Options and Option Strategies

Introduction

Options are derivative contracts whose value depends on an underlying asset such as a share, bond, currency, commodity, index, or futures contract. Unlike a forward or futures contract, an option gives its holder a right rather than an obligation. The buyer pays an upfront premium for this right, while the seller, or writer, receives the premium and assumes a contingent obligation.

Options are central to risk management because they can limit downside exposure while preserving some or all upside potential. They are also used for income generation, speculation, portfolio insurance, and the construction of complex positions whose risk depends on both the direction and the volatility of the underlying asset.

This chapter develops the mechanics of calls and puts, payoff and profit profiles, moneyness, put–call parity, and major option strategies. The analysis uses standard European options unless otherwise stated. A European option can be exercised only at expiration; an American option can be exercised at any time up to and including expiration.

Learning Objectives

By the end of this chapter, you should be able to:

1. Distinguish calls from puts and explain the rights and obligations of buyers and writers.
2. Calculate option payoffs and profits at expiration.
3. Interpret moneyness and the components of an option premium.
4. Apply put–call parity and identify arbitrage relationships.
5. Analyze protective puts, covered calls, spreads, straddles, and collars.
6. Select option strategies for hedging, income, or directional and volatility views.
7. Evaluate the risks, limitations, ethical obligations, and governance requirements associated with options.

1. Fundamental Structure of an Option

A call option gives its buyer the right to buy the underlying asset at a predetermined strike price or exercise price, denoted by K. A put option gives its buyer the right to sell the underlying asset at K. The date on which the option expires is the expiration date, and the premium is the price paid by the buyer to the writer.

The buyer of an option has limited downside: if exercise is disadvantageous, the buyer can allow the option to expire. The writer receives the premium but may face substantial or, for an uncovered call, theoretically unlimited losses.

For one share at expiration, the payoff of a long call is:

The payoff of a long put is:

Here, S_T is the underlying price at expiration, K is the strike price, and max means that the holder chooses the economically greater value between exercise value and zero. These formulas describe payoff, not profit. Profit must subtract the premium paid.

For a call buyer with premium c₀ and a put buyer with premium p₀, profits at expiration are:

If the contract represents N units, the total profit is the per-unit profit multiplied by N, subject to contract specifications and transaction costs.

Breakeven prices

A long call breaks even when the intrinsic value equals the premium:

A long put breaks even when the intrinsic value equals the premium:

These relationships assume one-for-one delivery, no transaction costs, and that the premium is stated per unit of the underlying.

2. Payoff Profiles and Risk–Return Characteristics

A payoff profile shows the value of a position at expiration as the underlying price changes. A long call has zero payoff below the strike and an increasing payoff above the strike. A long put has increasing payoff as the underlying falls below the strike and zero payoff above the strike.

Figure: Expiration payoffs of long call and long put positions, before premiums.

A long call has a maximum loss equal to the premium and potentially unlimited upside. A long put has a maximum loss equal to the premium and a maximum payoff of K per unit if the underlying price falls to zero. For writers, the signs are reversed: the maximum gain is normally the premium, while losses may be substantial.

Worked Example 1: Long call and long put

Assume a share is currently priced at $50. A three-month call with strike $52 costs $3, while a three-month put with strike $52 costs $4. Consider expiration prices of $45, $52, and $60.

For example, at S_T = 60, the call payoff is max(60−52,0)=8, and its profit is $8 − $3 = $5. The put payoff is zero, so its profit is −$4.

3. Moneyness and the Option Premium

Moneyness compares the underlying price with the strike price.

• A call is in the money (ITM) when S_T > K, and a put is ITM when S_T < K.
• An option is at the money (ATM) when the underlying price is approximately equal to the strike.
• A call is out of the money (OTM) when S_T < K, and a put is OTM when S_T > K.

Moneyness is not the same as profitability. An ITM option can still produce a loss if its intrinsic value is less than the premium paid.

An option premium can be divided into intrinsic value and time value:

For a call, intrinsic value is max(S₀−K,0). For a put, it is max(K−S₀,0), where S₀ is the current underlying price. Time value reflects the possibility that the option may become more valuable before expiration. It is generally non-negative for a standard option and tends to decline as expiration approaches, all else equal.

