scholes put premium \(P(S)\) where \(r=0.05\), \(q = 0.08\), \(\sigma=0.45\), \(T=1\) 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 \(\max(K e^{-r T} - S e^{-q T}, 0)\).

Figure made with TikZ

Example 14.3

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,

\[\begin{align*} d_{1} & = \frac{\ln(100/100) + (0.05 - 0.08 + 0.5(0.30)^{2})(10/12)}{0.30\sqrt{10/12}} = 0.0456 \\ d_{2} & = 0.0456 - 0.30\sqrt{10/12} = -0.2282 \end{align*}\]

Therefore, \(\cdf(d_{1}) = 0.5182\) and \(\cdf(d_{2}) = 0.4097\), which implies that:

\[\begin{align*} C & = 100e^{-0.08(10/12)}(0.5182) - 100e^{-0.05(10/12)}(0.4097) = \$9.18 \\ P & = 100e^{-0.05(10/12)}(1 - 0.4097) - 100e^{-0.08(10/12)}(1 - 0.5182) = \$11.54 \end{align*}\]

For an asset that a pays a continuous dividend yield \(q\), we have that for a European call option:

\[\begin{equation*} \frac{\partial C}{\partial S} = e^{-q T} \Phi(d_{1}) \end{equation*}\]

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 \(q\), which means that you need to buy less.

Similarly, for a European put option we have that:

\[\begin{equation*} \frac{\partial P}{\partial S} = - e^{-q T} \Phi(-d_{1}) \end{equation*}\]

Example 14.4

In the previous example, we found that \(\cdf(d_{1}) = 0.5182\) and \(\cdf(d_{2}) = 0.4097\). Hence,

\[\begin{align*} \frac{\partial C}{\partial S} & = e^{-0.08(10/12)} (0.5182) = 0.4848 \\ \frac{\partial P}{\partial S} & = - e^{-0.08(10/12)} (1 - 0.5182) = -0.4507 \end{align*}\]

This means that an OTC dealer who sells a call option needs to buy 0.4848 units of the asset while borrowing

\[\begin{equation*} 100 e^{-0.05 (10/12)} (0.4097) = \$39.30 \end{equation*}\]

at the risk-free rate. To hedge a put option, the dealer needs to short-sell 0.4507 units of the asset and invest

\[\begin{equation*} 100 e^{-0.05 (10/12)} (1 - 0.4097) = \$56.62 \end{equation*}\]

in the money-market account.

Property 14.1 (Black-Scholes Model for a Stock that Pays a Dividend Yield)

Consider a stock \(S\) that pays a continuous dividend yield \(q\) and that follows a GBM under the risk-neutral measure:

\[\begin{equation*} dS = (r - q) S dt + \sigma S dW \end{equation*}\]

The price of European call and put options with strike price \(K\) and time-to-maturity \(T\) are given by:

\[\begin{align*} C & = S e^{-q T} \cdf(d_{1}) - K e^{-r T} \cdf(d_{2}) \\ P & = K e^{-r T} \cdf(-d_{2}) - S e^{-q T} \cdf(-d_{1}) \end{align*}\]

where

\[\begin{align*} d_{1} & = \frac{\ln(S/K) + (r - q + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T} \end{align*}\]

Furthermore, the delta of the call is given by \(e^{-q T} \cdf(d_{1})\) whereas the delta of the put is computed as \(- e^{-q T} \cdf(-d_{1})\).

14.2. Options on Indices#

14.2.1. 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.

14.2.2. 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 14.5

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,

\[\begin{align*} d_{1} & = \frac{\ln(4251/4300) + (0.03 - 0.0133 + 0.5(0.17)^{2})(3/12)}{0.17\sqrt{3/12}} = -0.0432 \\ d_{2} & = -0.0432 - 0.17\sqrt{10/12} = -0.1282 \end{align*}\]

Hence, \(\cdf(d_{1}) = 0.4828\) and \(\cdf(d_{2}) = 0.4490\), which implies that:

\[\begin{align*} C = 4,251 e^{-0.0133(3/12)}(0.4828) - 4,300 e^{-0.03(3/12)}(0.4490) = \$129.193 \end{align*}\]

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.

14.3. Options on Currencies#

14.3.1. Exchange Rates#

The (nominal) exchange rate between two currencies is the number of domestic currency units per unit of foreign currency. We need to be careful, though, since the street market convention for the EUR/USD exchange rate implies:

  • The quote currency is the US dollar (USD)

  • The base currency is the Euro (EUR)

For example, the direct quotation of the EUR/USD could be $1.4380/€, and represents the price in USD of 1 EUR. Note that you could always define it the other way around (indirect-quotes), and this is done for many currency pairs as well.

The market convention of calling this exchange rate EUR/USD might be misleading. It is written EUR/USD, EUR-USD or EURUSD but it really represents the number of USD per EUR, i.e. \(\$1.4380 \Leftrightarrow \text{€}1.\) Be careful, though, as in some textbooks you might find it the other way around.

Example 14.6

If the EUR/USD exchange rate is $1.47/€, for a US investor, 1 Euro is worth $1.47, but in Europe, how many Euros is worth $1?

\[\begin{equation*} \$1 = \frac{1}{1.47} = \text{€} 0.68/\$. \end{equation*}\]

The exchange rate is a relative price.

Some currency pairs such as EUR/USD or GBP/USD use the USD as the quote currency. However, most currency pairs are expressed using the dollar as the base currency, i.e., USD/JPY, USD/CNY, USD/CLP, etc.

