Free cheat sheet
CIRE Derivatives Payoffs Cheat Sheet
This sheet covers every options and futures payoff calculation that appears on the CIRE under the CIRO Proficiency Model. Payoff profiles at expiry for all four basic positions, strategy combinations, put-call parity, futures cost-of-carry pricing, and the Greeks sign conventions. All formulas assume no dividends and European-style exercise unless stated otherwise. Last reviewed: 2026-05-08.
1. Long Call Payoff
Payoff at expiry = max(S - K, 0)
Profit = max(S - K, 0) - C
S = stock price at expiry; K = strike price; C = call premium paid.
Break-even:
Break-even = K + C
The stock must rise above K + C at expiry for the long call to profit.
Max gain / max loss:
Max gain = theoretically unlimited (stock can rise without bound). Max loss = C (the premium paid). Loss is fixed and known upfront.
Example:
Buy a call, K = $50, C = $3. Break-even = $53. If S at expiry = $60, profit = $60 - $50 - $3 = $7. If S = $48, profit = $0 - $3 = -$3 (full premium lost).
2. Short Call Payoff
Payoff at expiry = -max(S - K, 0)
Profit = C - max(S - K, 0)
Break-even:
Break-even = K + C
The writer profits as long as S stays below K + C at expiry.
Max gain / max loss:
Max gain = C (premium received). Max loss = theoretically unlimited on a naked (uncovered) short call. This is why naked short calls require margin approval and are the highest-risk single-leg option position.
Example:
Sell a call, K = $50, C = $3. If S = $45 at expiry, profit = $3. If S = $60, loss = $60 - $50 - $3 = $7.
Exam gotcha: the short-call writer assumes unlimited risk; the long-call buyer has limited risk. Never confuse the two on a risk/reward question.
3. Long Put Payoff
Payoff at expiry = max(K - S, 0)
Profit = max(K - S, 0) - P
P = put premium paid.
Break-even:
Break-even = K - P
Max gain / max loss:
Max gain = K - P (stock falls to zero). Max loss = P (premium paid). Long puts are used as portfolio insurance.
Example:
Buy a put, K = $40, P = $2. Break-even = $38. If S = $30 at expiry, profit = $40 - $30 - $2 = $8. If S = $45, loss = $2.
4. Short Put Payoff
Payoff at expiry = -max(K - S, 0)
Profit = P - max(K - S, 0)
Break-even:
Break-even = K - P
The writer profits as long as S stays above K - P.
Max gain / max loss:
Max gain = P. Max loss = K - P (stock falls to zero; writer must buy the stock at K regardless). Short puts obligate the writer to buy shares; requires margin or cash-secured coverage.
Example:
Sell a put, K = $40, P = $2. Max gain = $2. Max loss = $40 - $2 = $38. If S = $35 at expiry, loss = $40 - $35 - $2 = $3.
5. Covered Call and Protective Put
Covered call (long stock + short call)
Payoff = S + max(-(S - K), 0) = min(S, K)
Profit = min(S, K) - S0 + C
S0 = stock purchase price; C = call premium received.
Break-even: S0 - C (stock price at which cost basis is recovered after premium).
Max gain: K - S0 + C (capped at the strike, gains above K are given away). Max loss: S0 - C (stock falls to zero; premium cushions the fall).
Example: Buy stock at $48, sell call K = $50 for $2. Break-even = $46. Max gain = $50 - $48 + $2 = $4. If S = $55, profit is still $4 (calls are assigned, stock sold at $50).
Protective put (long stock + long put)
Profit = (S - S0) + max(K - S, 0) - P
Break-even: S0 + P (must recover the put premium).
Max loss: S0 - K + P (floor set by strike; premium is an additional cost). Max gain: theoretically unlimited minus the premium cost.
Example: Buy stock at $50, buy put K = $48 for $1.50. Break-even = $51.50. Max loss = $50 - $48 + $1.50 = $3.50. If S = $35 at expiry, put is exercised; net loss = $50 - $48 + $1.50 = $3.50 (floored).
6. Bull Call Spread and Bear Put Spread
Bull call spread (long lower-strike call + short higher-strike call)
Net premium paid = C(K1) - C(K2); K1 < K2
Max gain = K2 - K1 - net premium
Max loss = net premium paid
Break-even = K1 + net premium
Example: Buy call K1 = $45 for $4, sell call K2 = $55 for $1. Net premium = $3. Max gain = $55 - $45 - $3 = $7. Break-even = $45 + $3 = $48. Suitable when moderately bullish; reduces cost vs a plain long call.
Bear put spread (long higher-strike put + short lower-strike put)
Net premium paid = P(K2) - P(K1); K2 > K1
Max gain = K2 - K1 - net premium
Max loss = net premium paid
Break-even = K2 - net premium
Example: Buy put K2 = $55 for $4, sell put K1 = $45 for $1. Net premium = $3. Max gain = $55 - $45 - $3 = $7. Break-even = $55 - $3 = $52. Profits if stock falls moderately below $52.
7. Straddle and Strangle
Long straddle (long call + long put, same K, same expiry)
Net premium = C + P
Upper break-even = K + C + P
Lower break-even = K - C - P
Profits if the stock moves sharply in either direction. Max loss = C + P (stock stays exactly at K at expiry). Max gain unlimited on the upside; limited only by the stock floor (zero) on the downside.
Example: K = $50, C = $3, P = $3. Net premium = $6. Upper break-even = $56. Lower break-even = $44. Stock must move more than $6 in either direction to profit. Used before earnings announcements when direction is uncertain.
