🔄 Derivatives

Put-Call Parity Formula — Options Pricing Relationship Explained

Put-call parity formula explained: C + PV(K) = P + S. The fundamental options pricing relationship with derivation, examples, and arbitrage applications.

Key Concepts

No-Arbitrage Pricing

Derivatives priced so no riskless profit is possible. If mispriced, arbitrageurs act to restore equilibrium.

Replication

A derivative can be replicated by a portfolio of the underlying and risk-free asset. Replication cost = derivative price.

Cost of Carry

Forward price = Spot × (1+r)<sup>T</sup> + Storage - Convenience yield. For financial assets: F = S(1+r)<sup>T</sup>.

Put-Call Parity

c + PV(X) = p + S. Links call, put, bond, and stock prices. Violation = arbitrage opportunity.

Binomial Option Pricing

Model stock as up/down moves. Calculate option values at nodes. Discount back using risk-neutral probabilities.

Formulas

From this module

Forward Price (no income)

F0 = S0 × (1 + r)T

Where: S = spot, r = risk-free rate, T = time

Forward with Continuous Dividends

F0 = S0 × e(r-q)T

Where: q = continuous dividend yield

Put-Call Parity

c + PV(X) = p + S

Where: c = call, p = put, X = strike, S = stock

Risk-Neutral Probability

πu = (1 + r - d) / (u - d)

Where: u = up factor, d = down factor

Binomial Option Value

C = [πuCu + (1-πu)Cd] / (1+r)

Where: Discount expected payoff at risk-free rate

Master Formula Sheet -- Derivatives

Forward Price

F₀ = S₀ × (1 + r)T

No-arbitrage forward (no income)

Forward Payoff (Long)

Payoff = S_T - F₀

Gain if spot > forward

Call Payoff

max(0, S_T - X)

Right to buy at strike X

Put Payoff

max(0, X - S_T)

Right to sell at strike X

Put-Call Parity

c + PV(X) = p + S

European options only

Risk-Neutral Probability

πᵤ = (1 + r - d) / (u - d)

Binomial model probability

Binomial Option Value

C = [πᵤCᵤ + (1-πᵤ)C_d] / (1+r)

Discounted expected payoff

Decision Frameworks

When is put-call parity violated?

Use when:

  • If c + PV(X) ≠ p + S, arbitrage exists
  • Buy the cheap side, sell the expensive side

Avoid when:

  • Assuming put-call parity holds for American options (it holds exactly only for European)

Test Your Understanding

Spot price = $50, risk-free rate = 4%, 6-month forward price is:

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