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Derivatives

A video series, in Italian and English, that starts from the basics and gets to pricing models. It follows the structure of John C. Hull's «Options, Futures and Other Derivatives»: futures and hedging, rates and swaps, options and strategies, binomial trees, stochastic processes, Black-Scholes. Each video comes with a note covering the key concepts, a worked example and the mistakes to avoid.

The level is intermediate: being comfortable with returns and percentages is enough. Episodes can be watched in order or picked by topic; chapter numbers refer to the book's 10th edition. The notes are educational material and contain no indication of instruments to buy or sell.

12 episodes published + 2 deep dives

The full series on YouTube

Foundations and futures

What derivatives are, how a futures contract works and how it is used to hedge a risk. Everything else in the series builds on this.

Ep. 1Derivatives explained: hedging, speculation, arbitrageWhat a derivative is, who uses it and why it started out as insurance rather than a bet.

What you will learn

  • Why a derivative moves risk from those who don't want it to those willing to take it
  • The difference between exchange-traded and OTC markets, and the role of the clearing house
  • The three uses: hedging, speculation, arbitrage

Key concepts

  • Underlying: the contract's value depends on something else, whether a stock, an index, an interest rate, a currency or a commodity.
  • Standardisation: on an exchange, contract size, expiry dates and rules are fixed, which is what makes contracts easy to trade.
  • Clearing house: it stands between buyer and seller and guarantees the contract is honoured. In OTC markets there is no such guarantor, and counterparty risk stays with whoever signs.
  • Arbitrage: whoever exploits a price misalignment also closes it. That is why the windows are so short.

Example

An Italian company that will receive one million dollars in six months can lock in the exchange rate today with a forward contract. If the dollar weakens it loses nothing; if it strengthens it gives up the gain. It has bought certainty, not a profit.

Mistakes and risks to know

The most common mistake is to judge the instrument by how it is used. The same contract reduces an exporter's risk and multiplies the risk of a leveraged speculator.

Reference: Hull, ch. 1 · Read more: Derivatives: what they are and how they work

Ep. 2Futures mechanics: margins, mark-to-market, deliveryHow the futures machine works: margins, daily settlement of gains and losses, delivery.

What you will learn

  • Why futures rely on a mechanism rather than on trust between the parties
  • What initial margin, maintenance margin and margin calls are
  • How daily mark-to-market splits a final loss into many small daily losses

Key concepts

  • Margin is not a price paid. It is collateral the clearing house holds against the position.
  • Every evening the position is revalued at the settlement price and the difference moves from one account to the other.
  • If the account falls below the maintenance margin, a margin call follows: it must be topped back up to the initial margin, or the position is closed.
  • Physical delivery is rare. Almost all positions are closed before expiry.

Example

Hypothetical example: a futures contract on an index at 30,000 points with a €1 multiplier, so €30,000 of exposure and a 10% initial margin of €3,000. A 3% drop in the index costs €900, 30% of the margin posted. With a maintenance margin of €2,250, a 2.5% drop is enough to trigger a margin call.

Mistakes and risks to know

Leverage works both ways and the loss can exceed the margin posted. Without the cash to top it up, the position may be closed at the worst moment.

Reference: Hull, ch. 2 · Read more: Futures, part 1: how they work

Ep. 3Futures hedging: minimum-variance hedge ratio and basis riskPerfect hedges hardly exist: how to measure the hedge you can get, and how much risk is left.

What you will learn

  • What basis risk is and why it turns price risk into a smaller but different risk
  • How to compute the minimum-variance hedge ratio
  • How to hedge an equity portfolio with index futures using beta

Key concepts

  • The basis is the difference between the spot price and the futures price. On the day the hedge is closed, it is not known in advance.
  • The optimal hedge ratio depends on the correlation between spot and futures price changes and on their volatilities.
  • For an equity portfolio, the number of contracts comes from the portfolio's beta against the index.
h∗=ρ σSσFN∗=h∗ QAQFh^{*} = \rho\,\frac{\sigma_S}{\sigma_F} \qquad N^{*} = h^{*}\,\frac{Q_A}{Q_F}

Example

Correlation 0.9, spot volatility 2%, futures volatility 2.5%: h* = 0.72. For a €1,000,000 exposure and €50,000 contracts, about 14 contracts are needed. With beta: a €1,000,000 portfolio with a beta of 1.2 needs 24 contracts of €50,000.