Long strangle (long OTM call + long OTM put, different strikes, same expiry)
Kcall > Kput (call strike above put strike)
Upper break-even = Kcall + C + P
Lower break-even = Kput - C - P
Cheaper than a straddle (both options are OTM) but requires a larger move to profit. Max loss = C + P. Max gain unlimited to the upside.
Example: Kcall = $55 for $1.50, Kput = $45 for $1.50. Total premium = $3. Upper break-even = $58. Lower break-even = $42. Wider range required than the straddle for the same underlying stock.
8. Put-Call Parity
C - P = S - PV(K)
Equivalently: C + PV(K) = P + S
C = call price; P = put price; S = current stock price; PV(K) = present value of strike = K / (1 + r)^T where r = risk-free rate and T = time to expiry in years.
Use cases on the exam:
- Given three of the four variables (C, P, S, K), solve for the fourth.
- Identify an arbitrage if parity is violated (e.g., C is too cheap relative to P + S - PV(K)).
- Replicate a call using a put, stock, and borrowing.
Example:
S = $50, K = $50, T = 1 year, r = 5%, C = $6. What is P? P = C - S + PV(K) = $6 - $50 + $50/1.05 = $6 - $50 + $47.62 = $3.62.
When parity breaks down:
Put-call parity holds for European options on non-dividend-paying stocks. It does not hold exactly for: (a) American options (early exercise possible), (b) dividend-paying stocks (dividends affect call and put prices differently), (c) deep in-the-money American puts where early exercise may be optimal.
9. Futures Cost-of-Carry Pricing
Theoretical futures price (no dividends):
F = S x (1 + r)^T
S = spot price; r = risk-free rate; T = time to delivery in years.
With continuous dividend yield q:
F = S x e^((r - q) x T)
For the exam, use the discrete approximation: F = S x (1 + r - q)^T unless the question specifies continuous compounding.
Convenience yield and storage costs (commodities):
F = (S + storage costs) x (1 + r)^T - convenience yield
Storage costs increase the futures price; convenience yield (benefit of holding the physical commodity) decreases it. The CIRE tests oil and grain futures in this context.
Example:
Gold spot = $2,800/oz, r = 5% annual, storage = $20/oz/yr, T = 0.5 years. F = ($2,800 + $10) x (1.05)^0.5 = $2,810 x 1.0247 = $2,879.37.
Exam gotcha: if the actual futures price is below the theoretical F, there is a cash-and-carry arbitrage opportunity. If above, there is a reverse cash-and-carry.
10. Options Greeks: Sign Conventions
Direction only. The CIRE tests whether each Greek is positive or negative for a long call, long put, short call, or short put. Formulas are not required; understanding the sign is.
| Greek | What it measures | Long call | Long put | Short call | Short put |
|---|---|---|---|---|---|
| Delta | Change in option price per $1 change in stock price | + (0 to +1) | - (-1 to 0) | - (0 to -1) | + (0 to +1) |
| Gamma | Rate of change of delta; always positive for long options | + | + | - | - |
| Vega | Change in option price per 1% change in implied volatility | + | + | - | - |
| Theta | Change in option price per day passing (time decay) | - | - | + | + |
| Rho | Change in option price per 1% change in risk-free rate | + | - | - | + |
Delta intuition: A delta of +0.60 on a long call means the call price increases by $0.60 for each $1 increase in the stock. At-the-money options have a delta of approximately +0.50 for calls and -0.50 for puts.
Theta intuition: Long options lose value every day due to time decay (negative theta). Short option writers collect that decay (positive theta). Theta accelerates as expiry approaches.
Gamma/Theta trade-off: Long gamma (buying options) costs theta. Short gamma (selling options) earns theta but exposes the writer to sudden large moves. The CIRE may test which position benefits from high realized volatility (long gamma) vs. low volatility (short gamma).
Test Yourself: 5 Derivatives Questions
Q1. A client buys a call with K = $60 and pays a premium of $4. At expiry, the stock is at $67. What is the profit?
Show answer
Profit = $67 - $60 - $4 = $3. The break-even was $64; at $67 the position is profitable.
Q2. A client writes a covered call: buys stock at $45, sells a call K = $50 for a $3 premium. What is the maximum gain?
Show answer
Max gain = $50 - $45 + $3 = $8. The stock is called away at $50; the investor keeps the $3 premium. Gains above $50 are given away to the call buyer.
Q3. Using put-call parity: C = $5, S = $48, K = $50, r = 4%, T = 1 year. What is the put price?
Show answer
P = C - S + PV(K) = $5 - $48 + ($50 / 1.04) = $5 - $48 + $48.08 = $5.08.
Q4. Gold spot = $2,500. Risk-free rate = 4%. No storage costs. What is the theoretical 6-month futures price?
Show answer
F = $2,500 x (1.04)^0.5 = $2,500 x 1.0198 = $2,549.51.
Q5. A client holds a short straddle: short call K = $50 (premium $3) and short put K = $50 (premium $3). What are the break-even prices at expiry?
Show answer
Total premium received = $6. Upper break-even = $50 + $6 = $56. Lower break-even = $50 - $6 = $44. The writer profits if the stock stays between $44 and $56. Beyond those prices, losses are unlimited (on the upside) or large (on the downside).
Related Cheat Sheets
Last updated: 2026-05-08. All payoff formulas assume European-style options and no dividends unless stated. Put-call parity holds exactly for European options on non-dividend-paying stocks only.