Core Path · L3.2.05

FX and commodity risk

12 minMarket riskFXCommodities

By the end you will be able to

  1. 1Measure a book's net open currency position from spot holdings and forwards, and its P&L for a move in the rate.
  2. 2Explain hedge drift, and why a forward carries interest rate risk as well as spot risk.
  3. 3Read a commodity futures curve: contango, backwardation, storage, convenience yield and seasonality, and the roll they imply.
  4. 4Size a cross-hedge and say how much basis risk is left.

Why it matters

The desk bought £30m of US shares from a client and sold $26m forward to hedge the dollars. On day 205 of its history the pound rose 1.2% against the dollar: the basket lost £357,849 in pounds and the forward made back £238,566. The hedge worked, but only two thirds of the way, by design. Currencies and commodities are where a desk's hedges look simplest and leak in the most ways.

Core explanation

Currencies and commodities

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

Forward points, the futures curves, the storage figures and the jet fuel hedge are extras for this lesson, not part of the book.
PositionSizeValue todayRisk factors
Equities
US equity basket ($39m)$39m at 1.30 = £30m£30.00mUS index, GBP/USD
FX and commodities
Sell $26m forwardSell $26m at 1.3000, 3 months£0.00mGBP/USD
Brent futures (200,000 barrels)200,000 barrels × $80 = $16m£0.00mBrent, GBP/USD

Spot exposure: the net open position

A sterling bank holding dollar assets gains when the dollar strengthens. GBP/USD is quoted in dollars per pound (L1.6.01: How the FX market works), so a fall in GBP/USD is a stronger dollar. The basket is worth £30m today and its pound value moves one for one, inversely, with GBP/USD: −£300,000 for a 1% rise. The forward that sells $26m for £20m does the opposite: +£200,000 per 1%.

Net, the book is long $13m: its net open position. It loses £100,000 for each 1% rise in GBP/USD. With GBP/USD's daily volatility of 0.5%, that is £50,000 on a one-standard-deviation day. The hedge ratio, 66.7%, is a choice: the trader keeps some dollar exposure on purpose.

The hedge drifts

The forward sells a fixed $26m. The basket's dollar value moves with the US index. After a 10% rise the basket holds $42.9m, and $16.9m is now unhedged. That is hedge drift: a hedge sized on today's value must be topped up as the value changes. In the language of L3.2.02: Risk factors and position mapping, the basket's £ value depends on the US index and GBP/USD together, a cross effect that L3.2.06: Option Greeks across a book measures.

Forward exposure

A forward is not just spot. By covered interest parity (L1.6.02: FX forwards and covered interest parity), the forward rate is spot adjusted by the two currencies' interest rates. With an illustrative 3-month USD rate of 4.5% and GBP at 4.0%, 3-month GBP/USD would be 1.3016: 16.3 forward points above spot. A forward is therefore a spot trade plus a dollar deposit and a sterling loan, and it moves when either interest rate moves. The book ignores forward points (it treats the two rates as equal), so its forward has a pure spot exposure. With the points in, the forward's spot exposure would be £19.78m per 1.00 rather than £20m, and it would carry £494 per bp of USD rates and −£494 per bp of GBP rates: small here, large for a desk with billions of forwards.

Commodities: one price is not enough

The desk is long 200,000 barrels of Brent futures, $16m or £12.31m. At 2% daily volatility that is £246,154 on a one-standard-deviation day. On day 640, an oil supply shock lifted Brent 8.6% and the position made £1,105,309.

Commodities have no single price. Each delivery month trades as its own future, and the curve of those prices has a shape (L1.6.06: Commodity markets). For a storable commodity, buying now and storing it must cost about the same as buying a future, so futures sit at most at spot plus financing and storage: the contango limit. When the physical commodity is scarce, holders value having it now (a convenience yield) and the curve can fall into backwardation. Gas and power cannot be stored cheaply, so their curves show seasonality instead: winter gas costs more than summer gas.

The curve's shape creates risk a single Brent factor cannot see. A desk long the front month and short the sixth month looks flat to a parallel move, but if a backwardated curve flattens it loses: the spread was $2.00 a barrel, worth $400k on 200,000 barrels. And a position rolled from month to month earns or pays the roll: $80k a month in the backwardated curve below, −$100k in the contango one.

ExploreThree futures curves

Curve

70758085123456789101112Delivery month$ per barrel

Hold the second-month contract for a month with the curve unchanged and it becomes the front month: a gain of $0.40 a barrel, or $80k on the desk's 200,000 barrels (+£0.062m). In backwardation a long position rolls up the curve.

Basis risk

Hedges rarely match exactly. An airline's jet fuel is hedged with Brent futures because Brent trades in size; a client's US share portfolio is hedged with index futures; dollars due in five months are hedged with a 3-month forward. The gap between what is hedged and the hedge is the basis, and it moves. L2.6.02: Hedging with forwards and futures introduced the idea; the formal section sizes the best hedge and the risk left over.

