Lernzettel: Introduction to Derivatives and Hedging

📋 Course Outline

1. Derivatives and contract types
2. Forwards and arbitrage
3. Option positions and payoffs
4. Market participants and zero-sum trading
5. Futures markets and clearinghouses
6. Margining and mark-to-market
7. Hedging with futures and basis risk
8. Cross hedging and hedge ratios
9. Equity portfolio hedging
10. Options mechanics and valuation
11. Protective puts and option strategies

📖 1. Derivatives and contract types

🔑 Key Concepts & Definitions

• Derivative : A derivative is a contract whose value depends on an underlying asset, transferring risk between parties rather than creating value itself.
• Forward contract : A forward contract is an OTC agreement to buy or sell an asset at a specified forward price on a future date with both parties obligated to perform.
• Futures contract : A futures contract is an exchange-traded standardized forward that is settled daily via a clearing house using mark-to-market.
• Options contract : An options contract gives the holder a right but not an obligation to buy or sell at strike price K, with an upfront premium and an asymmetric payoff.
• Call option : A call option is the right to buy the underlying at strike price K, making it profitable when the underlying finishes above K plus the premium.

📝 Essential Points

• Long forward payoff equals $S_T-F_0$ and short forward payoff equals $F_0-S_T$, so gains and losses across the two sides offset to zero total payoff.
• A forward is typically settled only at maturity and can be settled by physical delivery or cash settlement with the same economic effect for the long and short.
• A futures contract is settled daily through a clearing house via mark-to-market, which nearly eliminates credit risk versus typical OTC forwards.
• An option holder pays an upfront premium and can choose whether to exercise, while the short receives the premium and must perform if the option is exercised.

💡 Memory Hook

Derivative = side bet on the underlying: the contract itself does nothing; the underlying outcome drives the payoff.

📖 2. Forwards and arbitrage

🔑 Key Concepts & Definitions

• Cash-and-carry arbitrage : Cash-and-carry arbitrage is a strategy that locks in a riskless profit by borrowing to buy spot and taking the opposite position in the overpriced forward.
• Forward price arbitrage : Forward price arbitrage is a riskless profit opportunity created when the quoted forward price conflicts with the spot price and cost of carry.

📝 Essential Points

• A forward is settled only at maturity and both parties are obligated to perform, so some credit risk remains.
• A futures contract nearly eliminates credit risk because the clearing house settles daily via mark-to-market.
• The zero-sum property of forwards means the long’s payoff $S_T-F_0$ is exactly offset by the short’s payoff $F_0-S_T$.
• With $F_0=S_0(1+r)$ when storage costs are zero, any quoted $F_0$ above this fair value makes the forward overpriced for a cash-and-carry arbitrageur.
• For the gold case $S_0=1200$, $r=0.03$, $F=1300$, the profit is $1300-1236=64$ per ounce after delivering under the forward and repaying the loan.

💡 Memory Hook

Fair forward: forward ≈ spot × (1 + interest) when carry is zero; if market forward is higher, borrow→buy spot→sell forward for the gap.

📖 3. Option positions and payoffs

🔑 Key Concepts & Definitions

• Call option buyer : A call option position grants the holder the right, without obligation, to buy the underlying at the strike price K at exercise time.
• Call option seller : A call option position obligates the seller to sell the underlying at strike K if the holder exercises.
• Put option buyer : A put option position grants the holder the right, without obligation, to sell the underlying at strike price K at exercise time.
• Protective put : A protective put is a put option bought on stock you own, used to insure against losses if the stock price falls below K.

📝 Essential Points

• Ignoring the option premium, the call seller’s payoff is never greater than 0 because exercise only helps the holder.
• Ignoring the option premium, the put buyer’s payoff is never less than 0 because the buyer can choose not to exercise when it is unfavorable.
• Buying a put on stock you already own acts like insurance: if the stock falls below K, you can still sell at K for protection.
• For a short put with strike K and premium c, break-even occurs at stock price $K - c per share.

💡 Memory Hook

Call seller is capped on the downside (never > 0), while put buyer is protected (never < 0), both “sign-locked” by who controls exercise.

