From Market to No-Arbitrage¶

An option is worth something — but how much? The previous sections defined payoffs, premiums, and trading mechanics. None of them answered the central question: what is the correct price of an option today? This question is not merely academic. If the market price deviates from the theoretically correct value, an arbitrageur can lock in a guaranteed profit with zero net investment. The purpose of option pricing theory is to determine the unique price at which no such opportunity exists.

This section shows, through concrete examples, that mispriced options create arbitrage. It then introduces the key idea — replicating an option's payoff with a portfolio of stock and cash — that transforms the pricing problem into a problem of differential equations.

Mispricing Creates Arbitrage¶

Key idea: The option price is determined by replication, not by belief about where the stock is headed.

Consider a European call option with strike $K = 100$ maturing in one period. Suppose the stock currently trades at $S_0 = 100$, the risk-free rate is $r = 5\%$ per period, and the stock can move to either $S_u = 120$ or $S_d = 90$ at maturity. The call payoff is

\[ C_T = \max(S_T - K, 0) = \begin{cases} 20 & \text{if } S_T = 120 \\ 0 & \text{if } S_T = 90 \end{cases} \]

We can replicate this payoff exactly by holding $\Delta$ shares of stock and lending $B$ dollars at the risk-free rate. The replication conditions are

\[ \Delta \cdot 120 + B \cdot 1.05 = 20 \]

\[ \Delta \cdot 90 + B \cdot 1.05 = 0 \]

Subtracting the second equation from the first gives $30\Delta = 20$, so $\Delta = 2/3$. Substituting back yields $B = -60/1.05 \approx -57.14$. The replicating portfolio costs

\[ C_0 = \Delta \cdot S_0 + B = \frac{2}{3} \cdot 100 - \frac{60}{1.05} \approx 9.52 \]

This is the only arbitrage-free price. If the market quotes any other price, a riskless profit is available:

If the market price $C^{\text{mkt}} > 9.52$: Sell the call for $C^{\text{mkt}}$ and buy the replicating portfolio for $9.52. The portfolio matches the call's payoff in every state, so the obligation is perfectly hedged. The difference $C^{\text{mkt}} - 9.52 > 0$ is a guaranteed profit.
If the market price $C^{\text{mkt}} < 9.52$: Buy the call for $C^{\text{mkt}}$ and sell the replicating portfolio (short $2/3$ shares, lend $57.14). At maturity, the call payoff exactly offsets the portfolio obligation. The difference $9.52 - C^{\text{mkt}} > 0$ is a guaranteed profit.

In both cases the profit is locked in at time zero with no risk. This is the hallmark of arbitrage: a self-financing strategy that starts with zero wealth and ends with positive wealth in at least one state, with no possibility of loss.

Replication as a Pricing Principle¶

The example above illustrates a profound idea: the option price is determined not by investors' views on whether the stock will go up or down, but by the cost of replicating the option's payoff using traded instruments. This replication perspective and the risk-neutral expectation $V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\text{Payoff}]$ introduced in the premium section are mathematically equivalent — two representations of the same no-arbitrage principle. The drift $\mu$ of the stock is irrelevant — only the volatility (which determines the range of possible outcomes) and the risk-free rate (which determines the cost of financing) matter.

Recall (see § Black-Scholes PDE Derivation): in continuous time, the replicating portfolio holds $\Delta_t = \partial V/\partial S$ shares (the option's delta), and the requirement that the delta-hedged portfolio earn the risk-free rate translates into the Black-Scholes PDE. The drift $\mu$ drops out — only $r$, $\sigma$, $K$, and the payoff determine the price.

Big Picture

Options derive their value from asymmetric payoffs. Their prices are determined not by beliefs about the future, but by the cost of replicating those payoffs in the market. This equivalence between replication and expectation transforms pricing into a mathematical problem — leading directly to the Black-Scholes equation.

Checking No-Arbitrage with Real Data

Put-call parity provides a simple consistency check on live market prices. With the S&P 500 near 6,600 and front-month options at the 6,600 strike, a call trades around 180 points and a put around 160 points. The difference $C - P \approx 20$ points should approximately equal $S_0 e^{-qT} - K e^{-rT}$, where $q$ is the continuous dividend yield of the index.

Plugging in typical values ($r \approx 4\%$, $q \approx 1.3\%$, $T \approx 1/12$) confirms that the observed call-put spread is consistent with no-arbitrage once dividends are accounted for. If this relationship were violated by more than the bid-ask spread, an arbitrageur could lock in a risk-free profit — exactly the mechanism described above.

Exercises¶

Exercise 1. In the one-period binomial model above ($S_0 = 100$, $S_u = 120$, $S_d = 90$, $r = 5\%$), find the arbitrage-free price of a European put with strike $K = 100$.