Sort itWhich risk is it?

Name the main risk left in each hedged position.

  • An airline hedges jet fuel with Brent futures; jet fuel's premium over Brent widens.

  • The desk's US basket rises 15%, but its forward still sells $26m.

  • A desk long December gas and short March gas sees a mild winter forecast.

  • A UK importer hedges euro payments with a GBP/USD forward because it is cheaper to trade.

The formal version

Spot. A holding of AA dollars is worth V=A/SV = A/S pounds, with SS in dollars per pound. With S=S0efS = S_0 e^{f}:

V(f)=AS0 e−f,∂V∂f∣f=0=−AS0net:∂Vbook∂f=−Abasket−AsoldS0=−$13m1.30=−£10.00m\begin{aligned} V(f) &= \frac{A}{S_0}\,e^{-f}, \qquad \frac{\partial V}{\partial f}\Big|_{f=0} = -\frac{A}{S_0} \\ \text{net:}\quad \frac{\partial V_{\text{book}}}{\partial f} &= -\frac{A_{\text{basket}} - A_{\text{sold}}}{S_0} = -\frac{\$13\text{m}}{1.30} = -\text{£}10.00\text{m} \end{aligned}

Forward. Covered interest parity with continuous rates, and the value of a contract to receive A/KA/K pounds and pay AA dollars at TT:

F=S e(r$−r£)T=1.30×e(0.045−0.040)×0.25=1.3016Vfwd=AK e−r£T−AS e−r$T∂Vfwd∂f=AS e−r$T,∂Vfwd∂r$=AS T e−r$T,∂Vfwd∂r£=−AK T e−r£T\begin{aligned} F &= S\,e^{(r_\$ - r_\text{£})T} = 1.30 \times e^{(0.045 - 0.040) \times 0.25} = 1.3016 \\ V_{\text{fwd}} &= \frac{A}{K}\,e^{-r_\text{£}T} - \frac{A}{S}\,e^{-r_\$T} \\ \frac{\partial V_{\text{fwd}}}{\partial f} &= \frac{A}{S}\,e^{-r_\$T}, \qquad \frac{\partial V_{\text{fwd}}}{\partial r_\$} = \frac{A}{S}\,T\,e^{-r_\$T}, \qquad \frac{\partial V_{\text{fwd}}}{\partial r_\text{£}} = -\frac{A}{K}\,T\,e^{-r_\text{£}T} \end{aligned}

Commodity futures. With financing rate rr, storage cost uu and convenience yield yy (all continuous, a year):

F(T)=S e(r+u−y)T≤S e(r+u)Ty=r+u−1Tln⁡F(T)S\begin{aligned} F(T) &= S\,e^{(r + u - y)T} \le S\,e^{(r + u)T} \\ y &= r + u - \frac{1}{T}\ln\frac{F(T)}{S} \end{aligned}

With Brent at $80, rr = 4.5% and uu = 2%, the 6-month future can be at most $82.64. The backwardated curve's 6-month price of $77.60 implies a convenience yield of 12.6%.

The minimum-variance hedge. Hedge an exposure with price change ΔS\Delta S by selling hh futures per unit, with futures change ΔF\Delta F:

var⁡(ΔS−h ΔF)=σS2−2hρ σSσF+h2σF2ddh=0  ⟹  h∗=ρ σSσFvar⁡min⁡=σS2(1−ρ2),hedge effectiveness=ρ2\begin{aligned} \operatorname{var}(\Delta S - h\,\Delta F) &= \sigma_S^2 - 2h\rho\,\sigma_S\sigma_F + h^2\sigma_F^2 \\ \frac{d}{dh} = 0 &\;\Longrightarrow\; h^* = \rho\,\frac{\sigma_S}{\sigma_F} \\ \operatorname{var}_{\min} &= \sigma_S^2(1 - \rho^2), \qquad \text{hedge effectiveness} = \rho^2 \end{aligned}

Try it

Move GBP/USD and the US index, and change how many dollars the desk sells forward. Then push the US index up 10% with GBP/USD up 5%, and watch the currency part grow beyond what the hedge was sized for.

Try itHedge the basket's dollars

US basket, in pounds+£1,578,947
FX forward−£1,052,632
Net+£526,316
Equity part (dollars at 1.30)£0
Currency part+£526,316
Dollars still unhedged after the move$13.0m
New GBP/USD1.2350

A falling GBP/USD is a stronger dollar, which helps a sterling holder of dollar assets. Raise the US index and the basket holds more dollars than the forward covers: the hedge drifts.