📖 4. Market participants and zero-sum trading

🔑 Key Concepts & Definitions

• Speculator : A speculator takes positions mainly to profit from price movements rather than to use the underlying asset.
• Hedger : A hedger uses derivatives to offset losses in an existing exposure to reduce overall risk.
• Option buyer : An option buyer pays a premium and receives a payoff only if the option finishes in the money.
• Option seller : An option seller receives the premium and takes on the obligation to cover the payoff if the option ends in the money.

📝 Essential Points

• Options typically have nonzero value at initiation because the holder pays a premium at trade time.
• For a call with strike $K and premium$p$, the call buyer breakeven is$K+p$, not$K$.
• Option payoffs create asymmetry: the buyer’s downside is limited to the premium, while the seller bears larger losses when the underlying moves against them.
• Open interest falls into the delivery month largely because many speculators close positions instead of taking or making delivery.

💡 Memory Hook

Premium paid by buyer ⇒ value at initiation; breakeven = strike + premium; seller owes payoff when in the money.

📖 5. Futures markets and clearinghouses

🔑 Key Concepts & Definitions

• Futures contracts : Futures contracts are standardised, exchange-traded agreements to buy or sell an asset at a fixed price on a specified future date.
• Clearinghouse daily settlement : A clearinghouse settles futures through daily settlement using the day’s settlement price rather than only settling at maturity.
• Convergence : Convergence is the requirement that the futures price approaches the spot price at delivery to prevent exploitable mispricing.
• Settlement price : The settlement price is the price set just before the close of trading each day to compute daily gains and losses.
• Open interest : Open interest is the total number of outstanding futures contracts, equal to the number of longs and also the number of shorts.

📝 Essential Points

• Futures prices must converge to the spot price at delivery, and if they diverge arbitrage pressures them back together.
• The settlement price is used to calculate daily mark-to-market variation margin gains and losses.
• Open interest increases only when both sides open new positions and decreases only when both sides close positions.
• Volume can exceed open interest because traders may open and close positions multiple times in a day.
• Because futures are exchange-traded and daily settled through a clearing house, their counterparty credit risk is virtually none.

💡 Memory Hook

Open interest moves only when both sides do the same thing: open+open raises it, close+close lowers it, open+close leaves it unchanged.

📖 6. Margining and mark-to-market

🔑 Key Concepts & Definitions

• Maintenance margin : Maintenance margin is the minimum margin balance required to keep a futures position open without triggering an action by the broker.
• Margin call : A margin call is the broker’s demand for additional funds when the margin account balance falls below the maintenance margin.
• Daily mark-to-market : Daily mark-to-market is the process of revaluing a futures position each day using the settlement price and updating the margin account for gains or losses.
• Initial margin : Initial margin is the initial amount required to establish a futures position and is the balance level the account is typically topped up to after a margin call.

📝 Essential Points

• A margin call is triggered when the margin account balance falls below the maintenance margin of $3,000, meaning the shortfall relative to$3,000 causes required top-up.
• For a short futures position, losses occur when the futures price rises and gains occur when the futures price falls.
• Futures payoffs are settled daily via mark-to-market, so interim losses can require financing even if the final outcome matches a forward payoff.
• Hedgers’ futures gains and losses are recognised in the same period as gains and losses on the hedged item, matching recognition across the hedge.
• Speculators’ profits and losses are recognised on a mark-to-market basis annually, even if the futures position is not yet closed.

💡 Memory Hook

Mark-to-market is a daily “true-up”: profits add to margin, losses deduct, and falling below maintenance causes a call to restore the initial margin.

📖 7. Hedging with futures and basis risk

🔑 Key Concepts & Definitions

• Basis risk : Basis risk is the hedging error that occurs because the final basis between spot and futures is uncertain.
• Basis : Basis is the difference between spot and futures prices at the same time.
• Effective price paid : Effective price paid is the realized purchase price for a long futures hedge after adjusting for how basis changes.
• Effective price received : Effective price received is the realized selling price for a short futures hedge after adjusting for how basis changes.

📝 Essential Points

• For the basis definition, $b=S-F$ where $S$ is spot and $F$ is futures price, and it can change from $b_1$ at setup to $b_2$ at close-out.
• For a LONG hedge, effective price paid equals $F_1+b_2$, so uncertainty in $b_2$ drives hedging incompleteness.
• For a SHORT hedge, effective price received equals $F_1+b_2$, so uncertainty in $b_2$ drives hedging incompleteness.
• When basis strengthens ($b_2>b_1$), it improves outcomes for a short hedge and worsens outcomes for a long hedge.
• When basis weakens ($b_2<b_1$), it worsens outcomes for a short hedge and improves outcomes for a long hedge.

