2. Derivative Valuation

Teaching rewrite of how commodity forwards, swaps, and options are priced. Original wording; all worked numbers recomputed. Prices (gold near USD 1,400, crude near USD 50) are teaching values, not today’s market. Not investment advice.

Introduction

Lesson 1 described what commodity derivatives do. This lesson explains where their prices come from. The recurring idea:

The price of a derivative is driven by the cost of hedging it — not by anyone’s forecast.

This lesson covers:

• Why commodities can’t be valued like bonds or shares
• What drives commodity prices: inventories, seasonality, inelasticity, marginal cost, mean reversion
• Forward curves: cash-and-carry, contango vs backwardation, convenience yield
• Arbitrage across place, time, and quality — with real cases
• Swap valuation: discount factors, fair fixed price, OIS discounting
• Option valuation: Black–Scholes–Merton, Black-76, Bachelier (negative prices), put-call parity
• The Greeks — delta, gamma, theta, vega — and volatility skews and term structures

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2.1 Three kinds of assets

Asset type	Examples	How valued
Capital assets	Bonds, shares	Discounted cash flows — coupons, dividends
Consumable / transformable (C/T)	Oil, metals, grains	No income stream → not NPV; valued from supply and demand in their own market
Store of value	Fine art (and, arguably, gold)	Neither consumed nor income-producing, yet valued

This is why finance techniques imported from bonds and equities often fail on commodities.

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2.2 Commodity prices through the cycle

flowchart LR
  A[Strong demand / deficit<br/>prices accelerate] --> B[Stocks drawing, limited new supply<br/>prices peak]
  B --> C[Demand destruction, capex accelerating<br/>prices stable/high]
  C --> D[Supply growth, stocks building<br/>prices falling, capex peaking]
  D --> E[High stocks, strong supply<br/>severe decline]
  E --> F[Supply cut, capex lagging<br/>prices stabilise low]
  F --> A

Because new mines, wells, and farms take years, supply lags price — producing long boom-and-bust cycles.

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2.3 What drives commodity prices

Driver	Idea	Example
Inventories	Large above-ground stocks cushion shocks; thin stocks amplify them	Gold: huge stocks → short-term price barely reacts to mine supply. Copper: thin stocks → sensitive
Seasonality	Demand/supply concentrated in parts of the year	Heating oil dearer for winter delivery
Inelastic demand	Demand doesn’t fall quickly when price rises	Small share of final cost (coffee in a latte), essential input (jet fuel), hard to substitute, brand loyalty
Inelastic supply	Output can’t rise quickly	Finite resources, land/water limits, weak infrastructure, geopolitics, slow technology
Marginal cost (“cost support”)	Long-run price floor ≈ cost of the marginal producer	Short term, prices can fall below cost — fixed costs are sunk; producers endure losses hoping for recovery
Mean reversion	Prices tend to drift back to a long-run average	Useful, but “long run” has no precise definition

Worked example: inelasticity

Price elasticity = % change in quantity ÷ % change in price. Jet fuel rises 40% and airline fuel demand falls 4% in the short run → elasticity = −4 / 40 = −0.1 (very inelastic). Fuel surcharges pass the cost to passengers.

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Checkpoint 1: price drivers

Quiz

Checkpoint — sections 2.1 to 2.3

1 / 5

Why can’t crude oil be valued with a discounted cash flow model like a bond?

Pick an answer to see if you are right.

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2.4 Forward price curves

A forward curve plots today’s prices for delivery at different future dates. Typical features:

• Contango: forward > spot (upward slope). Backwardation: forward < spot (downward slope).
• Short-dated prices are more volatile than long-dated ones.
• At low prices the curve tends to slope up; as prices rise it flattens and may invert.