Worked example

The FX and commodities book

Net position, scenarios, Brent and a cross-hedge

Step 1 of 6
  1. 1

    Net open position

    $39m − $26m = $13m long, or £10.00m per 1.00 log move of GBP/USD. One standard deviation (0.5%): £50,000. 99% one-day VaR from FX alone: 2.326×£50,000=£116,3172.326 \times \text{£}50{,}000 = \text{£}116{,}317.

Regulatory lens · looking ahead

Capital rules treat FX and commodity risk across the whole bank, not just the trading book: a bank's foreign-currency positions in its banking book count too. The Basel standardised approach uses each currency's net position against the reporting currency, and for commodities it uses risk factors per commodity, delivery date and delivery location, with correlations below one between distinct commodities (such as Brent and WTI), dates and locations to capture basis. Both are in M4.3: Market risk capital (FRTB) and CVA.

Spark Atoms

Spark AtomNegative oil, 20 April 2020
With storage at Cushing, Oklahoma nearly full, the May 2020 WTI crude future settled at −$37.63 a barrel on 20 April 2020, the day before it expired. Holders who could not take delivery paid others to take their oil. The June contract settled above $20 the same day: the same commodity, one month apart.
Spark AtomThe Swiss franc floor, 15 January 2015
The Swiss National Bank had held EUR/CHF at or above 1.20 since 2011. When it removed the floor without warning, EUR/CHF fell from 1.20 to as low as about 0.85 almost at once, before settling near parity. Several retail FX brokers whose clients could not cover their losses failed or needed rescuing. Pegs make FX volatility look low until the day they break.

Key takeaways

  • FX exposure is the net open position: foreign assets less forward sales. RiskAtoms Bank's desk is net long $13m, −£100,000 per 1% rise in GBP/USD.
  • A fixed hedge on a moving position drifts; the basket's £ value depends on the US index and GBP/USD together.
  • A forward is spot plus two interest rate positions: forward points come from the rate differential.
  • Commodity futures form a curve: contango up to full carry, backwardation when the convenience yield is high, seasonal shapes for gas and power. Roll and calendar spreads are curve risks one factor cannot see.
  • Cross-hedges leave basis risk: the minimum-variance hedge ratio is ρσS/σF and removes only ρ² of the variance.

Reaction

Check what stuck

Question 1 · objective 1

A sterling book holds $50m of US assets and has sold $35m forward. GBP/USD is 1.25. What does it make or lose if GBP/USD rises 1%, in £?

£

Question 2 · objective 3

Oil is $80 a barrel spot. The financing rate is 5%, storage costs 2% a year and there is no convenience yield. What is the 6-month futures price, in $?

$

Question 3 · objective 4

A hedger's exposure has daily volatility 3%; the futures contract has 2.5%; their correlation is 0.8. What is the minimum-variance hedge ratio?

Question 4 · objective 3

A desk holds a long Brent position and rolls it each month. The curve is in backwardation and does not move. What happens?

Question 5 · objective 2

The US index rises 10%. Why is the desk's dollar hedge now too small?

Glossary

Net open position
The amount of a foreign currency a book would gain or lose on if the exchange rate moved: assets less liabilities and forward sales in that currency.
Hedge drift
A hedge sized against a position's value today stops matching when that value changes; the unhedged amount drifts.
Forward points
The difference between the forward and the spot exchange rate, set by the two currencies' interest rates (covered interest parity).
Contango
A futures curve that slopes up: later delivery is dearer, by about the cost of financing and storing the commodity.
Backwardation
A futures curve that slopes down: later delivery is cheaper than near delivery.
Convenience yield
The benefit of holding the physical commodity now (keeping a refinery running, say) rather than a future; it pulls futures prices below the cost of carry.
Roll
The gain or loss from replacing an expiring future with a later one, or from a contract's price moving along an unchanged curve as time passes.
Seasonality
A regular pattern in prices through the year, such as dearer gas for winter delivery.
Basis risk
The difference between the price of what is hedged and the price of the hedging instrument; basis risk is the chance that it changes.

Deep Dives & Origins

Deep Dives

  • H. Geman, Commodities and Commodity Derivatives, Wiley, 2005. Advanced.
  • J. C. Hull, Options, Futures, and Other Derivatives, 11th ed., Pearson, 2021, the chapters on hedging with futures and on forward and futures prices. Intermediate.

Origins

  • N. Kaldor, “Speculation and Economic Stability”, Review of Economic Studies 7(1), 1939.
  • H. Working, “The Theory of Price of Storage”, American Economic Review 39, 1254–1262, 1949.
  • Swiss National Bank, “Swiss National Bank discontinues minimum exchange rate and lowers interest rate to −0.75%”, press release, 15 January 2015.
  • U.S. Commodity Futures Trading Commission, Interim Staff Report: Trading in NYMEX WTI Crude Oil Futures Contract Leading up to, on, and around April 20, 2020, 2020.

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