Reading 71

Derivatives · Pricing and Valuation of Futures Contracts

MODULE 71.1: FUTURES VALUATION

LOS 71.a

Compare the value and price of forward and futures contracts.

While the price of a forward contract is constant over its life when no mark-to-market gains or losses are paid, its value will fluctuate with changes in the value of the underlying. The payment at settlement of the forward reflects the difference between the (unchanged) forward price and the spot price of the underlying.

The price and value of a futures contract both change when daily mark-to-market (MTM) gains and losses are settled. Consider a futures contract on 100 ounces of gold at $1,870 purchased on Day 0. The following illustrates the changes in contract price and value with daily mark-to-market payments.

中文翻譯

遠期合約（forward）在沒有每日結算的情況下，其價格（price）於合約期間維持不變，但價值（value）會隨標的資產價格波動而變化。結算日的支付金額反映遠期價格（未變動）與標的物即期價格之間的差額。

期貨合約（futures）由於每日結算（mark-to-market），其價格與價值都會隨之變動。以 Day 0 以每盎司 $1,870 買入 100 盎司黃金期貨為例，下表展示每日 MTM 的價格與價值變動。

Figure 71.1: Gold Futures — Daily Mark-to-Market

Day	Sequence of Events
Day 0	Price = settlement price of 1,870; MTM value = 0
Day 1	Settlement price = 1,875; MTM value = +$500 $500 addition to margin account New futures price = 1,875; MTM value reset to 0
Day 2	Settlement price = 1,855; MTM value = −$2,000 $2,000 deduction from margin account New futures price = 1,855; MTM value reset to 0

The change in the futures price to the settlement price each day returns its value to zero. Prices of forward contracts for which mark-to-market gains and losses are settled daily will also be adjusted to the settlement price.

教授提醒

每日 MTM 的核心觀念：結算後價值歸零。每日將期貨價格更新到當日的 settlement price 並完成現金交割（margin 增減），合約對雙方的「未實現損益」隨之歸零，相當於每天「重新訂約」。少數帶有每日 MTM 的遠期合約亦會比照處理。

中文翻譯

每日結算的核心：當期貨價格被更新為當日的 settlement price 後，價值即歸零（multi-day 累積損益已透過保證金帳戶現金結算）。若遠期合約亦採每日 MTM，價格也會比照調整為當日結算價。

Day 0：建倉價 1,870，MTM 價值 = 0。
Day 1：結算價漲到 1,875，多方獲利 $500（100 盎司 × $5），保證金帳戶 +$500；隨後合約價格更新為 1,875，價值重置為 0。
Day 2：結算價跌至 1,855，多方虧損 $2,000（100 盎司 × $20），保證金帳戶 −$2,000；合約價格更新為 1,855，價值再次歸零。

Interest Rate Futures and Basis Point Value

Interest rate futures contracts are available on many market reference rates. We may view these as exchange-traded equivalents to forward rate agreements (FRAs). One key difference is that interest rate futures are quoted on a price basis. For a market reference rate from time A to time B, an interest rate futures price is stated as:

$\text{futures price} = 100 - (100 \times \text{MRR}_{A,\,B-A})$

For example, if the futures price for a 6-month rate six months from now is 97, then $\text{MRR}_{6m,6m} = 3\%$.

Like other futures contracts, interest rate futures are subject to daily mark-to-market. The basis point value (BPV) of an interest rate futures contract is defined as:

$\text{BPV} = \text{notional principal} \times \text{period} \times 0.01\%$

If the contract in our example is based on a notional principal of $\text{€}1{,}000{,}000$, its BPV is

$\text{€}1{,}000{,}000 \times \dfrac{0.0001}{2} = \text{€}50$

This means a one basis point change in the MRR will change the futures contract value by €50.

