Options strategies and greeks - Calls puts and put call pari...

Learning Outcomes

This article explains core option strategies and the main Greeks relevant for the CFA Level 1 exam, including:

• Describing the structure, payoff, and profit profiles for long and short calls and puts, and linking these to directional views, leverage, and downside risk control.
• Comparing risk–return characteristics such as maximum gain, maximum loss, and breakeven price across individual options and common exam-style scenarios.
• Identifying and evaluating combined strategies—including protective puts, covered calls, and long straddles—and interpreting their payoff shapes, capital requirements, and appropriate market outlooks.
• Interpreting delta, gamma, theta, vega, and rho qualitatively, and using them to assess price, volatility, time-decay, and interest-rate sensitivities for both long and short positions.
• Applying put–call parity to compute missing option or stock prices, construct synthetic stock, bond, call, and put positions, and determine when parity violations imply arbitrage opportunities.
• Recognizing frequent CFA Level 1 pitfalls, including confusing payoff with profit, misreading payoff diagrams, mishandling premiums on short positions, and misapplying parity to American or dividend-paying options.

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are required to understand options strategies and Greeks with a focus on the following syllabus points:

• Describe payoff profiles and strategies involving call and put options.
• Distinguish between long and short positions and their risk–return characteristics.
• Interpret and apply the main Greeks (delta, gamma, theta, vega, rho) in risk management.
• Explain the concept and uses of put–call parity for pricing, arbitrage, and synthetics.
• Apply option strategies and Greeks knowledge in practical calculation questions.

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. What is the maximum possible loss for an investor who buys a call option and holds it to expiration?
   1. Unlimited
   2. The option premium paid
   3. The strike price
   4. The asset price at expiration
2. A portfolio is long a European call and short a European put on the same non-dividend-paying stock, with the same strike and expiration. Ignoring interest, this position is economically closest to:
   1. A long position in the stock
   2. A short position in the stock
   3. A long straddle
   4. A risk-free bond
3. Which Greek primarily measures the sensitivity of an option’s value to changes in the price of the reference asset?
   1. Delta
   2. Vega
   3. Theta
   4. Rho
4. Under standard CFA Level 1 assumptions, put–call parity for options on individual stocks is most directly applicable to:
   1. American options on non-dividend-paying stocks
   2. European options on non-dividend-paying stocks
   3. American options on dividend-paying stocks
   4. European options on dividend-paying stocks

Introduction

Options provide tailored risk and return profiles for investors and risk managers. Fully understanding call and put option payoffs, the impact of option strategies, and the interpretation of the Greeks is essential for valuation, hedging, and detecting arbitrage opportunities. This article covers these key option strategies, the role and calculation of standard Greeks, and the importance of put–call parity in pricing and strategy construction.

Key Term: call option
The right, but not the obligation, to buy a reference asset at a specified strike price on or before a specified expiration date.

Key Term: put option
The right, but not the obligation, to sell a reference asset at a specified strike price on or before a specified expiration date.

Key Term: strike price
The fixed price specified in an option contract at which the reference asset can be bought (call) or sold (put) if the option is exercised.

Key Term: expiration date
The last date on which the option can be exercised. After this date, the option ceases to exist.

Key Term: option premium
The price paid by the buyer (or received by the seller) for the option, reflecting exercise value, time value, and risk.

A key feature of options is asymmetry: the buyer has limited downside (the premium) but potentially large upside; the seller has the opposite payoff. This asymmetry motivates many of the strategies and risk measures discussed in this article.

Key Term: exercise value
For a call, the maximum of zero and the asset price minus the strike. For a put, the maximum of zero and the strike minus the asset price. It is the value if the option were exercised immediately.

Key Term: time value
The portion of the option premium above exercise value, reflecting the probability that the option may move into the money before expiration.

Key Term: moneyness
A description of the relationship between the asset price and the strike: in the money (ITM), at the money (ATM), or out of the money (OTM).

Key Term: European option
An option that can be exercised only at expiration.

Key Term: American option
An option that can be exercised at any time up to and including expiration.

Level 1 questions often give you the option premium, strike price, and asset price and then ask for exercise value, time value, or payoff at expiration. Keeping the definitions clear avoids many errors.

