Derivatives
Pricing versus valuation
At initiation, a forward's price is the rate making its value zero: F0 = S0(1 + r)^T, adjusted for carry benefits and costs. Later, value is the present value of the difference between the current forward price and the locked-in price.
Swaps and options
An interest rate swap is a portfolio of forwards, or equivalently a long floating-rate note against a short fixed-rate note: swap value = PV(floating) - PV(fixed). Currency swaps combine interest rate and currency exposure. Options obey no-arbitrage: put-call parity, c + X/(1 + r)^T = p + S0, ties European calls and puts. The binomial model uses risk-neutral probabilities; Black-Scholes-Merton extends it to continuous time.
Greeks and hedging
The Greeks are delta, gamma, vega, theta, and rho. Delta-hedging neutralizes small moves but needs rebalancing as gamma takes over. Number of futures contracts = (beta or duration adjustment) x portfolio value / (futures price x multiplier).
Traps
- A swap's value is zero at initiation, but its fixed rate is not zero.
- Parity holds for European options on non-dividend-paying stocks only.
- Hedge ratios must reflect contract size and the beta or duration of the exposure.
- The Black model prices options on futures; Black-Scholes prices options on spot.
Worked example
S0 = $100, r = 5%, one year, X = $100. A call trades at $8. Parity implies p = 8 + 100/1.05 - 100 = $3.24. A put quoted at $4.00 is rich: buy the call, sell the put, short the stock, and lend $95.24 to lock in the difference.
Rule of thumb: value any derivative by replicating its payoff with the underlying and a bond, then checking that the replication cost equals the quoted price.