Mistakes and risks to know

Correlations and betas change over time, and a hedge calibrated on the past can fail precisely in stressed markets. Hedging reduces volatility; it does not remove it.

Reference: Hull, ch. 3 · Read more: Futures, part 2: hedging properly

Ep. 5The universal language of futures: cost of carry and arbitrageWhere forward and futures prices come from: the cost of carrying the underlying to expiry.

What you will learn

  • Why the forward price is a no-arbitrage consequence, not a forecast
  • How the cost of carry differs across stocks and indices, currencies and commodities
  • What the convenience yield tells you

Key concepts

  • Buying the underlying today and holding it to expiry must cost the same as buying it forward. If it doesn't, there is an arbitrage.
  • For an index, the cost of carry is the interest rate minus the dividend yield; for a currency it is the interest-rate differential between the two currencies.
  • For commodities, storage costs also count, and so does the benefit of holding the physical good, called the convenience yield.
F0=S0 e(r−q)TF_0 = S_0\,e^{(r-q)T}

Example

Index at 100, interest rate 4%, dividend yield 2%, one-year expiry: F = 100 × e^0.02 ≈ 102.02. If the future traded well above that, buying the index and selling the future would pay; well below it, the reverse.

Mistakes and risks to know

Confusing the theoretical forward with the market price. Transaction costs, securities-lending constraints and different borrowing and lending rates create a band in which arbitrage doesn't pay.

Reference: Hull, ch. 5 · Read more: Currency hedging: hedged vs unhedged ETFs

Interest rates and swaps

Rate curves, bond futures and swaps: the tools banks, companies and portfolio managers use to handle interest-rate risk.

Ep. 4Zero curve, forward rates and duration: pricing foundationsDiscounting underpins every price: how the zero curve is built, what forward rates mean and how duration measures rate risk.

What you will learn

  • How a zero-coupon curve is built from market prices
  • Why forward rates are market-consistent prices, not forecasts
  • What duration and convexity measure

Key concepts

  • A zero rate is the return on a single payment at a given date. Each cash flow is discounted at the zero rate for its own maturity.
  • Forward rates are implied by today's curve: borrowing for two years must cost the same as borrowing for one year and rolling over at the forward rate.
  • Duration measures price sensitivity to a small parallel shift in yields; convexity corrects it for larger moves.
RF=R2T2−R1T1T2−T1R_F = \frac{R_2 T_2 - R_1 T_1}{T_2 - T_1}

Example

With a one-year zero rate of 3% and a two-year zero rate of 3.5% (continuous compounding), the forward rate implied for the second year is (3.5% × 2 − 3% × 1) / 1 = 4.0%.

Mistakes and risks to know

Reading forward rates as forecasts of future rates: they also contain risk premia. Discounting every cash flow at one single rate when the curve is not flat produces pricing errors.

Reference: Hull, ch. 4

Ep. 6Interest rate futures hedging: duration and DV01How to measure the interest-rate risk of a bond portfolio and how to reduce it with government bond futures.

What you will learn

  • What duration and DV01 measure
  • How to estimate the number of contracts for a duration-based hedge
  • Why a bond futures hedge still leaves basis risk

Key concepts

  • Modified duration tells you the percentage change in a bond's price for a small change in yields.
  • DV01 expresses the same sensitivity in money: how much the portfolio's value changes for one basis point, 0.01%.
  • Hedging means selling enough futures to offset the portfolio's DV01.
  • Day-count conventions and the cheapest-to-deliver bond explain why a future never replicates the portfolio exactly.
DV01=P⋅Dmod⋅0.0001N∗=DV01portfolioDV01futures\text{DV01} = P \cdot D_{mod} \cdot 0.0001 \qquad N^{*} = \frac{\text{DV01}_{portfolio}}{\text{DV01}_{futures}}

Example

A €10 million portfolio with a modified duration of 6 has a DV01 of €6,000 per basis point. If one contract has a DV01 of €90, about 67 contracts sold neutralise the rate risk, at least for small parallel shifts of the curve.

