# Rho

**BSM-11 · Black–Scholes & Greeks · Core**

**Lesson question:** What does Rho measure, in which units, and what does it not predict?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain Rho in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-03

Rho changes the model's interest-rate input while keeping spot, time, volatility and dividend yield fixed. It does not predict how stock prices will react to a rate announcement.

Under standard European BSM, call Rho is positive and put Rho negative. A real event moves several inputs simultaneously.

## Read the derivative as a sentence

A rate move from 4% to 5% is +0.01 in decimals. Raw Rho is divided by 100 for a one-percentage-point display. This course uses continuously compounded annual rates.

Longer maturity often gives rates more influence, but the actual contract still needs calculation. A compounding or curve mismatch can create different outputs across systems.

## Use it within its scope

Separate this partial derivative from the total market response. Spot weakness or an IV change can dominate Rho. A call falling while rates rise does not contradict positive model Rho; it means the controlled experiment was not the whole event.

## The model boundary

The smooth formulas require positive spot, strike, volatility and remaining time. Zero time and zero volatility still admit explicit value limits, but smooth Greek availability needs separate rules. No invalid or zero input is silently replaced with a default. Continuous dividends, constant parameters and European exercise are modeling assumptions, not claims about every real option.

## One complete numerical experiment

The shared fictional European call uses spot 100, strike 100, 30 calendar days, sigma 25%, continuous rate 4% and yield 1%. Its BSM value is 2.977574155; d1=0.070239317 and d2=-0.001433455. Values are per option unit before contract multiplication.

Raw Rho is 4.091415409729. The displayed convention gives 0.040914154097 option-value change per percentage-point rate change. Isolate rate +1 percentage point while holding every other input fixed.

Starting option value is 2.977574155222. The local estimate after the shock is 3.018488309319; a full-model recomputation gives 3.018658789653. Approximation minus recomputation is -0.000170480334. The difference shows the limitation of a local derivative, not a claim that the market must follow either number.

The graph varies only its labeled horizontal input. It is a series of controlled states, not historical observations or a future trajectory. Read its units before interpreting sign or magnitude.

## Calculation popup: Calculate Rho

```text
Rho_call=K T e^(−rT)N(d2); Rho_put=−K T e^(−rT)N(−d2); displayed=raw/100
```

T is years remaining; calendar time has the opposite derivative sign. N is the standard normal CDF and phi its density.

1. d1=0.070239316937; d2=-0.001433455448.

2. Raw=4.091415409729; display multiplier=0.01.

3. Displayed=0.040914154097.

4. Local affected measure=3.018488309319.

5. Recomputed affected measure=3.018658789653.

**Result:** 0.040914154097 option-value change per percentage-point rate change

## Graph: Rho across controlled states

**Horizontal:** Calendar days remaining  
**Vertical:** option-value change per percentage-point rate change

Original BSM calculation with only the x-axis input varied. All other inputs are stated in the example; the curve is not a prediction.

Complete numerical data are embedded in `scripts/black-scholes-greeks/BSM-11.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is option-value change per percentage-point rate change. Mixing raw decimal derivatives with point shocks or calendar time with remaining time can magnify or reverse the result.

The controlled comparison also omits live spread, surface dynamics, unsupported exercise effects and execution. A local sensitivity is not a market forecast or a guarantee of realized P/L.

## Check your understanding

### What is the displayed Rho in this example?

**Answer:** 0.040914154097 option-value change per percentage-point rate change.

**Why:** Always state the unit.

### Does model agreement prove the same live-market change?

**Answer:** No.

**Why:** Actual inputs, instrument features and executable quotes can differ. **Critical risk check.**

## Takeaway

Translate Rho into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks)

- [OIC Vega](https://www.optionseducation.org/advancedconcepts/vega)


## Plain-English quick read

### Plain English

Rho is rate exposure embedded in option value.

### How to read 0.05

A one-percentage-point rate rise initially adds about $0.05.

### Typical sign

Calls positive, puts negative in the standard model.

### Where it matters

Rho generally grows with maturity.


## Try it in the working calculator

1. Set every scenario shock to zero except rates +1 percentage point.
2. Compare call versus put.
3. Explain why the displayed factor is /100, not /10,000.

The working lab uses fictional shared inputs; the printed worked example above stays fixed. Open it from the lesson's same-page Live Greeks Lab control. The engine prices European calls and puts with continuous yield and calendar days/365.