## **Learning Outcomes**

This article explains how plain vanilla options create non-linear payoffs, how common option strategies reshape risk exposures, and how no-arbitrage links calls and puts through put–call parity. After reading, you will be able to:

- Calculate option payoffs and profits for long and short calls and puts
- Analyze the payoff logic of key strategies (protective put, covered call, spreads, straddles)
- Apply put–call parity (with and without dividends) to value options and build synthetic positions

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand options as non-linear derivatives with a focus on the following syllabus points:

- Option payoff functions and profit/loss for option buyers and writers
- Common option strategies and their payoff diagrams (protective put, covered call, bull spreads, butterfly, straddle)
- Put–call parity for European options, including dividend adjustments and expressing parity using forward prices

## **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. A protective put strategy is best described as:  
   a) Long call and short put with the same strike  
   b) Long stock and long put on the same stock  
   c) Long stock and short put on the same stock  
   d) Short stock and long call on the same stock

2. A European call has strike \$50 and premium \$3. If $S_T = 57$, what is the call profit at expiration?

3. State the European put–call parity relationship for a non-dividend-paying stock and explain what $PV(X)$ represents.

4. True/False (brief explanation required): Put–call parity holds exactly (as an equality) for American options on dividend-paying stocks.

## **What Do You Know Already?**

- Can you write the payoff at expiration for a long call, a short call, a long put, and a short put?
- Can you identify the maximum loss and maximum gain for a covered call and a protective put?
- Can you rearrange put–call parity to synthetically replicate stock, calls, puts, or a risk-free bond?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Options are widely used in financial risk management because they can cap losses while preserving gains. That one-sided protection is not available with linear derivatives such as forwards and futures, which lock in outcomes symmetrically. For a risk manager, this matters when the institution wants insurance against adverse moves (equity drawdowns, FX depreciation, commodity spikes) but does not want to eliminate favorable outcomes.

For the FRM Part 1 exam, the core skills are mechanical and no-arbitrage based: (1) compute payoffs and profits, (2) recognize the standard strategy payoff shapes, and (3) apply put–call parity to relate calls, puts, stock, and the risk-free bond. Put–call parity is also the gateway to synthetic replication, index arbitrage intuition, and many “spot the mispricing” exam questions.

> **Key Term: Option premium**  
> The price paid by the option buyer (and received by the option writer) at initiation. It is denoted $C_0$ for a call and $P_0$ for a put, and it enters profits but not payoffs.

## **Option Payoffs and Profit at Expiration**

### Plain vanilla payoffs

Let $S_T$ be the asset’s price at expiration $T$, and let $X$ be the strike (exercise) price.

- Long European call payoff:
  $$C_T = \max(0, S_T - X)$$
- Long European put payoff:
  $$P_T = \max(0, X - S_T)$$

Short positions (writers) have the negative of these payoffs. Payoff describes the cash flow at expiration from exercising (or being assigned); it ignores the premium paid/received at time 0.

### From payoff to profit

Profit (or P\&L) includes the premium:

- Long call profit:
  $$\Pi_{\text{long call}} = C_T - C_0$$
- Short call profit:
  $$\Pi_{\text{short call}} = C_0 - C_T$$
- Long put profit:
  $$\Pi_{\text{long put}} = P_T - P_0$$
- Short put profit:
  $$\Pi_{\text{short put}} = P_0 - P_T$$

Break-even at expiration is where profit equals zero:

- Long call break-even: $S_T = X + C_0$
- Long put break-even: $S_T = X - P_0$

> **Key Term: Payoff vs profit**  
> Payoff is the value at expiration from exercise/assignment only. Profit equals payoff minus premium for buyers (and plus premium minus payoff for writers).

### Worked Example 1.1

A stock finishes at \$50. A call and put both have strike \$45. The call premium is \$3.50 and the put premium is \$2.50. Compute payoff and profit for call buyer, call writer, put buyer, and put writer.

> **Answer:**  
> Call buyer payoff:  
> $$C_T = \max(0, 50 - 45) = 5$$  
> Call buyer profit:  
> $$5 - 3.50 = 1.50$$
>
> Call writer payoff:  
> $$-5$$  
> Call writer profit:  
> $$3.50 - 5 = -1.50$$
>
> Put buyer payoff:  
> $$P_T = \max(0, 45 - 50) = 0$$  
> Put buyer profit:  
> $$0 - 2.50 = -2.50$$
>
> Put writer payoff:  
> $$0$$  
> Put writer profit:  
> $$2.50 - 0 = 2.50$$

## **Moneyness and Economic Interpretation**

Moneyness classifies whether exercise would be beneficial given the current spot price $S$ (or at expiration using $S_T$):

- Call:
  - In-the-money (ITM) if $S > X$
  - At-the-money (ATM) if $S \approx X$
  - Out-of-the-money (OTM) if $S < X$
- Put:
  - ITM if $S < X$
  - ATM if $S \approx X$
  - OTM if $S > X$

Moneyness matters because it shapes (1) the likelihood of exercise and (2) how sensitive the option’s value is to small moves in the market price of the asset.

