Derivatives Analysis and Valuation
The no-arbitrage foundation underlying every derivative in this chapter
The method chapter flagged no-arbitrage reasoning as one of the three tools recurring throughout AFM, and this chapter is where that reasoning does the most direct, load-bearing work: every pricing formula below is derived from the same basic principle — if a derivative's payoff can be exactly replicated by a portfolio of the underlying asset and risk-free borrowing or lending, the derivative must be priced identically to that replicating portfolio, or a risk-free arbitrage profit would be available to any market participant who noticed the mispricing and traded on it.
Forward and futures contracts
Forward contracts are customised, over-the-counter agreements to buy or sell an asset at a specified future date for a price agreed today, carrying counterparty risk (the risk the other party fails to honour the agreement) since no exchange or clearing house stands between the two parties. Futures contracts are standardised, exchange-traded equivalents, with a clearing house as the counterparty to every trade (eliminating direct counterparty risk between the original two parties), standardised contract sizes and expiry dates, and daily mark-to-market margining (gains and losses are settled in cash daily against each party's margin account, rather than accumulating until the contract's expiry as with a forward).
Forward/futures pricing under no-arbitrage (cost of carry model). For an asset with no income (dividends, interest) during the contract's life:
F = S₀ × e^(r×T) (continuous compounding) or F = S₀ × (1 + r)^T (discrete compounding)
where S₀ is the spot price, r is the risk-free rate, and T is time to expiry. For an asset paying a known income (dividend yield q for an equity index, or a specific cash dividend), the formula adjusts to net out that income, since holding the asset directly earns the income while holding the futures contract does not:
F = S₀ × e^((r−q)×T)
Why this formula must hold. If the actual futures price traded in the market is higher than this formula suggests, an arbitrageur can borrow money at the risk-free rate, buy the asset today (a "cash-and-carry" strategy), simultaneously sell the futures contract, and lock in a risk-free profit at expiry; if the actual futures price is lower, the reverse ("reverse cash-and-carry") strategy — selling the asset short, investing the proceeds at the risk-free rate, and buying the futures contract — locks in a risk-free profit instead. This arbitrage mechanism is precisely why the formula holds in genuinely liquid, efficient futures markets: any deviation invites arbitrage activity that pushes the price back toward the no-arbitrage value.
Hedging with futures. A firm holding (or expecting to acquire) the underlying asset and wanting to hedge against an adverse price movement takes an offsetting futures position — a short hedge (selling futures) protects a holder of the asset against a price decline; a long hedge (buying futures) protects a firm expecting to purchase the asset in future against a price increase. Basis risk — the risk that the difference between the spot price and the futures price (the "basis") does not remain constant or move as expected — means a futures hedge is rarely perfect, since the hedger typically cannot find a futures contract with an expiry and underlying exactly matching their specific exposure, leaving some residual, imperfectly-hedged risk even after the hedge is in place.
Options
Basic terminology. A call option gives its holder the right, but not the obligation, to buy the underlying asset at a specified strike price on or before expiry; a put option gives its holder the right, but not the obligation, to sell at the strike price. The option writer (seller) bears the corresponding obligation if the holder exercises. American options can be exercised any time up to expiry; European options can be exercised only at expiry itself.
Intrinsic value and time value. An option's premium (market price) splits into intrinsic value — the value if exercised immediately (Spot − Strike for a call, if positive, else zero; Strike − Spot for a put, if positive, else zero) — and time value — the additional premium reflecting the possibility that the option's intrinsic value could increase further before expiry, which is always non-negative and shrinks to zero exactly at expiry.
Put-call parity. For European options on the same underlying, with the same strike price and expiry:
Call premium + Present value of strike price = Put premium + Spot price
This relationship, again, follows directly from no-arbitrage reasoning: a portfolio of a long call and a risk-free bond paying the strike price at expiry produces exactly the same payoff, at every possible future spot price, as a portfolio of a long put and the underlying asset itself, so the two portfolios must cost the same today, or an arbitrage opportunity would exist. Put-call parity is tested both as a direct valuation tool (given three of the four variables, solve for the fourth) and conceptually (identifying a mispricing and constructing the arbitrage trade that would exploit it).
