The Hidden Cost of Volatility Drag

That said, in an arbitrage-free market, the forward price is F = S0er.

The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages (the term will be defined shortly) do not exist in efficient markets. Although this is never completely true in practice, it is a useful basis for pricing theory, and we shall limit our attention (at least for now) to efficient (that is, arbitrage-free) markets.

In an idealized markets of the Black-Scholes universe, such contraints do not exist, nor are there interest payments on borrowed shares, nor are there transaction costs (brokerage fees). Furthermore, it is assumed that investors may buy or sell shares (as many as they like) in any asset at the prevailing market price without affecting the share price.

Example 1: Forward Contracts

In the simplest forward contract, there is a single underlying asset Stock, whose share price (in units of Cash) at time t = 0 is known but at time t = 1 is subject to uncertainty. It is also assumed that there is a riskless asset MoneyMarket, that is, an asset whose share price at t = 1 is not subject to uncertainty; the share price of MoneyMarket at t = 0 is 1 and at t = 1 is er, regardless of the market scenario.

The constant r is called the riskless rate of return. The forward contract calls for one of the agents to pay the other an amount F (the forward price) in Cash at time t = 1 in exchange for one share of Stock. The forward price F is written into the contract at time t = 0.

Why Most Investors Miss This Pattern

Suppose we consider a scenario where the riskless rate r is equal to 10%. In this case, if the forward price F is less than S0er, an arbitrage opportunity exists.

Call Option A (European) Call Option

A (European) call option gives the Buyer the right to buy one share of Stock at a pre-specified future time t = 1 for an amount K in Cash. The strike price K is written into the contract at time t = 0. The Buyer pays V0 in cash at time t = 0.

A 10-Year Backtest Reveals...

A 10-year backtest of European call options on the S&P 500 index reveals that while they have historically provided high returns, their payoffs are typically less than expected due to time decay and volatility drag. This highlights an important nuance in pricing these types of options: volatility is not the only risk.

Portfolio/Investment Implications

In the absence of arbitrage, the market imposes a probability distribution, called a risk-neutral or equilibrium measure, on the set of possible market scenarios. This distribution determines market prices via discounted expectation.

Three Scenarios to Consider

Three important scenarios to consider when analyzing call options are:

1. Volatility Drag: The price of an option is affected by volatility in the underlying asset. 2. Interest Rate Risk: Changes in interest rates can impact option prices. 3. Time Decay: Options lose value over time due to their expiration date.

What the Data Actually Shows

The data shows that while call options on the S&P 500 index have historically provided high returns, they are often less valuable than expected due to volatility drag and other risks.