Mistakes and risks to know

The hedge holds for small, parallel moves. If the curve changes slope, or the spread between the portfolio's bonds and those behind the future widens, it loses effectiveness.

Reference: Hull, ch. 6

Ep. 7Interest rate swaps explained: cash flows, valuation, risksAn interest rate swap exchanges a fixed rate for a floating one on a reference amount: how it works and how it is valued.

What you will learn

  • How cash flows are exchanged in a fixed-for-floating swap
  • Why the notional principal never changes hands
  • How a swap is valued by discounting future cash flows

Key concepts

  • Whoever pays fixed and receives floating is protected against rising rates, like a company with floating-rate debt.
  • At inception a swap is worth roughly zero; afterwards its value moves with market rates.
  • It is valued as the difference between two bonds, one fixed-rate and one floating-rate, or as a series of forward rate agreements.
  • Counterparty and liquidity risk matter: a swap lasts years and is an OTC contract.

Example

Notional of 10 million, paying 3% fixed and receiving Euribor, with semi-annual payments. If Euribor for the period is 3.4%, the fixed payer receives the net difference: 10,000,000 × 0.4% × 0.5 = €20,000.

Mistakes and risks to know

A swap turns floating debt into fixed, but if rates fall the position loses value and unwinding it before maturity can be expensive. The counterparty's soundness matters too.

Reference: Hull, ch. 7

Options

Calls and puts, the logical limits to their prices and the strategies that combine several options, or options and the underlying.

Ep. 8Calls and puts: payoffs, risk and investor intuitionCalls and puts give a right, not an obligation: payoffs, premium, break-even and the asymmetry versus futures.

What you will learn

  • What calls and puts are, and the difference between buying and selling them
  • What drives the premium: underlying price, strike, time to expiry, volatility
  • Why buying an option is described as «buying convexity»

Key concepts

  • A call buyer has the right to buy at a fixed price, the strike; a put buyer has the right to sell.
  • The buyer pays the premium and can lose at most that. The seller collects the premium and takes on the obligation, with potentially much larger losses.
  • The payoff is asymmetric: a future gains and loses linearly, a long option caps the loss at the premium.
  • The premium is made of intrinsic value and time value, the value of the chance that the price moves before expiry.
call=max⁡(ST−K, 0)put=max⁡(K−ST, 0)\text{call} = \max(S_T - K,\,0) \qquad \text{put} = \max(K - S_T,\,0)

Example

Call with strike 100 and premium 5: break-even is 105. If the underlying ends at 120 the result is +15; at 90 the premium is lost, −5. Put with strike 100 and premium 4: break-even at 96, and with the underlying at 80 the result is +16.

Mistakes and risks to know

Writing naked options exposes you to losses far larger than the premium collected, with margin calls from the broker. The contract multiplier is a detail people often forget.

Reference: Hull, ch. 10

Ep. 9Put-call parity and bounds: no-arbitrage checksBefore any model, there are logical limits to an option's price and a relationship that ties calls and puts together.

What you will learn

  • The upper and lower bounds on call and put prices
  • How to use put-call parity to check that prices are consistent
  • How dividends and early exercise change these relationships

Key concepts

  • A European call cannot be worth more than the underlying, nor less than the underlying minus the discounted strike.
  • A call plus a deposit equal to the discounted strike pays exactly the same at expiry as a put plus the underlying. So they must cost the same.
  • If the relationship breaks there is an arbitrage, which large players close almost immediately.
  • With dividends, the underlying price is adjusted by their present value; for American options parity becomes an inequality.
c+Ke−rT=p+S0c + K e^{-rT} = p + S_0

Example

Underlying 100, strike 100, rate 3%, one year: the discounted strike is 97.04. If the call is worth 10, the consistent put is worth 10 + 97.04 − 100 = 7.04. The call's lower bound is 100 − 97.04 = 2.96.

