Why This Matters
When I first studied options, most textbook examples were equity-style: you pay a premium upfront, and you receive the payoff at expiry, or when you choose to exercise for American options. That framing was so ingrained that I assumed it was the general case.
When I started working on commodity derivatives, I encountered a different world. Many options are traded under futures-style margining. No premium changes hands at inception; instead, the option is margined daily like a futures contract. The convention tends to split along venue lines rather than by underlying. US exchanges are predominantly equity-style: options on WTI crude futures at the CME and options on corn and wheat futures at the CBOT all require an upfront premium. European venues lean the other way. Options on ICE Brent futures and options on EUA carbon futures at ICE Endex and EEX are both margined futures-style.
A standard assumption in practice is that American futures-style options are valued identically to their European counterparts. When I first went looking for an explanation on why early exercise has no benefit, the most common answer I found was something like:
Daily marking-to-market removes the time-value-of-money advantage that usually justifies early exercise for American options.
That statement makes some sense, but it never gave me the mathematical comfort I needed. To really understand why American and European options coincide under futures-style margining, I found it helpful to break the problem into smaller steps along two separate dimensions:
- Margining convention: futures-style margining (FSM) vs. equity-style margining (ESM), both applied to European options on futures.
- Exercise style: American vs. European.
The first step is to get a clear understanding of the margining dimension: what is the difference between a futures-style and an equity-style European option on a futures contract, and how does the change in margining convention affect the option’s valuation? The second step is the exercise dimension: what happens to the American early exercise feature once the option is margined futures-style? Understanding the first step is the key to answering this question.
European Options: Equity-Style vs. Futures-Style Margining
I want to compare the valuation difference from the PDE perspective. We will derive the PDE from scratch. Let $F_t$ denote the futures price at time $t$, assumed to follow geometric Brownian motion under the risk-neutral measure $\mathbb{Q}$:
$$dF = \sigma FdW^{\mathbb{Q}}$$There is no drift term. Under the risk-neutral measure, futures prices are martingales since entering a futures contract requires no net capital. Initial margin is set aside with the exchange, but it is returned when the position is closed and accrues interest in the meantime, so it is a collateral posting rather than a cost. The same is true of the margin requirement on a futures-style option position. The risk-free rate $r$ is taken to be deterministic, and margin balances are assumed to accrue at that same rate.
Consider a European option with value $V = V(F, t)$. We hedge it with a short position in $\Delta$ futures contracts and follow the P&L of the combined position, which we write as $d\Pi$. The futures leg contributes its daily settlement $-\Delta dF$.
The Equity-Style Case
In the equity-style world, $V$ is the cash premium paid upfront. Since the futures leg requires no net capital, the strategy’s entire initial outlay is the option premium $V$.
Applying Itô’s lemma to $V(F, t)$:
$$dV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial F}dF + \frac{1}{2}\frac{\partial^2 V}{\partial F^2}(dF)^2$$The P&L of the strategy is:
$$d\Pi = dV - \Delta dF = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2}\right)dt + \left(\frac{\partial V}{\partial F} - \Delta\right)\sigma F dW^{\mathbb{Q}}$$Setting $\Delta = \frac{\partial V}{\partial F}$ eliminates the stochastic term. The strategy is now instantaneously risk-free:
$$d\Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2}\right)dt$$No-arbitrage condition: a riskless position must earn the risk-free rate $r$ on the capital it ties up. That capital is the premium $V$, so we require $d\Pi = rV dt$. Setting the two expressions equal and rearranging:
$$\boxed{\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2} - rV = 0}$$with terminal condition $V(F, T) = $ option payoff.
The $-rV$ term is the cost of carry on the cash investment $V$. It is present because $V$ is the money the holder has paid out and it must earn the risk-free rate to break even. The solution is the Black formula:
$$V^{\text{equity}}(F, t) = e^{-r(T-t)}\left[F N(d_1) - K N(d_2)\right]$$where $d_1, d_2$ are the standard Black expressions.
The Futures-Style Case
Under futures-style margining, no cash premium is paid for the option at inception. Instead, the option is margined daily: if the exchange’s settlement price moves from $V_t$ to $V_{t+dt}$, the holder receives (or pays) $dV = V_{t+dt} - V_t$ through their margin account. How does this change the PDE and its valuation?
