Trading Strategies: formula sheet

Key formulas

Zero-coupon bond cost (continuous compounding)

B = P × e^(-rT)

P is the principal repaid at maturity. r is the continuously compounded rate. T is in years.

Money available for options

Budget = P - P × e^(-rT)

Per unit of principal at par issue. Use P × (1 - e^(-rT)).

Participation rate

Participation = Budget ÷ Call price per unit of index exposure

If the call is dear, participation falls below 100%.

Payoff at maturity

Payoff = P + P × participation × max(S_T - K, 0) ÷ S_0

With K = S_0 the call is at the money. Principal is returned if held to maturity.

Number of calls per note

N = Budget ÷ c

c is the price of one call. Scale by the notional so units match.

Covered call profit at expiry

Profit = (S_T − S_0) − max(S_T − K, 0) + C

S_0 is the purchase price, K the strike, C the call premium received. Maximum profit = K − S_0 + C. Breakeven = S_0 − C.

Protective put profit at expiry

Profit = (S_T − S_0) + max(K − S_T, 0) − P

P is the put premium paid. Maximum loss = S_0 − K + P. Breakeven = S_0 + P. Upside is unlimited.

Put-call parity (European, no dividends)

c + K·e^(−rT) = p + S_0

Use K ÷ (1 + r)^T if discrete compounding is given. With dividends, subtract the present value of dividends from S_0.

Protective put equivalence

S_0 + p = c + K·e^(−rT)

Stock plus put equals a call plus cash. This is why a protective put looks like a long call.

Covered call equivalence

S_0 − c = K·e^(−rT) − p

Stock minus call equals cash minus put. The covered call looks like a short put.

Bull call spread payoff at expiry

Payoff = max(S − K1, 0) − max(S − K2, 0), with K1 < K2

Ranges from 0 (S ≤ K1) to K2 − K1 (S ≥ K2). Profit = payoff − net premium.

Bull call spread: net cost, max profit, max loss

Net debit = C(K1) − C(K2); Max profit = (K2 − K1) − net debit; Max loss = net debit

C(K1) > C(K2) because the lower strike call is worth more.

Bull call spread breakeven

Breakeven = K1 + net debit

Profit is zero when S equals this level.

Bear put spread payoff at expiry

Payoff = max(K2 − S, 0) − max(K1 − S, 0), with K1 < K2

Long the high-strike put, short the low-strike put. Ranges from 0 (S ≥ K2) to K2 − K1 (S ≤ K1).

Bear put spread: net cost, max profit, max loss

Net debit = P(K2) − P(K1); Max profit = (K2 − K1) − net debit; Max loss = net debit

P(K2) > P(K1) because the higher strike put is worth more.

Bear put spread breakeven

Breakeven = K2 − net debit

Profit is zero when S equals this level.

Credit spread versions

Max profit = net credit; Max loss = (K2 − K1) − net credit

Applies to a bull put spread and a bear call spread. Breakeven for the bull put spread = K2 − credit. Breakeven for the bear call spread = K1 + credit.

Box spread payoff at expiry

Payoff = K2 − K1

Same for every final price S. Bull call spread plus bear put spread on strikes K1 < K2.

Box spread no-arbitrage value (European)

Value = (K2 − K1) × e^(−rT)

Use (K2 − K1) ÷ (1 + r)^T with discrete compounding. A price away from this value signals arbitrage.

Long call butterfly payoff

Payoff = max(S − K1, 0) − 2 × max(S − K2, 0) + max(S − K3, 0)

K2 = (K1 + K3) ÷ 2. Payoff is never negative.

Butterfly maximum payoff

Max payoff = K2 − K1, at S = K2

Max profit = K2 − K1 − net premium.

Butterfly breakevens

Lower = K1 + premium; Upper = K3 − premium

Profit only between these two prices. Maximum loss is the net premium.

Butterfly net premium

Cost = c(K1) − 2 × c(K2) + c(K3)

Positive for a long butterfly, given convex option prices in strike.

Long straddle payoff

Payoff = max(S_T − K, 0) + max(K − S_T, 0) = |S_T − K|

Profit = |S_T − K| − (c + p). Maximum loss = c + p, at S_T = K.