> **Key Term: Moneyness**  
> A classification of an option based on the relationship between the asset price and the strike: ITM, ATM, or OTM.

### Option value drivers you must know

For stock options, six inputs are standard determinants of value:

1. Current stock price $S_0$
2. Strike price $X$
3. Time to expiration $T$
4. Volatility of the stock price $\sigma$
5. Risk-free rate $r$
6. Dividends (expected during the option’s life)

Directionally (holding other inputs fixed):

- Higher $S_0$ increases call values and decreases put values
- Higher $X$ decreases call values and increases put values
- Higher $\sigma$ tends to increase both calls and puts (more dispersion increases convex payoffs)
- Higher $r$ tends to increase calls and decrease puts (the strike is paid later in present value terms)
- Higher dividends tend to decrease calls and increase puts (dividends reduce expected future stock price, all else equal)

These directional effects are tested frequently, including via bounds and early exercise logic.

## **Risk Management Strategies With Single Options**

Options are often combined with the stock or other asset to reshape risk.

### Protective put

A protective put is:

- Long stock + long put (usually with the same maturity; the strike sets the floor)

At expiration, the combined payoff is:
$$S_T + \max(0, X - S_T) = \max(S_T, X)$$

So the investor cannot end below $X$ (ignoring premium). This is a classic “downside floor with upside participation” structure.

> **Key Term: Protective put**  
> A position formed by owning the stock (or other asset) and buying a put on that same instrument. It limits losses below the put strike, at the cost of the put premium.

### Covered call

A covered call is:

- Long stock + short call

At expiration, the payoff is:
$$S_T - \max(0, S_T - X) = \min(S_T, X)$$

The call premium provides income up front, but upside above $X$ is given away. Downside remains substantial (stock can still fall sharply), partially offset by the premium received.

### Worked Example 3.1

An investor buys one share at \$30 and buys a 3-month put with strike \$30 for \$1.50. Compute profit if (i) $S_T=\$25$ and (ii) $S_T=\$35$.

> **Answer:**  
> The stock profit alone would be $S_T - S_0$.
>
> (i) If $S_T=\$25$:
>
> - Stock profit: \$25 - \$30 = -\$5
> - Put payoff: $\max(0, 30-25)=\$5$
> - Total profit: $-5 + 5 - 1.50 = -\$1.50$
>
> (ii) If $S_T=\$35$:
>
> - Stock profit: \$35 - \$30 = \$5
> - Put payoff: $\max(0, 30-35)=\$0$
> - Total profit: $5 + 0 - 1.50 = \$3.50$
>
> Interpretation: the put converts a large downside loss into a limited loss equal to the premium, while reducing upside by the same premium.

### Principal-protected note as “bond plus call”

A principal-protected note (PPN) is commonly described as:

- Buy a risk-free zero-coupon bond that grows to the principal at maturity, and
- Use the remaining funds to buy a call option on an equity, index, or other risky asset

The bond component targets return of principal at maturity; the call provides upside participation. The economic cost of principal protection is the opportunity cost of tying up capital in the bond (and any foregone income that would have been earned by investing in higher-yielding assets).

## **Spread and Combination Strategies**

Option strategies often use multiple options to shape exposure to price direction and dispersion.

### Vertical spreads

A vertical spread uses options with the same maturity but different strikes.

#### Bull call spread

Construction:

- Buy call at lower strike $X_L$
- Sell call at higher strike $X_H$ ($X_H > X_L$)

Profit at expiration:
$$\Pi = \max(0, S_T - X_L) - \max(0, S_T - X_H) - C_{L0} + C_{H0}$$

Key features:

- Limited downside: maximum loss is the net premium paid $(C_{L0}-C_{H0})$
- Limited upside: maximum gain is $(X_H - X_L) - (C_{L0}-C_{H0})$

#### Bear put spread

Construction:

- Buy put at higher strike $X_H$
- Sell put at lower strike $X_L$

Profit at expiration:
$$\Pi = \max(0, X_H - S_T) - \max(0, X_L - S_T) - P_{H0} + P_{L0}$$

Key features:

- Gains when the asset price falls, but with capped profit and reduced premium cost versus buying only the higher-strike put.

