No-arbitrage probabilities allow us to price options by replication
The insight embedded in Black-Scholes is that, under a certain set of assumptions, **the fair price of an option must be the cost of replicating its payoff under many scenarios. Any other price offers the opportunity for a risk-free profit.
Have you ever wondered why the Black Scholes “drift” term for a stock is the risk-free rate and not an equity risk premium (like you’d expect from another type of pricing model ie CAPM) or the stock’s WACC?**
A position in a derivative and an opposing position in its replication is a riskless portfolio. Therefore that portfolio only needs to be discounted by the risk-free rate. Option pricing derived from a no-arbitrage replication strategy means we should use the risk-free rate to model a stock’s return.
<aside> ‼️ **What seasoned option traders get wrong
**Outside of the option pricing context, the risk-free rate is the wrong assumption for drift!
From Philip Maymin’s *Financial Hacking:*
*One of the most common mistakes that even highly experienced practitioners make is to act as if the assumptions of Black-Scholes (lognormal, continuous distribution of returns, no transactions costs, etc.) mean that we can always arbitrarily assume the underlying grows at the riskfree rate r instead of a subjective guess as to its real drift μ. But this is not quite accurate. The insight from the Black-Scholes PDE is that the price of a hedged derivative does not depend on the drift of the underlying. The price of an unhedged derivative, for example, a naked long call, most certainly does depend on the drift of the underlying.
Let's say you are naked long an at-the-money one-year call on Apple, and you will never hedge. And suppose Apple has very low volatility. Then the only way you will profit is if Apple's drift is positive; suppose Apple has very low volatility. Then the only way you will profit is if Apple's drift is positive…if it drifts down, your option expires worthless. But if you hedge the option with Apple shares, then you no longer care what the drift is. You only make money on a long option if volatility is higher than the initial price of the option predicted.
The drift term of the underlying only disappears when your net delta is zero. In other words, an unhedged option cannot be priced with no-arbitrage methods*
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<aside> 💡 Takeaway: Arbitrage Pricing Theory
Sometimes called the Law of One Price, the idea contends that the fair price of a derivative must be equal to the cost of replicating its cash flows. If the derivative and cost to replicate are different then there is free money by shorting one and buying the other.
This approach is how arbitrageurs and market-makers price a wide range of financial derivatives in every asset class including:
These derivatives are the legos from which more exotic derivatives are constructed. </aside>
Let’s recap the logic:
Arbitrage ensures that the price of a derivative trades in line with the cost to replicate it.
A master portfolio comprising:
This master portfolio is riskless.
A riskless portfolio will be discounted to present value by a risk-free rate otherwise there is free money to be made.
The prevailing prices of derivatives imply probabilities.
Those probabilities are risk-neutral arbitrage-free probabilities.
But those probabilities don’t need to reflect real-world probabilities. They are simply an artifact of a riskless arbitrage if it exists.
This can lead to a difference in opinion where the arbitrageur and the speculator are happy to trade with each other.