Beyond Basic Black-Scholes: Dividends, Currencies, and More

The vanilla Black-Scholes model assumes a clean world: no dividends, constant parameters, one type of underlying. Real markets are messier. Chapter 8 of Wilmott’s book starts adding realism. Dividends, currencies, commodities, stock borrowing costs, time-dependent parameters. Each generalization is surprisingly straightforward once you understand the basic framework, which is the good news. The bad news is that you need to keep track of which adjustments apply to your specific situation.

The Dividend Problem

In the simplest Black-Scholes world, the underlying pays a continuous dividend yield D, proportional to the stock price. The holder of one share receives DSdt in each small time step dt. This works well for stock indices (where many individual dividends average out through the year) and for currencies (where D becomes the foreign interest rate).

But for individual stocks? Not so much. Real dividends are:

1. Not known until shortly before they are paid
2. A fixed dollar amount, not a percentage of the stock price
3. Paid at discrete dates, not continuously

Wilmott addresses the discrete, known-amount case. He assumes the amount and timing of dividends are known in advance, which is a reasonable approximation for the near term.

What Happens When a Stock Goes Ex-Dividend

This is a neat no-arbitrage argument. Suppose a stock pays a dividend of D_i on a specific date. If you buy the stock just before the ex-dividend date, you get the dividend. If you buy just after, you do not.

Is there a free lunch in buying just before? No. The stock price drops by exactly the dividend amount as it goes ex-dividend:

S(after) = S(before) - D_i

If it dropped by less, you could buy before, collect the dividend, and sell after for an arbitrage profit. If it dropped by more, you could sell before and buy after. The market eliminates both opportunities.

Jump Conditions for Options

The stock jumps, but what happens to the option? Here is the key insight: if the dividend amount and date are known in advance, there is no surprise. The option holder does not receive the dividend (it goes to the stock holder), and the fall in stock price is completely predictable.

The result: the option value is continuous across the dividend date, even though the stock price is not.

In formula terms:

V(S(before), t_before) = V(S(before) - D_i, t_after)

The option value at the stock price before the dividend equals the option value at the reduced stock price after the dividend. The function V itself changes shape, but the actual dollar value you hold does not jump.

This has a subtle consequence: even though the option value is continuous, the delta changes discontinuously across a dividend date. The slope of V versus S shifts, which means you need to adjust your hedge right at the ex-dividend moment.

Commodities: A Different Beast

Commodities introduce concepts that do not exist for stocks and bonds.

Investment commodities (like gold) are held for investment. Standard arbitrage arguments apply, and the relationship between spot and futures prices works the same as for financial assets.

Consumption commodities (like oil or wheat) are held because someone actually needs them. This breaks the usual arbitrage arguments because holders may not want to sell even when the price is high. You need the oil for heating or the wheat for making bread. This makes arbitrage one-sided: you can only establish an upper bound on the futures price, not an exact relationship.

Storage Costs and Convenience Yield

Commodities cost money to store (warehousing gold, refrigerating food, maintaining oil tanks). If storage costs are proportional to the spot price at rate u, it is mathematically equivalent to a negative dividend yield.

The convenience yield y captures the benefit of physically holding the commodity versus holding a futures contract. For consumption commodities, there is real value in having the stuff on hand. The convenience yield is always non-negative, and for investment commodities it is zero.

The relationship between spot and futures:

F = S * e^((r + u - y)(T-t))

Cost of Carry

The cost of carry bundles everything together: storage costs plus financing costs minus any income from the asset. Depending on the underlying, the drift coefficient in Black-Scholes changes:

• No dividends: drift = r
• Dividend yield D: drift = r - D
• Foreign exchange: drift = r - r_f (domestic minus foreign rate)
• Commodity with storage cost u: drift = r + u

Options on commodities satisfy a modified Black-Scholes equation with the appropriate cost of carry substituted in.

Stock Borrowing and Repo

When you sell a stock short for hedging, you first need to borrow it. This is not free. You typically pay interest at some rate R on the value of the borrowed stock.

Wilmott walks through the Black-Scholes derivation with this extra cost included. The twist: the borrowing cost only applies when delta is positive (meaning you are actually short the stock). This creates a max() function in the equation, which makes it non-linear.

The resulting PDE is:

dV/dt + (1/2)sigma^2S^2d2V/dS2 + rSdV/dS - rV - RSmax(dV/dS, 0) = 0

Non-linearity is a big deal because it means you cannot just add solutions together anymore. This is a preview of themes explored much later in the book, where transaction costs and uncertain parameters also create non-linear problems.

Time-Dependent Parameters

What if the risk-free rate, volatility, and dividend yield all change over time? In practice, people often have a view on where rates or volatility are headed. Can the model handle r(t), D(t), and sigma(t)?

Yes. Wilmott shows an elegant variable substitution that eliminates all time dependence from the coefficients. The result: the solution for time-dependent parameters is the same as the constant-parameter solution, but with averaged values:

r_c = (1/(T-t)) * integral from t to T of r(s) ds

sigma_c^2 = (1/(T-t)) * integral from t to T of sigma(s)^2 ds

D_c = (1/(T-t)) * integral from t to T of D(s) ds

So you average the interest rate over the remaining life of the option, average the dividend yield, and average the squared volatility (not the volatility itself, an important distinction).

For a European call with time-dependent parameters:

Call = S * e^(-D_c(T-t)) * N(d1) - E * e^(-r_c(T-t)) * N(d2)

with d1 and d2 using the averaged parameters.

Important caveat: this only works for European options with standard payoffs. If there is early exercise or exotic features, you need to check whether the boundary conditions survive the transformation. Often they do not.

Power Options and the Log Contract

Two interesting special cases round out the chapter.

Power options have payoffs that depend on S raised to some power alpha. Because of lognormality, you can price these by substituting S^alpha for S in the standard formulas and adjusting the parameters.

The log contract has payoff log(S/E) at expiry. Its value has a remarkable property: the dependence on S and the dependence on volatility completely separate. One term contains S but no sigma, the other contains sigma but no S. This makes the log contract potentially useful for hedging volatility exposure, though Wilmott notes it is not exactly a liquid instrument.

The Practical Takeaway

Most of these generalizations boil down to one principle: figure out the effective “cost of carry” for your underlying, plug it into the right place in the Black-Scholes equation, and solve. Dividends, foreign interest rates, storage costs, and borrowing fees all modify the drift term in similar ways.

The tricky parts are:

• Discrete dividends require jump conditions, not just parameter adjustments
• Stock borrowing creates non-linearity
• Time-dependent parameters require averaging (of the right quantities)
• Commodities need careful thought about whether arbitrage arguments work in both directions

None of this requires fundamentally new math. It is the same PDE framework with different coefficients. That is both the strength and the limitation of the approach.

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