Lesson 1 · Derivatives PricingCFA L2

Forward Markets and Contracts

A forward contract locks in today the price of a transaction that will settle in the future. This lesson builds the no-arbitrage framework for pricing and valuing forwards — at initiation, during their life, and at expiration — and applies it to equities, bonds, forward rate agreements, and currencies.

On this page

1 · Introduction 2 · The global forward market 3 · Types of forwards Day-count conventions 4.1 · Generic valuation 4.2 · Equity forwards 4.3 · Fixed-income forwards 4.3 · Forward rate agreements 4.4 · Currency forwards 5 · Credit risk

A forward contract is an agreement between two parties in which the buyer agrees to buy an underlying asset from the seller at a future date, at a price fixed today. The buyer holds the long position; the seller holds the short position. Unlike futures, forwards are customised, over-the-counter agreements — flexible, but exposing each side to the other's credit.

Learning outcomes

Explain how the value of a forward contract is determined at initiation, during the life of the contract, and at expiration.
Calculate and interpret the price and value of an equity forward contract, with dividends paid either discretely or continuously.
Calculate and interpret the price and value of (i) a forward on a fixed-income security, (ii) a forward rate agreement (FRA), and (iii) a currency forward.
Evaluate credit risk in a forward contract and explain how market value measures the credit exposure of each party.

1 · Introduction

The defining feature of a forward is that terms are agreed now but performance happens later. Three operational questions follow from that gap in time.

1.1 Delivery and settlement

At expiration the contract settles by physical delivery — the short delivers the asset and receives the agreed price — or by cash settlement, where only the difference between the market price and the contract price changes hands. Cash settlement dominates where physical delivery is impractical (an equity index, an interest rate).

1.2 Default (credit) risk

Because settlement lies in the future, each party risks the other failing to perform. As §5 shows, only the party to whom the contract currently has positive value is exposed: a forward's market value is itself a measure of credit exposure.

1.3 Terminating a contract early

A party can exit before expiration by entering an offsetting forward with the same expiration. With the original counterparty, exposure is extinguished; with a different party, price risk is removed but residual credit exposure to both counterparties remains.

2 · The structure of the global forward market

Context

The global derivatives market is vast — notional estimates run into the hundreds of trillions of dollars, with some analysts placing it at several times world GDP, a figure others argue is overstated because notional amounts dwarf the capital actually at risk. Forwards exist on virtually every asset class: equities, bonds, interest rates, currencies, and commodities.

3 · Types of forward contracts

3.1 Equity forwards

Forwards on individual stocks, stock portfolios, or stock indices. The key complication is the effect of dividends paid before expiration, handled in §4.2.

3.2 Bond and interest-rate forwards

Forwards on individual bonds and bond portfolios, and forwards on interest rates — forward rate agreements (FRAs). The underlying rate is historically a reference rate such as LIBOR/Euribor on a Eurocurrency time deposit.

3.3 Currency forwards

An agreement to exchange one currency for another at a fixed rate on a future date — priced by covered interest-rate parity (§4.4).

3.4 Other forwards

Commodity forwards (oil, precious metals, agricultural goods) and even weather forwards. Empirical work documents strong spillovers across crude-oil, precious-metal, and agricultural futures markets.

Day-count conventions

Every valuation below discounts over a time fraction, and that fraction depends on the market's day-count convention — the rule for counting days between two dates and days in a year. Using the wrong one is a common source of error.

Convention	Typical use	Example: 1 May → 1 Aug
30/360	Corporate, municipal, agency bonds; MBS	90/360 = 0.2500
Actual/360	Commercial paper, T-bills, most LIBOR (ex-GBP)	92/360 = 0.2556
Actual/365	US Treasuries; all GBP rates incl. LIBOR	92/365 = 0.2521
Actual/Actual	GBP & EUR bonds, US Treasuries, some USD swaps	92/366 = 0.2514

Watch out

Conventions are historical accidents, accepted as given. Confirm which one applies before discounting — a mismatch quietly corrupts every figure downstream.

4.1 · Generic pricing and valuation

First, three definitions that are easy to conflate:

Value — what an asset can be sold for, or what must be paid to acquire it.
Contract (forward) price — the fixed price $F(0,T)$ at which the future transaction occurs.
Cost — the amount that must be paid to produce, buy, or obtain the asset.

Set up a timeline. Today is time 0 (the contract is created); expiration is time $T$; $t$ is an arbitrary point in between.

0today

tmid-life

Texpiration

Let $S_0,\,S_t,\,S_T$ be the spot price at each date, $F(0,T)$ the forward price set at 0, and $V_t(0,T)$ the value at $t$ to the long.

Value at expiration (1)

$$ V_T(0,T) = S_T - F(0,T) $$

At $T$ the long buys at $F(0,T)$ an asset worth $S_T$; the contract is that difference.