Important determinants of option value include:

1. Underlying price: A higher underlying price generally increases call value and decreases put value.
2. Strike price: A higher strike generally decreases call value and increases put value.
3. Time to expiration: More time usually increases option value because it permits more possible favorable outcomes.
4. Volatility: Higher expected volatility generally increases both call and put values because option holders benefit from favorable extremes while losses are limited.
5. Interest rates: Higher risk-free rates generally increase call values and decrease put values, holding other factors constant.
6. Expected dividends or income: Dividends generally reduce call values and increase put values because the ex-dividend underlying price is expected to be lower.

These directional statements are comparative-static results, not guarantees for every market circumstance. Actual prices also reflect liquidity, supply and demand, credit risk, market conventions, and model assumptions.

4. Put–Call Parity

Put–call parity is a no-arbitrage relationship linking European calls, European puts, the underlying asset, the strike price, and a risk-free zero-coupon investment. For a non-dividend-paying asset:

Here, C₀ is the call price, P₀ is the put price, S₀ is the current asset price, K is the strike, r is the continuously compounded annual risk-free rate, and T is time to expiration in years. The term K e^−rT is the present value of the strike payment.

An equivalent form is:

The relationship follows from comparing two portfolios with identical expiration payoffs:

• Portfolio A: one call plus a risk-free investment paying K at expiration.
• Portfolio B: one put plus the underlying asset.

At expiration, both portfolios are worth max(S_T,K). If their prices differed today, an arbitrageur could buy the cheaper portfolio and sell the more expensive one, assuming frictionless markets and the ability to trade at stated prices.

With a known continuous dividend yield q, the relationship becomes:

For discrete known dividends, the present value of those dividends is subtracted from the spot asset in the appropriate parity expression. American options require additional consideration because early exercise can make the simple European equality an inequality or require an early-exercise premium.

Worked Example 2: Testing parity

Suppose S₀ = 100, K = 100, r = 5%, T = 1, and there are no dividends. The one-year call sells for $10 and the put for $5. The present value of the strike is:

The left side of parity is approximately $10 + $95.12 = $105.12. The right side is $5 + $100 = $105.00. The difference is about $0.12, which may be attributable to rounding, bid–ask spreads, and transaction costs. A much larger difference could indicate an arbitrage opportunity or a difference in contract assumptions.

5. Major Option Strategies

Strategies combine an underlying position with one or more options. They should be evaluated through four questions: What is the market view? What is the maximum gain? What is the maximum loss? At what price or prices does the strategy break even?

5.1 Protective put

A protective put combines ownership of the underlying with a long put. It is analogous to insurance: the investor retains upside participation but establishes a minimum effective sale value near the put strike, before considering the premium.

At expiration, the payoff is:

If the investor buys the asset at S₀ and pays put premium p₀, profit is:

The maximum loss is approximately S₀ + p₀ − K if the asset can fall to zero, assuming K < S₀. The upside is theoretically unlimited, but the premium reduces returns. A protective put is appropriate when an investor wants to remain invested but is unwilling to accept losses below a specified level.

5.2 Covered call

A covered call combines ownership of the underlying with a short call. The investor receives premium income but gives up gains above the strike because the shares may be called away.

The payoff at expiration is:

Profit, assuming purchase at S₀ and call premium c₀, is:

The maximum gain is approximately K − S₀ + c₀, while downside remains substantial if the underlying price falls sharply. The strategy is suitable for a moderately bullish or neutral investor who is willing to sell at the strike. It is not a complete downside hedge.

5.3 Vertical spreads

A vertical spread uses options of the same type and expiration but different strikes. It limits both gain and loss relative to a naked option.

A bull call spread buys a call at lower strike K₁ and sells a call at higher strike K₂, where K₁ < K₂. If the net premium paid is d, the maximum loss is d, maximum gain is K₂−K₁−d, and breakeven is K₁+d. It expresses a moderately bullish view.

A bear put spread buys a put at higher strike and sells a put at lower strike. If the net debit is d, the maximum gain is the strike difference minus d, and the maximum loss is d. It expresses a moderately bearish view.

Credit spreads reverse the financing: the investor receives a net premium but accepts limited potential loss. For example, a bear call spread sells a lower-strike call and buys a higher-strike call. The maximum gain is the credit received, while the maximum loss is the strike difference less that credit.

