Derivatives, Options, and Structured Products
Question: can we agree on a future purchase price, protect against a fall, or design an investment that pays only when a particular movement occurs?
A derivative is a contract whose value depends on something else, called its underlying. For example, a contract linked to a share price is a derivative; the share is its underlying. The contract specifies who must pay, how much, and when.
Start with a future purchase. An investment costs £100 today, but you want to buy it in one year. One arrangement commits you to a purchase price now. Another gives you the choice to buy later. The difference between an obligation and a choice will lead us from forwards to options and then to combinations of contracts.
Track status: the teaching sequence and final capstone are available. Follow the chapter links below from Forward Contracts through Building Structured Products, then complete Design, Price, and Explain a Product. This additional learning track follows Portfolio Management Fundamentals.
Where to Begin
You need percentages, averages, basic algebra, and a little Python. We will introduce the calculus needed for Itô's lemma beside the examples, before using it in the Black–Scholes derivation.
Useful foundations from Portfolio Management Fundamentals are:
- Returns and units: distinguish a pound amount from a percentage return.
- Compounding and discounting: £100 invested at 5% for one year becomes £105; £105 due in one year is worth £100 today at that rate.
- Asset prices and uncertainty: Strategies Beyond Diversification introduces simulated price paths.
- Dated payments: Fundamentals of Asset-Liability Management explains bonds, present value, and interest-rate sensitivity.
The recommended sequence continues from this track to Advanced Portfolio Construction and Analysis. The latter studies the estimates behind portfolio allocations; this track studies contractual payments and their replication. Both share the same cash-flow and return conventions. The Black–Scholes convention comparison distinguishes portfolio arithmetic means, GBM drift, quoted rates, and pricing probabilities. The portfolio course can also be studied directly after the foundations.
The Learning Sequence
Forward Contracts
Available: start the lesson.
Agree today on a price for a purchase next year. Define spot price, delivery price, maturity, and long/short positions. Price a forward by comparing it with buying and financing the asset. Then value an existing contract and use it to hedge a future purchase.
Experiment: change today's asset price while keeping the existing delivery price fixed. Explain why a contract that began with zero value can later have a positive or negative value.
Futures and Forwards Compared
Available: study futures and daily settlement.
Reuse the forward example to introduce futures, exchanges, clearing, initial margin, and daily variation margin. Follow the cash paid or received each day and compare that timing with a forward settled at maturity. Distinguish collateral from an option premium and a quoted futures price from a forecast.
Experiment: compare two price paths with the same endpoint. Examine cumulative settlement and the cash required along the way. Explain the assumptions under which forward and futures quotes coincide, and why uncertain interest rates can make them differ.
Options and Replication
Available: study options and replication.
Introduce calls and puts as rights rather than obligations. Build payoff tables before plotting them. Separate premium, maturity payoff, and financed profit. Establish put–call parity and solve a two-outcome example using an asset holding and cash.
Experiment: change the final asset price and identify which right is exercised. Reconcile the replicated payment in both outcomes before introducing a continuous-time model.
Random Motion and Itô's Lemma
Available: Random Motion and Itô's Lemma.
Begin with rates of change and curvature. Show why an unchanged average input can produce a changed average curved output. Introduce Brownian increments and quadratic variation, derive Itô's lemma from a second-order expansion, and apply it to a log price.
Experiment: compare transformations of simulated increments with the first-order and curvature-adjusted calculations. Explain what changes as the time step shrinks.
Deriving and Using Black–Scholes
Available: Deriving and Using Black–Scholes.
Match an option's exposure with the underlying asset and a cash account. Explain self-financing rebalancing, derive the pricing equation, and work through the normal integrals leading to European call and put prices. Separate the pricing distribution from a real-world return forecast. Connect put–call parity to a forward payoff and reconcile discrete stock and futures hedge ledgers.
Experiment: compare call and put prices under matched assumptions, verify parity, and inspect expiry and low-volatility limits.
Futures Basis, Dividends, and Implied Financing
Available: Futures Basis, Dividends, and Implied Financing.
Derive carry from dated cash flows, distinguish dividend entitlement from payment, and compare executable prices. Read financing and dividends backwards from futures under independent anchors; fit a small financing curve, extract option parity, and reconcile margin funding and calendar-spread profit.
Experiment: find several rate/dividend pairs that reproduce one quote, then investigate how independent observations identify a fit. Compare residual convergence with the actual position's P&L.
The Greeks and Hedging
Available: The Greeks and Hedging.