By a cash-and-carry argument, buying the asset today and shorting a forward is riskless and must earn the risk-free rate — else arbitrage exists. Setting the initiation value to zero gives:

Forward price (2)

$$ F(0,T) = S_0\,(1+r)^{T} $$

The forward price is the spot compounded at the risk-free rate. It is not a forecast of $S_T$.

Value during the life of the contract (3)

$$ V_t(0,T) = S_t - \frac{F(0,T)}{(1+r)^{\,T-t}} $$

A positive value means the asset is worth more than the present value of what the long must pay.

The three valuation points, together

At initiation $V_0(0,T)=S_0-\dfrac{F(0,T)}{(1+r)^{T}}=0$; during the life $V_t(0,T)=S_t-\dfrac{F(0,T)}{(1+r)^{T-t}}$; at expiration $V_T(0,T)=S_T-F(0,T)$. All three are the same idea: spot now, minus the present value of the price you locked in.

Worked example 1

Forward on an asset, no income

An investor holds an asset worth $125.72 and will sell it in 9 months, hedging with a short forward. The risk-free rate is 5.625%; day-count is 30/360.

A. The appropriate forward price.

Show solution

With $S_0=125.72,\ T=0.75,\ r=0.05625$:

$$ F(0,T)=125.72\,(1.05625)^{0.75}=\$130.99 $$ F(0,T) = $130.99

B. A counterparty offers the forward at $140. Identify the riskless strategy and its annualised return.

Show solution

Sell forward at the rich $140 while holding the asset. Locked-in 9-month return:

$$ \frac{140}{125.72}-1 = 0.1136 \;\Rightarrow\; (1.1136)^{12/9}-1 = 15.43\% $$

Above the 5.625% risk-free rate — a hedged position earning an arbitrage profit.

15.43% annualised > 5.625% risk-free

C. Two months later the asset trades at $118.875. Value the contract to the (short) investor.

Show solution

$$ V_t(0,T)=118.875-\frac{130.99}{(1.05625)^{7/12}}=-\$8.00 $$

Negative to the long → +$8.00 to the short investor.

−$8.00 to the long → +$8.00 to the short

D. At expiration the asset is $123.50. Value the contract and assess the hedged position.

Show solution

$$ V_T(0,T)=123.50-130.99=-\$7.49 $$

The short gains $7.49 on the forward but loses $125.72-123.50=\$2.22$ on the asset; net $\$5.27$, a 4.19% return over 9 months — $(1.0419)^{12/9}-1=5.625\%$, exactly the risk-free rate.

Net +$5.27 ≈ 5.625% annualised = risk-free

4.2 · Equity forwards (with dividends)

If the underlying pays dividends before expiration, the forward holder forgoes that income, so it must be stripped out of the spot price. With discrete dividends of present value

Present value of dividends

$$ PV(D,0,T)=\sum_{i=1}^{n}\frac{D_i}{(1+r)^{t_i}} $$

Equity forward price — discrete dividends (4)(5)

$$ F(0,T)=\big[S_0-PV(D,0,T)\big](1+r)^{T}=S_0(1+r)^{T}-FV(D,0,T) $$

Equity forward price — continuous dividends (6)

$$ F(0,T)=\big(S_0\,e^{-\delta^{c}T}\big)e^{\,r^{c}T},\qquad r^{c}=\ln(1+r) $$

$\delta^{c}$ is the continuously-compounded dividend yield; $r^{c}$ the continuously-compounded risk-free rate.

Value during the life (7)(8)

$$ V_t(0,T)=S_t-PV(D,t,T)-\frac{F(0,T)}{(1+r)^{T-t}} $$

$$ V_t(0,T)=S_t\,e^{-\delta^{c}(T-t)}-F(0,T)\,e^{-r^{c}(T-t)} $$

Worked example 2

Equity forward with two dividends

A manager will buy a stock in 200 days via a long forward. The stock is $62.50 and pays $0.75 dividends in 50 and 140 days. The risk-free rate is 4.2%.

A. The forward price.

Show solution

$$ PV(D,0,T)=\frac{0.75}{1.042^{50/365}}+\frac{0.75}{1.042^{140/365}}=\$1.48 $$ $$ F(0,T)=(62.50-1.48)(1.042)^{200/365}=\$62.41 $$ F(0,T) = $62.41

B. 75 days later the stock is $55.75. Value the contract.

Show solution

The first dividend is paid; only the 140-day dividend remains (65 days away):

$$ PV(D,t,T)=\frac{0.75}{1.042^{65/365}}=\$0.74 $$ $$ V_t(0,T)=55.75-0.74-\frac{62.41}{(1.042)^{125/365}}=-\$6.53 $$ −$6.53 to the long

C. At expiration the stock is $58.50. Value the contract.

Show solution

$$ V_T(0,T)=58.50-62.41=-\$3.91 $$ −$3.91 to the long

4.3 · Fixed-income forwards

A forward on a coupon bond is valued like an equity forward, with coupon interest $CI$ playing the role of dividends. Write $B_0^{c}(T+Y)$ for the bond price (incl. accrued) of a bond maturing $Y$ periods after the forward expires.

