Financial Risk Manager (FRM) Financial Markets and Products Syllabus

Define convenience yield and give the commodity forward pricing formula including it.

Convenience yield $y$ is the benefit of physically holding a commodity (e.g., keeping production running, meeting demand spikes) not available to a futures holder. $$F_0 = S_0 e^{(r+u-y)T}$$ where $u$ is the storage cost rate; a high convenience yield can push markets into backwardation.

Define contango and backwardation.

Contango: futures price exceeds the expected future spot (or futures price rises with maturity), typically when carrying costs dominate. Backwardation: futures price is below the expected future spot (or falls with maturity), often driven by a high convenience yield.

State the cost-of-carry formula for a currency forward (covered interest parity).

$$F_0 = S_0 e^{(r_d - r_f)T}$$ where $S_0$ is the spot exchange rate (domestic per foreign), $r_d$ the domestic risk-free rate, and $r_f$ the foreign risk-free rate. The foreign rate acts like a dividend yield on the foreign currency.

What is a Eurodollar futures contract and how does its price relate to the interest rate?

A Eurodollar future is based on the 3-month rate on a notional \$1 million deposit. Its quoted price is $100 - R$ (with $R$ the rate in percent), so price and rate move inversely. Each basis point change equals \$25 per contract ($\$1{,}000{,}000 \times 0.0001 \times \tfrac{3}{12}$).

Define a plain vanilla interest rate swap.

An agreement to exchange, on a notional principal, a stream of fixed-rate interest payments for a stream of floating-rate payments (e.g., tied to SOFR/LIBOR) over a set tenor. Only net interest is exchanged; the notional principal is not exchanged.

How can a plain vanilla interest rate swap be decomposed for valuation?

It can be valued as (1) a long/short position in a fixed-rate bond versus a floating-rate bond: $V_{swap} = B_{fix} - B_{flt}$ (for the floating-rate payer), or equivalently (2) as a portfolio of forward rate agreements (FRAs), one per payment date, each valued and discounted.

What is the value of the floating-rate side of a swap immediately after a payment date?

Immediately after a reset/payment, a floating-rate bond is worth its par (notional) value, because the next coupon is set at the current market rate. Thus $B_{flt} = \text{Notional}$ right after reset, simplifying swap valuation.

How is the swap (fixed) rate set at initiation of a plain vanilla swap?

It is set so the swap has zero value at inception, i.e., the present value of fixed payments equals the present value of expected floating payments. Equivalently, the fixed rate is the par yield that makes $B_{fix}=B_{flt}=\text{Notional}$.

Distinguish a currency swap from a cross-currency basis swap.

A currency swap exchanges principal and interest in two different currencies (often fixed-for-fixed), with principals typically exchanged at start and end at the same rate. A cross-currency basis swap exchanges floating-for-floating in two currencies, with the basis spread reflecting cross-currency funding/demand pressures.

How does principal exchange differ between an interest rate swap and a currency swap?

In a single-currency interest rate swap the notional is never exchanged (only net interest). In a currency swap the full principal amounts in each currency are typically exchanged at both the start and the maturity of the swap, since they are in different currencies.

What is an equity swap?

A swap where at least one leg pays the total return (price change plus dividends) on an equity or equity index, exchanged for a fixed or floating interest payment on a notional. It lets an investor gain or shed equity exposure synthetically without trading the underlying shares.

What is a commodity swap?

An agreement to exchange a fixed price for a floating (market) price on a notional quantity of a commodity over several periods. A consumer locks in cost (pays fixed, receives floating); a producer locks in revenue (receives fixed, pays floating), hedging commodity price risk.

Define a credit default swap (CDS) and identify the two cash-flow legs.

A CDS is a contract where the protection buyer pays a periodic premium (the CDS spread) to the protection seller, who compensates the buyer for losses if a reference entity defaults. Premium leg = periodic spread payments; protection/contingent leg = payout on a credit event.

What is the CDS spread and what does it indicate?