Worked Example 3: Bull call spread

A trader buys a $50 call for $6 and sells a $60 call for $2. The net debit is $4.

• Maximum loss: $4.
• Maximum gain: ($60 − $50) − $4 = $6.
• Breakeven: $50 + $4 = $54.

At expiration of $57, the lower call is worth $7 and the higher call is worth zero. The spread payoff is $7, and profit is $7 − $4 = $3. At expiration of $65, the long call is worth $15 and the short call costs $5, producing a payoff of $10 and profit of $6, the maximum gain.

5.4 Straddles

A long straddle buys a call and a put with the same strike and expiration. It profits from a sufficiently large movement in either direction and is therefore a volatility strategy rather than a simple bullish or bearish strategy.

If the strike is K, call premium is c₀, and put premium is p₀, the total cost is c₀+p₀. The two breakeven prices are:

The maximum loss is the total premium, occurring when the underlying finishes at the strike. Gains can be substantial on either a large rise or a large fall.

A short straddle sells both the call and put. It profits if the underlying remains close to the strike and implied volatility declines, but it has limited maximum gain—the premiums received—and potentially very large losses. It therefore requires strict risk limits, margin, and monitoring.

5.5 Collars

A collar combines a long underlying position, a long protective put, and a short call. The put establishes a downside floor, while the short call helps finance the put premium. The cost may be reduced substantially and can sometimes be near zero, but the call caps upside.

For a stockholder, a collar is useful when protecting a portfolio value over a defined period is more important than participating in gains above the call strike. The chosen strikes reflect a trade-off between protection and upside. A lower put strike offers cheaper protection but permits more loss; a higher call strike preserves more upside but generates less premium.

6. Integrated Illustrative Case

Case: Hedging a concentrated equity holding

This is an illustrative case. An investor owns 1,000 shares of a technology company currently priced at $80. The shares were acquired at $60, so the investor has a substantial unrealized gain but expects increased market uncertainty over the next six months. The investor does not want to sell immediately because of investment objectives and possible tax consequences, but cannot tolerate a decline below approximately $70.

A six-month $70 put costs $4 per share, and a six-month $90 call can be sold for $3 per share. The investor considers a protective put and a collar.

Protective put. The investor buys 1,000 puts and pays $4,000. The effective minimum value at expiration is approximately $70,000 before considering the premium, and the net floor relative to the current holding value is $66 per share. The investor retains unlimited upside, but pays the full insurance cost.

Collar. The investor buys the $70 put and sells the $90 call. The net premium is $1 per share, or $1,000. At expiration:

• If the share price is $60, the put is worth $10 per share and protects the position. The share-plus-put value is $70 per share before the net premium; after the $1 cost, the effective value is $69 per share.
• If the share price is $80, neither option is exercised. The investor retains the shares, less the $1 net premium.
• If the share price is $100, the short call requires the investor to sell at $90. The investor benefits from the rise up to $90 but not beyond it, less the $1 net premium.

The collar is preferable when the investor values downside protection and accepts a cap on gains. The protective put is preferable when preserving unlimited upside justifies the higher premium. The decision should also consider taxes, liquidity, early assignment risk for American options, dividends, contract size, and the investor’s legal authority to trade derivatives.

7. Sensitivities, Implementation, and Risk Management

Option positions are affected by more than the final underlying price. Common risk sensitivities, often called the Greeks, include:

• Delta: Approximate change in option value for a one-unit change in the underlying price. Long calls generally have positive delta; long puts generally have negative delta.
• Gamma: Change in delta for a one-unit change in the underlying. Long calls and puts generally have positive gamma.
• Theta: Sensitivity to the passage of time. Long options commonly have negative theta because time value tends to decline.
• Vega: Sensitivity to implied volatility. Long calls and puts generally have positive vega.
• Rho: Sensitivity to interest rates. Its sign depends on the option type and conventions.

These are local approximations and are not constant. They can change rapidly near expiration, around the strike, or during stressed markets. A hedge that appears effective under a small price movement may perform very differently after a large gap.