💡 Memory Hook

Think b=S−F: long buyers hate b rising; short sellers love b rising (strengthening basis flips the sign).

📖 8. Cross hedging and hedge ratios

🔑 Key Concepts & Definitions

• Cross hedging : Cross hedging uses a futures contract on a different underlying asset than the one being hedged.
• Minimum variance hedge ratio : Minimum variance hedge ratio chooses the hedge strength to reduce the variance of hedging error using correlation and volatilities.
• Hedge ratio h* : Hedge ratio h* is the minimum-variance ratio linking spot and futures price changes when cross hedging.
• Optimal number of contracts N* : Optimal number of contracts N* converts the hedge ratio into a contract count using the exposure size and contract size.

📝 Essential Points

• Cross hedging applies when the hedged asset differs from the futures underlying, such as hedging jet fuel with WTI crude oil futures.
• The minimum variance hedge ratio is $h^* = \rho \times (\sigma_S/\sigma_F)$ using correlation and standard deviations of spot and futures price changes.
• Ignoring daily settlement, the optimal number of futures contracts is $N^* = h^*\times(Q_A/Q_F)$ where $Q_A$ is the exposure size and $Q_F$ is the size per contract.
• For a cross-hedge where you buy the exposure, you take a long futures position; where you sell the exposure, you take a short futures position.

💡 Memory Hook

Use h* = correlation × volatility ratio, then N* = h* × (exposure ÷ contract size).

📖 9. Equity portfolio hedging

🔑 Key Concepts & Definitions

• Equity portfolio hedge : An equity hedge using index futures reduces market risk so the portfolio’s return is less driven by broad index movements.
• Target beta β* : The chosen beta level after hedging, where setting β* = 0 aims to remove the portfolio’s exposure to systematic market risk.
• Index futures systematic risk hedge : A futures position primarily offsets the portion of stock returns predicted by CAPM, leaving mostly idiosyncratic (stock-specific) returns.
• Market-neutral position : A hedged equity-plus-futures stance designed so returns are less sensitive to overall market moves, typically by neutralizing beta.

📝 Essential Points

• With target beta β* = 0, a portfolio with β = 1.3 and value $1,500,000 is hedged to remove systematic market risk.
• Given a futures contract value $75,000, the number of contracts to short is$N^=(β-β^),(P/F)=1.3\times(1,500,000/75,000)=26$.
• The hedge is profitable if the stock’s realized return exceeds the CAPM-implied return based on the index’s return.
• Shorting index futures turns the stock holding into a market-neutral exposure that retains only the stock’s alpha (excess return versus CAPM).

💡 Memory Hook

β* = 0 means “erase market swing” with futures; profit remains from alpha, not from index moves.

📖 10. Options mechanics and valuation

🔑 Key Concepts & Definitions

• At-the-money option : An option is at-the-money when the current stock price is equal to (or very close to) the strike price $K$, so exercise and non-exercise are not strongly favored.
• Intrinsic value : Intrinsic value is the payoff the option would have if exercised immediately, computed as $\max(S-K,0)$ for calls and $\max(K-S,0)$ for puts.
• Time value : Time value is the option premium minus intrinsic value, capturing the added worth from the possibility the option becomes more valuable before expiry.
• Option series : An option series is the set of options in the same class with the same expiration date and the same strike price.
• European vs American options : European options can be exercised only at expiration, while American options can be exercised anytime up to and including expiration.

📝 Essential Points

• For a call, intrinsic value is $\max(S-K,0)$ and for a put it is $\max(K-S,0)$, so intrinsic value is always nonnegative.
• Time value equals option price minus intrinsic value and is positive for a live option.
• A European option’s exercise is allowed only at expiration, while an American option can be exercised any time through expiration.
• In a stock split of $n$-for-$m$, the strike becomes $mK/n$ and the number of shares per contract becomes $nN/m$ to preserve total economic value.
• After a $K=60$ call holder’s 2-for-1 split, the option becomes one to buy 200 shares at $K=30$, because $K' = 60/2$ and shares per contract $= 2\times100$.
• Breakeven levels are $K+\text{premium}$ for a long call and $K-\text{premium}$ for a long put.

