Option Volatility & Pricing

1. Options are Insurance Contracts with Unique Risk/Reward Profiles.

In option trading all rights lie with the buyer and all obligations with the seller.

Fundamental nature. Options are financial contracts granting the buyer the right, but not the obligation, to buy (call) or sell (put) an underlying asset at a fixed price (strike) by a specified date (expiration). This inherent asymmetry defines their unique risk/reward: buyers face limited risk (the premium paid) and potentially unlimited reward, while sellers face limited reward (the premium received) and potentially unlimited risk. This contrasts sharply with futures contracts, where both parties have obligations.

Key terminology. Understanding the language of options is paramount.

• Call Option: Right to buy the underlying.
• Put Option: Right to sell the underlying.
• Underlying: The asset (stock, commodity, index, futures) the option controls.
• Exercise Price (Strike): The fixed price at which the transaction occurs.
• Expiration Date: The last day the option can be exercised.
• American vs. European: American options can be exercised anytime before expiration; European, only at expiration.
• Intrinsic Value: The immediate profit if exercised (e.g., a $100 call when the stock is $105 has $5 intrinsic value).
• Time Value: The portion of the premium beyond intrinsic value, reflecting the chance of future price movement.
• In-the-money: Has intrinsic value.
• Out-of-the-money: No intrinsic value.
• At-the-money: Strike equals underlying price.

Elementary strategies. Basic option strategies involve simply buying or selling calls and puts. A long call profits from rising prices, a long put from falling prices. Conversely, a short call profits from falling prices, a short put from rising prices. Expiration graphs illustrate these profit/loss profiles, highlighting the limited risk for buyers and unlimited risk for sellers, a crucial distinction from direct ownership of the underlying asset.

2. Theoretical Models Quantify Option Value, but Volatility is the Elusive Key.

The concept of speed is vital in trading options.

Quantifying expectations. Option trading demands more than just predicting market direction; it requires anticipating market speed. Theoretical pricing models, like the Black-Scholes model, translate these expectations into a quantifiable "theoretical value" for an option. This value helps traders identify mispriced options by comparing it to the market price, much like an actuary prices an insurance policy based on probabilities.

Model inputs. The Black-Scholes model, in its various forms (for stocks, futures, currencies), requires five core inputs:

• Exercise Price: Fixed by contract.
• Time to Expiration: Fixed by contract, but constantly decreasing.
• Underlying Price: Current market price.
• Risk-Free Interest Rate: Cost of carrying funds over the option's life.
• Volatility: The expected speed of price movement in the underlying asset.

Volatility's dominance. Of all inputs, volatility is the most critical and challenging to estimate, as it cannot be directly observed. It dictates the likelihood of the underlying asset reaching or surpassing the strike price. The model assumes that underlying price changes follow a random walk, resulting in a lognormal distribution of prices at expiration, where volatility acts as the standard deviation of these changes. This lognormal assumption accounts for unlimited upside potential while bounding downside risk at zero.

3. The "Greeks" are Your Compass for Navigating Market Changes.

Without a very healthy respect for the risks of option trading and a full understanding of risk management techniques, today's profits can quickly turn into tomorrow's losses.

Beyond direction. Option values are influenced by multiple factors beyond just the underlying asset's price. The "Greeks" are sensitivities that quantify how an option's theoretical value changes with respect to these factors, providing essential tools for risk management and strategy adjustment.

Key sensitivities:

• Delta (Δ): Measures the option's price sensitivity to a one-point change in the underlying asset. It also represents the hedge ratio (how many underlying units to offset an option) and the approximate probability of an option expiring in-the-money. Calls have positive deltas (0-100), puts have negative deltas (0 to -100).
• Gamma (Γ): Measures the rate at which an option's delta changes. High gamma means delta changes rapidly, indicating high directional risk. At-the-money options have the highest gamma, especially near expiration. Long options have positive gamma; short options have negative gamma.
• Theta (Θ): Measures the option's time decay—how much value it loses per day, all else being equal. Long options have negative theta (lose value over time); short options have positive theta (gain value over time). High gamma often correlates with high theta.
• Vega (ν): Measures the option's sensitivity to a one-percentage-point change in implied volatility. Long options have positive vega (benefit from rising volatility); short options have negative vega (benefit from falling volatility). At-the-money options have the highest vega in absolute terms.
• Rho (ρ): Measures the option's sensitivity to a change in interest rates. Generally less significant than other Greeks, its impact varies by underlying asset type (stocks vs. futures) and settlement method.

