Estimated study time: 60 minutes
Content:
Options pricing and strategies at CFA Level 2 focus on the Black-Scholes-Merton (BSM) model, put-call parity, the Greeks, and the construction and analysis of options strategies in vignette scenarios. The BSM model provides a closed-form solution for European option prices: Call = S_0*N(d1) - X*e^(-rT)*N(d2); Put = X*e^(-rT)*N(-d2) - S_0*N(-d1), where d1 = [ln(S/X) + (r + sigma^2/2)*T] / (sigma*sqrt(T)) and d2 = d1 - sigma*sqrt(T). N(.) is the cumulative standard normal distribution. The five inputs to BSM are: S (underlying price), X (strike price), r (risk-free rate), T (time to expiration), and sigma (volatility). Notably, the expected return of the underlying asset does not appear in the BSM formula — options can be priced by constructing a risk-free hedge, eliminating the need for a risk premium.
Put-call parity is a no-arbitrage relationship between European calls, puts, the underlying stock, and a risk-free bond: C + X*e^(-rT) = P + S_0, or equivalently P - C = X*e^(-rT) - S_0. This relationship holds because both sides represent portfolios with identical payoffs at expiration. A protective put (P + S_0) has the same payoff as a fiduciary call (C + X*e^(-rT)) — the portfolio consisting of a call plus an investment of the present value of the strike. Violations of put-call parity create arbitrage opportunities. At Level 2, put-call parity is used to derive the price of one option type given the other, and to construct equivalent positions synthetically.
The Greeks measure the sensitivity of option prices to changes in underlying variables. Delta (Δ) = dC/dS, the sensitivity of the option price to changes in the underlying price. For a call option, 0 < Delta < 1; for a put, -1 < Delta < 0. A delta-neutral position (delta = 0) is insensitive to small price changes in the underlying. Gamma (Γ) = d(Delta)/dS, the rate of change of delta — positive for both calls and puts, and highest for at-the-money options near expiration. Gamma exposure is the main risk of delta-hedged positions (dynamic hedging requires frequent rebalancing). Theta (Θ) is the time decay — options lose value as expiration approaches (negative for long option positions). Vega (v) measures sensitivity to implied volatility — long option positions benefit from rising implied volatility (positive vega). Rho measures sensitivity to interest rates — calls increase and puts decrease with rising rates (rho is most important for long-dated options).
Options strategies at Level 2 are evaluated in terms of their profit and loss profiles, breakeven points, maximum gain, maximum loss, and the market view they express. Key strategies include: covered calls (long stock + short call, generates income, caps upside); protective puts (long stock + long put, provides downside protection at the cost of the premium); bull call spread (long lower-strike call, short higher-strike call, net debit, limited gain and limited loss); bear put spread (long higher-strike put, short lower-strike put, net debit, profits from stock decline); straddle (long call + long put at same strike and expiry, benefits from large moves in either direction — profits from high volatility); strangle (long OTM call + long OTM put, cheaper than straddle but requires larger move); and calendar spread (short near-term option, long longer-dated option at same strike, profits from time decay differential and stable price).
Implied volatility and the volatility surface are important advanced topics. Implied volatility (IV) is the volatility backed out from market option prices using BSM — it represents the market's consensus forecast of future realized volatility. The volatility smile (or smirk) describes the empirical observation that BSM implied volatility varies across strike prices — out-of-the-money puts on equity indices have higher implied volatility than at-the-money options (implied volatility skew), reflecting demand for downside protection. The term structure of implied volatility describes how IV varies across expiration dates. At Level 2, candidates must interpret changes in implied volatility (VIX measures equity index IV), understand the relationship between IV and options pricing, and recognize when IV is high or low relative to historical realized volatility as an input to options strategy selection.
Key Terms:
Quiz Questions:
Q1. A European call option on a non-dividend-paying stock has the following parameters: S = $50, X = $50, r = 5%, T = 0.5 years, sigma = 30%. Using the BSM model, d1 = 0.2298 and d2 = 0.0178. N(d1) = 0.5908 and N(d2) = 0.5071. What is the BSM call price?
A) C = 50 * 0.5908 - 50 * e^(-0.05*0.5) * 0.5071 = 29.54 - 50 * 0.9753 * 0.5071 = 29.54 - 24.73 = $4.81. B) C = 50 * N(d1) - 50 * N(d2) = 50 * 0.5908 - 50 * 0.5071 = 29.54 - 25.36 = $4.18. C) C = 50 - 50 * e^(-0.025) = $1.24. D) C = (S - X) * N(d1) = 0 * 0.5908 = $0.
Answer: A — BSM call price: C = S*N(d1) - X*e^(-rT)*N(d2) = 50*(0.5908) - 50*e^(-0.05*0.5)*(0.5071) = 29.54 - 50*(0.97531)*(0.5071) = 29.54 - 50*(0.49476) = 29.54 - 24.74 = $4.80. The calculation uses the present value of the strike (X*e^(-rT) = 50*0.97531 = 48.77) multiplied by N(d2).