Derivation of the Futures-Style PDE
Follow the same delta-hedged strategy. Setting $\Delta = \frac{\partial V}{\partial F}$ eliminates the stochastic term as before, and the instantaneous risk-free P&L is:
$$d\Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2}\right)dt$$Now apply the no-arbitrage condition. The option is margined futures-style and carries no premium, and the futures hedge requires no upfront net capital either. Since the strategy is instantaneously riskless, any non-zero deterministic drift would imply arbitrage. Its drift must vanish:
$$d\Pi = 0$$$$\boxed{\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2} = 0}$$with the same terminal condition. The $-rV$ term is gone because there is no capital tied up to carry.
$V$ as a Martingale Under the Risk-Neutral Measure
The absence of the $-rV$ term has a direct probabilistic interpretation. Applying Itô’s lemma to $V(F, t)$ under the risk-neutral measure and substituting $dF = \sigma F dW^{\mathbb{Q}}$:
$$dV = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2}\right)dt + \frac{\partial V}{\partial F}\sigma F dW^{\mathbb{Q}}$$The $dt$ term is exactly the left-hand side of the futures-style PDE, so it vanishes, leaving:
$$dV = \frac{\partial V}{\partial F}\sigma F dW^{\mathbb{Q}}$$Therefore $V$ is a local martingale under $\mathbb{Q}$, and under the assumed dynamics a true martingale. This is the direct counterpart to the futures price $F$ itself being a martingale under $\mathbb{Q}$. Neither position commits net capital: no premium changes hands, and the initial margin posted against each is collateral rather than money spent. With no capital committed, neither quantity carries a discount factor. Being a martingale, $V$ satisfies:
$$V(F_t, t) = \mathbb{E}^{\mathbb{Q}}\left[V(F_T, T) \middle|\mathcal{F}_t\right] = \mathbb{E}^{\mathbb{Q}}\left[\text{Payoff}(F_T) \middle|\mathcal{F}_t\right]$$where
$$\text{Payoff}(F_T) = \begin{cases} \max(F_T - K, 0) & \text{call} \\ \max(K - F_T, 0) & \text{put} \end{cases}$$That is, the futures-style MTM at any point in time is the risk-neutral expectation of the terminal payoff with no discount factor applied.
Comparing the Two Worlds
We can now contrast the two margining conventions clearly.
| Equity-Style | Futures-Style | |
|---|---|---|
| Premium at inception | Paid upfront in cash | None, no premium and no net capital outlay |
| PDE | $V_t + \frac{1}{2}\sigma^2F^2V_{FF} - rV = 0$ | $V_t + \frac{1}{2}\sigma^2F^2V_{FF} = 0$ |
| What $V$ represents | Present value of the option | Exchange MTM settlement price |
| Probabilistic form | $e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[\text{payoff}]$ | $\mathbb{E}^{\mathbb{Q}}[\text{payoff}]$ |
| Cash flow to holder | Premium $V$ paid at $t_0$, payoff received at $T$ | Daily margin flows $dV$, summing to payoff at $T$ |
In the equity-style world, $V(F, t)$ is the fair cash amount to exchange today for the right to receive the option payoff at expiry. It is a present value in the traditional sense.
In the futures-style world, $V(F, t)$ is the exchange’s mark-to-market settlement quote, used to compute each day’s margin flow. It is not paid or received as a lump sum.
Since $V$ is the undiscounted expectation of the payoff, the closed-form solution is Black’s formula with the discount factor removed:
$$V^{\text{futures}}(F, t) = F N(d_1) - K N(d_2)$$Comparing with the equity-style solution:
$$V^{\text{futures}} = e^{r(T-t)} V^{\text{equity}}$$The futures-style MTM exceeds the equity-style present value by exactly $e^{r(T-t)}$. The futures-style holder collects the same economic cash flows as the equity-style holder but without paying anything upfront, so the quoted price is scaled up by the cost of carry that the equity-style holder effectively prepays.
American Options Under Futures-Style Margining
We now turn to the central question. In the equity-style world, American options can be worth more than European options. Early exercise can be optimal when the intrinsic value in hand, reinvested at $r$, exceeds the value of waiting (discussed in Early Exercise of American Options: Call Equivalence and the Put Premium). Does the same logic apply under futures-style margining?
Setup and Notation
Consider an American put option on a futures contract, traded under futures-style margining, with strike $K$, expiry at time $T$, and current time $t_0$. The exchange publishes a daily MTM settlement price for the option, which we denote $V_i = V(F_i, t_i)$ on day $i$. The holder receives $V_i - V_{i-1}$ on day $i$ through their margin account.
