No-Arbitrage Principle and Risk-Neutral Valuation › Pricing by Replication

The replicating portfolio methodology provides a systematic approach to derivative pricing. Rather than trying to assess the probability of various outcomes or the appropriate discount rate for risky cash flows, we sidestep these thorny issues entirely by constructing a portfolio that mimics the derivative's behavior exactly. The approach proceeds as follows:

1. Identify the payoff of the derivative you want to price
2. Construct a portfolio of traded assets that produces exactly this payoff
3. Calculate the cost of this replicating portfolio
4. Conclude that the derivative price equals this cost

This approach is powerful because it converts the problem of pricing an exotic derivative into the simpler problem of finding the right combination of basic instruments. Once we have identified a replicating portfolio, no further analysis is needed. The Law of One Price guarantees that the derivative must trade at the same price as the replication strategy, regardless of investor preferences, probability assessments, or discount rates.

As we discussed in the chapter on Forward and Futures Pricing, we already used this logic implicitly. To price a forward contract on a stock, we showed that buying the stock today and financing it with borrowing replicates the forward's payoff. This replication argument gave us the forward price $F = S_0 e^{rT}$ without any assumptions about the stock's expected return. We never needed to estimate whether the stock was likely to go up or down. The no-arbitrage condition forced the forward price to take a specific value determined solely by the current stock price, the interest rate, and the time to expiration.

Consider European call and put options with the same strike $K$ and expiration $T$ on a non-dividend-paying stock. This setting allows us to derive one of the most famous relationships in option pricing, purely through no-arbitrage arguments. At expiration, consider two portfolios:

Portfolio A: One call option plus $Ke^{-rT}$ in cash (invested at the risk-free rate)

Portfolio B: One put option plus one share of stock

At expiration, let us trace through what each portfolio is worth in both possible scenarios: when the stock finishes above the strike, and when it finishes below. Portfolio A is worth:

$$\max(S_T - K, 0) + K = \max(S_T, K)$$

where:

• $S_T$: stock price at expiration
• $K$: strike price

If the stock price exceeds the strike, the call pays $S_T - K$ and the cash has grown to $K$, giving a total of $S_T$. If the stock price falls below the strike, the call expires worthless but the cash is worth $K$. Either way, the portfolio is worth the larger of $S_T$ and $K$.

Portfolio B is worth:

$$\max(K - S_T, 0) + S_T = \max(K, S_T)$$

where:

• $S_T$: stock price at expiration
• $K$: strike price

If the stock price falls below the strike, the put pays $K - S_T$ and the stock is worth $S_T$, giving a total of $K$. If the stock price exceeds the strike, the put expires worthless but the stock is worth $S_T$. Once again, the portfolio is worth the larger of $K$ and $S_T$.

Both portfolios have identical payoffs in all states of the world. By the Law of One Price:

$$C_0 + Ke^{-rT} = P_0 + S_0$$

where:

• $C_0$: European call option price
• $P_0$: European put option price
• $S_0$: current stock price
• $K$: strike price
• $r$: risk-free interest rate
• $T$: time to expiration

This is put-call parity which you encountered in Part II. Notice we derived it purely from no-arbitrage, without any assumptions about how the stock price moves or what probability distribution governs returns. We did not need to specify a model for stock price dynamics, estimate volatility, or make assumptions about investor risk preferences. The relationship holds simply because both portfolios produce the same future cash flows, and therefore they must cost the same today.