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Q2. A European put option on the same stock (S=$50, X=$50, r=5%, T=0.5, sigma=30%) is trading at $4.10. The call option computed in Q1 is $4.80. Using put-call parity, what should the put price be, and is the market pricing consistent with no-arbitrage?
A) Theoretical put = C + X*e^(-rT) - S = $4.80 + $48.77 - $50 = $3.57; the market put of $4.10 is overpriced, creating an arbitrage opportunity. B) Theoretical put = C - S + X*e^(-rT) = $4.80 - $50 + $48.77 = $3.57; market put at $4.10 is overpriced by $0.53; sell the put, buy the call, buy bonds (PV of X), and sell the stock short to earn a riskless $0.53. C) The market put of $4.10 is correct; put-call parity does not apply to at-the-money options. D) Theoretical put = S - X = $50 - $50 = $0.
Answer: B — Put-call parity: P = C + X*e^(-rT) - S = $4.80 + $48.77 - $50.00 = $3.57. The market put at $4.10 exceeds the theoretical put by $0.53. This creates an arbitrage: sell the overpriced put ($4.10 received), buy the call ($4.80 paid), buy risk-free bonds worth X*e^(-rT) = $48.77, and short the stock ($50 received). Net cash flow today = $4.10 - $4.80 - $48.77 + $50.00 = $0.53 (riskless profit). At expiration, all positions offset perfectly regardless of stock price.
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Q3. An investor holds 1,000 shares of stock at $60 per share. She sells 10 call option contracts (100 shares per contract) with a strike of $65 and receives $2.50 per share in premium. What is the maximum gain and maximum loss for this covered call position?
A) Maximum gain = unlimited; maximum loss = $57.50 per share. B) Maximum gain = ($65 - $60 + $2.50) * 1,000 = $7,500; maximum loss = ($60 - $2.50) * 1,000 = $57,500 (if stock goes to zero). C) Maximum gain = $2.50 * 1,000 = $2,500 (premium only). D) Maximum gain = $65 * 1,000 = $65,000; maximum loss = unlimited.
Answer: B — Covered call: long stock + short call. Maximum gain occurs when stock price at expiration equals or exceeds the call strike ($65): gain = (Strike - Purchase Price + Premium) = ($65 - $60 + $2.50) * 1,000 shares = $7.50 * 1,000 = $7,500 (the upside is capped because the call is sold). Maximum loss occurs if stock goes to zero: loss per share = $60 - $2.50 = $57.50 (initial cost minus premium received); total loss = $57,500. The premium received reduces the downside but does not eliminate it — the covered call is not a hedge for large stock declines.
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Q4. An investor expects high volatility in a stock over the next 3 months but is uncertain about direction. She buys a straddle: a $50 call for $3.50 and a $50 put for $3.00, both with 3-month expiry. What are the two breakeven prices and the maximum loss?
A) Breakeven = $50 ± $6.50 = $43.50 and $56.50; maximum loss = $6.50. B) Breakeven = $50 ± $3.50 = $46.50 and $53.50; maximum loss = $3.50. C) Breakeven = $50 - $3.00 = $47 and $50 + $3.50 = $53.50; maximum loss = $6.50. D) Breakeven = $50 ± $3.00 = $47 and $53; maximum loss = $3.00.
Answer: A — Total premium paid = $3.50 + $3.00 = $6.50. Upper breakeven = Strike + Premium = $50 + $6.50 = $56.50. Lower breakeven = Strike - Premium = $50 - $6.50 = $43.50. Maximum loss = total premium paid ($6.50) if the stock expires exactly at $50 (both options expire worthless). The straddle profits if the stock moves more than $6.50 in either direction — a bet on high realized volatility exceeding the implied volatility embedded in the premium.
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Q5. An analyst observes that a stock's 30-day implied volatility (IV) is 45% while its historical 30-day realized volatility over the past 6 months has averaged 25%. Which options strategy is best positioned to profit from this apparent discrepancy if IV reverts toward historical levels?
A) Buy a straddle to profit from high volatility. B) Sell a straddle (or strangle) to collect inflated option premium — if realized volatility is lower than implied volatility, short vega positions (short options) profit as IV falls toward realized levels. C) Buy out-of-the-money calls to profit from the IV differential. D) Buy protective puts because high IV indicates the market expects a large decline.
Answer: B — When IV (45%) substantially exceeds recent realized volatility (25%), options are "expensive" relative to likely future realized volatility. Selling options (selling the straddle or strangle) collects premium that overstates expected realized volatility. If IV reverts to ~25% and the stock doesn't make a large move, the sold options expire worthless or are bought back at lower prices, generating profit. This is a classic volatility mean-reversion trade. The risk is that IV continues to rise (short vega loss) or the stock makes a very large move (short gamma loss), causing losses exceeding the premium received.
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