14.3.2. The Risk-Neutral Process for a Currency#

When working with currencies, it is usually convenient to denote by \(r\) the risk-free rate of the quote currency and by \(r^{*}\) the risk-free rate of the base or foreign currency. The risk-neutral process for an exchange-rate \(S\) expressed with direct-quotes is then:

\[\begin{equation*} dS = (r - r^{*}) S dt + \sigma S dW \end{equation*}\]

Using the previous notation, the forward price with maturity \(T\) for the currency is:

\[\begin{equation*} F = S e^{(r - r^{*}) T} \end{equation*}\]

Example 14.7

The EUR/USD currently trades at $1.18663. The continuously compounded 9-month risk-free rates in USD and EUR are 1.5% and 0.5% per year, respectively. The 9-month EUR/USD forward rate is then:

\[\begin{equation*} F = 1.18663 e^{(0.015 - 0.005) (9/12)} = \$1.19556 \end{equation*}\]

or \(+89.3\) forward-points.

14.3.3. Options on Currencies#

Options on currencies reveal an interesting relationship between the underlying asset and the numeraire used to express the price of the asset. Consider an American investor analyzing a call option on the EUR/USD with maturity 1-year, strike price $1.25 over a notional of €1 million. From the point of the view of a European investor, that option is really a put on the USD/EUR with same maturity, strike price €0.80 over a notional of $1.25 million. Hence, it is convenient to be explicit about the currency being bought and the one being sold when specifying the contract, i.e., we will talk about a EUR call/USD put when describing the previous contract.

Property 14.2 (Black-Scholes Model for Currencies)

It is common to express the Black-Scholes formulas for options on currencies as a function of the corresponding forward price:

\[\begin{align*} C & = F e^{-r T} \cdf(d_{1}) - K e^{-r T}\cdf(d_{2}) \\ P & = K e^{-r T}\cdf(-d_{2}) - F e^{-r T}\cdf(-d_{1}) \end{align*}\]

where

\[\begin{equation*} F = S e^{(r-r^{*}) T} \text{, } d_{1} = \frac{\ln(F/K) + \frac{1}{2} \sigma^{2} T}{\sigma \sqrt{T}} \text{ and } d_{2} = d_{1} - \sigma \sqrt{T}. \end{equation*}\]

An option with a strike price equal to its corresponding forward price is called at-the-money-forward (ATMF). Remember put-call parity for currencies:

\[\begin{equation*} C - P = S e^{-r^{*} T} - K e^{-r T} \end{equation*}\]

When \(K = F = S e^{(r - r^{*}) T}\) we have that \(C - P = 0\), i.e., when the strike price is equal to the forward price a call and a put with the same maturity are worth the same.

14.4. Appendix#

14.4.1. General PDE for Dividend-Paying Derivatives#

Consider a derivative \(H\) expiring at time \(T\) and written on an asset that pays a dividend yield \(q\). Let us assume that the derivative itself also pays a dividend yield \(\delta\). The risk-neutral process for the stock is given by:

\[\begin{equation*} dS = (r - q) S dt + \sigma S dW \end{equation*}\]

whereas the risk-neutral process of the derivative is given by:

(14.7)#\[\begin{equation} dH = (r - \delta) H dt + \sigma_{H} H dW \label{dividend_paying_derivative} \end{equation}\]

Moreover, according to Ito’s lemma, the risk-neutral process of the derivative must also satisfy:

(14.8)#\[\begin{align} dH & = \frac{\partial H}{\partial S} dS + \frac{1}{2} \frac{\partial^{2} H}{\partial S^{2}} (dS)^{2} + \frac{\partial H}{\partial t} dt \notag \\ & = \left( \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} H}{\partial S^{2}} + (r - q) S \frac{\partial H}{\partial S} + \frac{\partial H}{\partial t} \right) dt + \sigma S \frac{\partial H}{\partial S} dW \label{ito_dividend_paying_derivative} \end{align}\]

Since the drift in \(\eqref{dividend_paying_derivative}\) and \(\eqref{ito_dividend_paying_derivative}\) is the same, we have that:

(14.9)#\[\begin{equation} \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} H}{\partial S^{2}} + (r - q) S \frac{\partial H}{\partial S} + \frac{\partial H}{\partial t} - (r - \delta) H = 0 \label{generalized_black_scholes_pde} \end{equation}\]

In particular, the stock itself satisfies \(\eqref{generalized_black_scholes_pde}\) if we take \(H(S) = S\) and \(\delta = q.\)

14.5. Practice Problems#

Exercise 14.1

Calculate the value of a three-month at-the-money European call option on a stock index when the index is at 250, the risk-free interest rate is 10% per annum, the volatility of the index is 18% per annum, and the dividend yield on the index is 3% per annum.

Exercise 14.2

Calculate the value of an eight-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free interest rate is 4% per annum, and the foreign risk-free interest rate is 8% per annum.

Exercise 14.3

The S&P 100 index currently stands at 696 and has a volatility of 30% per annum. The risk-free rate of interest is 7% per annum and the index provides a dividend yield of 4% per annum. Calculate the value of a three-month European put with strike price 700.

Exercise 14.4

Suppose that the spot price of the Canadian dollar (CAD) is USD $0.75 and that the CAD/USD exchange rate has a volatility of 4% per annum. The risk-free rates of interest in Canada and the United States are 9% and 7% per annum, respectively. Calculate the value of a European call option to buy one Canadian dollar for USD $0.75 in nine months. Use put-call parity to calculate the price of a European put option to sell one Canadian dollar for U.S. $0.75 in nine months.