Bond forward price (9)(10)

$$ F(0,T)=\big[B_0^{c}(T+Y)-PV(CI,0,T)\big](1+r)^{T}=B_0^{c}(T+Y)(1+r)^{T}-FV(CI,0,T) $$

Bond forward value (11)

$$ V_t(0,T)=B_t^{c}(T+Y)-PV(CI,t,T)-\frac{F(0,T)}{(1+r)^{T-t}} $$

Worked example 3

Forward on a coupon bond

A 5-year bond pays $50 semi-annual coupons (days 181, 365, 547, 730). It is now day 150, bond at $1,010.25 (incl. accrued). The owner forward-sells the day after the 4th coupon (day 731). Risk-free 8%.

A. The forward price.

Show solution

$T=(731-150)/365=581/365$. PV of the four coupons (31, 215, 397, 580 days out):

$$ PV(CI,t,T)=\frac{50}{1.08^{31/365}}+\frac{50}{1.08^{215/365}}+\frac{50}{1.08^{397/365}}+\frac{50}{1.08^{580/365}}=\$187.69 $$ $$ F(0,T)=(1010.25-187.69)(1.08)^{581/365}=\$929.76 $$ F(0,T) = $929.76

B. 365 days later (day 515) the rate is 7% and the bond is $1,025.375. Value the contract.

Show solution

Two coupons remain (32 and 215 days out); time to expiration $216/365$:

$$ PV(CI,t,T)=\frac{50}{1.07^{32/365}}+\frac{50}{1.07^{215/365}}=\$97.75 $$ $$ V_t(0,T)=1025.37-97.75-\frac{929.76}{(1.07)^{216/365}}=\$34.36 $$ +$34.36 to the long

4.3 · Forward rate agreements (FRAs)

An FRA fixes today the interest rate on a future deposit. The notation $a \times b$ means the FRA expires in $a$ months on a deposit maturing in $b$ months — the underlying rate covers $b-a$ months.

Notation	Expires (h)	Underlying rate
1 × 4	1 month	3-month rate
1 × 7	1 month	6-month rate
3 × 6	3 months	3-month rate
3 × 9	3 months	6-month rate
6 × 12	6 months	6-month rate

Let the FRA expire on day $h$ on an $m$-day deposit (maturing at $h+m$). $L_0(\cdot)$ is today's LIBOR for a tenor; $g$ is a valuation date during the FRA's life.

FRA rate (13)

$$ FRA(0,h,m)=\left[\frac{1+L_0(h+m)\frac{h+m}{360}}{1+L_0(h)\frac{h}{360}}-1\right]\frac{360}{m} $$

The rate that makes lending for $h+m$ days equivalent to lending for $h$ then $m$ days — pure no-arbitrage.

Value of the FRA on day g (14)

$$ V_g(0,h,m)=\frac{1}{1+L_g(h-g)\frac{h-g}{360}}-\frac{1+FRA(0,h,m)\frac{m}{360}}{1+L_g(h+m-g)\frac{h+m-g}{360}} $$

Payoff at expiration (day h)

$$ \text{Payoff}=\frac{\big[L_h(m)-FRA(0,h,m)\big]\frac{m}{360}}{1+L_h(m)\frac{m}{360}} $$

Worked example 4

FRA on 180-day Euribor

A treasurer hedges 180-day Euribor 30 days from now. Today: 30-day Euribor 5.75%, 210-day Euribor 6.15%. Notional €20m.

A. Identify the FRA.

Show solution

Expires in 1 month on a 6-month deposit → a 1 × 7 FRA ($h=30,\ m=180$).

1 × 7 FRA

B. The FRA rate.

Show solution

$$ FRA(0,30,180)=\left[\frac{1+0.0615\frac{210}{360}}{1+0.0575\frac{30}{360}}-1\right]\frac{360}{180}=0.0619 $$ FRA rate = 6.19%

C. 20 days later rates fall: 10-day Euribor 5.45%, 190-day Euribor 5.95%. Value the long FRA.

Show solution

$$ V_{20}=\frac{1}{1+0.0545\frac{10}{360}}-\frac{1+0.0619\frac{180}{360}}{1+0.0595\frac{190}{360}}=-0.0011 $$

On €20m: $20{,}000{,}000\times(-0.0011)=-\text{€}22{,}000$.

