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Duan GARCH option pricing model

Duan’s Gaussian GARCH pricing model preserves one-step conditional variance while shifting innovations under the pricing measure. Follow the shock transformation and a checked option-price example.

The Duan GARCH option pricing model prices options when asset returns have GARCH conditional volatility, using a locally risk-neutral valuation relationship (LRNVR) to connect the historical return model to a pricing measure. Its practical contribution is a disciplined way to carry volatility clustering into option valuation: changing the expected stock return also changes how shocks enter future variance.
Duan’s 1995 paper establishes the relationship under specified preference and distribution assumptions. Estimating GARCH from historical returns alone does not select a unique pricing kernel. The LRNVR supplies an additional pricing restriction.

Start with the physical return equation

Consider a non-dividend-paying stock in equally spaced discrete periods, with a constant continuously compounded risk-free rate rr per period. Define its decimal log return Rt=log⁡(St/St−1)R_t=\log(S_t/S_{t-1}). A Gaussian GARCH(1,1) specialization is
Rt=r+λht−12ht+htzt,ht+1=ω+αhtzt2+βht,zt∼PN(0,1).\begin{aligned} R_t&=r+\lambda\sqrt{h_t}-\frac12h_t+\sqrt{h_t}z_t,\\ h_{t+1}&=\omega+\alpha h_tz_t^2+\beta h_t, \qquad z_t\overset{\mathbb P}{\sim}\mathcal N(0,1). \end{aligned}
Here hth_t is the conditional variance of RtR_t given information Ft−1\mathcal F_{t-1}, and is known before that return arrives. Under P\mathbb P, the innovations are iid standard normal and independent of past information. Use ω>0\omega>0, α,β≥0\alpha,\beta\ge0 and a positive starting variance. Variance has squared decimal-return units; ω\omega has those same units, and α\alpha, β\beta and λ\lambda are dimensionless.
The premium convention matters: the premium is λht\lambda\sqrt{h_t}, not λht\lambda h_t. Conditional lognormality gives
EP[St/St−1∣Ft−1]=exp⁡(r+λht).\mathbb E^{\mathbb P}[S_t/S_{t-1}\mid\mathcal F_{t-1}] =\exp(r+\lambda\sqrt{h_t}).

Thus λ\lambda is a unit risk-premium parameter for this specification. A Gaussian quasi-likelihood fit that remains statistically useful with nonnormal residuals does not, by itself, justify this Gaussian pricing construction.

Preserve local variance; shift the shock

The LRNVR uses an equivalent pricing measure Q\mathbb Q under which one-period gross returns remain conditionally lognormal, their conditional expectation is ere^r, and their conditional log-return variance is still hth_t. Consequently,
Rt=r−12ht+htzt∗,zt∗∼QN(0,1).R_t=r-\frac12h_t+\sqrt{h_t}z_t^*, \qquad z_t^*\overset{\mathbb Q}{\sim}\mathcal N(0,1).

Matching the two representations of the same realized return gives zt=zt∗−λz_t=z_t^*-\lambda. Substitution into the variance recursion is essential:

ht+1=ω+αht(zt∗−λ)2+βht.\boxed{h_{t+1}=\omega+\alpha h_t(z_t^*-\lambda)^2+\beta h_t.}
The Alfaro–Inzunza model setup displays this transformation. The martingale requirement concerns gross returns: EQ[eRt∣Ft−1]=er\mathbb E^{\mathbb Q}[e^{R_t}\mid\mathcal F_{t-1}]=e^r. Expected log return is r−ht/2r-h_t/2.

“Variance is preserved” means the one-step conditional variance agrees at the same information state. It does not mean that future variance has the same distribution under both measures. For example, direct expectation gives

EQ[ht+1∣Ft−1]=ω+{α(1+λ2)+β}ht.\mathbb E^{\mathbb Q}[h_{t+1}\mid\mathcal F_{t-1}] =\omega+\{\alpha(1+\lambda^2)+\beta\}h_t.

The expected-variance persistence coefficient is therefore α(1+λ2)+β\alpha(1+\lambda^2)+\beta, rather than α+β\alpha+\beta. A finite stationary mean variance under Q\mathbb Q requires the former to be below one in this model; positivity alone does not establish that property.