Option Strategies: Calls and Puts

Long and Short Calls

A long call lets the holder benefit from rises in the asset price. The potential gain is unlimited. The loss is capped at the premium paid, even if the asset falls to zero.

At expiration (time T):

• Payoff of a long call:
  $\max(0, S_T − K)$
• Profit of a long call:
  $\max(0, S_T − K) − \text{premium}$

Where $S_T$ is the asset price at expiration and K is the strike.

Key features:

• Breakeven price at expiration: $S_T = K + \text{premium}$.
  Above this, the position earns a profit; below this, it loses money (up to the premium).
• Directional view: Bullish. The buyer expects the price of the asset to rise significantly.
• Risk profile: Limited downside (premium) and theoretically unlimited upside.

A short call is the opposite: the writer is obligated to sell the asset if exercised. This strategy risks unlimited loss if the asset price soars, with the premium as the maximum gain.

• Payoff of a short call:
  $−\max(0, S_T − K)$
• Profit of a short call:
  $\text{premium} − \max(0, S_T − K)$

Key points for short calls:

• Maximum profit: The premium received, achieved if the option expires out of the money.
• Maximum loss: Theoretically unlimited if the asset price rises without bound.
• Margin: Short calls usually require margin because of potential large losses.
• Directional view: Bearish or at least neutral; the writer expects the price not to rise above the strike by much.

Covered calls (discussed later) are a way to limit this risk by holding the asset.

Long and Short Puts

A long put profits when the asset falls below the strike. The upside is capped (the asset price cannot fall below zero); the maximum loss is the premium.

At expiration:

• Payoff of a long put:
  $\max(0, K − S_T)$
• Profit of a long put:
  $\max(0, K − S_T) − \text{premium}$

Key features:

• Maximum profit: $K − \text{premium}$, achieved if the asset price falls to zero.
• Maximum loss: The premium paid.
• Breakeven price at expiration: $S_T = K − \text{premium}$.
• Directional view: Bearish. The buyer expects the price to fall significantly.

A short put is the obligation to buy the asset if exercised. Potential gain is the premium; the maximum loss occurs if the asset falls to zero (strike minus premium).

• Payoff of a short put:
  $−\max(0, K − S_T)$
• Profit of a short put:
  $\text{premium} − \max(0, K − S_T)$

Key points for short puts:

• Maximum profit: Premium received.
• Maximum loss: Approximately $K − \text{premium}$ if the asset price goes to zero.
• Directional view: Bullish or at least neutral; the writer expects the price to stay at or above the strike.
• Economic interpretation: A short put resembles a commitment to buy the asset at the strike if the market falls, in return for the premium.

Combined Strategies

Options can be combined with positions in the asset to create tailored payoff profiles.

Key Term: protective put
A strategy of holding the asset and buying a put on the same asset, providing downside protection while retaining upside potential.

Key Term: covered call
A strategy of holding the asset and selling a call on the same asset, generating income while capping upside.

Key Term: straddle
A strategy of buying (or selling) a call and a put on the same asset with the same strike and expiration, to profit from large (or small) price movements regardless of direction.

Protective Put

• Components: Long stock + long put.
• Intuition: The put acts as insurance. Losses below the strike are limited because the put pays off.
• Payoff at expiration:
  • For stock: $S_T$
  • For put: $\max(0, K − S_T)$
  • Combined payoff: $S_T + \max(0, K − S_T)$

Below the strike, the combined payoff is roughly K; above the strike, the payoff rises with the stock.

• Maximum loss: Approximately $K + \text{put premium} − S_0$ if K is below the initial stock price, or limited to the put strike minus initial stock price plus premium. With K set near current price, downside is largely capped.
• Use case: Investors who are long the stock but want to limit downside risk without selling.

Covered Call

• Components: Long stock + short call.
• Intuition: The call premium provides income, but if the stock rises above the strike, gains are capped because the stock will be called away.
• Payoff at expiration:
  • Stock: $S_T$
  • Short call: $−\max(0, S_T − K)$
  • Combined payoff: $S_T − \max(0, S_T − K)$

Above the strike, the combined payoff is approximately K; below the strike, the payoff falls with the stock, partially cushioned by the premium collected.