Risk management imperative. Understanding the Greeks allows traders to assess and manage multi-dimensional risks. A delta-neutral position aims to remove directional bias, but gamma, theta, and vega risks remain. Traders must constantly monitor these sensitivities, as they change with market conditions, time, and volatility, to avoid "paralysis through analysis" while still respecting the inherent risks.

4. Spreading is the Art of Risk Reduction and Leveraging Market Views.

Spreading is simply a way of enabling an option trader to take advantage of theoretically mispriced options, while at the same time reducing the effects of short-term changes in market conditions so that he can safely hold an option position to maturity.

Beyond naked positions. While buying or selling single options (naked positions) offers high leverage, it comes with significant risk and limited margin for error. Spreading involves taking simultaneous, opposing positions in different instruments (options or underlying assets) to reduce overall risk, increase the margin for error in market forecasts, and leverage specific views on volatility or direction.

Why spread?

• Risk Reduction: Mitigates the impact of adverse short-term market fluctuations.
• Increased Margin for Error: Allows for profitable outcomes even if forecasts aren't perfectly accurate.
• Leveraging Specific Views: Enables traders to profit from opinions on volatility (e.g., expecting high or low volatility) or relative price movements between assets, rather than just outright direction.

Types of volatility spreads: Spreads are categorized by their risk/reward profiles and sensitivities:

• Backspreads (Long Straddles/Strangles, Short Butterflies): Positive gamma, negative theta, positive vega. Profit from large market movements and rising volatility.
• Ratio Vertical Spreads (Short Straddles/Strangles, Long Butterflies): Negative gamma, positive theta, negative vega. Profit from quiet markets and falling volatility.
• Time Spreads (Calendar Spreads): Negative gamma, positive theta, positive vega. Profit from quiet markets and rising implied volatility (as short-term options decay faster than long-term ones, and long-term options are more sensitive to volatility changes).

Choosing the best spread. The "best" spread isn't always the one with the highest theoretical edge. It's the one that offers the most favorable risk/reward tradeoff given a trader's market outlook and risk tolerance. This involves analyzing the spread's implied volatility (break-even volatility) and its sensitivities to various market changes.

5. Arbitrage Exploits Synthetic Relationships for "Riskless" Profit.

There are no riskless strategies. There are only strategies with greater or lesser risk.

Synthetic positions. Options can be combined with other options or the underlying asset to create "synthetic" positions that mimic the characteristics of another instrument. The fundamental relationship, known as put-call parity, states: call price - put price = underlying price - exercise price + carrying costs - dividends. If this equality doesn't hold, an arbitrage opportunity exists.

Conversions and reversals. These are classic arbitrage strategies:

• Conversion: Buy underlying + sell call + buy put. Profits when the synthetic underlying (call - put) is overpriced relative to the actual underlying.
• Reversal: Sell underlying + buy call + sell put. Profits when the synthetic underlying is underpriced relative to the actual underlying.
  These strategies aim to lock in a small, virtually risk-free profit by simultaneously buying the cheaper asset and selling the more expensive, expecting the prices to converge.

Advanced arbitrage: Boxes and jelly rolls.

• Box: A combination of a conversion at one strike and a reversal at another strike (e.g., long a call/short a put at strike A, short a call/long a put at strike B). It eliminates underlying exposure, effectively becoming a pure interest rate play. Its value at expiration is the difference between the strike prices, discounted by interest.
• Jelly Roll (Roll): A combination of synthetic positions across different expiration months but the same strike (e.g., long a call/short a put for near month, short a call/long a put for far month). Its value reflects the cost of carry between the two expiration periods.

Arbitrage risks. While often called "riskless," arbitrage strategies carry risks:

• Interest Rate Risk: Changes in borrowing/lending rates can erode profits.
• Execution Risk: Difficulty in simultaneously executing all legs of the trade at favorable prices.
• Pin Risk: At expiration, if the underlying price is exactly at the strike, uncertainty about exercise/assignment can lead to unwanted open positions.
• Settlement/Dividend Risk: Differences in settlement procedures (cash vs. physical) or unexpected dividend changes can impact profitability, especially in futures or stock options.

