Difficulty: Intermediate to Advanced | Prerequisites: time value of money, basic probability, fixed-income pricing (Ch. 14-16).
Derivatives are financial instruments whose value is derived from an underlying asset (a stock, a bond, a currency, an index). This material covers three core areas: option pricing theory (put-call parity, the binomial model, Black-Scholes), hedging with futures, and interest rate parity in currency markets. These chapters tie together the no-arbitrage principle that runs through all of modern finance. If you understand why two portfolios with the same payoffs must have the same price, the rest follows.
TL;DR
Options are priced by replication: build a portfolio of the underlying asset and borrowing/lending that mimics the option's payoff, and its cost must equal the option price (no arbitrage). The binomial model does this step by step; Black-Scholes does it in closed form. Futures let you hedge risk without selling the underlying. Interest rate parity links currency forwards to interest rate differentials.
Call option
The right (not obligation) to buy the underlying asset at the exercise (strike) price on or before the expiration date.
Put option
The right (not obligation) to sell the underlying asset at the exercise price on or before the expiration date.
European option
An option that can be exercised only at expiration, not before.
American option
An option that can be exercised at any time up to and including expiration. Worth at least as much as its European counterpart.
Put-call parity
A no-arbitrage relationship linking the prices of European puts and calls with the same strike and expiration: C - P = S0 - PV(X). If this relationship is violated, an arbitrage opportunity exists.
In simple terms: the difference between a call and a put price must equal the stock price minus the present value of the strike. If it does not, free money is on the table.
Arbitrage
A risk-free profit from simultaneous transactions that exploit a pricing discrepancy. In efficient markets, arbitrage opportunities are fleeting.
Binomial option pricing model
A model that prices options by building a tree of possible future stock prices (up or down each period), computing option payoffs at each terminal node, and working backwards using risk-neutral probabilities and discounting.
Delta (hedge ratio)
The number of shares needed to replicate or hedge one option. In the binomial model: delta = (C_u - C_d) / (S_u - S_d), where u and d denote the up and down states.
Think of it as: how many shares of stock you need to hold per option written to create a riskless position.
Risk-neutral probability (p)
The probability of an up move in the stock price that makes the expected return on the stock equal to the risk-free rate. Used to price options without knowing the true probabilities or investor risk preferences.
Black-Scholes model
A closed-form formula for pricing European options on non-dividend-paying stocks, based on five inputs: stock price, strike price, time to expiration, risk-free rate, and volatility.
Futures contract
A standardised, exchange-traded agreement to buy or sell an asset at a specified price on a future date. Unlike options, both parties are obligated.
Hedge ratio (futures)
The number of futures contracts needed to offset the risk of a portfolio position. For equity portfolios: N = (beta × Portfolio value to hedge) / (Futures price × Multiplier).
Covered interest rate parity (CIP)
The no-arbitrage relationship between spot exchange rates, forward exchange rates, and interest rate differentials between two countries: F0/E0 = (1 + r_domestic) / (1 + r_foreign).
For European options with the same strike X and expiration T:
C - P = S0 - PV(X)
where PV(X) = X / (1 + r_f)^T (or X / (1 + r_f) for a single-period holding period return).
This says: buying a call and selling a put with the same strike is equivalent to buying the stock and borrowing the present value of the strike price.
Example: stock at $42, call at $3.90, put at $1.45, strike $40, risk-free 6-month return 4%.
Left side: C - P = 3.90 - 1.45 = $2.45.
Right side: S0 - PV(X) = 42 - 40/1.04 = 42 - 38.46 = $3.54.
Since $2.45 is not equal to $3.54, put-call parity is violated. The call is cheap relative to the put (or equivalently, the put is expensive relative to the call).
When C - P < S0 - PV(X), the strategy is:
Buy the call (cheap side)
Sell the put (cheap side)
Short the stock (expensive side)
Invest the proceeds at the risk-free rate
This locks in a riskless profit equal to the parity discrepancy. At expiration, the option and stock positions cancel out regardless of where the stock ends up, and you keep the arbitrage gain.
The binomial model assumes the stock can move to one of two prices next period: S_u (up) or S_d (down). To price a call:
Compute the option payoff in each state: C_u = max(S_u - X, 0) and C_d = max(S_d - X, 0)
Find the hedge ratio (delta): H = (C_u - C_d) / (S_u - S_d)
Form a riskless portfolio: buy H shares and sell 1 call. The portfolio has the same value in both states.
Discount that certain payoff at the risk-free rate to get the portfolio value today.
Solve for the call price.
Alternatively, compute the risk-neutral probability p = [(1 + r_f) - d] / (u - d), where u = S_u/S0 and d = S_d/S0. Then: C = [p × C_u + (1 - p) × C_d] / (1 + r_f).
For a three-period tree (as in the problem set), work backwards from the terminal nodes. At each node, either compute delta and the replicating portfolio, or use risk-neutral pricing. The option value at each node is the discounted expected value (under risk-neutral probabilities) of the two successor nodes.