Cash-and-carry: the financial benchmark

A bank sells a client a 1-year forward on an equity index (spot 100). It hedges by buying the shares today:

Item	Index points
Spot	100
+ Financing at 3%	+3
− Dividends received (1%)	−1
− Securities-lending fee earned	−0.5
Fair forward	101.5

This is a breakeven, not a forecast. If another bank quotes 101, an arbitrageur buys the forward at 101, borrows and sells the shares (100), deposits the cash (+3), pays the dividend to the lender (−1) and a borrow fee (−0.5): receives 101.5, pays 101 → 0.5 riskless profit.

Gold forward: the same logic

A producer asks for a 6-month (182-day) gold price. The bank:

1. Sells gold spot at USD 1,425.40 (hedging the gold it will receive).
2. Borrows gold from a central bank at a lease rate of 0.1157% to deliver now.
3. Deposits the dollars at 3.39%.

Cash flow (per oz)	Calculation	USD
Spot proceeds		1,425.40
+ Deposit interest	1,425.40 × 3.39% × 182/360	+24.43
− Gold lease cost	1,425.40 × 0.1157% × 182/360	−0.83
Fair forward		≈ 1,449.00

Forward = spot + cost of carrying the hedge. As maturity approaches, carry shrinks and forward converges to spot.

If the market quotes 1,440 (“cheap to fair value”), an arbitrageur buys the forward at 1,440 and does the spot sale / lease / deposit trades — locking in about USD 9/oz.

Benchmark rate update

The source text uses LIBOR. USD LIBOR ceased in June 2023; today USD deals use SOFR, EUR deals €STR (which replaced EONIA in 2022). The mechanics are identical — only the reference rate changes.

Contango and backwardation

With storage and insurance:

Forward = spot + interest + storage & insurance

…which implies forward should always exceed spot. Yet crude oil and base metals are often backwardated. The textbook fix is the convenience yield — the benefit of physically having the commodity now (no stock-out, keep the plant running):

Forward = spot + interest + storage − convenience yield

• Near zero when supply is normal; no upper limit when supply is tight.
• Practitioners are sceptical (“it’s the number that makes the formula work”), but it’s a useful proxy for scarcity: research comparing 1989–2004 data found higher convenience yields (crude ≈ 36%) where inventories were thin (≈ 20 days) and negative ones for gold (≈ −0.8%, inventories ≈ 16,500 days) — and higher volatility where convenience yield was high.

Why can’t arbitrage remove backwardation? You’d need to borrow the commodity to sell spot — and in a scarce market there is none to borrow.

An alternative view: producers need more protection than consumers, so they accept a risk premium:

Forward = expected future spot − risk premium

E.g. a producer (marginal cost USD 45) and trader both expect USD 50 in six months; the trader pays USD 49 — the USD 1 is the trader’s risk premium.

For index investors: total return = spot return + roll yield + collateral yield, and convenience yield ≈ roll yield + storage + interest.

Backwardation	Contango
Scarcity, low inventories	Glut, high inventories
Volatile, strongly rising prices	Stable, weak prices
“Fear” premium	“Complacency” discount
Typical: crude, base metals	Typical: precious metals

Crude is costly to store (roughly a fifth of a barrel’s value), so demand spikes hit spot prices first; gold is cheap to store and abundant above ground, so it usually sits in contango.

Worked example: implied convenience yield

Spot crude USD 80, 1-year forward USD 76, interest 5% (USD 4), storage USD 3.

Convenience yield = (spot − forward) + storage + interest = 4 + 3 + 4 = USD 11 (≈ 13.75% of spot) — a tight market.

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Checkpoint 2: forward curves

Quiz

Checkpoint — section 2.4 (curves)

1 / 6

Spot 200, financing 6, income 2, lending fee 1. Cash-and-carry forward?

Pick an answer to see if you are right.

