# American Option Price using GARCH(1,1) (Monte Carlo)

#### Hob

Hi All,

I am attempting a MC simulation using GARCH(1,1) volatility model. I am not trying to estimate parameters as these have been given (Ritchken & Trevor (1999)). I am just trying to achieve the correct result. I have obtained the expected variance using the information in (Hull, 2005) shown below:

From: Options, Futures and Other Derivatives (Hull, 2005) ( Evolution of sigma(t+1) )
GARCH(1,1) Volatility:

$$\sigma^2_{n} = \gamma V_{L} + \alpha u^{2}_{n-1} +\beta \sigma^{2}_{n-1}$$

where:
($$\gamma = 1 - \alpha - \beta$$)

$$\sigma^2_{n} = (1 - \alpha -\beta)V_{L} + \alpha u^{2}_{n-1} +\beta \sigma^{2}_{n-1}$$

Where the long term variance ($$V_{L}$$):

$$V_{L} = \frac{\omega}{1 - \alpha -\beta}$$

and the Return ($$u_{n-1}$$):

$$u_{n-1} = \frac{S_{n-1} - S_{n-2}}{S_{n-2}}$$

The expected variance ($$E[\sigma^2_{n+1}]$$):

$$E[\sigma^2_{n+1}] = V_{L} + (\alpha + \beta)(\sigma^{2}_{n} - V_{L}$$)
I am then trying to use the equation given by Duan(1995) for the stock price evolution:
From: Duan (1995) ( Evolution of the stock price S(t+1) )

Using the Locally Risk-Neutral Valuation Relationship (LRNVR), the dynamics in the equivalent martingale measure is given by:

$$ln \frac{S_{t+1}}{S_{t}} = r - \frac{1}{2}\sigma_{n} +\widetilde{\varepsilon}_{t}$$

where:

$$\widetilde{\varepsilon}_{t}$$ ~ $$N(0,\sigma_{n})$$
However, this does not produce the correct result. My question is, is the equation above the correct one for the evolution of the stock price under GARCH(1,1) volatility? I cannot seem to find any other equation.

From: Ritchken & Trevor (1999) (The option I am trying to price)

American Put Option Price:

Interest rate (r) is fixed at 10% (annualized using 365 days a year).
Stock price (S) is 100.
Time is (T = 100 days).

ω = 0.06575 (as we are working with returns in percentage terms)
α = 0.04
β = 0.90
γ = 0.00
λ = 0.00

Option Price: 3.143
Many thanks,

Hob

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