Speculating with Options: The Non-Hedger's Perspective

Almost everything in quantitative finance is built around one assumption: you hedge. You buy the option, you delta hedge, you eliminate risk, and the drift of the stock does not matter. Beautiful theory. But Chapter 59 of Wilmott’s book asks an uncomfortable question: what if you are not hedging?

A few minutes on any trading floor will show you that plenty of people buy options to speculate, not to hedge. They have a view on where the stock is going. They think it will go up 15% next year. They want to leverage that view with options. For these people, the Black-Scholes price is not really the “value” of the option. Their value depends on where they think the stock is headed.

Why Black-Scholes Does Not Care About Drift

Let me remind you of something remarkable about Black-Scholes. Two people can disagree completely about the expected return of a stock and still agree on the price of an option on that stock. One person thinks the stock will return 30% a year. Another thinks 5%. If they agree on the volatility, they agree on the option price. The drift mu does not appear anywhere in the Black-Scholes equation.

This happens because of hedging. When you continuously delta hedge, you eliminate the exposure to the stock’s drift. What remains is driven only by volatility. It is elegant and powerful. But it is also irrelevant to the speculator who has no intention of hedging.

Wilmott makes this point clearly: there is nothing wrong with the Black-Scholes framework for the market as a whole. If an option is mispriced relative to Black-Scholes, someone will arbitrage it. But for an individual speculator making a risky bet, the option’s worth to them personally is a different number than the Black-Scholes price.

The Present Value of Expected Payoff

If you are speculating, the natural question is: what is my expected profit? In option terms, this means the present value of the expected payoff under the real probability distribution (not the risk-neutral one).

The math is similar to Black-Scholes but with one crucial difference. Instead of the risk-free rate r appearing in front of the delta term in the PDE, you get the real drift mu. This seemingly tiny change is fundamental. It means the option “value” to a speculator depends on the drift, which in turn means you need a model for where the stock is going.

Wilmott works through a concrete example. A call option struck at 100, one year to expiry, 20% volatility, drift of 15%, and interest rate of 5%. The Black-Scholes value is the lower curve when you plot value against stock price. The present value of the real expected payoff is the upper curve. When mu is greater than r, the real expected payoff of a call exceeds the Black-Scholes value. This makes financial sense: if you believe the stock will outperform the risk-free rate, a call option is worth more to you than to someone who just hedges.

Of course, “expect” is doing a lot of heavy lifting here. This is an average over many possible outcomes. Some of those outcomes involve the call expiring worthless. The higher expected return is compensation for the risk of speculation that is absent in a hedged position.

Measuring the Risk

Expected return alone is not enough. You also need to know the risk. For this, Wilmott calculates the standard deviation of the payoff around its mean.

He introduces a function G(S,t) that gives the expected value of the square of the present value of the payoff. This satisfies its own PDE (a backward Kolmogorov equation with the real drift). From G and the expected value V, you can compute the standard deviation.

With both return and risk in hand, you can make a risk/reward plot. The reward is defined as the log of the ratio of expected payoff to Black-Scholes cost, scaled by time. The risk is the standard deviation, similarly scaled. These definitions connect to the standard measures used in Modern Portfolio Theory.

For a call option with an upward-drifting stock, the risk/reward plot looks appealing. For a put option with the same upward drift, the plot shows decreasing return for increasing risk. Obviously, if you believe the stock is going up, calls are more attractive than puts. This is not profound, but having quantitative measures lets you compare specific options and choose the one with the best profile for your view.

When Your View Changes: Models for Drift

Real speculators know their view might change. Wilmott introduces two models for a randomly varying drift rate.

Diffusive drift. The drift itself follows a stochastic differential equation. The option value depends on S, mu, and time, giving a two-factor PDE. Flexible but possibly overkill for how traders actually think.

Jump drift. Simpler and more practical. The drift can be in one of two states: high or low, with jumps between states governed by a Poisson process. You believe the stock has 15% drift now, but it might drop to 0% at some random point. This gives two coupled PDEs, one per drift state.

Wilmott’s numerical example uses 15% high drift, 0% low drift, 20% volatility, 5% interest rate. The option value in the high-drift state is above Black-Scholes, and the low-drift state is below. Black-Scholes sits neatly between the two: a view-independent price between the optimistic and pessimistic speculator valuations.

Early Closing: Sell When Your View Turns Bad

Here is where the chapter gets really practical. You bought a call because you believed in a 15% upward drift. Six months later, the stock has dropped and you no longer believe in that drift. What do you do?

The basic models above price in the possibility of a drift change, but they assume you hold the option until expiry. In reality, if your view turns negative, you would sell the option and close the position. Why sit on a losing bet?

Wilmott models this as an early exercise problem, very similar to American options. You impose a constraint: the option value to you must always be at least as much as the Black-Scholes market value. If your speculative value falls below what the market will pay, you sell. This generates a free boundary problem.

For the two-state jump drift model, something interesting happens. The option value in the low-drift state turns out to be exactly the Black-Scholes value. The interpretation is clean: sell the option as soon as you believe the drift has switched to the low state. Do not wait around hoping it will come back.

The ability to close the position adds value. With the diffusive drift model and the same parameters, the option value at S=100 and zero drift goes from 11.64 (holding to expiry) to 11.84 (with early closing). That difference comes entirely from the optionality of being able to exit.

To Hedge or Not to Hedge

The final section brings hedging and speculation together. Pure speculation means no hedging at all. Pure Black-Scholes means full delta hedging. What about something in between?

Wilmott proposes an intuitively appealing strategy: speculate when you expect to make money, hedge when you would expect to lose. If the option’s delta is positive and you believe the stock will drift up faster than the risk-free rate, do not hedge. You expect the unhedged position to outperform. But when the delta is negative, hedge, because without hedging you expect to lose relative to the risk-free rate.

Mathematically, this means choosing a hedge ratio that bounces between 0 (no hedge, full speculation) and the Black-Scholes delta (full hedge). The resulting PDE for the option value is non-linear because the choice depends on the sign of the delta. The speculator-hedger does better (in expectation) than pure Black-Scholes hedging by capturing upside from drift while limiting downside through hedging.

The Big Picture

Chapter 59 is a reality check. Most of quant finance pretends that everyone hedges all the time. In practice, a huge portion of option activity is speculative. People have views. Those views matter for their personal valuation of a contract.

The tools here let a speculator quantify the value of their view. Expected payoff, standard deviation, risk/reward profiles, optimal exit strategies, and even a hybrid approach that speculates when conditions are favorable and hedges when they are not. None of this replaces Black-Scholes for market pricing. But for the person putting on a directional bet, it adds a quantitative framework to what would otherwise be pure gut feeling.

One mustn’t get too hung up on delta hedging and risk neutrality, as Wilmott puts it. The concepts are critical, but there are times when their relevance is questionable. And speculation is one of those times.

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