Derive delta, gamma, theta, vega, rho, dividend sensitivity, vanna, volga, charm, speed, zomma, color, veta, and ultima. Translate risk readings into shares and pounds, explain delta neutrality, and build a simple gamma hedge before combining delta/gamma/vega targets. Explain futures delta, delta-one desks, and volatility quoting. Separate carry from discounting and reconcile actual stock trades and funding.
Experiment: compare exact repricing with local approximations; expose a same-expiry gamma/vega rank limit and a target-only volatility shock. Follow the cash account and compare hedge errors and costs across shared synthetic paths at different rebalancing frequencies.
Spreads and Basic Option Combinations
Available: Spreads and Basic Option Combinations.
Construct bull and bear spreads, covered calls, protective puts, and collars one position at a time. Explain signed legs, debit and credit, payoff regions, financing, and break-even calculations. Use parity to build synthetic positions and boxes; add Greeks, reprice early exits, and account for entry bid/ask costs.
Experiment: change a strike and explain the change in entry cost, payoff, and aggregate Greeks.
Reading the Options Market: Implied Volatility and Surfaces
Available: Reading the Options Market: Implied Volatility and Surfaces.
Invert option bid/mid/ask prices using explicit forward and discount inputs. Explain vega, quote conditioning, strike/moneyness/delta coordinates, smiles, and total variance before fitting a bounded surface. Compare weighted-mid, interval-distance, prior-regularized, and anchored joint fits. Reprice held-out and later quotes, check strike and normalized calendar consistency, and distinguish implied pricing from forecasts and executable profit.
Experiment: change the forward input and observe the inferred smile. Compare a smooth fit with quote-consistent prices; explain why differences in ATM total variance are not automatically executable forward-variance trades.
Straddles, Strangles, Butterflies, and Condors
Available: Straddles, Strangles, Butterflies, and Condors.
A short application of the existing signed-leg tools: compare the movement required by a straddle or strangle with the finishing range purchased by a butterfly or condor. Work through each payoff region and financed break-even, then compare local exposures and a six-month exit. One shared exercise applies execution costs to all four positions.
Experiment: open a butterfly into a condor by widening its plateau, reprice the premium, and select a final share price to explain the trade-off.
Butterflies, Implied Distributions, and Tail Pricing
Available: Butterflies, Implied Distributions, and Tail Pricing.
Read pricing tail probabilities from call-spread ramps and recover discrete masses from butterflies. Distinguish mass from smooth density, check total probability, the forward mean, and repricing, then expose quote-noise amplification and nonunique missing tails. An optional moments extension separates terminal-price skewness/kurtosis from volatility-quote conventions.
Experiment: narrow the strike spacing while perturbing one premium. Compare negative midpoint readings and quote-box bounds, then construct two tail laws that reproduce all observed prices but disagree about unobserved outcomes.
Building Structured Products
Available: Building Structured Products.
Allocate a £100 budget between a maturity bond and option participation. Compare uncapped and capped designs with downside-funded income, then reconcile fees, an early sale with fixed original holdings, and an explicit issuer-recovery stress. Distinguish nominal gains from performance against safe cash and contractual maturity promises from current resale value.
Experiment: change the safe cash rate and volatility for a new issuance, repricing bond, forward, and option inputs consistently. Complete an executable capped-product budget with fees and bid/ask costs.
Design, Price, and Explain a Product
Available: Design, Price, and Explain a Product.
Compare two executable £100-floor products against a stated investor brief. Reconcile the bond, signed option legs, fees, and entry spreads, then calculate maturity payments, fixed-holdings Greeks, six-month exit scenarios, and a locally delta-neutral hedge. Finish with a plain-language investor explanation.
Exercise: vary the cap and determine whether the target payment still holds. Derive the largest fee that can fund the promised floor and target, and explain what must change when costs exceed it.
How to Study Each Chapter
Follow everyday explanation → numbers → calculation → explained graph → Python → experiment → worked exercise. Run cells in order, then predict the effect of one changed assumption before changing it. Python and charts run in your browser.
Keep three questions separate throughout the track:
- What does the contract promise to pay? Read its terms and payment dates.
- What is that promise worth today? State the pricing assumptions and show the replication or discounting calculation.
- What does the investor gain or lose? Include the cost of entering and financing the position, using a common valuation date.
Check your reasoning: a contract pays £20 in one year but costs £12 today. Is £20 the profit? No: it is the maturity payoff. At a 5% annual financing rate, the £12 cost grows to £12.60, leaving a maturity-date profit of £7.40 before other costs.
Begin with Forward Contracts.