6. American Options Offer Early Exercise Value, but Not Without Nuance.

In a market where options are subject to early exercise, no option ought to be trading at less than parity.

The right of early exercise. American options grant the holder the right to exercise at any time before expiration, a feature not present in European options. This right adds value, making American options theoretically more valuable than their European counterparts. However, early exercise is only optimal under specific conditions.

Conditions for early exercise:

• Futures Options: Early exercise is primarily driven by the desire to capture interest on the option's intrinsic value, but only if the option is subject to stock-type settlement. If futures-type settlement, there's no economic benefit.
• Stock Option Calls: Exercised early to capture an upcoming dividend. This is typically considered only on the day before the stock goes ex-dividend.
• Stock Option Puts: Exercised early to capture interest on the proceeds from selling the stock at the strike price. This is more likely when interest rates are high or after a stock goes ex-dividend.

Key indicators for optimal early exercise:

• Trading at Parity: The option's market price equals its intrinsic value. If it trades above parity, selling the option is better; if below, an immediate arbitrage exists.
• Delta Close to 100: Indicates the option is deep in-the-money, with minimal time value or "insurance" left to forfeit.

American vs. European models. Models like Cox-Ross-Rubenstein and Whaley are designed to value American options more accurately than Black-Scholes by accounting for early exercise. However, their complexity and reliance on precise dividend/interest inputs mean that practical advantages over simpler models can be marginal, especially for short-term or futures options.

7. Hedging with Options Transfers Risk, Balancing Cost and Protection.

The cost may be immediately apparent because it requires an immediate cash outlay. But the cost may also be more subtle, either in terms of lost profit opportunity, or in terms of additional risk under some circumstances.

Purpose of hedging. Hedging uses options to transfer or mitigate the risk associated with an existing position in an underlying asset. It's akin to buying insurance: you pay a premium (cost) to protect against adverse price movements, accepting that you might forgo some potential profit.

Basic hedging strategies:

• Protective Puts/Calls:
  • Protective Put: Buy a put to protect a long underlying position. Limits downside risk while retaining upside potential. Cost is the put premium. (Synthetic long call).
  • Protective Call: Buy a call to protect a short underlying position. Limits upside risk while retaining downside potential. Cost is the call premium. (Synthetic long put).
• Covered Writes:
  • Covered Call (Buy/Write): Own underlying + sell a call. Generates income (premium) and offers limited downside protection up to the premium received, but caps upside profit potential. (Synthetic short put).
  • Covered Put: Short underlying + sell a put. Generates income (premium) and offers limited upside protection, but caps downside profit potential. (Synthetic short call).

Fences (Collars): A popular strategy combining a protective option with a covered option to reduce net cost, sometimes even for a credit. For a long underlying position, this involves buying an out-of-the-money put (protection) and selling an out-of-the-money call (income). This limits both downside risk and upside potential, creating a defined profit/loss range.

Portfolio insurance: A dynamic hedging strategy where a portfolio's exposure to an underlying asset is continuously adjusted (by buying/selling the asset or its futures) to mimic the payoff of a protective put. This creates a synthetic put, insuring the portfolio against losses below a certain level. While theoretically elegant, it can incur high transaction costs and is vulnerable to market gaps.

8. Volatility is Mean-Reverting, but Implied Volatility Reflects Market Consensus.

Volatility may rise well above 12%, or fall well below 11%, but eventually it always seems to return to this area.

Volatility's nature. Unlike asset prices, which can trend indefinitely, volatility tends to be mean-reverting. It fluctuates around a long-term average, eventually returning to that mean. Volatility also exhibits serial correlation, meaning current volatility is a good predictor of near-term future volatility.

Forecasting volatility. Traders use various methods to forecast future volatility, which is the true input for option valuation:

• Historical Volatility: Calculated from past price movements over specific periods (e.g., 30-day, 60-day). Longer periods reveal mean volatility; shorter periods show recent trends.
• Weighted Averages: Assigning more weight to recent data or data corresponding to the option's remaining life.
• Implied Volatility: Derived from current option prices using a theoretical model. It represents the market's consensus expectation of future volatility.