Effective implementation requires:

1. Defining the exposure, risk tolerance, hedge horizon, and target protection.
2. Matching contract notional, expiration, currency, and underlying precisely.
3. Comparing premiums, spreads, liquidity, margin, and execution quality.
4. Stress-testing large price changes, volatility changes, interest-rate changes, and early exercise.
5. Monitoring hedge effectiveness and rebalancing when exposure changes.
6. Recording valuation methods, approvals, collateral, and counterparty exposures.

For exchange-traded options, clearing arrangements can reduce bilateral counterparty risk, although they do not eliminate market, liquidity, operational, or margin risk. Over-the-counter options may be customized but introduce greater documentation and counterparty considerations.

8. Limitations, Ethics, and Governance

Option models rely on assumptions about volatility, distributions, interest rates, dividends, liquidity, and exercise behavior. Actual returns may exhibit jumps, skewness, fat tails, and changing correlations. Implied volatility is a market price, not a guaranteed forecast. A position can lose money even when the directional forecast is correct if the move is too small, occurs too late, or is offset by time decay and transaction costs.

Professionals must explain leverage, assignment, margin calls, liquidity limitations, and worst-case outcomes clearly. Selling uncovered options without appropriate capital and controls can expose an institution to catastrophic losses. Misstating option values, concealing losses, manipulating settlement prices, or recommending unsuitable strategies violates ethical and often legal duties.

Governance should include documented mandates, independent valuation, model validation, segregation of duties, position limits, stress testing, collateral procedures, and escalation thresholds. Technology can support scenario analysis and automated monitoring, but artificial intelligence or algorithmic systems do not remove the need for human accountability. Data quality, model drift, explainability, and controls over automated execution remain essential.

Contemporary Developments

Modern options markets increasingly offer weekly and short-dated contracts, exchange-traded options on indexes and exchange-traded funds, and products linked to volatility. These instruments improve precision but can encourage excessive trading and amplify the effects of time decay. Electronic execution and portfolio analytics permit continuous monitoring of Greeks and scenario exposures. At the same time, rapid market movements, fragmented liquidity, and concentrated short-volatility positions can create procyclical losses. Sound risk management therefore combines quantitative tools with judgment, liquidity planning, and clear governance.

Chapter Summary

Options give buyers rights without obligations and impose contingent obligations on writers. Calls provide the right to buy, while puts provide the right to sell. At expiration, a call payoff is max(S_T−K,0) and a put payoff is max(K−S_T,0). Profit equals payoff less the premium for a buyer and has the reverse sign for a writer.

Moneyness classifies options as in the money, at the money, or out of the money. Premiums consist conceptually of intrinsic value and time value and are influenced by the underlying price, strike, maturity, volatility, interest rates, and dividends. Put–call parity links European calls, puts, the underlying, and the present value of the strike through a no-arbitrage relationship.

Protective puts insure an underlying position while preserving upside. Covered calls generate premium income but cap upside and do not eliminate downside. Vertical spreads limit both gain and loss. Straddles express a view on volatility and large price movements. Collars combine a protective put with a covered call to establish a downside floor while limiting upside. Each strategy should be assessed through its market view, payoff, profit, breakeven points, maximum gain, maximum loss, liquidity, margin, and governance requirements.

Key Terms

Option: A derivative contract giving the buyer a right, but generally not an obligation, to buy or sell an underlying asset.

Call: An option giving the holder the right to buy the underlying at the strike price.

Put: An option giving the holder the right to sell the underlying at the strike price.

Strike price: The predetermined price at which the underlying may be bought or sold.

Premium: The price paid by the option buyer to the writer.

Expiration date: The date on which the option ceases to exist.

European option: An option exercisable only at expiration.

American option: An option exercisable at any time up to expiration.

Payoff: The value of an option position at expiration before considering the premium.

Moneyness: The relationship between the underlying price and the strike price.

Intrinsic value: The immediate exercise value of an option.

Time value: The portion of premium attributable to the possibility of future favorable value.

Breakeven price: The underlying price at which a strategy’s profit is zero.

Put–call parity: A no-arbitrage relationship connecting European calls, puts, the underlying, and the discounted strike.

Protective put: A long underlying position combined with a long put.

Covered call: A long underlying position combined with a short call.

Vertical spread: Options of the same type and expiration with different strikes.