### Worked Example 4.1

A stock is at \$100. A 3-month call with strike \$95 costs \$9. A 3-month call with strike \$105 costs \$4. Consider a bull call spread: long \$95 call and short \$105 call.

1. What is the net premium?
2. What are max profit, max loss, and break-even at expiration?

> **Answer:**
>
> 1. Net premium paid = \$9 - \$4 = \$5.
> 2. At expiration:
>
> - Max loss occurs if $S_T \le 95$:
>   - Both calls expire worthless, spread payoff = \$0
>   - Profit = \$0 - \$5 = -\$5
> - Max profit occurs if $S_T \ge 105$:
>   - Long \$95 call payoff = $S_T - 95$
>   - Short \$105 call payoff = $-(S_T - 105)$
>   - Spread payoff = $(S_T-95) - (S_T-105) = 10$
>   - Profit = \$10 - \$5 = \$5
> - Break-even solves payoff = net premium:
>   - For $95 < S_T < 105$, payoff = $S_T - 95$
>   - Set $S_T - 95 = 5 \Rightarrow S_T = 100$

### Butterfly spread with calls

A call butterfly is built using three strikes $X_L < X_M < X_H$:

- Long call at $X_L$
- Short two calls at $X_M$
- Long call at $X_H$

Profit at expiration:
$$\Pi = \max(0, S_T - X_L) - 2\max(0, S_T - X_M) + \max(0, S_T - X_H) - C_{L0} + 2C_{M0} - C_{H0}$$

Economic meaning: the position benefits if the stock finishes near the middle strike $X_M$, and it loses (in a limited way) if the stock finishes far below $X_L$ or far above $X_H$.

### Volatility-oriented combinations

#### Straddle

A long straddle is:

- Long call + long put with the same strike $X$ and maturity

Profit at expiration:
$$\Pi = \max(0, S_T - X) + \max(0, X - S_T) - C_0 - P_0$$

This position gains from large moves in either direction and loses if the stock finishes near the strike (because both options expire with little value from exercise compared with the premiums paid).

#### Strangle

A long strangle is:

- Long OTM call at higher strike $X_H$
- Long OTM put at lower strike $X_L$

Profit at expiration:
$$\Pi = \max(0, S_T - X_H) + \max(0, X_L - S_T) - C_0 - P_0$$

Compared with a straddle, a strangle is typically cheaper but needs a larger move to become profitable.

### Worked Example 5.1

A 1-month ATM straddle uses strike \$100. The call premium is \$4 and the put premium is \$3.

1. What is the total premium paid?
2. What are the two break-even prices at expiration?

> **Answer:**
>
> 1. Total premium = \$4 + \$3 = \$7.
> 2. Break-even occurs when the value from exercising the options equals \$7:
>
> - Upside break-even: $S_T - 100 = 7 \Rightarrow S_T = 107$
> - Downside break-even: $100 - S_T = 7 \Rightarrow S_T = 93$

## **Put–Call Parity and Synthetic Replication**

Put–call parity is a no-arbitrage equality linking European calls and puts with the same strike $X$ and maturity $T$.

### Core parity result for non-dividend-paying stocks

For a European call $c$ and European put $p$ on the same non-dividend-paying stock:

$$c + PV(X) = p + S_0$$

where $PV(X)$ is the present value today of a risk-free payment of $X$ at time $T$.

If the risk-free rate is compounded annually (discrete compounding), a common exam form is:
$$PV(X) = \frac{X}{(1+r)^T}$$

(Continuous compounding may also be used in other FRM readings: $PV(X)=Xe^{-rT}$.)

### Why parity holds (payoff argument)

Parity is shown by comparing two portfolios that have the same payoff at expiration:

- Fiduciary call: long call + risk-free zero-coupon bond paying $X$ at $T$
- Protective put: long stock + long put

At expiration:

- If $S_T \le X$:
  - Call expires worthless, fiduciary call pays $X$
  - Put is exercised, protective put delivers $X$ (sell stock for $X$)
- If $S_T > X$:
  - Call is exercised, fiduciary call ends with one share (pay $X$ using the bond proceeds)
  - Put expires worthless, protective put is simply one share

So both portfolios pay $\max(S_T, X)$ at $T$ and must have the same price today.

> **Key Term: Put–call parity**  
> A no-arbitrage relationship for European options with the same strike and maturity: $c + PV(X) = p + S_0$ (for a non-dividend-paying stock).