中文翻譯

利率期貨可視為交易所版的 FRA（遠期利率協議）。最大差別在於利率期貨採價格報價（不是直接報利率）：

期貨價格 = 100 − (100 × MRR)

例：6 個月後起算的 6 個月期利率，若期貨價格報 97，則隱含 MRR = 3%。

基點價值（BPV）＝名目本金 × 期間 × 0.01%。若名目本金為 €1,000,000、期間 6 個月，則 BPV = €1,000,000 × (0.0001 / 2) = €50，亦即 MRR 每變動一個基點，合約價值就變動 €50。

LOS 71.b

Explain why forward and futures prices differ.

For pricing, the most important distinction between futures and forwards is that with futures, mark-to-market gains and losses are paid each day. Gains above initial margin can be withdrawn from a futures account, and losses that reduce margin deposits below the maintenance level require payments into the account. Forwards most often have no MTM cash flows, with gains or losses settled at contract expiration. Forwards typically do not require or provide funds in response to fluctuations in value during their lives.

If interest rates are constant or uncorrelated with futures prices over time, the prices of futures and forwards are the same. A positive correlation between interest rates and the futures price means that (for a long position) daily settlement provides funds (excess margin) when rates are high and they can earn more interest, and requires funds (margin deposits) when rates are low and the opportunity cost of deposited funds is less. Because of this:

Futures > Forwards (more attractive) when interest rates and futures prices are positively correlated.
Futures < Forwards (less attractive) when interest rates and futures prices are negatively correlated.

Because of the short maturity of most forwards and the availability of funds at near risk-free rates, differences between equivalent forwards and futures are not observed in practice. Additionally, derivative dealers in some markets with central clearing are required to post margin and may require derivative investors to post mark-to-market margin payments as well.

教授提醒

記憶口訣：「正相關 → 期貨贏」。利率漲時期貨價格也漲（多頭收 MTM 現金，可在高利率下投資），利率跌時期貨價格也跌（多頭付 MTM 現金，但機會成本低）——對多頭一律有利，所以多頭願意為期貨多付一點。負相關則相反。

中文翻譯

期貨與遠期最大的定價差異來自每日結算（MTM）。期貨保證金有獲利可提領、不足須補繳；遠期通常到期才一次結算。

若利率與期貨價格無相關或恆定，兩者價格相同。
若正相關：對多頭而言，利率高時收到 MTM 現金可賺較高利息；利率低時付出 MTM 現金的機會成本也低 → 期貨優於遠期。
若負相關：情況相反 → 遠期優於期貨。

實務上，由於多數遠期到期短，且資金可以接近無風險利率取得，兩者價差通常不顯著。此外，許多有中央結算的衍生品市場已要求交易商與投資人提供 MTM 保證金。

Convexity Bias for Interest Rate Forwards vs. Futures

A separate issue arises for interest rate forwards and futures settlement payments. Recall that the payoff on an interest rate forward is the present value (at the beginning of the forward period) of any interest savings (at the end of the forward period) from the difference between the realized MRR and the forward MRR. Because the realized MRR is itself the discount rate, the payment for an increase in the MRR will be smaller in magnitude than the payment for an equal decrease in the MRR — as the following example illustrates.

Consider a $1{,}000{,}000$ interest rate future on a 6-month MRR priced at 97.50 (an MRR of 2.5%) that settles six months from now. Each basis point change in the (annualized) MRR will change the value of the contract by

$0.0001 \times \dfrac{6}{12} \times \$1{,}000{,}000 = \$50$

If the MRR at settlement is either 2.51% or 2.49%, the payoff on the future at the end of one year is either $50 higher or $50 lower than when the MRR at settlement is 2.5%. Symmetric.

Compare this with the payoffs for an otherwise equivalent forward, $F_{6m,6m}$, priced at 2.5%:

If MRR at settlement is 2.51%, the long receives $\dfrac{50}{1 + 0.0251/2} = \$49.3803$.
If MRR at settlement is 2.49%, the long must pay $\dfrac{50}{1 + 0.0249/2} = \$49.3852$.

The value of forwards exhibits convexity: an increase in rates decreases the forward's value by less than a decrease in the interest rate increases the forward's value — just as we saw with bonds. Also as with bonds, the convexity effect on forward value increases for longer periods. The convexity of forwards is termed convexity bias, and forwards and futures prices can be significantly different for longer-term interest rates.

