Derivative foundations

Part 1 gave present value, no-arbitrage replication, and risk-neutral pricing as a discounted expectation over a finite set of states. This part applies those results to the first derivative contracts. Replication fixes the forward delivery price and put-call parity, and the risk-neutral expectation values an option, first on a one-period tree and then node by node on a multi-period lattice.

This part covers forwards and European options on one underlying. The first chapter separates a forward’s obligation, its fair delivery price, and the value of an existing contract. The second chapter separates an option’s right and expiry payoff from its current value, premium, and profit. The third chapter matches call and put portfolios to derive put-call parity. The last two chapters value an option: on a one-period binomial tree by replication and by risk-neutral weights, and on a multi-period lattice by backward induction with local discount factors.

Part 4 of 75 chapters1 numbered equations5 knowledge checks

1. Forward contracts, delivery price, and value

   Separate a forward's obligation, fair delivery price, and changing signed contract value.

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2. European option contracts and payoffs

   Separate call and put rights from expiry payoff, current value, premium, and profit.

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3. Put-call parity and synthetic forwards

   Match European call and put portfolios state by state before solving a missing value.

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4. One-period binomial option valuation

   Price two-state option payoffs by both exact replication and risk-neutral weighting.

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5. Multi-period lattice valuation

   Extend one-period risk-neutral valuation to node-by-node backward induction with local discount factors.

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The argument

1. A forward is an obligation, and its delivery price makes a new contract worth zero
2. An option is a right, and its payoff at expiry is not its value today
3. Matched call and put portfolios give put-call parity
4. A two-state payoff is replicated, or weighted by risk-neutral probabilities
5. Backward induction repeats the one-period step node by node

After this part you can

• Interpret the long and short obligations, delivery time, delivery price, and signed expiry payoff of a forward contract.
• Calculate the fair delivery price of a newly struck forward from spot value, present value of deterministic income before delivery, and the delivery-date discount factor.
• Calculate the signed value to the long of an existing forward from the current forward price, its fixed delivery price, quantity, and the delivery-date discount factor.
• Identify the underlying, call or put right, holder and writer, strike, expiry, and settlement perspective of a European option.
• Calculate the expiry payoff of a long or short European call or put from the underlying expiry value, strike, and quantity.
• Distinguish an option's current value or premium from its expiry payoff and from profit after accounting for the premium under a stated financing convention.
• Explain European put-call parity by matching the expiry cash flows of a call-minus-put portfolio and a forward-style position under stated income and discounting assumptions.
• Calculate a missing European call or put value from the other option value, prepaid forward value, strike, and expiry discount factor.
• Calculate the underlying units and risk-free cash position that replicate a claim's two one-period state payoffs.
• Calculate a European option's one-period binomial value using model-implied risk-neutral weights and deterministic discounting.
• Calculate node values on an input multi-period recombining pricing lattice by risk-neutral expectation and one-period node discounting.

Builds on

• Foundations

Leads to

• Bonds
• Bond options