The CDS spread is the annual premium (in basis points of notional) the protection buyer pays. It reflects the market's assessment of the reference entity's credit risk; a wider spread implies higher perceived default probability and/or lower expected recovery.

Why does counterparty risk arise in swap transactions, and when is exposure largest?

Each party faces the risk the other defaults when the swap has positive value to the non-defaulting party (it is in the money to them). For an interest rate swap, expected exposure is typically largest in the middle of the swap's life, balancing rate-diffusion (growing) against amortization (shrinking remaining payments).

What is a Credit Valuation Adjustment (CVA)?

CVA is the market value of expected loss due to counterparty default — the adjustment that reduces the risk-free value of a derivative to its fair value accounting for counterparty credit risk. Conceptually $\text{CVA} = \sum (\text{PD} \times \text{LGD} \times \text{Discounted Expected Exposure})$ over time buckets.

State the payoffs at expiration for long call and long put options.

Long call payoff: $\max(S_T - K, 0)$. Long put payoff: $\max(K - S_T, 0)$, where $S_T$ is the underlying price at expiration and $K$ the strike. Short positions have the negated payoffs.

State the put-call parity relationship for European options on a non-dividend stock.

$$c + K e^{-rT} = p + S_0$$ where $c$ and $p$ are European call and put prices on the same strike $K$ and maturity $T$, and $S_0$ is the spot price. It says a fiduciary call equals a protective put.

How does put-call parity change for a stock paying continuous dividend yield $q$?

$$c + K e^{-rT} = p + S_0 e^{-qT}$$ The spot is discounted by the dividend yield. For discrete dividends with present value $D$: $c + Ke^{-rT} = p + S_0 - D$.

Define intrinsic value and time value of an option.

Intrinsic value is the immediate exercise payoff: $\max(S-K,0)$ for a call, $\max(K-S,0)$ for a put. Time value is the option premium minus intrinsic value, reflecting the chance of further favorable moves before expiry; it is always nonnegative and decays to zero at maturity.

Describe a bull spread using calls and its risk/reward profile.

Buy a call at lower strike $K_1$ and sell a call at higher strike $K_2$ ($K_2>K_1$), same expiry. Net cost (debit) is paid upfront. Both profit and loss are capped: maximum profit $= K_2 - K_1 - \text{net premium}$; maximum loss $=$ net premium. It profits from a moderate rise.

Describe a straddle and when it is used.

A long straddle buys a call and a put at the same strike and expiry. It profits from a large move in either direction (high volatility), losing the most if the underlying stays near the strike. Cost = both premiums; breakevens are $K \pm (\text{total premium})$.

What is a butterfly spread (long, with calls)?

Buy one call at $K_1$, sell two calls at $K_2$, buy one call at $K_3$, with equal spacing $K_2-K_1=K_3-K_2$. It is a low-cost bet that the underlying stays near $K_2$; maximum profit at $S_T=K_2$, limited loss equal to the net premium, profiting from low volatility.

What is a collar and what is it used for?

Holding the underlying, buy a protective put (strike below spot) and sell a covered call (strike above spot). The call premium offsets the put cost, creating a low/zero-cost band that caps both downside loss and upside gain — used to lock in a value range.

State the lower and upper bounds for a European call on a non-dividend stock.

$$\max(S_0 - K e^{-rT},\,0) \le c \le S_0$$ The call is worth at least the stock price minus the present value of the strike (and at least zero), and never more than the stock itself.

State the lower and upper bounds for a European put on a non-dividend stock.

$$\max(K e^{-rT} - S_0,\,0) \le p \le K e^{-rT}$$ The put is worth at least the present value of the strike minus the stock (and at least zero), and at most the present value of the strike.

Why is it never optimal to exercise an American call on a non-dividend-paying stock early?

Because $C \ge c \ge S_0 - Ke^{-rT} > S_0 - K$ (intrinsic value) for $r>0$, the call is worth more alive than exercised — early exercise forfeits remaining time value and the interest earned by delaying payment of $K$. Thus $C = c$ in this case.