• Maximum profit: Approximately $K − S_0 + \text{call premium}$, achieved if the stock ends at or above the strike.
• Maximum loss: Substantial; if the stock goes to zero, loss is about $S_0 − \text{call premium}$ (same as holding the stock minus the premium).
• Use case: Investors who are mildly bullish or neutral and willing to give up upside in exchange for premium income.

Long Straddle

• Components: Long call + long put with the same strike and expiration.
• Intuition: You profit if the asset price moves significantly up or down. You lose if the price stays near the strike because both options expire with low or no exercise value.
• Payoff at expiration:
  • Call: $\max(0, S_T − K)$
  • Put: $\max(0, K − S_T)$
  • Combined payoff: $\max(0, S_T − K) + \max(0, K − S_T)$

The payoff is V-shaped, with minimum at $S_T = K$.

• Total premium paid: Sum of call and put premiums.
• Breakeven prices at expiration:
  • Upper breakeven: $K + \text{total premium}$
  • Lower breakeven: $K − \text{total premium}$
• Directional view: Expectation of high volatility, but uncertainty about direction.

Option Payoffs: Summary

At expiration:

• Long call payoff: $\max(0, S_T − K)$; maximum loss = premium.
• Short call payoff: $−\max(0, S_T − K)$; maximum gain = premium.
• Long put payoff: $\max(0, K − S_T)$; maximum loss = premium.
• Short put payoff: $−\max(0, K − S_T)$; maximum gain = premium.

Remember that payoff ignores the initial premium and profit includes it. Level 1 questions may ask for either, so read carefully.

The Option Greeks

Options respond in complex ways to market changes. The Greeks are partial derivatives capturing option value sensitivities to various risk factors.

Key Term: delta
The sensitivity of an option’s price to a small change in the price of the reference asset.

Key Term: gamma
The rate of change of delta with respect to the asset price; a measure of the curvature (convexity) of the option value with respect to the asset.

Key Term: theta
The sensitivity of the option value to the passage of time, usually expressed as the change in option value per day (time decay).

Key Term: vega
The sensitivity of the option value to a change in the volatility of the reference asset.

Key Term: rho
The sensitivity of the option value to a change in the risk-free interest rate.

Delta (Δ)

Delta measures the change in option value for a small change in the asset price:

• For calls, delta lies between 0 and 1.
  • Deep out-of-the-money (OTM) call (S much less than K): delta close to 0.
  • At-the-money (ATM) call: delta around 0.5.
  • Deep in-the-money (ITM) call (S much greater than K): delta close to 1.
• For puts, delta lies between −1 and 0.
  • Deep ITM put: delta close to −1.
  • ATM put: delta around −0.5.
  • Deep OTM put: delta close to 0.

Interpretations:

• Hedge ratio: A call delta of 0.6 means that the call behaves like 0.6 units of the asset. To hedge one short call, you might buy 0.6 units of the asset.
• Directional sensitivity: A call with delta 0.6 gains approximately 0.6 per 1 unit increase in the asset price, all else equal; a put with delta −0.4 loses about 0.4.

Gamma (Γ)

Gamma measures how delta changes when the asset price moves:

• High gamma means delta changes quickly; the option’s sensitivity to the asset price is unstable.
• Gamma is highest for ATM options close to expiration.
• Deep ITM or deep OTM options have low gamma.

Gamma is important for hedgers:

• A position that is delta-neutral but has high gamma will soon become unhedged if the asset price moves.

Theta (Θ)

Theta captures time decay: the rate at which an option’s value falls as time passes, holding other factors constant.

• For long calls and long puts, theta is usually negative.
• Time decay accelerates as expiration approaches, especially for ATM options.
• Short option positions generally benefit from negative theta (they gain as time passes, if nothing else changes).

On the exam, you may be asked:

• Which position benefits from the passage of time? (Short options.)
• Which is most hurt by time decay? (Long, ATM options close to expiry.)

Vega (V)

Vega measures sensitivity to volatility:

• Higher volatility increases the likelihood that the option finishes ITM, so vega is positive for both calls and puts.
• Options far from expiration and near the money have the highest vega.
• Deep ITM or OTM options, or very near-expiry options, have low vega.