Binomial option pricing. The one-period binomial model values an option by constructing a replicating portfolio — a specific combination of the underlying asset and risk-free borrowing or lending that produces exactly the same payoff as the option in both the "up" and "down" states of the world — and pricing the option identically to this replicating portfolio's known cost today, again a direct application of no-arbitrage reasoning:
Option value = [p × Cu + (1−p) × Cd] ÷ (1 + r), where p = [(1+r) − d] ÷ (u − d)
with u and d the up and down price movement factors, Cu and Cd the option's payoff in the up and down states, and p the risk-neutral probability — a probability derived purely from the no-arbitrage condition, not from any actual real-world assessment of how likely the up or down move genuinely is, which is precisely why this valuation approach works regardless of investors' actual risk preferences or real-world probability beliefs.
The Black-Scholes model extends this replicating-portfolio logic to continuous time, providing a closed-form formula for a European option's value as a function of the spot price, strike price, time to expiry, risk-free rate, and the underlying asset's volatility — the single input, unlike the other four, that cannot be directly observed and must be estimated, making volatility estimation (historical volatility from past price data, or implied volatility backed out from other options' observed market prices) a genuinely distinct analytical exercise in its own right.
The Greeks. Delta measures an option's price sensitivity to a small change in the underlying spot price; gamma measures the rate of change of delta itself (the option's own convexity); vega measures sensitivity to a change in volatility; theta measures the rate of time decay in the option's value as expiry approaches (time value's steady erosion); and rho measures sensitivity to a change in the risk-free rate. These are used both for understanding an option's risk exposure and for constructing delta-hedged positions, where an options market-maker holds an offsetting position in the underlying asset (or other options) sized precisely to neutralise the portfolio's overall delta, a hedge that must be continuously rebalanced as delta itself changes with the spot price (captured by gamma) and the passage of time.
Option strategies. Combinations of options (and sometimes the underlying asset itself) construct specific risk-return payoff profiles — a covered call (holding the underlying asset and writing a call against it, generating income at the cost of capping upside), a protective put (holding the underlying asset and buying a put, insuring against downside at the cost of the put premium), a straddle (buying both a call and a put at the same strike, profiting from a large price movement in either direction, at the cost of losing both premiums if the price stays flat), and a bull/bear spread (buying one option and selling another at a different strike, limiting both potential profit and potential loss relative to holding a single option outright).
Swaps
Interest rate swaps exchange one party's fixed-rate interest payment obligation for another party's floating-rate obligation (or vice versa), on a notional principal that is never itself exchanged — used to convert a firm's existing floating-rate debt exposure to fixed (or vice versa) without needing to actually refinance the underlying debt itself. Currency swaps exchange principal and interest payments in one currency for principal and interest payments in another currency, used to convert a firm's foreign-currency-denominated debt exposure into its own domestic currency exposure (or vice versa), combining both an interest rate exchange and a currency exchange within a single instrument.
Swap valuation rests on decomposing a swap into a portfolio of simpler, already-familiar instruments: an interest rate swap can be valued as the difference between a fixed-rate bond and a floating-rate bond (whose value, immediately after each reset date, always equals its own face value, making floating-rate bond valuation straightforward); a currency swap can be valued as the difference between a bond denominated in one currency and a bond denominated in the other, converted to a common currency at the current spot exchange rate — the same "replicate with simpler instruments and value the replicating portfolio" logic underlying every other derivative in this chapter.
Why derivatives carry the paper's heaviest weightage
Every derivative instrument in this chapter is, at its analytical core, an application of the same no-arbitrage, replicating-portfolio logic — a forward/futures price replicated by spot-plus-financing, an option replicated by a specific combination of the underlying and risk-free borrowing, a swap replicated by a portfolio of two bonds. Recognising this shared structure, rather than treating forwards, options and swaps as three unrelated instrument types each requiring separate memorisation, is precisely the kind of cross-topic pattern recognition the method chapter identified as the genuine skill this entire paper rewards, and it is why this chapter, more than any other single chapter in AFM, deserves the largest share of your practice time.