Rebalancing: the hedge ratio changes at every node. If you are maintaining a riskless portfolio of stock and options, you must adjust the number of shares at each step as the stock price and delta change.
In a 3-period binomial tree with risk-neutral probability p of an up move, the probability of reaching each terminal node is:
3 ups (uuu): p³
2 ups, 1 down: 3p²(1-p) (three paths lead here)
1 up, 2 downs: 3p(1-p)² (three paths)
3 downs (ddd): (1-p)³
Two approaches:
Direct: compute put payoffs at each terminal node, max(X - S, 0), and work backwards using risk-neutral probabilities.
Put-call parity: once you have the call value from the binomial model, P = C - S0 + PV(X).
Both must give the same answer. If they do not, you have made an error.
The Black-Scholes formula for a European call on a non-dividend-paying stock uses:
S0: current stock price
X: strike price
T: time to expiration (in years)
r_f: continuously compounded risk-free rate
sigma: annualised standard deviation of returns (volatility)
C = S0 × N(d1) - X × e^(-r_f × T) × N(d2)
where d1 = [ln(S0/X) + (r_f + sigma²/2) × T] / (sigma × sqrt(T)) and d2 = d1 - sigma × sqrt(T).
When the stock pays known dividends before expiration, replace S0 with S0 minus the present value of dividends paid during the option's life:
S0* = S0 - PV(dividends)
Use S0* in place of S0 throughout the formula. This reduces the call value (the stock drops on the ex-dividend date, making the call less likely to finish in the money) and increases the put value.
Example: ARB stock at $75, two dividends of $2 (at 91 and 182 days). PV of first dividend = 2 × e^(-0.09 × 0.25) = $1.955. Only the first dividend matters for a 91-day option. S0* = 75 - 1.955 = $73.045.
Use put-call parity: P = C - S0* + X × e^(-r_f × T).
Higher volatility → higher call and put values. More dispersion means more chance of large payoffs in the money, while losses are capped at zero (you just do not exercise).
Higher risk-free rate → higher call value, lower put value. The PV of the strike falls, benefiting the call holder and hurting the put holder.
Dividends reduce call values and increase put values. The stock drops by the dividend amount, which hurts call holders.
Longer time to expiration → higher call value (European). More time for favourable moves. For European puts, the effect is ambiguous because the longer wait also reduces the PV of the strike you receive.
A portfolio manager who expects a market decline can hedge with index futures rather than selling stocks (which incurs transaction costs and tax events).
Number of contracts = beta × (Value to hedge / (Futures price × Multiplier))
Example: $15 million to hedge, beta 0.88, S&P 500 futures at 1,000, multiplier 250.
N = 0.88 × (15,000,000 / (1,000 × 250)) = 0.88 × 60 = 52.8, so approximately 53 contracts.
The hedge is short (sell futures), because you want to profit from futures if the market falls, offsetting losses in the portfolio.
Two other methods that replicate a short futures position:
Equity swaps: enter a swap to pay the equity return and receive a fixed or risk-free return on the notional amount. Economically similar to selling equities and investing in bonds, but without physically selling stock.
Protective puts: buy put options on the index. This provides downside protection while preserving upside, unlike futures which lock in both sides. The cost is the put premium.
CIP links forward and spot exchange rates to interest rate differentials:
F0 / E0 = (1 + r_domestic) / (1 + r_foreign)
where E0 is the spot exchange rate (domestic per foreign), F0 is the forward rate, r_domestic is the domestic interest rate, and r_foreign is the foreign interest rate.
Example: r_US = 4%, r_UK = 7%, E0 = $2.00/£, F0 = $1.98/£.
Parity forward = 2.00 × (1.04 / 1.07) = $1.9439/£.
The actual forward ($1.98) exceeds the parity forward ($1.9439). This means the pound is overpriced in the forward market relative to what CIP implies.
You would lend in the US (domestic, 4%) and borrow in the UK (foreign, 7%), using the forward market to eliminate currency risk:
Borrow pounds at 7%.
Convert to dollars at the spot rate ($2.00/£).
Invest dollars at 4%.
Sell dollars forward for pounds at $1.98/£ to repay the pound loan.
The forward rate is too high relative to CIP, so this strategy earns a risk-free profit. You are effectively earning the US rate while locking in a favourable forward conversion back to pounds, more than covering the UK borrowing cost.
C - P = S_0 - \frac{X}{(1 + r_f)^T} \quad \text{(put-call parity)}\Delta = \frac{C_u - C_d}{S_u - S_d} \quad \text{(binomial hedge ratio)}p = \frac{(1 + r_f) - d}{u - d} \quad \text{(risk-neutral probability)}C = \frac{p \cdot C_u + (1 - p) \cdot C_d}{1 + r_f} \quad \text{(one-period binomial call value)}C = S_0 N(d_1) - X e^{-r_f T} N(d_2) \quad \text{(Black-Scholes)}d_1 = \frac{\ln(S_0/X) + (r_f + \sigma^2/2)T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}N_{\text{futures}} = \beta \times \frac{\text{Value to hedge}}{F \times \text{Multiplier}}\frac{F_0}{E_0} = \frac{1 + r_{\text{domestic}}}{1 + r_{\text{foreign}}} \quad \text{(covered interest rate parity)}"Put-call parity holds for American options." It holds exactly only for European options. American options have early-exercise value that can break the strict equality.
