Derivatives Pricing

Definition

Derivatives pricing determines the fair value of a contract whose payoff depends on an underlying asset. In a discrete-time model (initial time t=0, final t=1, states of nature Ω), the core insight is that a derivative’s price equals the cost of a portfolio that replicates its payoff — no probabilities of up/down moves required. Course notes here follow Derivatives (Università di Napoli Federico II) and the Mathematical Methods for Quantitative Finance (MITx 15.455x) syllabus.

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Core Ideas

Forwards vs futures

A forward is a private contract with no daily adjustment; a future is exchange-traded and marked-to-market daily (price can change day to day).

Market model and replication

A market is (Ω, p, S): states Ω, current prices p, and payoff matrix S (asset i’s payoff in state ω). A portfolio x costs p·x and pays S·x. To price a target payoff W̄, solve x = S⁻¹ W̄.

• Complete market — enough independent assets (A ≥ N, S non-singular) that any payoff can be replicated.
• Incomplete market — A < N; only some payoffs are replicable.

Law of One Price (LOP)

No second-type arbitrage ⇒ two portfolios with identical payoffs in every state must have the same price. Therefore a replicated asset and its replicating portfolio share a price — the basis of the pricing method: (1) replicate the payoff, (2) assign the replicating portfolio’s cost.

One-period option pricing (binomial)

With a risk-free asset (1+r) and a risky asset going uS₀ (up) or dS₀ (down), d < 1+r < u, a European call is replicated by solving a 2×2 system. Crucially, the option price does not depend on the up/down probabilities — only on replicating the payoff. The call’s Delta is the number of shares in the replicating portfolio:

$\Delta =\frac{C_u-C_d}{S_{0}u-S_{0}d}$

Risk-neutral valuation

Define the risk-neutral probability p = (1+r − d)/(u − d). Then price by discounting the expected payoff under p:

$C_0=\frac{1}{1+r}\left[p{C}_u+(1-p)C_d\right]$

Example: S₀=10, u=1.1, d=0.98, K=10, r=0.04 → p=0.5, C_u=1, C_d=0 → C₀ ≈ 0.481.

Arrow-Debreu securities and the FTAP

An Arrow-Debreu security ADω pays 1 unit if state ω occurs, 0 otherwise. If an AD exists for every state, the market is complete and any wealth pattern is replicated by buying Wω units of each. The Fundamental Theorem of Asset Pricing (FTAP): no arbitrage ⇔ existence of positive state prices π = (π₁,…,π_Ω), πω > 0, with

$p_a={\sum }_{\omega }{\pi }_{\omega }{S}_{\omega ,a}$

State prices are the discounted risk-neutral probabilities — the same valuation, viewed through linear algebra (dual spaces, the one-period model). This is the entry point to continuous-time finance: Brownian motion, Itô’s lemma, and the Black-Scholes-Merton PDE.

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Relationships

• Options Greeks — Delta emerges directly from the replicating portfolio
• Options Strategies — put-call parity and synthetic positions rest on replication
• Implied Volatility — Black-Scholes inverts this pricing to back out volatility
• Bonds and Fixed Income — the risk-free leg and discounting
• Trading & Finance — parent topic