≈ −€22,000 to the long

D. At expiration 180-day Euribor is 5.72%. Find the settlement payment.

Show solution

$$ \text{Payoff}=\frac{(0.0572-0.0619)\frac{180}{360}}{1+0.0572\frac{180}{360}}=-0.0023 $$

On €20m: $-\text{€}46{,}000$ — paid by the long, since the realised rate came in below the FRA rate.

−€46,000 paid by the long

4.4 · Currency forwards

A currency forward is priced by covered interest-rate parity: holding the foreign currency earns the foreign rate $r^{f}$, so the forward offsets that yield difference against the domestic rate $r$. $S_0$ is quoted as domestic per unit of foreign.

Currency forward price (15)(16)

$$ F(0,T)=\frac{S_0}{(1+r^{f})^{T}}(1+r)^{T} \qquad\text{or}\qquad F(0,T)=\big(S_0\,e^{-r^{fc}T}\big)e^{\,r^{c}T} $$

Currency forward value (17)(18)

$$ V_t(0,T)=\frac{S_t}{(1+r^{f})^{T-t}}-\frac{F(0,T)}{(1+r)^{T-t}} $$

$$ V_t(0,T)=S_t\,e^{-r^{fc}(T-t)}-F(0,T)\,e^{-r^{c}(T-t)} $$

Worked example 5

Over-priced currency forward (GBP)

Spot GBP = $1.7600; US rate 5.1%, UK rate 6.2% (annual). One-year forwards quoted at $1.7500.

A. Find the riskless arbitrage and its return.

Show solution

$$ F(0,T)=1.7600\cdot\frac{1.051}{1.062}=\$1.7418 $$

The quote $1.7500 is over-priced: sell the pound forward and replicate — take $1.7600/1.062=\$1.6573$, convert to £0.9416, invest in the UK at 6.2% to grow to £1.0000, deliver into the forward for $1.7500.

$$ \frac{1.7500}{1.6573}-1 = 5.59\% > 5.1\% = r_\$ $$ 5.59% riskless > 5.1% domestic rate

B. One month later, rates unchanged but spot = $1.72. Value the short position.

Show solution

$$ V_t(0,T)=\frac{1.72}{1.062^{11/12}}-\frac{1.75}{1.051^{11/12}}=-\$0.0443 $$

−$0.0443 to the long → +$0.0443 to the short.

+$0.0443 per £ to the short

C. At expiration the pound is $1.69. Value to the short.

Show solution

$$ V_T(0,T)=1.69-1.75=-\$0.06 \text{ (long)} \;\Rightarrow\; +\$0.06 \text{ (short)} $$ +$0.06 per £ to the short

Worked example 6

Under-priced currency forward (EUR)

Spot EUR = $1.2015; US rate 4.5%, EC rate 5.2% (annual). One-year forwards quoted at $1.1920.

A. Find the arbitrage and its return.

Show solution

$$ F(0,T)=1.2015\cdot\frac{1.045}{1.052}=\$1.1935 $$

The quote $1.1920 is under-priced: buy the euro forward and fund it — start with €0.7965, convert to $0.9569, invest in the US at 4.5% to reach $1.0000, use the forward to buy €0.8389.

$$ \frac{0.8389}{0.7965}-1 = 5.33\% > 5.2\% = r_\euro $$ 5.33% riskless > 5.2% foreign rate

B. Two months later, rates unchanged but spot = $1.2200. Value the long position.

Show solution

$$ V_t(0,T)=\frac{1.2200}{1.052^{10/12}}-\frac{1.1935}{1.045^{10/12}}=\$0.0086 $$

Measured against the quoted forward of $1.1920, this is a gain of about $0.0205 to the long.

≈ +$0.0205 per € to the long

C. At expiration the euro is $1.1950. Value to the long.

Show solution

$$ V_T(0,T)=1.1950-1.1920=\$0.0030 $$ +$0.0030 per € to the long

5 · Credit risk and forward contracts

Credit risk arises when the party that owes the greater amount at expiration cannot pay — for instance through bankruptcy before settlement. Three principles follow:

The market value of a forward reflects the current value of the expiration claim — so it directly measures credit exposure.
At any moment only one party faces credit risk: the one to whom the contract has positive value.
Marking to market mitigates this — periodically one party pays the other the current market value, and the contract is re-struck at the prevailing price, resetting exposure to zero.

Key takeaway

Every forward valuation here is one sentence in different clothes: the value to the long is the current worth of the underlying minus the present value of the price locked in. Income on the underlying (dividends, coupons, a foreign interest rate) is subtracted from the spot leg; the forward price is whatever makes the initiation value zero. Master that, and equities, bonds, FRAs, and currencies are one framework.

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