"The binomial model requires you to know the real probability of an up move." It does not. The risk-neutral probability p is derived from the risk-free rate and the up/down factors. Real-world probabilities are irrelevant for pricing.
"Black-Scholes gives exact option prices." Black-Scholes gives a theoretical price based on its assumptions (constant volatility, no transaction costs, continuous trading, etc.). Market prices deviate, hence the volatility smile.
"Selling futures is the same as selling the portfolio." Economically similar in terms of risk reduction, but futures do not trigger capital gains taxes, require less capital, and can be adjusted easily. The hedge is also imperfect: basis risk remains because the portfolio may not track the index perfectly (captured by the R² of the regression).
"CIP always holds in practice." CIP is a no-arbitrage condition. Small deviations exist due to transaction costs, credit risk, and capital constraints, but large violations are rare and short-lived.
⚠️ Put-call parity: know the formula, how to test it, and how to construct the arbitrage. This is a near-guaranteed exam question.
⚠️ Binomial model: be able to work through a multi-period tree, computing delta at each node and the option value by working backwards. Show your work clearly.
⚠️ Risk-neutral pricing: understand that p is not the real probability of an up move. It is the probability that makes the expected return equal to the risk-free rate.
⚠️ Black-Scholes with dividends: remember to subtract PV(dividends) from S0 before plugging in. This is where students lose marks.
⚠️ Futures hedging: the exam may ask for the number of contracts, whether the hedge is long or short, and why futures hedging is imperfect (basis risk, R² < 1).
⚠️ CIP arbitrage: know which country to borrow in, which to lend in, and how the forward market eliminates currency risk.
True or False: If C - P > S0 - PV(X), the call is overpriced relative to parity. Answer: True. You would sell the call and buy the put to exploit the discrepancy.
Fill in the blank: In the binomial model, the risk-neutral probability p depends on the ______ rate and the up/down factors, not on investors' risk preferences. Answer: Risk-free.
True or False: Increasing volatility increases the value of both calls and puts in the Black-Scholes model. Answer: True.
Fill in the blank: A short futures hedge profits when the market ______. Answer: Falls (declines).
True or False: Under covered interest rate parity, the currency with the higher interest rate trades at a forward premium. Answer: False. It trades at a forward discount (the forward rate is lower than the spot rate for that currency).
Q: Stock at $42, call at $3.90, put at $1.45, strike $40, risk-free 6-month return 4%. Is put-call parity satisfied?
A: C - P = $2.45. S0 - PV(X) = 42 - 40/1.04 = $3.54. Not equal, so parity is violated.
Q: In a binomial model, S0 = $40, S_u = $42, S_d = $38.40, X = $38, r_f = 6% per year (approximately 1.5% per quarter). What is the risk-neutral probability of an up move (per period)?
A: u = 42/40 = 1.05, d = 38.40/40 = 0.96. Per-period r_f ≈ 1.5%. p = (1.015 - 0.96) / (1.05 - 0.96) = 0.055 / 0.09 ≈ 0.611.
Q: ARB stock at $75, sigma = 20%, r_f = 9%, 91-day call with strike $70, one $2 dividend in 91 days. How do you adjust S0 for Black-Scholes?
A: PV of dividend = 2 × e^(-0.09 × 0.25) = $1.955. S0* = 75 - 1.955 = $73.045. Use S0* in the Black-Scholes formula.
Q: How many S&P 500 futures contracts to hedge $15M of a portfolio with beta 0.88? Futures at 1,000, multiplier 250.
A: N = 0.88 × (15,000,000 / 250,000) = 0.88 × 60 = 52.8 ≈ 53 contracts, short.
Q: r_US = 4%, r_UK = 7%, E0 = $2.00/£, F0 = $1.98/£. Where do you lend, where do you borrow, and why?
A: Lend in the US (the domestic, lower-rate country) and borrow in the UK (the foreign, higher-rate country). The forward rate ($1.98) is above the CIP-implied forward ($1.9439), so you profit by converting borrowed pounds to dollars, investing at 4%, and selling dollars forward at $1.98 to repay the pound loan.
Put-call parity and the binomial model both rest on the no-arbitrage principle, the same logic that prices bonds via the term structure (Ch. 14-16). If two things give you the same payoff, they must cost the same.
Black-Scholes extends the binomial model to continuous time. The hedge ratio (delta) in the binomial model is the discrete analogue of N(d1) in Black-Scholes.
Futures hedging uses the portfolio's CAPM beta (Ch. 9, 18), linking derivatives back to the risk-return framework. The number of contracts depends on systematic risk, not total risk.
Covered interest rate parity is the currency-market version of the same no-arbitrage reasoning. It parallels the relationship between spot and forward rates in the term structure of interest rates.
Related Terms / Search Tags
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