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2.5 Commodity arbitrage

Type	Exploits	Example
Geographic	Price gaps between locations	LME vs Shanghai Futures Exchange (SHFE)
Time	Forward curve away from cash-and-carry value	Buy spot, store, sell forward
Technical / quality	Blending or processing	Blending ethanol into US gasoline

Geographic: LME vs SHFE

Arbitrage level = [(LME ± curve adjustment + CIF premium & freight) × USD/RMB × (1 + VAT) × (1 + tariff) + admin] − SHFE

Everything in brackets is the all-in cost of moving LME metal into China. If SHFE is higher than that, ship metal from LME to SHFE; reverse when negative.

Worked example: is the import arb open?

LME copper USD 9,000/t, curve adjustment +20, CIF premium & freight +80, USD/RMB 7.1, VAT 13%, tariff 0%, admin RMB 50, SHFE RMB 73,000/t.

All-in import cost = (9,000 + 20 + 80) × 7.1 × 1.13 + 50 = 9,100 × 7.1 × 1.13 + 50 = RMB 73,059.3. SHFE 73,000 − 73,059.3 ≈ −59 → import arb slightly closed.

Time arbitrage: aluminium

Cash USD 2,000/t; one week’s storage 3.00 and finance 0.50 → fair 1-week forward 2,003.50. Market forward 2,005.

Buy cash, sell forward. Whatever happens to price:

Cash price after a week	Physical P&L incl. carry	Short forward P&L	Net
2,010	10 − 3.50 = +6.50	2,005 − 2,010 = −5.00	+1.50
1,990	−10 − 3.50 = −13.50	2,005 − 1,990 = +15.00	+1.50

Profit = 2,005 − 2,003.50 = 1.50 — locked in.

Case: the 2010 aluminium carry trade

LME aluminium stocks quadrupled to ~4.5 Mt; much of the world’s stock sat in financing deals:

Item	USD/t
Sell 3-month forward	2,108
− Buy spot	−2,076
Spread	32.00
− Finance: 2,076 × 0.5% × 92/360	−2.65
− Warehousing: 0.15 × 92 days	−13.80
Net profit	15.55 (≈ 3% annualised)

It persisted for years: producers kept selling surplus metal spot while consumers bought forward, and traders were paid to intermediate between them.

Case: oil in floating storage (2008–09)

Spot crude USD 47, 12-month forward USD 60. A supertanker holds 2 million barrels:

Item	USD
Forward sale: 2 M × 60	120,000,000
− Purchase: 2 M × 47	−94,000,000
− Storage: 2 M × 7.20	−14,400,000
− Financing: (94 M + 14.4 M) × 0.25%	−271,000
Net	≈ 11,329,000

As others copied the trade (100–120 M barrels on ~56 ships by April 2009), spot rose and forwards fell; by September the spread had shrunk to ~USD 5.45 and the trade faded — arbitrage closing itself.

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2.6 Swap valuation

A swap is fair on day one: PV(fixed) = PV(floating) → NPV = 0. The price of a swap is its fixed rate.

1. Floating leg — value each period using today’s futures curve (the trader hedges the floating leg with futures, so futures prices are the hedge cost).
2. Discount each cash flow: discount factor DF = 1 / (1 + r × days/360) for periods under a year (360 for USD/EUR, 365 for GBP); 1 / (1 + r)ⁿ for longer.
3. Solve for the fixed price that makes NPV zero:

Fixed = Σ(Fᵢ × DFᵢ) / Σ DFᵢ — a discount-weighted average of the futures prices.

Worked example: 6-month crude swap (10,000 bbl/month)

Assume 31-day months; each cash flow discounted to its own date.

Month	Futures (USD)	Rate	Days	DF	PV fixed	PV floating
1	50.00	1.00%	31	0.999140	505,798.85	499,569.81
2	50.25	1.25%	62	0.997852	505,146.93	501,420.55
3	50.50	1.50%	93	0.996140	504,280.32	503,050.68
4	50.75	1.75%	124	0.994008	503,201.22	504,459.23
5	51.00	2.00%	155	0.991462	501,912.38	505,645.83
6	51.25	2.25%	186	0.988509	500,417.05	506,610.65
Total					3,020,756.76	3,020,756.76

Fair fixed price ≈ USD 50.62/bbl; NPV = 0.

