Estimating GARCH Under the Physical Measure for Option Pricing

# Which measure should I use in the estimation of GARCH model for option pricing, the phsical one or the neutral one?


# Which measure should I use in the estimation of GARCH model for option pricing, the phsical one or the neutral one?












In the paper "The GARCH OPTION PRICING MODEL", Duan(1995) developed a pricing model for options on an asset whose returns follow GARCH process.

Let $X_t$ be the asset price at time t, its process is modeled under physical measure P as GARCH(1,1)-M: $$ ln(\frac{X_t}{X_{t-1}})=r+\lambda \sqrt{h_t} -0.5h_t+\epsilon_t \quad (2.1)\\ \epsilon_t|\phi_{t-1} \sim N(0,h_t) \\ h_t = \alpha_0 +\alpha_1 \epsilon_{t-1}^2+ \beta_1 h_{t-1} \quad (2.2) $$

For option pricing, Duan then introduce the concept of locally risk-neutral valuation relationship. Under the neutral measure Q, the process changes into $$ ln(\frac{X_t}{X_{t-1}})=r -0.5h_t+\xi_t \quad (2.3)\\ \xi_t|\phi_{t-1} \sim N(0,h_t) \\ h_t = \alpha_0 +\alpha_1 (\xi_{t-1}-\lambda \sqrt{h_{t-1}})^2+ \beta_1 h_{t-1} $$

The following is my question:

- To price an option, first of all, is to estimate the parameters in GARCH(1,1) using the data of underlying asset. Which measure should I use?

At the beginning, I thought to estimate the neutral one. Just as a precausion, I have checked many codes in Github to veryfy this idea.

However, in Duan's paper, the original words (in page 20) are "The GARCH-M model specified in (2.1) and (2.2) with p=1 and q=1 is fitted to the S&P 100 daily index series from January 2,1986 to December 15,1989". The estimation seems to hold under the physical measure! Is this true?

- I'm wonderring maybe the correct procedure is the following (1) Estimate the Garch model under the physical measure to get the parameters, the history data is occured in physical world after all. (2) Substitute the values of parameters into the model (2.3), based on which to make simulations (3) calculate the terminal value of options from simulations and then use the risk-free interest rate to discount

Please help me out. I would really appreciate it if you could provide a detailed explanation.

## Answer by QuantCalc.net (score 0)

https://quant.stackexchange.com/a/85321

The standard approach is to estimate the GARCH model under the physical measure $\mathbb P$ and price options under the risk-neutral measure $\mathbb Q$.

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- GARCH model under the physical measure $\mathbb P$

Returns are modeled as

$r_{t+1} = \mu + \sqrt{h_{t+1}}\, z^{\mathbb P}_{t+1}, \qquad z^{\mathbb P}_{t+1} \sim N(0,1),$

with conditional variance dynamics

$h_{t+1} = \omega + \alpha \varepsilon_t^2 + \beta h_t, \qquad \varepsilon_t = \sqrt{h_t}\, z^{\mathbb P}_t .$

The parameters $(\omega,\alpha,\beta)$ are estimated by maximum likelihood using historical returns, since observed data are generated under the physical measure P.

⸻

- Option pricing requires the risk-neutral measure $\mathbb Q$

Option prices are expectations under $\mathbb Q:$

$\mathbb E_t^{\mathbb Q} \!\left[ e^{-r(T-t)} \,\text{Payoff}(S_T) \right].$

Under $\mathbb Q, $discounted asset prices must be martingales.

⸻

- Risk-neutral dynamics (Duan, 1995)

Under the risk-neutral measure, returns satisfy $r_{t+1} = r_f - \tfrac12 h_{t+1} + \sqrt{h_{t+1}}\, z_{t+1}^{\mathbb Q}.$

$z^{\mathbb Q}_{t+1} \sim N(0,1).$

The conditional variance follows the same GARCH recursion: $h_{t+1}= \omega + \alpha \varepsilon_t^2 + \beta h_t .$

⸻

- Link between $\mathbb P $and $\mathbb Q$ when using NGARCH

The two measures are connected via a market price of volatility risk $\lambda:$

$z_{t+1}^{\mathbb Q} = z_{t+1}^{\mathbb P} + \lambda \sqrt{h_{t+1}},$

This transformation preserves the volatility dynamics while adjusting the return distribution to ensure no-arbitrage pricing. If GARCH is used, $\lambda$ term can be ignored ($\lambda=0$).