FINM6041 · Applied Derivatives
Options: Properties, Parity & Trading Strategies
Options: Properties, Parity & Trading Strategies pulls together Lecture 4 and Workshop 4 — the theory of what an option is worth before any model, plus the payoff engineering built on top of it. You start with the four building-block payoffs (long/short call and put), moneyness (ITM/ATM/OTM) and intrinsic value, then the factor sign table showing how a call and put respond as S₀, X, T, σ, r and dividends move. From there the chapter derives the no-arbitrage bounds (a European call is worth at least max(S₀ − Xe^{−rT}, 0)) and the central identity, put-call parity: c + Xe^{−rT} = p + S₀. The second half is strategy fluency — bull and bear spreads, butterflies, straddles, strangles, strips and straps, and hedged positions like the protective put — each read as a bet on direction and/or volatility, including the exam-favourite zero-cost structures.
What this chapter covers
- 01The four payoffs: long/short call and put, and reading them off a diagram
- 02Moneyness (ITM/ATM/OTM) and intrinsic value for calls and puts
- 03The factor sign table: how c, p, C, P move with S₀, X, T, σ, r and dividends
- 04No-arbitrage bounds: c ≥ max(S₀ − Xe^{−rT}, 0), p ≥ max(Xe^{−rT} − S₀, 0)
- 05Put-call parity c + Xe^{−rT} = p + S₀ — and the dividend and American versions
- 06Building an arbitrage when parity or a bound is violated (zero out-of-pocket, cash-flow table)
- 07Spreads and combinations: bull/bear spreads, butterfly, straddle, strangle, strip, strap
- 08Why an American call on a non-dividend stock is never exercised early
Put-call parity arbitrage with zero out-of-pocket cost
- 2 marksWrite put-call parity for a non-dividend stock: c + Xe^{−rT} = p + S₀, and value each side. Left: 4.20 + 45·e^{−0.05×0.25} = 4.20 + 45·e^{−0.0125} = 4.20 + 44.4410 = 48.6410. Right: 2.10 + 45 = 47.10.
- 2 marksThe left side (call + present value of the strike) is worth more than the right (put + stock), so the call side is overpriced. Sell the expensive side and buy the cheap synthetic: write the call, buy the put, and buy the stock.
- 2 marksTime-0 cash flow: +4.20 (write call) − 2.10 (buy put) − 45 (buy stock) = −42.90. Borrow $42.90 at 5% so the net time-0 outlay is exactly zero.
- 2 marksTime-T value: long stock + long put − short call is worth exactly X = $45 in every state (if S_T ≥ 45 the call is exercised against you at 45; if S_T < 45 you exercise the put at 45). Repay the loan: 42.90·e^{0.0125} = 43.4396. Riskless profit = 45 − 43.4396 = $1.56.
Key terms
- Put-call parity
- For European options on a non-dividend stock, c + Xe^{−rT} = p + S₀. It ties a call, a put, the stock and a bond together, so any three replicate the fourth; the dividend version subtracts PV(D) or uses S₀e^{−qT}.
- Moneyness
- Whether exercising now would pay: a call is in-the-money when S > X, at-the-money when S = X, out-of-the-money when S < X; a put is the mirror image.
- Intrinsic value
- The payoff from immediate exercise: max(S − X, 0) for a call and max(X − S, 0) for a put. An option's price is intrinsic value plus time value.
- No-arbitrage bound
- The price range an option must respect regardless of any model. For a European call on a non-dividend stock, c ≥ max(S₀ − Xe^{−rT}, 0) and c ≤ S₀; for a put, p ≥ max(Xe^{−rT} − S₀, 0).
- Bull / bear spread
- A position in two options of the same type and expiry but different strikes. A bull spread profits from a moderate rise (buy the low strike, sell the high); a bear spread from a moderate fall. The call and put versions differ by whether the trade is a net debit or a net credit.
- Straddle
- Buying a call and a put at the same strike and expiry — a bet on a large move in either direction. A strangle is the cheaper version with the call strike above the put strike; strips and straps tilt the bet bearish or bullish.
Options: Properties, Parity & Trading Strategies FAQ
Does the FINM6041 exam use put-call parity?
Yes — put-call-parity (and lower-bound) arbitrage is one of the most reliable question types, appearing in both sample finals. You are typically asked to detect a mispricing, build a zero-out-of-pocket arbitrage, and show the full time-0 and time-T cash flows. Watch the compounding convention: a rate quoted 'compounded quarterly' must be converted to continuous before you use e^{−rT}.
What option trading strategies do I need to know for FINM6041?
The spreads (bull, bear, butterfly, calendar), the combinations (straddle, strangle, strip, strap) and the hedged positions (covered call, protective put). For each, be able to draw the payoff, state the market view it encodes (direction and/or volatility), and compute the time-0 and time-T cash flows — especially the zero-cost bear spread read off an option chain.
How do I know whether to use put-call parity or a bound?
Use put-call parity when both a call and a put with the same strike and expiry are quoted. When only one option is quoted you cannot form parity, so you test the no-arbitrage bound instead — for example c ≥ max(S₀ − Xe^{−rT}, 0) — and build the arbitrage from a violation of that bound.
Why would you build a bull spread with puts rather than calls?
This is a classic 'explain/why' prompt. The two versions give the same payoff shape but different cash flows: the call bull spread is a net debit (you pay up front) while the put bull spread is a net credit (you receive cash up front). The choice hinges on whether you want an initial outlay or an initial inflow, and on the relative pricing of the two strikes.
Are American options ever exercised early?
An American call on a non-dividend-paying stock is never exercised early — you sacrifice no income, you keep the time value of deferring the strike payment, and the call still insures you if the stock falls — so it is valued as a European call. American puts, and options on dividend payers, can be worth exercising early.
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Exam move
Treat this chapter as two linked skills. First, model-free pricing: memorise put-call parity and the four no-arbitrage bounds, and drill the arbitrage recipe — value both sides, sell the rich side, buy the cheap synthetic, borrow or lend to zero the time-0 cash, then tabulate the guaranteed time-T profit. Always convert any non-continuous rate to continuous first, and lay the cash flows out in a time-0/time-T table because working earns the marks (a bare number scores zero). Second, strategy fluency: for every spread and combination, be able to sketch the payoff, name the market view, and read cash flows off an option chain, ready for the zero-cost bear-spread question. Finish by rehearsing the 'why puts not calls' and early-exercise reasoning out loud — the exam rewards the motivation, not just the diagram.