• The fixed payer profits only if actual futures rise faster than the curve already implies.
• Futures +1 across the curve → for the bank receiving fixed, NPV ≈ −10,000 × ΣDF ≈ −USD 59,671.
• One month later, curve unchanged, five flows left → NPV ≈ −USD 6,253 to the fixed receiver (the cheap early months have passed).

Note on the source table

The source applies a single 31-day period to every month’s discount factor; discounting each cash flow to its own date (as above) is the standard method. The fair price is about USD 50.62 either way.

OIS discounting

Collateralised swaps pay overnight interest on collateral, so they should be discounted at overnight index swap (OIS) rates (SOFR, €STR today), not term rates.

Worked example: you will receive USD 50,000 in one year. Term rate 5%, OIS 4%.

Collateral held	Grows at OIS 4% to	Shortfall?
50,000 / 1.05 = 47,619 (term discounting)	49,524	476 short
50,000 / 1.04 = 48,077 (OIS discounting)	50,000	None

Under OIS the fair fixed price barely changes, but sensitivities differ — large across big, long-dated portfolios.

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Checkpoint 3: arbitrage and swaps

Quiz

Checkpoint — sections 2.5 and 2.6

1 / 5

Cash 1,500, carry cost 6 for one month, one-month forward 1,512. Locked-in arbitrage per tonne?

Pick an answer to see if you are right.

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2.7 Option valuation

An option premium reflects the expected payout and its probability.

Black–Scholes–Merton (BSM) inputs

Spot, strike, time to maturity, cost of carry (interest − income), implied volatility.

Gold example: spot 1,400, strike 1,400, 6 months, interest 3%, lease 0.10%, vol 15% → 6-month forward ≈ 1,420.44, BSM call premium ≈ USD 69.38/oz, delta ≈ 0.58.

Looks at-the-money (spot = strike), but a European option is compared with the forward: buying at 1,400 when the forward is 1,420.44 means it is in-the-money.

Intrinsic value and time value

Component	Definition	Driven by
Intrinsic	Call: MAX(0, underlying − strike); put: MAX(0, strike − underlying) — underlying = forward for European, spot for American	Underlying price
Time value	Premium − intrinsic: a charge for future uncertainty; decays to zero by expiry	Time and implied volatility

Gold call: intrinsic ≈ 1,420.44 − 1,400 = 20.44 (≈ 20.14 present-valued); time value ≈ 69.38 − 20.14 ≈ 49.24.

Early exercise is rarely optimal: exercising captures only intrinsic value; selling the option captures intrinsic and time value.

Black-76 and Bachelier

Model	Underlying input	Distribution	Negative prices?	Volatility quoted as
BSM	Spot + carry	Lognormal	No	%
Black-76	Futures price directly	Lognormal	No	%
Bachelier (c. 1900)	Futures price	Normal	Yes	Money (e.g. USD 1.00)

Negative prices are real in commodities: power and gas, calendar spreads, freight indices — and WTI crude futures settled at about −USD 37/bbl on 20 April 2020. Exchanges switched some options to Bachelier afterwards.

Bachelier consequences:

• OTM puts worth more than under BSM (prices can go below zero); OTM calls relatively less.
• ATM delta is exactly 0.5.
• Example: 3-month put, strike 0, underlying 0, dollar vol USD 40 → premium = σ√T × 0.3989 = 40 × 0.5 × 0.3989 ≈ USD 7.98; Black-76 gives 0.
• With dollar volatility, a USD 1 range stays USD 1 whether the price is 10 or 5.

Put-call parity

C − P = PV(F − K)

Same strike, maturity, and amount. As a strategy identity: +C − P = +F — buy a call and sell a put = a long forward.