At expiry on day $n$, the settlement price converges to intrinsic value:
$$V_n = \max(K - F_n, 0)$$Exercise Mechanics
When the holder of a futures-style American put exercises on day $m$, the following happens in sequence:
- The regular daily margin flow $V_m - V_{m-1}$ is settled as usual through the margin account. This happens regardless of exercise.
- The option position is submitted for exercise. The exchange assigns the holder a short futures position at the strike price $K$. If the current futures settlement price is $F_m$, the resulting futures position carries a gain of $K - F_m$ for an in-the-money put. This amount is credited through the futures margin account. The holder may then close out the short futures position at $F_m$ at no further cost, or carry it forward.
- The option is extinguished. No further option margin flows occur from day $m+1$ onward.
Early Exercise on Day $m$
Suppose the holder reaches day $m$, before expiry on day $n$, and considers exercising.
| As of day $m$ | Hold | Exercise |
|---|---|---|
| Day $m$ margin flow | $V_m - V_{m-1}$ | $V_m - V_{m-1}$ |
| Exercise payoff | none | $\max(K - F_m, 0)$ |
| Days $m+1$ to $n$ | flows telescoping to $V_n - V_m$ | none, the option is terminated |
| Value as of day $m$ | $V_m$ | $\max(K - F_m, 0)$ |
The question is how the intrinsic value received at exercise compares to the option value $V_m$ given up. The European section already gives us the relationship we need:
$$V(F_t, t) = \mathbb{E}^{\mathbb{Q}}\left[\text{Payoff}(F_T) \middle|\mathcal{F}_t\right]$$What stands in the way is the expectation itself, and we can use Jensen’s inequality to remove it since the payoff is a convex function of $F_T$.
$$\mathbb{E}^{\mathbb{Q}}\left[\text{Payoff}(F_T) \middle| \mathcal{F}_t\right] \geq \text{Payoff}\left(\mathbb{E}^{\mathbb{Q}}\left[F_T \middle| \mathcal{F}_t\right]\right) = \text{Payoff}(F_t)$$Two separate facts are at work in the equation above. Jensen moves the expectation inside the payoff function, and the martingale property of $F$ then replaces the expected terminal futures price with $F_t$ itself. As a result, the comparison lands on intrinsic value at today’s price, which is the quantity exercise actually delivers.
What Jensen gives is a weak inequality, that holding is worth at least as much as exercising. For $V_m$ and intrinsic value to be worth the same, one of two conditions has to hold.
- $F_T$ is no longer random. With $\sigma = 0$ the futures price is already known, and there is no time value left to give up.
- The payoff is linear over every value $F_T$ can reach. Suppose $F_T$ were certain to land below $K$, so the put finishes in the money in every state. The payoff is then $K - F_T$ throughout, and averaging it gives $K$ minus the expected futures price, which is $K - F_m$. That is intrinsic value today. How the probability is spread out below the strike makes no difference, because averaging a straight line and evaluating it at the average are the same operation.
However, neither condition holds in practice. Volatility is not zero, and under lognormal dynamics $F_T$ can land anywhere in $(0, \infty)$, so the range of outcomes always straddles $K$ and the payoff always bends somewhere inside it. The inequality is strict, and on day $m$:
$$V_m > \max(K - F_m, 0) \quad \text{for all } m \lt n$$So the American feature has no value. We worked with a put throughout, but the argument rests only on convexity of the payoff, so the call case needs no separate treatment.
$$\boxed{V^{\text{American, futures-style}} = V^{\text{European, futures-style}}}$$Takeaway
Return to the explanation quoted at the start:
Daily marking-to-market removes the time-value-of-money advantage that usually justifies early exercise for American options.
It points to a mechanic rather than the root reason, which is what makes it unsatisfying. Daily marking-to-market is how the exchange makes a position with no premium workable; what actually removes the incentive is that no premium is paid upfront. With no capital tied up, the $-rV$ term drops out of the PDE, $V$ becomes a martingale, and Jensen’s inequality makes holding beat exercising. The daily mark-to-market mechanic is not the root reason; the absence of a premium is.
An American futures-style option can be valued with the Black model with discounting removed. The usual machinery for equity-style American options, a numerical PDE solve or a closed-form American approximation, is unnecessary here. That matters most for calibration. Backing out an implied volatility still needs a root finder, but against a numerical pricer each iteration becomes a closed-form evaluation rather than a numerical solve, which makes building a vol surface from these quotes considerably cheaper.