Straddle breakevens

K + (c + p) and K − (c + p)

Two breakevens, symmetric around K.

Long strangle payoff

max(K1 − S_T, 0) + max(S_T − K2, 0), with K1 < K2

Payoff is zero for K1 ≤ S_T ≤ K2. Profit = payoff − (p + c).

Strangle breevens

K2 + (p + c) and K1 − (p + c)

Premium p is for the put at K1, c for the call at K2.

Strip (1 call, 2 puts)

Payoff = max(S_T − K, 0) + 2 × max(K − S_T, 0)

Upside breakeven = K + (c + 2p). Downside breakeven = K − (c + 2p) ÷ 2.

Strap (2 calls, 1 put)

Payoff = 2 × max(S_T − K, 0) + max(K − S_T, 0)

Upside breakeven = K + (2c + p) ÷ 2. Downside breakeven = K − (2c + p).

Short positions

Short position profit = − (long position profit)

Maximum gain is the premium received. Short straddle and strangle have very large potential losses.

Quick revision

• Covered call = long asset + short call; it caps upside and the premium cushions small falls.
• Protective put = long asset + long put; it sets a floor at the strike, less the premium paid.
• Protective put payoff resembles a long call plus a bond, by put-call parity.
• Bull call spread = long low-strike call + short high-strike call; maximum profit = strike difference − net premium.
• Bear spreads profit when the price falls; a bear put spread buys the higher strike put and sells the lower strike put.
• Long straddle = long call + long put at the same strike; profit needs a large move either way. Two breakevens: K ± total premium.
• A strangle uses different strikes (out-of-the-money put and call), so it costs less but needs a bigger move.
• Long butterfly (calls) = long low and high strike calls, short two middle strike calls; profits if the price stays near the middle.
• A box spread combines a bull call spread and a bear put spread with the same strikes; its payoff is fixed at the strike difference.
• A box spread's fair value is the strike difference discounted at the risk-free rate; a different price implies arbitrage.
• Principal-protected note = zero-coupon bond + option; the option budget is the amount left after buying the bond.
• Always add premiums to the payoff to get profit, and check the maximum loss against the net premium.

Common mistakes

• Using the principal itself as the option budget. Fix: Always subtract the bond price P × e^(-rT). Only the remainder buys options.
• Saying higher volatility helps the investor in a PPN. Fix: Higher volatility makes the call dearer. At a fixed budget the participation rate falls. The note gets worse for a given structure.
• Treating a covered call as protection against large falls. Fix: The premium only offsets the first few points of loss. The downside is nearly the full stock loss, reduced by the premium.
• Forgetting the premium in breakeven. Fix: Covered call breakeven is S_0 − C. Protective put breakeven is S_0 + P.
• Reversing the legs, for example buying the high-strike call in a bull call spread. Fix: For a debit spread you always buy the more expensive option. Calls: the lower strike costs more. Puts: the higher strike costs more.
• Forgetting the net premium and quoting the payoff as the profit. Fix: Always write profit = payoff − net debit (or + net credit). Maximum profit is the gap minus the debit.
• Forgetting to discount the box payoff Fix: The payoff comes at expiry. Compare the price with (K2 − K1) discounted at the risk-free rate.
• Selling one middle call instead of two in a butterfly Fix: Write the pattern buy 1, sell 2, buy 1 before you start.
• Using only one premium when finding breakevens. Fix: Always add both premiums. The breakeven distance from the strike (or band edge) is the total premium.
• Putting the strangle strikes the wrong way round, with the call strike below the put strike. Fix: A standard strangle has K1 (put) < K2 (call). Both options are out of the money at the start.

Exam tips

• Always split the note into bond plus option before doing anything else.
• For 'what if' questions, trace the effect on the bond price first, then on the call price. Then state the net effect on participation.
• Check the units: per $1,000,000, per unit of index, or per ₹. Many errors come from scale.
• Remember that low rates and high volatility are the hard case for issuers. They force low participation, caps or a higher strike.
• If asked about risk, name issuer credit risk and liquidity before market risk.
• Draw the profit line at three prices before answering. It catches most sign errors.
• Know that protective put = long call + cash, and covered call = short put + cash. Parity links are a favourite.
• Check whether the question asks for payoff or profit, and per share or for a contract size.