CFA Derivatives Syllabus

State the value of a long forward position at time $t$ before expiration.

$$V_{t}(T) = S_{t} - \frac{F_{0}(T)}{(1+r)^{T-t}}$$ i.e., the current spot minus the present value of the originally contracted forward price (adjusting for any income).

Give the value of a long forward at time $t$ in terms of forward prices.

$$V_{t}(T) = \frac{F_{t}(T) - F_{0}(T)}{(1+r)^{T-t}}$$ the present value of the difference between the current forward price and the original contracted forward price.

Why does the forward price generally not equal the expected future spot price?

Under no-arbitrage the forward price equals the cost-of-carry future value of the spot, not the expected spot. Differences reflect a risk premium; forward = expected spot only if the underlying's risk is unpriced.

What is contango versus backwardation?

Contango: the forward/futures price is above the current spot price ($F_{0} > S_{0}$). Backwardation: the forward/futures price is below the current spot price ($F_{0} < S_{0}$), often associated with a high convenience yield.

How does a positive convenience yield affect the forward price?

A convenience yield is a benefit of holding the physical asset; it acts like income, lowering the forward price: $$F_{0}(T) = S_{0}\,e^{(r+c-y)T}$$ where $y$ is the convenience yield.

What is a Forward Rate Agreement (FRA)?

An OTC forward contract on an interest rate: the long agrees to pay a fixed (contract) rate and receive a floating reference rate on a notional principal for a future period. It locks in a borrowing/lending rate.

In FRA notation, what does a '2×5' FRA mean?

An FRA that expires (settles) in 2 months and is based on an underlying interest rate for the period from month 2 to month 5, i.e., a 3-month rate beginning 2 months from now.

Does the long (fixed-rate payer) in an FRA gain when rates rise or fall?

The long gains when the floating reference rate at expiration is above the contracted FRA rate (rates rise). The long effectively locks in borrowing and benefits if market rates exceed the fixed rate.

State the settlement payment formula for an FRA at expiration.

$$\text{Payment} = \frac{(R_{ref} - R_{FRA}) \times \frac{d}{360} \times NP}{1 + R_{ref} \times \frac{d}{360}}$$ where $d$ is days in the underlying period and $NP$ is notional. It is discounted because it settles at the start of the period.

Why is the FRA settlement amount discounted at expiration?

FRAs settle in advance (at the beginning of the underlying loan period) rather than in arrears, so the interest differential that would accrue at period-end is discounted back to the settlement date.

How is the FRA (fixed) rate determined at initiation?

As the forward interest rate implied by the current term structure, so the FRA has zero value at inception: $$(1+R_{FRA})^{?}$$ found from $ (1+z_{2})^{t_2} = (1+z_{1})^{t_1}(1+f) $, i.e., the no-arbitrage forward rate between the two spot rates.

Compute the forward rate $f$ implied by a 1-year spot rate $z_{1}$ and a 2-year spot rate $z_{2}$.

$$1+f = \frac{(1+z_{2})^{2}}{(1+z_{1})^{1}}$$ so $$f = \frac{(1+z_{2})^{2}}{1+z_{1}} - 1.$$

State the put-call parity relationship for European options.

$$c + \frac{X}{(1+r)^{T}} = p + S_{0}$$ A fiduciary call (call plus PV of strike) equals a protective put (put plus underlying).

What is put-call-forward parity?

$$c + \frac{X}{(1+r)^{T}} = p + \frac{F_{0}(T)}{(1+r)^{T}}$$ Replacing the underlying with the present value of the forward price; useful when the underlying itself is not directly traded.

State the payoff at expiration of a long call and a long put.

Long call: $$\max(0,\; S_{T} - X).$$ Long put: $$\max(0,\; X - S_{T}).$$ where $S_{T}$ is the spot at expiration and $X$ is the strike.

What are the two components of an option's value before expiration?

Intrinsic (exercise) value plus time value. $$\text{Option value} = \text{Intrinsic value} + \text{Time value},$$ where intrinsic value is the payoff if exercised now (floored at zero) and time value is the remainder.

State the one-period binomial model formula for a call option value.

$$c = \frac{\pi c^{+} + (1-\pi) c^{-}}{1+r}$$ where the risk-neutral (up) probability is $$\pi = \frac{(1+r) - d}{u - d},$$ with $u$ and $d$ the up/down factors and $c^{+},c^{-}$ the option payoffs.

