> ## Documentation Index
> Fetch the complete documentation index at: https://docs.joyride.exchange/llms.txt
> Use this file to discover all available pages before exploring further.

# Leverage & Lambda

> How Joyride calculates the Leverage and Lambda (λ) columns in the options chain.

The options chain offers two related but different multiples, each as an optional column: **Leverage** and **Lambda (λ)**. Both are derived from the option's **model price** — the theoretical Black-Scholes value Joyride computes from BlockScholes implied volatility — so neither depends on order-book depth.

They answer different questions. Showing both under distinct names follows the convention used in retail warrant markets, where "gearing" and "effective gearing" are displayed side by side.

## Leverage

**Leverage** is how many dollars of notional exposure each dollar of premium controls:

```text theme={null}
leverage = spot_price / model_price
```

A leverage of `50×` means one dollar of premium controls fifty dollars of the underlying. This matches the leverage figure shown on other major options venues.

Leverage is a **cost ratio, not a margin requirement**. Buying an option costs the full premium up front — there is no additional margin posted or borrowed against a long option.

Because the denominator is the option's model price, cheap deep out-of-the-money options show very large leverage. A high number means the option is cheap relative to spot — it does not describe how the option's value will move.

## Lambda

**Lambda** — also called *effective gearing* — is a sensitivity measure: approximately how many percent the option's model value moves for a 1% move in the underlying:

```text theme={null}
lambda = |delta| × spot_price / model_price
```

Lambda is Leverage scaled by the option's delta. Deep in-the-money options (delta near 1) have λ close to their leverage; deep out-of-the-money options (delta near 0) have λ far below their leverage, because most of a small underlying move does not reach the option's value.

Near expiry, λ for far out-of-the-money strikes can grow very large while the probability of any payoff shrinks. A triple-digit λ is a signal that the option is a long shot, not a promise of amplified returns.

## Leverage vs. Lambda

|                   | Leverage                                | Lambda (λ)                       |
| ----------------- | --------------------------------------- | -------------------------------- |
| Formula           | `spot / model_price`                    | `\|delta\| × spot / model_price` |
| Answers           | "How much exposure per premium dollar?" | "How much does my P\&L amplify?" |
| Uses delta        | No                                      | Yes                              |
| Deep OTM behavior | Very large                              | Large, but delta-damped          |
| Display           | Raw                                     | Raw                              |

## Inputs

| Input          | Source                                                                                         |
| -------------- | ---------------------------------------------------------------------------------------------- |
| Spot price     | Oracle index price                                                                             |
| Model price    | Black-Scholes value from BlockScholes implied volatility, computed server-side (`model_price`) |
| Delta (λ only) | Server-side Black-Scholes delta (`greeks.delta`)                                               |

The model price is a display-only theoretical value. It is not an executable quote and is not the mark used for margin or liquidation.

## Missing Values

Both columns render `--` when any input is unavailable or stale — for example while price feeds warm up or if the pricing service is degraded. Joyride never substitutes a placeholder number for a value it cannot compute.

## Interpretation

Neither number is a promise of returns, and neither describes margin. For a **bought** option, loss is limited to the premium paid — nothing else is posted or at risk. **Selling (writing) an option is different**: you post margin, and your loss is not capped at the premium, so neither Leverage nor Lambda reflects a seller's risk. Use Leverage to compare how much exposure your premium buys across strikes, and Lambda to compare how strongly each option's value reacts to the underlying.