### Synthetic positions from parity

Rearranging parity gives “synthetics” (portfolios that replicate another security’s payoff):

- Synthetic stock:
  $$S_0 = c - p + PV(X)$$
- Synthetic put:
  $$p = c - S_0 + PV(X)$$
- Synthetic call:
  $$c = S_0 + p - PV(X)$$
- Synthetic bond:
  $$PV(X) = S_0 + p - c$$

The signs tell you long (+) or short (−) positions.

### Worked Example 6.1

A stock is \$52. The risk-free rate is 5% (annual, discrete). A 3-month European put with strike \$50 trades at \$1.50. Estimate the 3-month European call price with the same strike.

> **Answer:**  
> Use parity: $c = p + S_0 - PV(X)$.
>
> Compute $PV(X)$ with $T=0.25$:  
> $$PV(50)=\frac{50}{1.05^{0.25}} \approx 47.39$$
>
> Then:  
> $$c = 1.50 + 52 - 47.39 = 6.11$$
>
> So the call price implied by parity is approximately \$6.11.
>
> (If your calculator uses more precision, your final cents may differ slightly. The exam skill is setting up parity correctly and discounting the strike.)

### **Exam Warning on Parity Conditions**

> Put–call parity as an equality requires European options, the same strike, and the same expiration. If strikes or maturities differ, parity does not apply in this simple form. Also, discount $X$ to $PV(X)$ (do not use $X$ itself).

## **Dividends, Early Exercise, and Forward-Price Parity**

### Parity with known dividends

If the stock pays known cash dividends during the option’s life, parity adjusts by the present value of dividends, denoted $D$:

$$p + S_0 = c + D + PV(X)$$

Intuition: holding the stock delivers dividends; a portfolio that does not hold the stock must be compensated with cash equal to the dividends’ present value to match payoffs.

### American options and why equality may fail

American options can be exercised early. That early exercise feature can add value, and it weakens equality-based relationships:

- For a non-dividend-paying stock, it is never optimal to exercise an American call early (you would give up time value and you do not gain dividends). Therefore, the American call often equals the European call in that special case.
- American puts can be exercised early if sufficiently ITM (especially when interest rates are not trivial), because exercising converts a low-risk future cash flow into immediate cash.

As a result, for American options you typically work with bounds (inequalities) rather than strict parity equality.

### Expressing parity using forward prices

Put–call parity can also be written in a forward-based form:

$$p + PV(F) = c + PV(X)$$

Here, $F$ is the forward price for delivery at option expiration, and $PV(F)$ is the amount of cash that, when invested at the risk-free rate, grows to $F$ at time $T$. This representation emphasizes replication: combining cash and a put can replicate a call plus a forward-style exposure, under no-arbitrage.

> **Key Term: Fiduciary call**  
> A portfolio consisting of a European call and a risk-free zero-coupon bond with face value equal to the strike. Its payoff matches a protective put’s payoff at expiration.

## **Summary**

Options create asymmetric payoffs: buyers have limited loss (the premium) and potential gains tied to favorable moves in the asset price, while writers have the opposite exposure. For FRM Part 1, you must be able to compute payoffs and profits mechanically, then interpret how those shapes change when options are combined into strategies.

Protective puts and covered calls are foundational risk-management structures that combine stock with an option to set floors or cap upside. Spreads (bull call, bear put, butterfly) and combinations (straddles, strangles) further tailor exposures to direction and dispersion. Put–call parity ties the entire framework together: for European options with matching strike and maturity, calls, puts, stock, and the risk-free bond are linked by no-arbitrage, enabling valuation of missing prices and construction of synthetic positions.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Payoffs at expiration: $C_T=\max(0,S_T-X)$ and $P_T=\max(0,X-S_T)$
- Profit equals payoff adjusted for the premium paid/received
- Moneyness definitions for calls and puts (ITM, ATM, OTM)
- Protective put payoff $\max(S_T,X)$ and covered call payoff $\min(S_T,X)$
- Vertical spread mechanics and capped profit/loss logic
- Straddle and strangle profit formulas and break-even points
- European put–call parity: $c + PV(X) = p + S_0$
- Synthetic replication identities derived by rearranging parity
- Dividend-adjusted parity: $p + S_0 = c + D + PV(X)$
- Parity stated using forward prices: $p + PV(F) = c + PV(X)$

## **Key Terms and Concepts**

- Option premium
- Payoff
- Profit (P\&L)
- Moneyness (ITM, ATM, OTM)
- Protective put
- Covered call
- Bull call spread
- Bear put spread
- Butterfly spread
- Straddle
- Strangle
- Put–call parity
- Fiduciary call
- Synthetic position
- Present value of the strike $PV(X)$
- Dividend-adjusted parity
- Forward-price form of parity


Options - Payoffs, strategies, and put–call parity