教授提醒

重點對照：利率期貨支付對稱（每基點 ±$50），因為每日 MTM 不需折現；利率遠期支付不對稱，因為到期支付需用實現的 MRR 折現回起算日，導致升降幅相同的利率變動造成的支付絕對值不等 → 凸性偏誤（convexity bias）。期間越長，凸性效果越大，兩者價差越明顯。

中文翻譯

除了 MTM 因素外，利率遠期與利率期貨還有一個結算面的差異：

利率期貨：每基點波動帶來的 P&L = 名目 × 期間 × 0.0001。例：$1,000,000 × 6/12 × 0.0001 = $50。利率上升 1bp 收 $50，下降 1bp 付 $50，完全對稱。
利率遠期：到期支付＝期末利息差的現值，需以實現的 MRR 折現。
- MRR 升至 2.51%：多方收 50 / (1 + 0.0251/2) ≈ $49.3803。
- MRR 降至 2.49%：多方付 50 / (1 + 0.0249/2) ≈ $49.3852。
升幅造成的「收款」小於降幅造成的「付款」，呈現凸性（convexity）。

這個不對稱稱為凸性偏誤（convexity bias），期間越長凸性影響越大，導致長天期利率合約的「遠期價格」與「期貨價格」可能顯著不同。

Module Quiz 71.1

1. For a forward contract on an asset that has no costs or benefits from holding it to have zero value at initiation, the arbitrage-free forward price must equal the:

A. expected future spot price.
B. future value of the current spot price.
C. present value of the expected future spot price.

B — For an asset with no holding costs or benefits, the forward price must equal the future value of the current spot price, compounded at the risk-free rate over the term of the forward, for the contract to have zero value at initiation. Otherwise an arbitrage opportunity would exist. (LOS 71.a)

2. For a futures contract to be more attractive than an otherwise equivalent forward contract, interest rates must be:

A. uncorrelated with futures prices.
B. positively correlated with futures prices.
C. negatively correlated with futures prices.

B — If interest rates are positively correlated with futures prices, interest earned on cash from daily settlement gains will be greater than the opportunity cost of interest on daily settlement losses, making the futures contract more attractive than an otherwise equivalent forward that does not feature daily settlement. (LOS 71.b)

Key Concepts — Reading 71

LOS 71.a

For a forward contract on which no MTM gains or losses are paid, the forward price is constant over its life, but the contract's value fluctuates with changes in the value of the underlying.

For a futures contract, both price and value change as daily MTM gains and losses are settled. The change in the futures price to the settlement price each day returns its value to zero.

Unlike forward rate agreements, interest rate futures are quoted on a price basis:

$\text{futures price} = 100 - (100 \times \text{MRR}_{A,\,B-A})$

The basis point value of an interest rate futures contract is $\text{BPV} = \text{notional} \times \text{period} \times 0.01\%$.

LOS 71.b

Because gains and losses on futures contracts are settled daily, prices of forwards and futures with the same terms can differ when interest rates are correlated with futures prices:

Positive correlation → futures more valuable than forwards.
Negative correlation → futures less valuable than forwards.
Constant or uncorrelated → futures and forwards prices are equal.

Convexity bias can result in price differences between interest rate futures contracts and otherwise equivalent forward rate agreements — the effect is larger for longer maturities.

中文翻譯 — 重點整理

【LOS 71.a】遠期合約若無 MTM：價格恆定、價值隨標的物變動。期貨合約因每日結算：價格與價值都會被更新，每日結算後價值歸零。利率期貨採價格報價：期貨價格 = 100 − (100 × MRR)；BPV = 名目本金 × 期間 × 0.01%。

【LOS 71.b】因每日結算，期貨與遠期價格可能因利率與期貨價格的相關性而不同：正相關 → 期貨較有價值；負相關 → 遠期較有價值；無相關或常數 → 價格相同。利率衍生品還會因到期支付折現造成凸性偏誤（convexity bias），期間愈長差異愈顯著。