One negative shock, two different updates

Use an illustrative daily model, not fitted market data:

ht=0.0004,ω=0.00001,α=0.1,β=0.85,λ=0.3.h_t=0.0004,\quad\omega=0.00001,\quad \alpha=0.1,\quad\beta=0.85,\quad\lambda=0.3.

The known daily standard deviation is 2%. For a pricing innovation zt∗=−1z_t^*=-1, the physical-residual coordinate is zt=−1.3z_t=-1.3. The next daily variance is

ht+1=0.00001+0.1(0.0004)(−1.3)2+0.85(0.0004)=0.0004176.h_{t+1}=0.00001+0.1(0.0004)(-1.3)^2+0.85(0.0004) =\boxed{0.0004176}.

Using (−1)2(-1)^2 instead gives 0.00039000.0003900, because it omits the measure-change shift. A positive pricing innovation zt∗=+1z_t^*=+1 gives 0.00036960.0003696. This asymmetry emerges under Q\mathbb Q even though the physical GARCH recursion was symmetric in ztz_t.

Illustrative daily GARCH variance response to pricing shock z*. The correctly shifted curve gives next variance 0.0004176 at minus one and 0.0003696 at plus one. Omitting the shift gives 0.0003900 at either shock. Variance is shown in units of ten to the minus four squared decimal daily returns.
Illustrative daily GARCH variance response to pricing shock z*. The correctly shifted curve gives next variance 0.0004176 at minus one and 0.0003696 at plus one. Omitting the shift gives 0.0003900 at either shock. Variance is shown in units of ten to the minus four squared decimal daily returns.
Original deterministic daily-model comparison, not market observations. The solid curve shifts the pricing shock by 0.3; the dashed comparison omits the shift. Prior variance and GARCH coefficients are identical.
More generally, for a>0a>0, the difference between next variances after pricing shocks −a-a and +a+a is 4αhtaλ4\alpha h_ta\lambda. With positive λ\lambda, negative pricing news generates the larger next variance. This is a model response associated with the leverage effect, not evidence identifying its economic cause.

Price the whole conditional payoff distribution

For a European call expiring after nn periods, risk-neutral valuation gives
Ct=e−rnEQ[(St+n−K)+∣Ft].C_t=e^{-rn}\mathbb E^{\mathbb Q}[(S_{t+n}-K)^+\mid\mathcal F_t].
Initialize the stock and its known next-period variance. For each future period, draw a fresh standard normal pricing innovation, update the stock using the current variance, then update variance using the same innovation with the −λ-\lambda shift. Average terminal call payoffs and discount. Chaudhury and Wei distinguish this path-dependent valuation from Black–Scholes approximations using average forecast variance.

There is a useful one-step check. At the start of one remaining day, set S=K=100S=K=100, known variance h=0.0004h=0.0004 and r=0.0001r=0.0001 per day, meaning 0.01% continuously compounded for that day. The terminal stock is conditionally lognormal, so

C=SΦ(d1)−Ke−rΦ(d2),d1=log⁡(S/K)+r+h/2h,d2=d1−h.C= S\Phi(d_1)-Ke^{-r}\Phi(d_2),\qquad d_1=\frac{\log(S/K)+r+h/2}{\sqrt h},\qquad d_2=d_1-\sqrt h.
Here Φ\Phi is the standard normal CDF. This gives C=0.80284109C=0.80284109 currency units per one-share call, also obtained by directly integrating its Gaussian payoff. The current variance fixes this one-step distribution; the recursion affects later steps. Over multiple steps, future variance responds to intervening returns, so inserting an average historical or forecast volatility into Black–Scholes generally does not reproduce the Duan price.
The model’s Gaussian innovations, premium specification and LRNVR are substantive restrictions. They do not guarantee that a return-only fit matches an observed volatility surface. Nor can the displayed Gaussian shift be carried unchanged into an arbitrary non-Gaussian GARCH model.
For the statistical side, ARCH and GARCH develops the variance recursion and estimation diagnostics. For pricing foundations, Market Model and Risk-Neutral Valuation explains discounted expectations and completeness. Both offer public previews; full lessons require membership.

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