Level 1 questions often ask conceptually:

• If volatility increases, what happens to option prices?
  Both calls and puts increase in value.

Rho (Ρ)

Rho measures sensitivity to the risk-free rate:

• For calls on non-dividend-paying stocks, rho is positive:
  • Higher interest rates make delaying payment (by using options) more attractive, increasing call value.
• For puts on such stocks, rho is negative:
  • Higher interest rates reduce the present value of receiving the strike in the future, making puts less valuable.

Rho is generally less important in short-dated options and is sometimes negligible in exam questions unless specifically mentioned.

Interpreting the Greeks Together

The main effects of an increase in each parameter (for European options on a non-dividend-paying stock) can be summarized qualitatively:

• Asset price:
  • Call value increases, put value decreases (delta effect).
• Volatility:
  • Both call and put values increase (vega effect).
• Time to expiration:
  • Usually increases the value of both calls and puts (more optionality), although deep ITM American puts can be an exception in practice.
• Risk-free rate:
  • Call value increases, put value decreases (rho effect).

Delta and gamma relate to price risk, theta to time risk, vega to volatility risk, and rho to interest rate risk.

Worked Example 1.1

A call option on a stock has a delta of 0.55. If the stock moves up 2 units in price, how much does the option price increase, approximately?

Answer:
The option value increases by about 1.10, computed as 0.55 × 2. Actual changes may differ slightly because delta itself may change when the price moves (gamma effect).

Worked Example 1.2

An at-the-money call and put both have high (in absolute value) theta. What risk does the holder of these options face if they hold to expiration and the asset price does not move?

Answer:
High theta implies significant time decay. If the asset price stays near the strike, the options will lose most or all of their time value as expiration approaches, even though the asset price did not change. The holder faces the risk of losing much of the premium paid purely due to the passage of time.

Put–Call Parity and Arbitrage

For European options on non-dividend-paying stocks, there is a fixed price relationship between calls and puts with the same strike and expiration. This is put–call parity.

Key Term: put-call parity
The fundamental relationship stating that a European call plus the present value of the strike equals a European put plus the asset, for the same strike and expiration, under no-arbitrage conditions.

The standard formula (time 0) is:

Where:

• $C_0$ = price of the European call.
• $P_0$ = price of the European put.
• $S_0$ = current spot price of the asset.
• $PV(K)$ = present value of the strike price discounted at the risk-free rate over the option’s life.

For a continuously compounded risk-free rate r and time to expiration T:

The intuition:

• A call plus enough cash to pay the strike at expiration (i.e., PV(K)) replicates a protective put position (stock plus put). Both positions guarantee you will either own the stock or receive K at expiration.

Rearranging Put–Call Parity

You can rearrange the parity relation to solve for any one of the four instruments:

• Call price:
  $C_0 = P_0 + S_0 − PV(K)$
• Put price:
  $P_0 = C_0 + PV(K) − S_0$
• Stock (synthetic stock):
  $S_0 = C_0 + PV(K) − P_0$
• Present value of strike (synthetic bond):
  $PV(K) = P_0 + S_0 − C_0$

These relationships are frequently tested. The exam may give you three quantities and ask for the fourth.

Synthetic Positions

Using put–call parity, you can create synthetic positions.

Key Term: synthetic position
A position in one or more instruments constructed to replicate the payoff of another instrument.

Examples (European, non-dividend-paying stock):

• Synthetic long stock:
  Long call + short put + long PV(K) of borrowing (or simply long call and short put and borrow money appropriately). Using parity:
  $C_0 − P_0 = S_0 − PV(K)$.
  If you also borrow PV(K), you replicate S.
• Synthetic long call:
  Long stock + long put − (risk-free bond) where the bond has face value K.
  Rewriting parity: $C_0 = P_0 + S_0 − PV(K)$.
• Synthetic long put:
  Long call + PV(K) − long stock.

In Level 1, the main requirement is to recognize these relationships and identify which combination replicates a given payoff.