When can early exercise of American options be optimal?

Early exercise of an American put can be optimal (deep in the money, to capture/reinvest the strike sooner). Early exercise of an American call can be optimal just before a dividend date, if the dividend captured exceeds the lost time value/interest.

Define and contrast knock-in and knock-out barrier options.

Barrier options activate or extinguish based on the underlying hitting a barrier $H$. A knock-in only comes into existence if the barrier is breached; a knock-out ceases to exist if breached. Relationship: knock-in + knock-out (same terms) = corresponding vanilla option.

What is an Asian option and why is it typically cheaper than a vanilla option?

An Asian option's payoff depends on the average price of the underlying over a period rather than the terminal price (e.g., call payoff $\max(\bar{S}-K,0)$). Averaging lowers effective volatility, making it cheaper than an otherwise identical European option and reducing manipulation/expiry-spike risk.

Define lookback and digital (binary) options.

A lookback option's payoff uses the maximum or minimum underlying price over its life (e.g., floating-strike call pays $S_T - S_{min}$). A digital/binary option pays a fixed amount (or nothing) if the underlying is beyond the strike at expiry — an all-or-nothing payoff.

Define a bond's clean price versus dirty (full) price.

The dirty (full) price is what the buyer actually pays — the present value of all remaining cash flows, including accrued interest. The clean price is the dirty price minus accrued interest, the figure typically quoted: $\text{Clean} = \text{Dirty} - \text{Accrued Interest}$.

How is accrued interest on a bond calculated?

$$\text{AI} = \text{Coupon} \times \frac{\text{days since last coupon}}{\text{days in coupon period}}$$ The day-count convention (e.g., actual/actual for Treasuries, 30/360 for corporates) determines how the day fractions are computed.

Define Macaulay duration and modified duration.

Macaulay duration is the cash-flow-weighted average time to receive a bond's payments (in years). Modified duration measures price sensitivity to yield: $$D_{mod} = \frac{D_{Mac}}{1 + y/m}$$ so that $\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y$, with $m$ the compounding frequency.

Define the term structure of interest rates and name three common theories explaining its shape.

The term structure (yield curve) is the relationship between yield and maturity for bonds of equal credit quality. Theories: (1) Pure/Unbiased Expectations — long rates reflect expected future short rates; (2) Liquidity Preference — investors demand a term premium for longer maturities; (3) Market Segmentation/Preferred Habitat — supply/demand within maturity segments set rates.

Distinguish spot rates, forward rates, and par yields.

A spot (zero) rate is the yield on a zero-coupon bond maturing at time $T$. A forward rate is the rate, agreed today, for borrowing/lending over a future period implied by two spot rates. A par yield is the coupon rate that prices a coupon bond at par given the spot curve.

How is a forward rate computed from two spot rates under continuous compounding?

The forward rate $f$ between times $T_1$ and $T_2$ satisfies $$f = \frac{r_2 T_2 - r_1 T_1}{T_2 - T_1}$$ where $r_1, r_2$ are the continuously compounded spot rates for maturities $T_1$ and $T_2$ respectively.

What is a forward rate agreement (FRA) and its settlement payoff?

An FRA is an OTC contract locking in an interest rate $R_K$ on a notional $L$ for a future period $[T_1,T_2]$. To the rate-payer, the value/payoff is proportional to $L \times (R_M - R_K) \times (T_2 - T_1)$, where $R_M$ is the realized reference rate; it is typically settled (discounted) at $T_1$.

What does duration-based hedging of a bond portfolio with futures require?

The number of interest-rate futures to immunize against parallel yield shifts is $$N = \frac{P \, D_P}{V_F \, D_F}$$ where $P$ is the portfolio value with duration $D_P$, and $V_F$ and $D_F$ are the futures contract value and the duration of its underlying (cheapest-to-deliver) bond.