Options Lab is a quantitative research tool for learning how option pricing theory turns into a concrete trading decision. An option is a contract whose payoff depends on where the underlying finishes relative to a strike, so its fair value is a probability-weighted average of those payoffs. Knowing that value is useful. It still does not tell you which combination of options best expresses a view on direction, on volatility and on risk.
The core idea is that strategies should be judged at the structure level, not one contract at a time. A long call, a vertical spread and an iron condor can share the same underlying and the same expiry while behaving completely differently. They differ in delta, gamma, theta, vega, probability of profit and maximum loss. Options Lab is built to put those trade-offs side by side, tie each number back to the model input that produced it, and explain the reasoning as it goes.
The lab runs on live Yahoo Finance data or as a manual pricing sheet where spot, rate, dividend yield, implied volatility, drift and expiry are set by hand. Every figure is a model output for research and education. None of it is a trade recommendation.
IResearch Question
The question this project works through is simple to state. How can an options research tool connect pricing theory to a real strategy choice without hiding its assumptions? Most tools show a chain, a payoff diagram or a single probability and stop there. Options Lab instead links the pricing model, the Greek sensitivities, the scenario analysis and the ranking logic into one workflow you can read end to end.
The aim is to test whether keeping the model in the open changes how strategies get compared. Each structure is evaluated across six things: its theoretical value, its local sensitivities, its payoff at expiry, its profit and loss under shocks, its expected value, and its reward relative to risk. Nothing is reduced to a score you cannot take apart.
IIWhat Makes It Different
The point of difference is not that it computes Black-Scholes-Merton. Most tools can do that. The difference is treating pricing as one layer in a stack rather than the answer. A fair value under the risk-neutral measure tells you what the model thinks the option is worth. It does not tell you whether the structure is an efficient way to express your view, given your risk budget and the payoff shape you actually want.
The whole model is on screen
The valuation model, its assumptions, the Greek sensitivities and the ranking criteria are all shown as part of the output. Nothing important is hidden behind a single number.
Q-measure kept apart from P-measure
Risk-neutral pricing and real-world expected return are computed separately and labelled, so a model price is never read as a forecast of where the stock will go.
The structure is the unit of analysis
Each comparison is over a complete multi-leg payoff, with its net Greeks, its defined risk and the capital it ties up, not over one contract in isolation.
Research, not an execution prompt
The output is a model-based comparison of structures for learning and research. It is built to explain the trade-offs, not to push a trade.
IIIResearch Objective And Methodology
The main claim is that strategy comparison is cleaner when no-arbitrage valuation is kept separate from subjective assumptions. The first layer is the pricing layer, where every leg is valued under a risk-neutral probability measure. The second is the decision layer, where your own assumptions about drift, volatility, horizon and loss tolerance do the comparing. Mixing the two is the usual way people end up trusting a forecast they have quietly labelled a price.
Under the risk-neutral measure Q, the underlying is assumed to grow at the risk-free rate, and Black-Scholes-Merton returns the theoretical value of each leg. The Greeks are the partial derivatives of that same value, so each one isolates a different source of risk. They are worth knowing by name, because the rest of the lab is built on them.
- ΔDelta
- First-order exposure to the underlying. How much the position value moves for a small move in spot.
- ΓGamma
- Curvature. How fast delta itself changes as spot moves, so it measures how unstable your hedge is.
- ΘTheta
- Time decay. How much value the position gives up as one day passes, holding everything else fixed.
- νVega
- Volatility exposure. How much value changes for a one-point move in implied volatility.
- ρRho
- Rate sensitivity. How much value changes for a move in the risk-free interest rate.
Under the real-world measure P the question changes. The terminal distribution now carries your assumed drift, and the structure is judged by expected profit and loss, probability of profit and reward against maximum loss. Options Lab keeps the Q and P labels on screen so a model price is never mistaken for a forecast of where the stock will go.
IVCalculation Pipeline
The workbench runs as one continuous pipeline. A view becomes parameters, parameters become prices and Greeks, and those exposures feed the scenario and ranking stages. Each step is shown below with the formula it uses, so the calculation stays inspectable. Read the formulas as definitions, not decoration. Every symbol in them is an input you control.
Write down the research view
Every analysis starts with a written view, not a formula. You state the direction you expect, what you think volatility will do, the horizon you are trading over, and the largest loss you are willing to take. These four choices are the economic objective. Everything downstream is scored against them, so making them explicit is what keeps the comparison honest.
Separate market inputs from assumptions
Spot, the risk-free rate, the dividend yield, implied volatility and the expiry define the pricing problem. The expected drift is deliberately held to one side. Drift is your real-world forecast, and no-arbitrage pricing is not allowed to use it. Time to expiry is converted to a year fraction, and the dividend yield discounts the spot into a forward-style term. Keeping the drift mu apart from the rate r is the line between a price and a forecast.
Price each leg under the Q-measure
Each leg is valued with Black-Scholes-Merton under the risk-neutral measure, where the underlying is assumed to grow at the risk-free rate. The terms d1 and d2 measure how far in-the-money the option is in standard-deviation units, and N() is the standard normal cumulative distribution, the probability mass sitting below that point. The same value function, differentiated once, gives every Greek. A multi-leg position is just the quantity-weighted sum of its legs, so portfolio delta, gamma and vega add up leg by leg.
Map the risk surface
A single fair value tells you almost nothing about risk. The lab moves spot, time and volatility and re-prices the structure at each point. The change in value is read through a second-order Taylor expansion: delta and gamma carry the move in spot, theta carries the passage of time, and vega carries the shift in volatility. This is how you see which Greek is actually driving a given profit or loss, instead of guessing.
Rank against the objective
The last stage switches to the real-world measure P, where your assumed drift is allowed back in. The terminal distribution of the underlying is integrated against each payoff to give an expected value and a probability of profit. Position size is set by dividing the risk budget by the worst-case loss of one lot, rounded down. The final score combines expected profit and loss, probability of profit, maximum loss and how well the structure fits the stated view, so the ranking reflects your assumptions rather than a number you cannot take apart.
VWorked Examples
The same engine draws every payoff. Each curve below is a one-lot structure on a $100 underlying at 25% volatility, plotted at expiry against the spot price. The shapes are the point. They show why the lab compares whole structures instead of isolated contracts, because the risk lives in the geometry of the combined position.
VITheoretical Foundations
Black and Scholes (1973) and Merton (1973) derived the closed-form value of a European option under geometric Brownian motion and no-arbitrage. That work gives the risk-neutral framework and the dividend-adjusted extension used here. Cox, Ross and Rubinstein (1979) then showed the same model as the limit of a discrete binomial tree. The tree is the most intuitive way to see where the risk-neutral probabilities come from.
Later work on the volatility smile, including Derman and Kani (1994), explains why market prices rarely behave as if one flat volatility governs every strike and expiry. For that reason Options Lab treats implied volatility as an input to inspect and stress, not as a constant of nature.
- [1]Black, F. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
- [2]Merton, R.C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
- [3]Cox, J.C., Ross, S.A. & Rubinstein, M. (1979). Option Pricing: A Simplified Approach. Journal of Financial Economics, 7(3), 229-263.
- [4]Derman, E. & Kani, I. (1994). Riding on a Smile. Risk, 7(2), 32-39.
- [5]Hull, J.C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
Educational and research project only. Options Lab is not investment advice, not a financial product, and not a recommendation to buy, sell or trade any instrument. All prices, Greeks and probabilities are theoretical model outputs computed from the inputs provided. They are not tradeable quotes or evidence of expected future performance.