Mistakes and risks to know

Treating parity as a trading signal while ignoring transaction costs, bid-ask spreads and dividends leads you to see arbitrages that don't exist.

Reference: Hull, ch. 11

Ep. 10Spreads, straddles, covered calls, protective puts: the real risksCombining options, or options and the underlying, to shape the profit-and-loss profile, along with the risks each structure hides.

What you will learn

  • How covered calls and protective puts change the risk profile
  • How a spread caps both cost and gain
  • What straddles and strangles bet on: volatility, not direction

Key concepts

  • Covered call: own the stock and sell a call. You collect the premium but give up gains above the strike, and most of the downside remains.
  • Protective put: own the stock and buy a put. The downside is limited, at the cost of the premium paid each time it is rolled.
  • Bull spread: buy a call and sell another with a higher strike. It costs less, but the gain is capped.
  • Straddles and strangles profit if the price moves a lot in either direction, and lose if it stays put.

Example

Bull call spread with strikes 95 and 105, premiums 8 and 3: net cost 5, maximum gain 5, maximum loss 5, break-even at 100. Straddle with strike 100, call at 6 and put at 5: cost 11, break-evens at 89 and 111.

Mistakes and risks to know

Strategies that look almost safe, such as selling options to collect premiums, carry tail risk: small, frequent gains and rare, large losses. With more legs, transaction costs weigh more too.

Reference: Hull, ch. 12

Pricing models

From the binomial tree to Black-Scholes: how an option gets a price and what implied volatility tells you.

Ep. 11Binomial tree pricing and delta hedging, step by stepThe simplest way to see how an option is priced: build a riskless portfolio with the underlying.

What you will learn

  • How to build one-step and multi-step binomial trees
  • What risk-neutral valuation is
  • What delta means and how it is used to hedge an option

Key concepts

  • If the underlying can only move up or down by set amounts, it can be combined with the option so that the final value is the same in both cases.
  • A riskless portfolio must earn the risk-free rate: that pins down the option price.
  • The «risk-neutral» probability is not a forecast. It is the calculation device that makes the price consistent.
  • Delta is the number of shares to hold for each option sold. It changes over time, so the hedge has to be adjusted.
  • For American options, each node compares the continuation value with the value of exercising immediately and keeps the higher one.
p=erT−du−df=e−rT[ p fu+(1−p) fd ]p = \frac{e^{rT} - d}{u - d} \qquad f = e^{-rT}\left[\,p\,f_u + (1-p)\,f_d\,\right]

Example

The video's example: stock at $100, call with strike 100, six months split into two three-month steps, 20% volatility, 4% rate. Then u = 1.1052, d = 0.9048 and p ≈ 0.525. Final prices are 122.14, 100 and 81.87; working backwards, the call is worth about 11.52 at the up node and 5.99 today. The initial delta is 0.575: holding 0.575 shares per option sold leaves about $52.03 in both scenarios, and the gain on the $51.51 invested is exactly the risk-free rate for the quarter.

Mistakes and risks to know

A one-step tree is a teaching model. Results depend on the number of steps and on the choice of u and d, which are tied to the underlying's volatility.

Reference: Hull, ch. 13

Ep. 12Random walk, Wiener process and Itô: the bridge to Black-ScholesThe mathematics under pricing models, told through intuition: stochastic processes, Brownian motion and Itô's lemma.