Example: gold forward 1,420.44, strike 1,400, 6 months at 3% → C − P ≈ 20.44 × e^(−0.015) ≈ 20.14. If the call is 69.38, the put should be ≈ 49.24.

Hedging application: a buyer wanting a better rate than the forward (K < F) must cheapen the call relative to the put. Options:

• Ratio forward — buy a call on N, sell a put on 2N
• Buy a reverse knock-out call with a barrier unlikely to trade
• Buy a short-dated call, finance with a longer-dated put at the same strike

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2.8 The Greeks

Greek	Measures	Buyer	Seller
Delta	Δ premium / Δ underlying	Long call +, long put −	Opposite
Gamma	Δ delta / Δ underlying	+	−
Theta	Δ premium / Δ time	− (time decay)	+
Vega	Δ premium / 1% change in implied vol	+ (long vega)	−

Delta

Four readings: rate of change of premium; slope of the premium-vs-price curve; rough probability of exercise; hedge ratio.

Gold call (delta ≈ 0.575): underlying +0.50 → premium +0.29 (69.38 → ~69.67). Only valid for small moves — the curve is convex.

Underlying	1,325	1,350	1,375	1,400	1,425	1,450	1,475	1,500	1,525
Premium	33.89	44.00	55.84	69.38	84.56	101.29	119.43	138.83	159.33
Delta	0.37	0.44	0.51	0.58	0.64	0.70	0.75	0.80	0.84

Delta equivalent: long calls on 50,000 oz (delta 0.58) = 29,000 oz; add 60,000 oz of calls with delta 0.40 = 24,000 oz → total 53,000 oz long. Delta hedge: sell 53,000 oz (forward market for European options) → delta neutral.

Delta bleed: as expiry nears, deltas drift toward 0 (OTM) or 1 (ITM).

Gamma

Gamma = how fast a delta hedge goes stale; exposure to realised volatility. Gold call: gamma 0.0026 → a small rise moves delta to ≈ 0.5826, a small fall to ≈ 0.5774. Buyers are long gamma, sellers short.

Theta

Years to expiry	1.0	0.8	0.6	0.5	0.4	0.2	0.1	0.01
Premium (USD)	103.94	91.10	77.05	69.38	61.10	41.53	28.53	8.58

Decay accelerates toward expiry — the last tenth of a year costs ~20.

Vega and implied volatility

Implied volatility = the market’s best guess of future volatility to expiry — the only truly unknown input, so interbank options are quoted in volatility. Spot and implied vol are independent inputs: no rule ties a spot move to a vol move.

Implied vol	OTM call (K 1,450)	ATM-forward call (K 1,420)	ITM call (K 1,400)
0%	0.00	0.00	20.00
5%	8.67	19.73	31.29
10%	27.00	39.46	50.07
15%	46.36	59.18	69.38
20%	65.99	78.88	88.80
30%	105.43	118.20	127.69

• ATM-forward premium is roughly proportional to volatility (double vol → double premium).
• At zero vol, OTM and ATM options are worthless; ITM keeps only intrinsic value — volatility is what makes an option an option.
• OTM premiums are non-linear in vol at low levels.
• Higher vol pushes deltas toward 0.5.

Non-constant volatility

BSM assumes one volatility for all strikes and maturities. Markets disagree.

Skew (by strike):

Market	Shape	Reading
WTI crude	Skewed to the downside (OTM puts dearer)	Demand for crash protection; puts often financed by selling OTM calls
Gold	Near-symmetrical smile / smirk	Traded like a currency (XAU): fear of a USD crash or a gold crash
Natural gas, power	Skewed to the upside (OTM calls dearer)	Fear of price spikes
Copper / base metals	Mild skew	Less pronounced historically

WTI example (3-month future 55.73): ATM vol 29% → premium 3.19; 10-delta call (strike 66.59) at 26.47% → 0.33; 10-delta put (strike 45.06) at 35.80% → 0.51.