Arbitrage When Put–Call Parity Is Violated

If the parity relationship does not hold, an arbitrage opportunity exists: you can construct the cheaper side, sell the more expensive side, and lock in a riskless profit.

The steps:

• Identify which side is overpriced.
• Sell (short) the overpriced side and buy (go long) the underpriced side.
• At expiration, the positions offset, and you keep the initial difference as profit.

Worked Example 1.3

A 3‑month European call and put with strike 50 on a non-dividend-paying stock are trading at 3.80 (call) and 1.10 (put). The stock is 52. The annual continuously compounded risk-free rate is 2%. Does put–call parity hold, and if not, which side is overpriced?

Answer:
First compute the present value of the strike. Time to expiration T is 0.25 years.

$PV(K) = 50 × e^{−0.02 × 0.25} \approx 50 × e^{−0.005} \approx 49.75$.

Left-hand side (LHS): $C_0 + PV(K) = 3.80 + 49.75 = 53.55$.
Right-hand side (RHS): $P_0 + S_0 = 1.10 + 52 = 53.10$.

The call-plus-bond side (LHS) costs 53.55, which is 0.45 more than the stock-plus-put side (RHS) at 53.10. The combination of call and PV(K) is overpriced relative to put plus stock.

Ignoring transaction costs, an arbitrageur could:

• Short the expensive side (call plus bond), and
• Buy the cheap side (stock plus put),
  locking in approximately 0.45 today. In practice, bid–ask spreads and transaction costs can explain small deviations.

Adjustments and Exam Assumptions

In reality:

• Dividends and carrying costs modify the parity relationship.
• For dividend-paying stocks, you subtract the present value of expected dividends from the stock side.

For Level 1, unless otherwise stated:

• Assume European options.
• Assume non-dividend-paying asset.
• Apply the simple parity form: $C_0 + PV(K) = P_0 + S_0$.

Worked Example 1.4

A non-dividend-paying stock trades at 40. The risk-free rate is 5% (annual, continuous), and time to expiration is 1 year. A European put with strike 42 costs 4.50. What is the no-arbitrage price of the corresponding European call with strike 42?

Answer:
Use put–call parity:
$C_0 = P_0 + S_0 − PV(K)$.
Here, $P_0 = 4.50$, $S_0 = 40$, $K = 42$, r = 5%, T = 1.

Compute $PV(K) = 42 × e^{−0.05 × 1} \approx 42 × e^{−0.05} \approx 42 × 0.9512 ≈ 39.95$.

Then $C_0 ≈ 4.50 + 40 − 39.95 = 4.55$.

The no-arbitrage call price is approximately 4.55.

Summary

Options are powerful tools for both speculation and risk management. Understanding calls, puts, the Greeks, and put–call parity is essential for accurately pricing options, constructing synthetic positions, and identifying arbitrage opportunities. CFA Level 1 questions often involve:

• Calculating payoffs and profits at expiration for single and combined option positions.
• Matching option strategies (protective put, covered call, straddle) to investor views and risk tolerance.
• Interpreting the impact of changes in price, volatility, time, and interest rates via the main Greeks.
• Applying the put–call parity formula to solve for a missing option price or to detect mispricing.

Careful attention to sign conventions (especially for puts and short positions), the distinction between payoff and profit, and the assumptions behind put–call parity will help avoid common exam errors.

Key Point Checklist

This article has covered the following key knowledge points:

• Define and describe call and put option payoffs, profits, and risk exposures.
• Distinguish standard single and combined option strategies (protective puts, covered calls, straddles) and their directional views.
• Explain and apply delta, gamma, theta, vega, and rho qualitatively.
• State and use the put–call parity formula in its European, non-dividend-paying form.
• Recognize uses of put–call parity for synthetic positions (synthetic stock, calls, and puts) and arbitrage.
• Identify common CFA exam pitfalls: mixing up payoff and profit, ignoring premiums, misinterpreting Greeks, and misapplying parity (e.g., using it for American options).

Key Terms and Concepts

• call option
• put option
• strike price
• expiration date
• option premium
• exercise value
• time value
• moneyness
• European option
• American option
• protective put
• covered call
• straddle
• delta
• gamma
• theta
• vega
• rho
• put-call parity
• synthetic position