Everything Wrong with Black-Scholes (And What to Do About It)

Before we tear Black-Scholes apart, Wilmott wants to make something clear. This model is a triumph. It changed finance forever. Two of its three creators won the Nobel Prize. Everyone in derivatives uses it, from salesmen to traders to quants. Option prices are often quoted not in dollars but in volatility terms, with the understanding that you plug that number into Black-Scholes to get the price.

You use a different model only with extreme caution, and you had better be very convincing about why.

That said, Chapter 46 catalogs everything wrong with it. Each defect points to a later chapter where Wilmott shows how to fix it. This chapter is basically a roadmap for the rest of the book.

Defect 1: Discrete Hedging

Black-Scholes assumes: Delta hedging is continuous.

The entire derivation relies on Ito’s lemma in continuous time. You must rebalance your hedge continuously, every instant, forever. Obviously you cannot do that. In real life, you hedge at discrete intervals. Maybe every hour, maybe every day. Between rehedges, your portfolio is exposed to risk that the model says does not exist.

This gap between theory and practice is not small. Even if every other assumption held perfectly, the impossibility of continuous hedging would have an enormous effect on actual profit and loss.

Defect 2: Transaction Costs

Black-Scholes assumes: Delta hedging is continuous AND free.

Not only can you not hedge continuously, but every time you do hedge, it costs money. Bid-offer spreads eat into your P&L. In liquid markets, this cost is small and you can hedge often. In illiquid markets, the cost can be so large that you cannot afford to hedge as frequently as you would like.

You are caught between two problems: hedge too rarely and you have unhedged risk. Hedge too often and you bleed transaction costs. Finding the optimal balance is a real problem, and Black-Scholes says nothing about it.

Defect 3: Volatility Is Not Constant

Black-Scholes assumes: Volatility is a known constant.

This is probably the most famous defect. Volatility is the single most important input to an option price, and we do not know what it is. Every time series of historical volatility shows it bouncing around wildly. It is not constant, it is not deterministic, and it is not predictable.

Wilmott covers multiple approaches to this problem in later chapters.

Deterministic volatility surfaces. Maybe volatility is a function of both time and the stock price. If the market prices all traded options consistently with some volatility function $\sigma(S, t)$, you can back out that function from market prices. This gives you an “implied volatility surface” and a “local volatility surface.” Popular for pricing exotics consistently with vanilla market prices.

Stochastic volatility. Model volatility itself as a random variable with its own stochastic differential equation. This is the natural approach since volatility is clearly both variable and unpredictable. Very popular for contracts that are sensitive to volatility behavior, like barrier options.

Uncertain parameters. Do not model volatility at all. Just say it lies within some range $[\sigma_{min}, \sigma_{max}]$. Do not specify a probability distribution within that range. Let it jump from one extreme to the other as often as it wants. Then price in a worst-case scenario. This is the Avellaneda-Levy-Paris model. You never need to know the actual value of volatility, just its bounds.

Empirical analysis. Look at actual data. Measure volatility of volatility. Figure out what stochastic differential equation fits best. Use that to inform your stochastic volatility model, or to determine how likely your uncertain volatility range is to be breached.

Asymptotic analysis. Sometimes you can have your cake and eat it. If volatility mean-reverts fast and has large vol-of-vol (both true in equity markets), you can use asymptotic techniques to get approximate solutions for option prices under very general stochastic volatility models. Fast and accurate.

Defect 4: The Asset Path Is Not Continuous

Black-Scholes assumes: The underlying price path is continuous.

Markets jump. Prices gap. Especially downward. These jumps are too large and too frequent to be explained by a Normal distribution. And they are too sudden for continuous hedging.

Jump diffusion. Add Poisson jumps to the lognormal random walk. This captures the observed discontinuities but introduces a problem: you can no longer eliminate all risk. Delta hedging only works if the underlying moves continuously. When it jumps, your hedge is wrong. You must price in an expected sense rather than a risk-free sense.

Crash modeling. If hedging against jumps is impossible, try worst-case analysis. Assume the worst jump happens and price accordingly. This removes the need for jump probability assumptions and connects back to the CrashMetrics ideas from Chapter 43.

Defect 5: Hedging Affects the Underlying

Black-Scholes assumes: The underlying price is unaffected by trading in the option.

In the Black-Scholes world, some cosmic random number generator sets stock prices, and option prices follow. In reality, buying and selling the underlying to delta hedge moves the underlying price. A large hedge trade can move the market significantly.

This creates a chicken-and-egg problem. The option price depends on the underlying, but the underlying moves because of hedging activity driven by the option price. Near expiry, when gamma is large and you need to hedge aggressively, this feedback can make the underlying move in dramatic and unexpected ways.

In extreme cases, unscrupulous traders deliberately move the underlying to trigger barrier options. A small push on the stock price creates a huge payoff change in the exotic option. The model completely ignores this possibility.

Defect 6: Risk Cannot Be Fully Eliminated

Black-Scholes assumes: Delta hedging eliminates all risk.

If we cannot hedge continuously, if there are transaction costs, if volatility is random, if prices jump, then there is always residual risk. We need a framework for valuing that risk. This leads to utility theory, where you assign a value to a random outcome based on your risk preferences.

Defect 7: American Exercise Is Not “Optimal”

Black-Scholes assumes: American options are exercised optimally.

But optimal for whom? The optimal exercise time for the option holder is not the same as what the writer assumes. The holder might exercise at a time that seems “non-optimal” from the writer’s perspective but is actually quite rational from the holder’s viewpoint. This mismatch can have a large impact on the writer.

Defect 8: Dividends Are Not Deterministic

Black-Scholes assumes: Dividends are deterministic.

Companies can change their dividend policies. The amount and timing of dividends are uncertain. For options with long maturities, this uncertainty can significantly affect pricing.

Defect 9: Returns Are Not Independent

Black-Scholes assumes: There is no serial autocorrelation in returns.

Each day’s return is supposed to be independent of every previous day. But data shows some autocorrelation in returns. Today’s price move is not completely independent of yesterday’s. The standard random walk assumption does not capture this.

Non-Linearity and Static Hedging

Many of the improved models (transaction costs, uncertain parameters, crash modeling, speculation) produce non-linear pricing equations. This means the value of a portfolio is not the sum of the values of individual contracts. The value of a contract depends on what else is in your portfolio, and specifically on what you hedge it with.

This leads to a beautiful idea: optimal static hedging. Instead of dynamic delta hedging (which costs money and is imperfect), you can statically hedge your exotic with traded vanilla contracts. If the hedge is perfect, you do not even need a model. The exotic’s price is determined entirely by the cost of the hedge portfolio.

The formula for the marginal value of a contract statically hedged with $\lambda$ units of another contract is:

$$V_{NL}(\text{contract} + \lambda \cdot \text{hedge}) - V_{NL}(\lambda \cdot \text{hedge}) - \lambda \cdot \text{cost}$$

Optimize over $\lambda$ to get the best price. Incorporate bid-offer spread. Use as many hedging contracts as you want. If you can perfectly replicate all cash flows, the model drops out entirely and the price is model-independent.

The Honest Summary

Black-Scholes has many faults. It is easy to build models that improve on it from a mathematical perspective. But it is nearly impossible to improve on its commercial success. The model is used everywhere, confidently, in situations it was never designed for. And it usually works well enough.

The rest of the book is about those cases where “well enough” is not good enough, and what you can do about it.

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Next: Discrete Hedging