Chapter 17: Options on Stock Indices and Currencies

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Options on Stock Indices

Major stock index options:

S&P 500 (SPX) — traded on CBOE, European-style
S&P 100 (OEX) — American-style
NASDAQ 100 (NDX)
FTSE 100, Nikkei 225, Euro Stoxx 50

Index options are typically cash-settled: the payoff is paid in cash rather than by delivering the portfolio. Contract size = $100× \times$ index level (for S&P 500).

Currency Options

Currency options are traded on exchanges (e.g., Philadelphia Stock Exchange) and OTC. They give the right to buy (call) or sell (put) one currency for another at a specified exchange rate.

A call option on £1 at strike $K$ (USD/GBP) gives the right to buy £1 for $K$ USD. If the spot rate $S_T > K$ , payoff = $S_T - K$ .

Put–call parity for currency options:

$c + K e^{-rT} = p + S_0 e^{-r_f T}$

where $r$ = domestic risk-free rate, $r_f$ = foreign risk-free rate.

Options on Stocks Paying Known Dividend Yields

A stock index is treated as a stock paying a continuous dividend yield $q$ . The lower bound for a European index call becomes:

$\geq S_0 e^{-qT} - K e^{-rT}$

Valuation Formulas

European Index Options (Merton's Extension)

Replace $S_0$ with $S_0 e^{-qT}$ in the Black–Scholes–Merton formula:

$c = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$

$p = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)$

where:

$d1=ln⁡(S0/K)+(r−q+σ2/2)TσT,d2=d1−σTd_1 = \frac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$

European Currency Options (Garman–Kohlhagen Model)

Replace $S_0$ with $S_0 e^{-r_f T}$ in the Black–Scholes–Merton formula:

$c = S_0 e^{-r_f T} N(d_1) - K e^{-rT} N(d_2)$

$p = K e^{-rT} N(-d_2) - S_0 e^{-r_f T} N(-d_1)$

$d1=ln⁡(S0/K)+(r−rf+σ2/2)TσT,d2=d1−σTd_1 = \frac{\ln(S_0/K) + (r - r_f + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$

This is the same form as the index option formula, with $q$ replaced by $r_f$ .

American Options

When Early Exercise is Optimal

Index call options: Can be optimal to exercise early if the dividend yield is sufficiently high (similar logic as an individual stock paying dividends)
Currency call options: Early exercise may be optimal when the foreign interest rate is high relative to the domestic rate
In general, an American call on an asset with a yield may be exercised early, unlike a call on a non-dividend-paying stock

Approximation Methods

Exact analytic solutions don't exist for American options with dividends/yields. Common approaches:

Binomial/trinomial trees (most versatile)
Barone-Adesi and Whaley quadratic approximation — analytic approximation
Finite difference methods — more accurate for complex cases

The "Stocks Beat Bonds in the Long Run" Fallacy

A call option with a very long maturity (e.g., 100 years) on a stock index is not nearly as cheap as one might expect. While the expected stock return exceeds the risk-free rate, the distribution of outcomes is relevant. The Black–Scholes–Merton formula shows that the price of a call option depends on $r$ , not $μ\mu$ . Even with a 100-year horizon, the risk-neutral probability of the index finishing above the strike is limited by volatility.