# Monte Carlo American Option Pricing under GARCH(1,1) volatitliy

I am attempting to price a couple of at-the-money American option using the LSM algorithm and GARCH(1,1) volatility. The LSM code I have works correctly for constant volatility, however, when I switch to the GARCH(1,1) model the option price is incorrect.

The option I am attempting to price is the example given in Ritchken and Trevor (1999) and it is also reproduced in (Pricing American options when the underlying asset follows GARCH processes by Lars Stentoft).

The variables given in both papers are:

The interest rate (r) is fixed at 10% (annualized using 365 days a year).

Stock price (S0) = 100

Strike price (K) = 100

Contract Time ($T$) = (2,10,50 and 100 days)

Option can be exercised daily ($dt$) = $\frac{1}{365}$

$\omega$ = 0.06575 (as we are working with returns in percentage terms) Note: $\omega$ = 6.575 x $10^{-6}$ in Ritchken and Trevor.

$\alpha$ = 0.04

$\beta$ = 0.90

c = 0

$\lambda$ = 0.

I am using Monte Carlo simulation to calculate the option price using the following (from Duan 1995):

$\sigma^2_{0} = \frac{\omega}{1 - \alpha - \beta}$

$\sigma_{t}^2 = \omega + \alpha\sigma_{t-1}(\varepsilon_{t-1})^2 + \beta \sigma^2_{t-1}$

Where: $\varepsilon_{t-1}$ ~ $N(0,1)$

and the evolution of the stock price is given by LRNVR condition:

$ln(S_{t+1}) = ln(S_{t}) + [(r - \frac{1}{2}\sigma_{t}^2) dt + (\varepsilon_{t}\sigma_{t}\sqrt{dt})]$

The answer for the LSM algorithm should be: (0.5589,1.1930,2.3984 and 3.1443) for (T = 2,10,50 and 100 days, respectively).

My answers, however, do not match those above. Is there anything incorrect in the stock price or variance evolution?

Many thanks,

Hob

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