Term structure (by maturity): in most financial markets vol rises with maturity; in crude and natural gas it often falls (short end most volatile; mean reversion damps long-run dispersion). Gold slopes up like a financial asset; copper is less defined.

Maturity	Constant vol 15%	Rising (15→18%)	Falling (15→12%)
1 month	1.73	1.73	1.73
3 months	2.98	3.18	2.79
6 months	4.21	4.77	3.93
12 months	5.92	7.10	5.53

A downward-sloping term structure makes long-dated options relatively cheap — useful in structured products.

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Checkpoint 4: options and Greeks

Quiz

Checkpoint — sections 2.7 and 2.8

1 / 7

Spot 1,400, strike 1,400, 6-month forward 1,420. A European call is…

Pick an answer to see if you are right.

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Rank: valuation principles

Quiz

Valuation principles

1 / 2

Rank the steps to price a commodity swap.

OrderDrag the handle to rearrange — top should be first.

1. ⋮⋮1Read the futures curve for each settlement period.
2. ⋮⋮2Compute floating cash flows = volume × futures price.
3. ⋮⋮3Solve for the fixed price that makes PV(fixed) = PV(floating).
4. ⋮⋮4Build discount factors from (OIS) rates.

Drag items into the correct order (or use ↑↓), then check.

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Check: derivative valuation

Quiz

Derivative valuation

1 / 6

The key maxim of derivative pricing is…

Pick an answer to see if you are right.

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Calculate: numeric practice

Type the number (no units; minus sign for losses). Use 360-day years unless stated.

Formula sheet

Item	Formula
Cash-and-carry forward	spot + financing + storage − income
Gold forward	spot × (1 + (r_cash − r_lease) × days/360)
Convenience yield	(spot − forward) + storage + interest
Discount factor (< 1 yr)	1 / (1 + r × days/360)
Swap fixed price	Σ(F × DF) / ΣDF
Put-call parity	C − P = PV(F − K)
Delta equivalent	notional × delta
Bachelier ATM premium	σ_$ × √T × 0.3989

Compute

Calculate: derivative valuation

1 / 12

Story problem — figure out which numbers are the formula inputs, then compute.

Index spot 250; financing 2.5% for one year; dividend yield 1.2%; lending fee 0.3% (all on spot). Fair 1-year forward?

Enter a number (at most 2 decimal places).

Randomized practice

Compute

Practice: valuation drills

1 / 6

Story problem — figure out which numbers are the formula inputs, then compute.

Fair (cash-and-carry) forward is USD 1894.5; the market forward is USD 1899.5. Riskless arbitrage profit per unit?

Enter a number (at most 2 decimal places).

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Takeaways

Remember

• Commodities are consumable/transformable assets — valued from supply and demand, not DCF
• Price drivers: inventories, seasonality, inelasticity, marginal cost, mean reversion
• Forward = spot + carry (financing, storage, minus income/lease); a breakeven, not a forecast
• Contango = glut; backwardation = scarcity, explained by convenience yield — and not arbitrageable without borrowable supply
• Arbitrage across place, time, quality keeps prices aligned — until it closes itself
• Swap fixed price = discount-weighted average of futures; discount collateralised swaps at OIS
• Options: BSM / Black-76 / Bachelier (negative prices); intrinsic + time value; put-call parity
• Greeks: delta (direction), gamma (realised vol), theta (time decay), vega (implied vol)
• Volatility varies by strike (skew) and maturity (term structure) — and differs by commodity

Further Reading

• 1. Fundamentals of commodities and derivatives
• International Business home
• Next: 3. Risk management principles
• Background: N. Schofield, Commodity Derivatives: Markets and Applications (Wiley) — used as a study outline; H. Geman, Commodities and Commodity Derivatives (2005); S. Natenberg, Option Volatility and Pricing (2014)