Multi-Leg Payoff & Probability of Profit Methodology

Overview

Implementation spec for SOPHIE's SPX Payoff Builder — multi-leg P&L at expiration, the "T+0" mark-to-market curve, net position Greeks, and risk-neutral probability of profit (POP). These are the standard tools professional and retail options traders alike use to plan a trade before placing it: what it pays off, what it's worth today if the market doesn't move, how exposed it is, and how likely it is to work.

P&L at Expiration ($T=0$)

The payoff of an $N$-leg position once every option has settled:

$$\text{PnL}(S_T) = \sum_{i=1}^N \text{Quantity}_i \times \text{Sign}_i \times \left[ \text{IntrinsicValue}(S_T, K_i, \text{Type}_i) - \text{Premium}_i \right] \times 100$$

where $\text{Sign}_i = +1$ for a long leg and $-1$ for short. This is the "hockey stick" diagram every options course starts with — a straight-line, closed-form payoff that requires no volatility model at all, because at expiration every option is worth exactly its intrinsic value.

How it's used: this is how a trade is sized before it's opened. Max loss and max profit are read directly off the diagram; breakeven strikes are where it crosses zero. Defined-risk strategies (spreads, condors, butterflies) are specifically built so this curve is bounded on both sides — the diagram is the risk-management case for the trade, not just an illustration of it.

"T+0" Mark-to-Market Curve

The expiration diagram answers "what does this pay off eventually." It says nothing about what the position is worth tomorrow, which is usually the more urgent question for anyone managing a live position. That requires actually pricing each leg today, not just at expiry:

1. Solve for implied volatility. Each leg's entry premium is inverted to its exact Black-Scholes $\sigma_i$ by bisection on the mid price — the position's IV assumptions come from what was actually paid, not a smile model.
2. Reprice at hypothetical spot levels. $$\text{Value Today}(S') = \sum_{i=1}^N \text{Quantity}_i \times \text{Sign}_i \times \left[ C_{\text{BS}}(S', K_i, T, r, q, \sigma_i) - \text{Premium}_i \right] \times 100$$

How it's used: this is the curve a trader actually watches while a position is open. It shows current unrealized P&L across a range of prices with today's remaining time value still priced in — which is why it sits inside the expiration diagram (time value cushions losses and caps gains before expiry). Comparing the two curves is how a trader decides whether to hold, adjust, or take profit early: a position near its T+0 profit target with weeks of time value left is a very different decision than the same P&L two days before expiration.

Net Position Greeks

Per-leg Greeks aggregated by position direction and contract size:

$$\text{Net Delta} = \sum \pm Q_i \cdot 100 \cdot \Delta_i, \quad \text{Net Gamma} = \sum \pm Q_i \cdot 100 \cdot \Gamma_i$$ $$\text{Net Theta} = \sum \pm Q_i \cdot 100 \cdot \Theta_i, \quad \text{Net Vega} = \sum \pm Q_i \cdot 100 \cdot \mathcal{V}_i$$

How it's used: this is the standard risk dashboard for a multi-leg book, and it's genuinely how professional desks manage exposure — not by tracking each leg individually, but by watching the net numbers.

• Net delta near zero is the entry condition for most credit-selling strategies (iron condors, straddles): the position starts direction-neutral.
• Net theta is the daily carry — positive for premium sellers, negative for premium buyers — and is what a theta-decay strategy is explicitly harvesting.
• Net gamma measures how fast delta will move if the underlying does. A large short-gamma position (short straddles, naked short options) needs active delta-hedging as price moves, which is the origin of "gamma scalping" as a trading style.
• Net vega is exposure to implied vol itself, independent of direction — a long-vega position profits if IV rises even with the underlying unchanged, which is the whole basis of buying volatility ahead of an event.

Risk-Neutral Probability of Profit

The probability the position finishes with $\text{PnL} > 0$ at expiration, under risk-neutral lognormal drift $\mu = r - q - \sigma^2/2$:

1. Find every breakeven (zero-crossing) strike $B_1, \dots, B_m$.
2. Partition the price domain into intervals between them.
3. For each profitable interval $[L_k, U_k]$, integrate the lognormal CDF: $$P(L_k < S_T < U_k) = \Phi\!\left( \frac{\ln(S/L_k) + \mu T}{\sigma\sqrt{T}} \right) - \Phi\!\left( \frac{\ln(S/U_k) + \mu T}{\sigma\sqrt{T}} \right)$$
4. $\text{POP} = \sum_{k \in \text{Profitable}} P(L_k < S_T < U_k)$.

How it's used, and its central caveat: POP is the single most commonly cited number in retail options education for comparing trade structures — a 70% POP iron condor "sounds" safer than a 50-delta strangle. But POP and expected value are different things: a high-POP credit spread typically has a small max profit against a large max loss, so it can have a high win rate and still lose money over time if the rare loss is large enough. POP measures how often a trade wins, not how much. This is also the risk-neutral (option-implied) probability, not the real-world one — the two coincide only if the market's volatility risk premium is zero, which the VRP methodology page shows is not generally true.

Implied Price Ranges (Confidence Intervals)

Two models for "where is the underlying likely to be" at a given confidence level $c$.

Flat ATM model — symmetric, using a single volatility for both tails:

$$\text{Boundary}_{\text{lower/upper}} = S \cdot \exp\!\left( \mu T \pm \sigma_{\text{ATM}}\sqrt{T} \cdot z_{(1+c)/2} \right)$$

For $68\%$ confidence ($1\sigma$), $z = 1.000$; for $80\%$, $z = 1.28155$.

Skew-adjusted model — asymmetric, repricing each boundary with the actual market implied volatility at that strike rather than one flat number, iterated to a fixed point:

$$\text{Upper}^{(n+1)} = \text{Boundary}\left(S, T, r, q, \sigma(\text{Upper}^{(n)}), +z\right), \qquad \text{Lower}^{(n+1)} = \text{Boundary}\left(S, T, r, q, \sigma(\text{Lower}^{(n)}), -z\right)$$

Because index options carry negative skew (puts trade at higher IV than calls — see Volatility Smile & Skew), this produces a wider downside boundary and a tighter upside boundary than the flat model.

How it's used: this is the market-implied version of "how far could this move," and the skew-adjusted range is the more honest one for index products specifically — it's the direct numerical expression of the market pricing crashes as more likely than equivalent rallies. Traders selling strangles or setting protective-put strikes reference the skew-adjusted range rather than the symmetric one for exactly this reason: a symmetric range built on ATM vol alone understates downside and overstates upside for something like SPX.

Limitations

• POP is risk-neutral, not physical. It reflects the market's pricing, including its volatility risk premium, not a forecast of actual outcomes.
• IV solved from entry premium is frozen at entry. The T+0 curve does not itself model how IV might change as spot moves (no vol skew dynamics), only how the position reprices at the entry-implied vol.
• Every model here assumes European exercise and no early assignment — correct for SPX (cash-settled index options), not for American-style single-name equity options.

Key Takeaways

• The expiration diagram is closed-form and model-free; the T+0 curve requires solving each leg's IV and repricing with Black-Scholes.
• Net Greeks are the real risk dashboard professional desks use — delta for direction, theta for carry, gamma for hedging urgency, vega for vol exposure.
• POP measures win rate, not expected value — a high-POP trade can still have negative expectancy.
• Skew-adjusted ranges are wider on the downside for index options, reflecting real market-priced crash risk.

Related Reading

• Options Viewer Methodology — index of all viewer specs
• Option Greeks Overview
• Volatility Smile & Skew
• Navigating the Options Strategy Taxonomy