Solution to Exercise 1

The put payoff is $P_T = \max(K - S_T, 0)$: it pays $0$ if $S_T = 120$ and $10$ if $S_T = 90$.

The replication conditions are

\[ \Delta \cdot 120 + B \cdot 1.05 = 0 \]

\[ \Delta \cdot 90 + B \cdot 1.05 = 10 \]

Subtracting the first from the second: $-30\Delta = 10$, so $\Delta = -1/3$. Substituting back: $B = (120/3)/1.05 = 40/1.05 \approx 38.10$.

The put price is

\[ P_0 = \Delta \cdot S_0 + B = -\frac{1}{3}\cdot 100 + \frac{40}{1.05} = -33.33 + 38.10 \approx 4.76 \]

Exercise 2. Using the call and put prices from Exercise 1 and the text, verify that put-call parity holds: $C_0 - P_0 = S_0 - K\,e^{-rT}$, where $e^{-rT} = 1/1.05$ in this one-period model.

Solution to Exercise 2

From the text, $C_0 \approx 9.52$. From Exercise 1, $P_0 \approx 4.76$.

\[ C_0 - P_0 \approx 9.52 - 4.76 = 4.76 \]

On the other side:

\[ S_0 - \frac{K}{1.05} = 100 - \frac{100}{1.05} = 100 - 95.24 = 4.76 \]

The two sides agree, confirming put-call parity. This is not a coincidence — it follows from the linearity of the replication argument. Any pair of call and put prices that violate this relation admits arbitrage.

Exercise 3. Suppose the market quotes the call from the text at $12 instead of $9.52. Describe the exact arbitrage strategy and compute the profit in each state.

Solution to Exercise 3

The call is overpriced ($12 > 9.52$), so we sell the call and buy the replicating portfolio.

At time 0: Sell the call, receive $12. Buy $2/3$ shares at $100 each, costing $66.67. Borrow $54.67 at $5\%$ (net cash flow: $12 - 66.67 + 54.67 = 0$).

More precisely, the replicating portfolio costs $C_0 = 9.52$, so we invest $9.52 in the replicating portfolio and pocket $12 - 9.52 = \$2.48$, which we invest at the risk-free rate.

If $S_T = 120$: The call is exercised against us; we owe $20$. The replicating portfolio pays $(2/3)(120) + (-60/1.05)(1.05) = 80 - 60 = 20$. Net from hedge: $0$. We still have $2.48 \times 1.05 = \$2.60$ from the initial profit.

If $S_T = 90$: The call expires worthless. The replicating portfolio pays $(2/3)(90) + (-60/1.05)(1.05) = 60 - 60 = 0$. We still have $2.48 \times 1.05 = \$2.60$.

In both states, the profit is $2.60. This is a riskless arbitrage.

Exercise 4. Explain in your own words why the stock's expected return $\mu$ does not appear in the arbitrage-free option price. What parameters do determine the price?

Solution to Exercise 4

The arbitrage-free price is determined by replication: we find a portfolio of stock and bond that matches the option payoff in every state. The cost of this portfolio depends on (i) the current stock price $S_0$, (ii) the range of possible future stock prices (determined by $\sigma$), (iii) the risk-free rate $r$ (which determines the cost of borrowing in the replicating portfolio), and (iv) the strike $K$ and maturity $T$ (which determine the payoff).

The drift $\mu$ does not appear because the replicating portfolio matches the option in every state, regardless of which state is more likely. An optimistic investor (high $\mu$) and a pessimistic investor (low $\mu$) must agree on the option price, because any deviation from the replication cost creates an arbitrage that neither investor's beliefs can override.

The parameters that determine the price are: $S_0$, $K$, $T$, $r$, and $\sigma$.

Exercise 5. In the continuous-time setting, the delta of a European call satisfies $0 \leq \Delta_t \leq 1$. Give an intuitive explanation for each bound. What does $\Delta_t \approx 1$ mean for the replicating portfolio?

Solution to Exercise 5

The call payoff $(S_T - K)^+$ is a non-decreasing function of $S_T$ (higher stock price means higher payoff), so the option price $V$ is non-decreasing in $S$. This gives $\Delta_t = \partial V / \partial S \geq 0$.

The call payoff increases at most dollar-for-dollar with the stock price (the slope of $(S_T - K)^+$ is at most $1$), so the option price cannot increase faster than the stock price itself. This gives $\Delta_t \leq 1$.

When $\Delta_t \approx 1$, the option is deep in-the-money and behaves almost like the stock itself. The replicating portfolio holds close to one full share and a large short bond position (borrowing nearly $Ke^{-r(T-t)}$), reflecting the near-certainty that the option will be exercised.