CIR Zero-Coupon Bond Pricing and Parameter Calibration

# How do I pricing a ZCB using CIR (Cox-Ingersoll-Ross) model


# How do I pricing a ZCB using CIR (Cox-Ingersoll-Ross) model












Please see the codes below

My question is about input parameters (a, b and sigma)and their calculation.

For the long term mean "b", do we use effective Fed Fund rates? or 3m T-bills?

Also, how do I calculate the mean reversion speed 'a'?

```

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import math

import timeit

start = timeit.default_timer()

def inst_to_ann(r):
    return np.expm1(r)

def ann_to_inst(r):
    return np.log1p(r)

def cir(n_years = 1, n_scenarios=1, a=0.05, b=0.03, sigma=0.05, steps_per_year=52, r_0=None):

    if r_0 is None: r_0 = b
    r_0 = ann_to_inst(r_0)
    dt = 1/ steps_per_year
    num_steps = int(n_years * steps_per_year) + 1  # because n_years might be a float

    I = np.random.normal(0, scale=np.sqrt(dt), size=(num_steps, n_scenarios))
    rates = np.empty_like(I)
    rates[0] = r_0

    ## For Price Generation
    h = math.sqrt(a ** 2 + 2 * sigma ** 2)
    prices = np.empty_like(I)

 

    def price(ttm, r):
        _A = ((2 * h * math.exp((h + a) * ttm / 2)) / (2 * h + (h + a) * (math.exp(h * ttm) - 1))) ** (
                    2 * a * b / sigma ** 2)
        _B = (2 * (math.exp(h * ttm) - 1)) / (2 * h + (h + a) * (math.exp(h * ttm) - 1))
        _P = _A * np.exp(-_B * r)
        return _P

    prices[0] = price(n_years, r_0)
    

    for step in range(1, num_steps):
        r_t = rates[step - 1]
        d_r_t = a * (b - r_t) * dt + sigma * np.sqrt(r_t) * I[step]
        rates[step] = abs(r_t + d_r_t)
        # generate prices at time t as well ...
        prices[step] = price(n_years - step * dt, rates[step])

    rates = pd.DataFrame(data=inst_to_ann(rates), index=range(num_steps))
    ### for prices
    prices = pd.DataFrame(data=prices, index=range(num_steps))
  
    return rates, prices

dfrates,dfprices = cir(n_scenarios=1000)
dfprices.plot()
plt.show()

print(dfprices)

stop = timeit.default_timer()
print('Time: ', stop - start)
```

## Answer by oliversm (score 1)

https://quant.stackexchange.com/a/57873

## Monte Carlo simulation for pricing a zero coupon bond

To assess the value $V_t$ of a zero coupon bond $B_{T, t}$ issued today at time $t$ with expiry at time $T$, you can do so using the formula $$ V_t = \mathbb{E}\left(\exp\left(-\int_t^T r_s \mathrm{d}s\right)\underbrace{B_{T, T}}_{=1}\right). $$ Now if you're using the CIR model the nice thing is that the distribution of $r_T$ given $r_t$ is known (it's a non-central $\chi^2$), and hence you can use the relation $$ r_T = r_s + \int_t^T r_s \mathrm{d}s, $$ and re-arrange this to know the distribution of $\int_t^T r_s \mathrm{d}s$.

#### Don't use the Euler-Maruyama scheme if you know the exact distribution

As we know the exact distribution of $\int_t^T r_s \mathrm{d}s$, there is no-need to use the Euler-Maruyama scheme to approximate a sample from this, as an exact sample can be drawn. This then makes using Monte Carlo to compute the expectation much easier (albeit the non-central $\chi^2$ is not cheap!), and thus pricing a zero coupon bond easy.

## Calibrating the model

You ask how to find the values for $a$, and what to use for $b$. The best way to do this is to calibrate your model, which is a trickier question, which I will let other answers address.

## Answer by rvignolo (score 0)

https://quant.stackexchange.com/a/57872

> For the long term mean "b", do we use effective Fed Fund rates? or 3m T-bills?

I don't think so. For this particular short rate model, you have to provide 3 input parameters, namely: $a$, $b$ and $\sigma$ (using your notation).

I believe that $a$ and $b$ are obtained by fitting the current term structure of zero coupon bonds $P^M(0,T)$ present in the market and $\sigma$ comes from swaption prices.

For example, the Gaussian Short Rate model (GSR) has an analytical expression for $b(t)$, that depends on derivatives of $P^M(0,T)$, by means of the forward instantaneous rate $f(0, T)$ and its derivative. On the other hand, the CIR model does not have such analytical result.

Constructing $P(0, T)$ and $f(0, T)$ is a whole subject itself (commonly referred as bootstrapping the yield curve). I can elaborate on this if it is desired.

Hope it helps! Thanks!