Implied vs. historical. Implied volatility often tracks historical volatility but tends to fluctuate less severely due to mean-reversion. It also incorporates market expectations of future events (e.g., economic reports, earnings announcements) that might not be reflected in historical data. When implied volatility is significantly higher than historical, options are considered "expensive" (and vice-versa).

Practical approach. Instead of seeking a perfect forecast, traders often assess the "volatility climate":

• Long-term mean volatility: The historical average.
• Recent historical volatility: Current trend relative to the mean.
• Implied volatility: Market's current expectation and its trend.
  This holistic view helps determine if a long-volatility (positive vega) or short-volatility (negative vega) strategy is appropriate, balancing the mean-reverting tendency with short-term trends and market sentiment.

9. Index Derivatives Offer Broad Market Exposure, with Unique Arbitrage Challenges.

For all of the foregoing reasons, no book on options would be complete without at least some discussion of index options.

Broad market access. Stock index futures and options allow traders to gain exposure to an entire market (e.g., S&P 500) without buying individual stocks. This simplifies portfolio management and hedging for institutional investors. Indices are calculated either price-weighted (higher-priced stocks have more impact) or capitalization-weighted (larger companies have more impact).

Index futures valuation. Index futures are cash-settled, not physically delivered. Their fair value is determined by the underlying index price, adjusted for the cost of carry (interest saved by not holding stocks) and dividends forgone (by not owning stocks). Futures Price = Index Price + Carrying Costs - Dividends. Any deviation from this fair value creates an arbitrage opportunity.

Index arbitrage (program trading). Traders exploit mispricings between index futures and the underlying basket of stocks. A "buy program" involves buying undervalued stocks and selling overvalued futures; a "sell program" is the reverse. However, practical challenges exist:

• Transaction Costs: Can erode small arbitrage profits.
• Execution Risk: Difficulty in simultaneously trading a large basket of stocks.
• Short-Selling Restrictions: Limits on selling stocks not owned.
• Settlement Risk: Futures' daily cash settlement vs. stocks' paper profits can create cash flow issues.

Index options nuances. Options on cash indices (e.g., OEX) are also cash-settled. Their valuation is complex due to the aggregate nature of dividends and the "phantom variable" – the extra early exercise value from the ability to exercise after the index is fixed but before other related markets close. This can lead to systematic biases, where implied volatility often exceeds historical volatility, reflecting hedgers' willingness to pay a premium for portfolio protection.

10. Models are Tools, Not Oracles: Understand Their Flaws and Use Common Sense.

Only a trader who fully understands what a model can and cannot do will be able to make the model his servant rather than his master.

Inherent model limitations. Traditional option pricing models, while powerful, rest on several simplifying assumptions that often diverge from real-world market behavior:

• Frictionless Markets: Assumes zero transaction costs, unlimited borrowing/lending at a single rate, and no tax consequences. Reality is far more complex.
• Constant Interest Rates/Volatility: Assumes these inputs remain fixed over the option's life, which is rarely true. Volatility, in particular, is dynamic and can cluster.
• Continuous Trading: Assumes prices move smoothly without gaps, ignoring overnight gaps or sudden jumps due to news. This is a major flaw, especially for short-term, high-gamma options.
• Volatility Independent of Price: Assumes volatility doesn't change based on whether the underlying asset is rising or falling.
• Lognormal Distribution: Assumes underlying price changes are normally distributed, leading to a lognormal distribution of prices at expiration. Real-world distributions often show "fat tails" (more extreme moves) and "leptokurtosis" (more small moves, fewer intermediate moves).

Volatility skews. The most visible evidence of model flaws is the volatility skew, where options with different strike prices have different implied volatilities. This "smile" or "smirk" in implied volatility suggests the market assigns higher probabilities to extreme (out-of-the-money) moves than a simple lognormal distribution would predict. Traders often adjust their models by incorporating these skews, shifting them up/down or side-to-side to align with market sentiment.

Practical wisdom. Despite their imperfections, models are invaluable tools. However, slavish adherence without critical thought is dangerous. Successful traders understand:

• Risk Management: Prioritize managing risk over maximizing theoretical profit.
• Liquidity: Trade in liquid markets to ensure easy entry/exit.
• Common Sense: Supplement model outputs with intuition, market feel, and experience, especially as options approach expiration when model assumptions are most strained.
• Adaptability: Be prepared to adjust strategies when market conditions invalidate model assumptions.

Last updated: September 8, 2025