💡 Memory Hook

Intrinsic Value = Immediate Exercise Payoff; Time Value = Extra Premium for Future Opportunity.

📖 11. Protective puts and option strategies

🔑 Key Concepts & Definitions

• Stop-loss order : A stop-loss order is an instruction to sell the asset once the price reaches a set level, without paying an option premium upfront.
• Put breakeven price : A put breakeven price is the underlying level where a long put’s payoff offsets the premium paid, yielding zero profit at expiration.
• Long straddle : A long straddle is a position combining a long call and a long put on the same stock that benefits from large moves in either direction.

📝 Essential Points

• A long put is insurance-like because it caps losses by giving the right to sell at strike K for a premium you pay upfront.
• Compared with a stop-loss, a put can prevent selling during a temporary dip if the price recovers, so you keep upside while paying for flexibility.
• Stop-loss downside is not guaranteed because the market can gap below the trigger level, forcing a sale at an unfavorable price.
• Long put profit is computed as $(K-S_T)-p$ and the put breakeven is $K-p$ where profit is zero at expiration.
• Holding both a long call and long put creates a straddle-style position whose profit increases when the stock moves far enough in either direction.

💡 Memory Hook

Protective insurance intuition: puts have a floor at K (minus premium), stop-loss has a trigger but no floor because of price gaps.

📅 Key Dates

📊 Synthesis Tables

Forward vs. Futures

Call vs. Put (buying)

⚠️ Common Pitfalls & Confusions

1. Confusing the direction in a short forward/futures: short positions lose when the underlying price rises and gain when it falls, not the other way around.
2. Forgetting that options breakeven include premium: long call breakeven is K + premium (not K), and long put breakeven is K − premium (not K).
3. Mixing up basis definition: basis is b = S − F, so a positive basis means spot is above futures, which flips the hedging effect for longs vs shorts.
4. Forgetting what triggers a futures margin call: a call occurs when the margin account balance falls below the maintenance margin, and after a call the account is topped up to the initial margin level (not maintenance).
5. Misstating zero-sum: forward long payoff S_T − F_0 is exactly offset by short payoff F_0 − S_T, so total payoff across both parties is always zero.
6. Assuming forward and futures interim financing are the same: futures mark-to-market can force financing from interim losses even if the final price ends up matching the forward outcome.
7. Ignoring that option ITM/OTM depends on S vs K, so time value can be entirely nonzero for an OTM option with zero intrinsic value.

✅ Exam Checklist

1. State the definition of a derivative and explain how forwards, futures, and options transfer risk differently.
2. Compute long vs short forward payoff using S_T − F_0 and F_0 − S_T, and explain the zero-sum implication.
3. Explain why a futures contract nearly eliminates credit risk (daily clearinghouse settlement and mark-to-market).
4. Identify the correct option payoff direction: call buyer’s payoff is max(S_T − K, 0) − premium and put buyer’s is max(K − S_T, 0) − premium.
5. For a short put, compute max gain, max loss (to S_T = 0), and the breakeven stock price K − c.
6. Use cash-and-carry logic to decide when a forward is overpriced/underpriced and compute the example profit (gold: 1300 vs fair 1236 gives 64/oz).
7. Track open interest correctly: it increases only when both sides open positions, decreases only when both sides close, and can’t be inferred from volume alone.
8. Perform daily mark-to-market and margin call logic for a long or short futures position, including topping up to initial margin after a margin call.
9. Compute hedge basis b = S − F, then use effective price formulas for long vs short hedges (effective price = F_1 + b_2) and describe how basis strengthening/weakening changes outcomes.
10. For cross hedging, compute h* = ρ × (σ_S/σ_F) and N* = h* × (Q_A/Q_F), and determine long vs short futures based on whether you are buying or selling the exposure.
11. For an equity portfolio hedge with index futures, compute contracts using N* = (β − β*) × (P/F) with β* = 0 to remove systematic risk and state when the hedge is profitable (stock return exceeding CAPM-implied return).
12. For options mechanics, compute intrinsic value, time value, option moneyness (ITM/ATM/OTM), and adjust strike/shares correctly after a stock split (New K = mK/n and New contract shares = nN/m).