Black and Bachelier Caplet Pricing

Article Quant Q&A · Author: Add

Summary

The document compares an attempted Black 76 caplet valuation with a Bachelier, or normal-model, implementation for an interest rate caplet. The central modeling distinction in the replies is that Black volatility is quoted as a relative, lognormal change, while Bachelier volatility is an absolute change in the rate. Thus, using the same numerical volatility in both models does not generally represent the same market assumption.

A second reply identifies a parenthesization issue in the Black d-plus calculation. The supplied Bachelier code also appears to use a cumulative normal probability where the normal density is needed in the volatility term, so the volatility-units observation alone does not make that implementation correct. The discussion gives no validated normal-model price, volatility conversion, or practical SABR implementation. It is a limited coding discussion, and the example’s inputs should not be treated as a market calibration.

Key ideas

Black 76 treats the forward rate as lognormal and uses relative volatility.
Bachelier models absolute rate changes, so its volatility input has different units and scale.
The Black implementation’s d-plus expression is flagged for a parenthesization correction.
The displayed Bachelier formula should be checked carefully because its volatility term appears to require the normal density.

Tags

Fixed income Options Derivatives pricing

Full text

# Black and Normal Model for Caplet using Python


# Black and Normal Model for Caplet using Python












I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ?

I am valuing caplet that caps interest rate on 10000 loan at 8% p.a. quarterly compounding for three months starting in one year.

The zero curve is flat at 6.9394% p.a. one year volatility is 20% p.a.

Also, If you can suggest best place to see practical SABR model implementation using python that would be great.

Code for Black in Python:

```
from scipy.stats import norm
import math

def black(F_0,y,expiry,vol,rfr,expiry_1,isCall):
    option_value = 0
    if expiry * vol == 0.0:
        if isCall:
            option_value = max(F_0 - y, 0.0)
        else:
            option_value = max(y - F_0 , 0.0)
    else:
        d1 = dPlusBlack(F_0 = F_0 , y = y, expiry = expiry ,vol = vol)
        d2 = dMinusBlack(F_0 = F_0 , y = y, expiry = expiry ,vol = vol)
        if isCall:
             option_value = (math.exp((-rfr)*expiry_1))*(F_0 * norm.cdf(d1) - y *norm.cdf(d2))
        else:
             option_value = (math.exp((-rfr)*expiry_1))*(y * norm.cdf(-d2) - F_0 *norm.cdf(-d1))
    return option_value
def dPlusBlack(F_0 , y, expiry , vol):
    d_plus = ((math.log(F_0 / y) + 0.5 * vol * vol * expiry)/ vol / math.sqrt(expiry))
    return d_plus
def dMinusBlack(F_0 , y, expiry , vol):
    d_minus = (dPlusBlack(F_0 = F_0 , y = y, expiry = expiry ,vol = vol ) - vol * math.sqrt(expiry))
    return d_minus
a = black(0.07,0.08,1,0.20,0.069394,1.25,"isCall")
a = 0.0020646470930435683
```

Code for bachelier

```
from scipy.stats import norm
import math

def bachelier(F_0,y,expiry,vol,rfr,expiry_1,isCall):
    option_value = 0
    if expiry * vol == 0.0:
        if isCall:
            option_value = max(F_0 - y, 0.0)
        else:
            option_value = max(y - F_0 , 0.0)
    else:
        d1 = dPlusBachelier(F_0 = F_0 , y = y, expiry = expiry ,vol = vol)
        if isCall:
             option_value = (math.exp((-rfr)*expiry_1))*((F_0 - y)* norm.cdf(d1) + vol * math.sqrt(expiry) * norm.cdf(d1))
        else:
             option_value = (math.exp((-rfr)*expiry_1))*((y-F_0) * norm.cdf(-d1) + vol * math.sqrt(expiry) * norm.cdf(-d1))
    return option_value

def dPlusBachelier(F_0 , y, expiry , vol):
    d_plus = (F_0 - y)/ (vol * math.sqrt(expiry))
    return d_plus

a = bachelier(0.07,0.08,1,0.20,0.069394,1.25,"isCall")
```

## Answer by Kiann (score 5)

https://quant.stackexchange.com/a/44558

are you using the same volatility 20% for both black76 and Bachelier?

The black76 is a lognormal model, where volatilities are quoted as relative price changes. The bachelier/normal model quotes volatilities as absolute changes.

That might be what you're missing?

Kind regards

## Answer by Varun (score 0)

https://quant.stackexchange.com/a/46603

Sorry I am bit late to the party. Just saw your post while trying to write my own black model. I am going to the mistake is a typo in dplus

```
d_plus = ((math.log(F_0 / y) + 0.5 * vol * vol * expiry)/ vol / math.sqrt(expiry))
```

Should be:

```
d_plus = ((math.log(F_0 / y) + 0.5 * vol * vol * expiry)/( vol * math.sqrt(expiry)))
```

Warm Regards, Varun

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Black and Bachelier Caplet Pricing Use Different Volatility Inputs

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