• C++ Programming for Financial Engineering
    Highly recommended by thousands of MFE students. Covers essential C++ topics with applications to financial engineering. Learn more Join!
    Python for Finance with Intro to Data Science
    Gain practical understanding of Python to read, understand, and write professional Python code for your first day on the job. Learn more Join!
    An Intuition-Based Options Primer for FE
    Ideal for entry level positions interviews and graduate studies, specializing in options trading arbitrage and options valuation models. Learn more Join!

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

Hob

Joined
10/14/13
Messages
14
Points
13
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
 
Back
Top