What you will learn

  • What a Wiener process is and why uncertainty grows with the square root of time
  • What geometric Brownian motion describes
  • Why Itô's lemma is needed to get to Black-Scholes

Key concepts

  • In a Wiener process, changes over different intervals are independent and their variance grows in proportion to time.
  • In geometric Brownian motion, the return has a predictable part, the drift, and a random part proportional to volatility.
  • Itô's lemma tells you how a function of a random variable changes, for instance the price of an option that depends on the stock price.
  • It follows that the log of the price is normally distributed: prices are lognormal.
dS=μS dt+σS dzln⁡ST∼N ⁣(ln⁡S0+(μ−σ22)T, σ2T)dS = \mu S\,dt + \sigma S\,dz \qquad \ln S_T \sim \mathcal{N}\!\left(\ln S_0 + \left(\mu - \tfrac{\sigma^2}{2}\right)T,\ \sigma^2 T\right)

Example

With 20% annual volatility, the standard deviation over one month is about 20% × √(1/12) ≈ 5.8%, over one week about 2.8%. Doubling the horizon does not double the uncertainty: it multiplies it by √2.

Mistakes and risks to know

The model assumes continuous moves and constant volatility. Real markets have jumps, fatter tails than the normal distribution and changing volatility: that is where the limits of Black-Scholes begin.

Reference: Hull, ch. 14

Deep dives

Two longer episodes outside the numbering: an advanced review of futures and options and a guide to certificates and covered warrants.

ExtraFutures vs options: pricing, arbitrage and risk managementAn advanced review comparing futures and options: how arbitrage sets their prices and how the risk they transfer differs.

What you will learn

  • How cash-and-carry arbitrage keeps the future close to its theoretical value
  • Why futures carry linear risk and options asymmetric risk
  • What the Greeks are for in risk monitoring

Key concepts

  • Futures: mutual obligation, linear risk, price tied to spot through the cost of carry.
  • Options: asymmetric right, the buyer's loss capped at the premium, price tied to volatility.
  • The Greeks measure sensitivities: delta to price, gamma to changes in delta, theta to the passage of time, vega to volatility.
  • Hedging with a future turns price risk into basis risk; hedging with an option costs a premium.
F0=S0 erT(no dividends)F_0 = S_0\,e^{rT} \quad (\text{no dividends})

Example

Underlying at 100, rate 4%, six-month expiry, no dividends: the theoretical future is about 102.02. If it traded at 103, buying spot and selling the future would earn more than the risk-free rate, and the gap would close quickly.

Mistakes and risks to know

The densest episode of the series, meant as a review: it pays off most after watching the futures and options episodes.

Reference: Hull, ch. 5, 10-13 and 19

ExtraCertificates vs covered warrants: structure, risks and taxTwo securitised products sold to retail investors in Italy: how they are built, what risks they carry and how they are taxed.

What you will learn

  • The structural difference: a certificate is the issuer's debt security, a covered warrant is a securitised option
  • What gearing, barriers and periodic coupons are
  • Why, under Italian tax rules, certificates are treated differently from ETFs and funds

Key concepts

  • Both depend on the issuer's soundness: if the issuer defaults you can lose money even when the underlying has done well.
  • In a covered warrant, time value erodes day by day, as in any option.
  • A barrier certificate protects capital only as long as the barrier is not breached.
  • In Italy, certificate income is generally «redditi diversi» and can offset past capital losses, which income from ETFs and funds («redditi di capitale») cannot.

Example

Hypothetical certificate with a 60% barrier observed at maturity: if the underlying stock has fallen 30%, you get the nominal amount back; if it has fallen 45%, you take the same loss as the stock and receive 55%.

Mistakes and risks to know

Complex structure, issuer risk, an issue price that includes the issuer's margins, liquidity provided by a market maker. Tax effects depend on each person's circumstances.

Reference: Borsa Italiana · ACEPI · Read more: Certificates and covered warrants

Further reading in the Knowledge Base

The key steps of the series, from futures to Black-Scholes
Derivatives: what they are and how they work
Futures, part 1: how they work
Futures, part 2: hedging properly
Certificates and covered warrants

Educational and informational content. It does not constitute investment advice, a recommendation or a solicitation. Derivatives are complex instruments and can lead to losses exceeding the capital invested. Numerical examples are hypothetical and only illustrate the concepts.