In the binomial model, what is the risk-neutral probability and what does 'risk-neutral' mean here?

$$\pi = \frac{1+r-d}{u-d}.$$ Risk-neutral means we price by discounting expected payoffs (under $\pi$) at the risk-free rate, valid because a hedged (arbitrage-free) portfolio earns the risk-free rate regardless of actual probabilities.

State the Black-Scholes-Merton formula for a European call (no dividends).

$$c = S_{0}N(d_{1}) - X e^{-rT}N(d_{2})$$ where $$d_{1} = \frac{\ln(S_{0}/X) + (r + \sigma^{2}/2)T}{\sigma\sqrt{T}}, \quad d_{2} = d_{1} - \sigma\sqrt{T}.$$

State the Black-Scholes-Merton formula for a European put (no dividends).

$$p = X e^{-rT}N(-d_{2}) - S_{0}N(-d_{1})$$ with $d_{1}$ and $d_{2}$ as defined for the BSM call. It can also be derived from put-call parity.

List the six inputs to the Black-Scholes-Merton option pricing model.

Underlying price $S_{0}$, exercise price $X$, time to expiration $T$, risk-free rate $r$, volatility $\sigma$ of the underlying's returns, and the yield/dividend on the underlying. Only volatility is not directly observable.

What key assumptions underlie the Black-Scholes-Merton model?

The underlying follows geometric Brownian motion with constant volatility, returns are lognormally distributed, no taxes/transaction costs, continuous trading, constant known risk-free rate, and no arbitrage; it prices European options.

How does the BSM model incorporate a continuous dividend yield $\delta$?

$$c = S_{0}e^{-\delta T}N(d_{1}) - X e^{-rT}N(d_{2})$$ with $$d_{1} = \frac{\ln(S_{0}/X) + (r - \delta + \sigma^{2}/2)T}{\sigma\sqrt{T}}.$$ Carry benefit $\delta$ lowers the call value.

What is the Black model used for?

Valuing European options on futures/forwards. $$c = e^{-rT}\left[F_{0}(T)N(d_{1}) - X N(d_{2})\right]$$ with $$d_{1} = \frac{\ln(F_{0}/X) + (\sigma^{2}/2)T}{\sigma\sqrt{T}}.$$

What is implied volatility?

The value of $\sigma$ that, when input into the BSM model, makes the model price equal the observed market option price. It represents the market's forward-looking estimate of the underlying's volatility.

Define Delta ($\Delta$) of an option.

The sensitivity of the option price to a small change in the underlying price: $$\Delta = \frac{\partial c}{\partial S}.$$ For a BSM call $\Delta = N(d_{1})$ (range $0$ to $1$); for a put $\Delta = N(d_{1}) - 1$ (range $-1$ to $0$).

Define Gamma ($\Gamma$) and what it measures.

The rate of change of delta with respect to the underlying price: $$\Gamma = \frac{\partial^{2} c}{\partial S^{2}}.$$ It measures the curvature/convexity of the option value; gamma is largest for at-the-money options near expiration.

Define Vega and how option value responds to it.

The sensitivity of option value to a change in volatility: $$\text{Vega} = \frac{\partial c}{\partial \sigma}.$$ Both calls and puts increase in value as volatility rises, so vega is positive for long options.

Define Theta ($\Theta$) of an option.

The sensitivity of option value to the passage of time (time decay): $$\Theta = \frac{\partial c}{\partial t}.$$ It is generally negative for long options, as value erodes as expiration approaches, all else equal.

Define Rho ($\rho$) of an option.

The sensitivity of option value to a change in the risk-free interest rate: $$\rho = \frac{\partial c}{\partial r}.$$ Calls increase in value as rates rise ($\rho > 0$); puts decrease as rates rise ($\rho < 0$).

What is a delta-neutral hedge and how many shares are needed to hedge a short call?

A position with zero net delta, insensitive to small underlying moves. To hedge a short call on $N$ options, hold $$N \times \Delta = N \times N(d_{1})$$ shares of the underlying long; the hedge must be rebalanced dynamically as delta changes (due to gamma).