Straddle: A call and put with the same strike and expiration.

Collar: A long underlying, long put, and short call.

Implied volatility: The volatility value implied by an option’s market price under a pricing model.

Review Questions

Short Answer Questions

1. What right does a call buyer obtain, and what obligation does a call writer assume?
2. Distinguish payoff from profit for an option buyer.
3. When is a put in the money?
4. State the no-dividend European put–call parity relationship.
5. Why does a covered call not provide complete downside protection?

Long Answer Questions

1. Explain the determinants of an option premium and distinguish intrinsic value from time value.
2. Compare protective puts and covered calls in terms of objectives, upside, downside, and cash flows.
3. Derive the two breakeven prices of a long straddle and explain its volatility exposure.
4. Explain how put–call parity follows from identical expiration payoffs.
5. Discuss the governance, ethical, liquidity, and model risks of option trading.

Analytical / Application Questions

1. An investor buys a call with strike $40 for a premium of $3. Calculate payoff and profit when the underlying expires at $35, $40, and $48. Identify the breakeven price.
2. A stockholder buys a $90 put for $5 and sells a $110 call for $3 while holding the stock. Analyze the collar’s net premium, downside floor, upside cap, and effective value at expiration prices of $70 and $125.
3. A trader buys a $100 call for $8 and sells a $115 call for $3. Calculate the maximum gain, maximum loss, breakeven, and profit when the underlying expires at $110 and $125.

Multiple Choice Questions with Answers

Question 1

Which statement correctly describes a long call?

A. It gives the holder the right to sell the underlying at the strike price.

B. It gives the holder the right to buy the underlying at the strike price.

C. It obliges the holder to buy the underlying regardless of price.

D. It guarantees a profit whenever the underlying price rises.

Correct Answer: B. It gives the holder the right to buy the underlying at the strike price.

Question 2

A call has a strike price of $50 and a premium of $4. What is its breakeven price at expiration?

A. $46

B. $50

C. $54

D. $200

Correct Answer: C. $54

Question 3

A long put with strike $60 has an underlying price of $52 at expiration. What is its payoff per unit before the premium?

A. $0

B. $8

C. $52

D. $112

Correct Answer: B. $8

Question 4

Which condition describes an in-the-money call?

A. The underlying price is below the strike price.

B. The underlying price equals zero.

C. The underlying price is above the strike price.

D. The premium is zero.

Correct Answer: C. The underlying price is above the strike price.

Question 5

For a non-dividend-paying asset, which equation represents European put–call parity?

A. C₀ + S₀ = P₀ + Ke^(−rT)

B. C₀ + Ke^(−rT) = P₀ + S₀

C. C₀ − P₀ = S₀ + Ke^(−rT)

D. C₀ + P₀ = S₀ − Ke^(−rT)

Correct Answer: B. C₀ + Ke^(−rT) = P₀ + S₀

Question 6

Which strategy combines a long stock position with a long put?

A. Covered call

B. Short straddle

C. Protective put

D. Bear call spread

Correct Answer: C. Protective put

Question 7

An investor buys a call for $6 and sells a higher-strike call for $2. What is the net initial position?

A. A $4 net debit

B. A $4 net credit

C. A $8 net debit

D. No premium exposure

Correct Answer: A. A $4 net debit

Question 8

A long straddle generally benefits most from which outcome, assuming other factors are appropriate?

A. The underlying remains exactly at the strike until expiration.

B. The underlying makes a sufficiently large move in either direction.

C. Time passes with no change in volatility or price.

D. The underlying price is permanently fixed at zero.

Correct Answer: B. The underlying makes a sufficiently large move in either direction.

Question 9

What is the principal trade-off in a collar?

A. It eliminates both downside and upside.

B. It provides downside protection while capping upside.

C. It creates unlimited downside and no premium cost.

D. It requires selling the underlying and buying two calls.

Correct Answer: B. It provides downside protection while capping upside.

Question 10

Which risk is especially associated with a short uncovered call?

A. A maximum loss limited to the premium paid

B. No obligation to deliver the underlying

C. Potentially unlimited loss if the underlying price rises substantially

D. A guaranteed payoff equal to the strike price

Correct Answer: C. Potentially unlimited loss if the underlying price rises substantially