Certified Valuation of Arithmetic Asian Options via Common Gaussian Smoothing

Author: Theodore Ouyang

Computable error bounds for projected Euler, joint weak expansions, and posterior quantile transfer

Keywords: Arithmetic Asian options; Heston model; projected Euler; conditional Gaussian smoothing; validated numerics; Bayesian calibration.

Abstract

This study develops computable bounds for the difference between continuous-model arithmetic Asian prices and prices under a specified projected Euler implementation. A Gaussian factor shared by all fixing prices smooths the payoff conditionally; a weighted second moment bounds the remainder around a geometric reference. Finite transform enclosures, projection corrections, integration tails, and rounding are combined into a deterministic error budget. For a one-year Heston call spread with twelve monthly fixings and step size 1/768, the absolute discretization bias is below 0.011025 price units. Two further results establish a common first-order weak expansion for ten payoffs on a deterministic-variance family and bound every posterior price-quantile displacement under a specified continuous prior. Each conclusion retains its own assumptions and numerical domain.

1. Research question and model

An accurate estimate of a discrete-model price does not by itself establish an accurate continuous-model valuation. Sampling uncertainty and discretization bias are different errors. The research question is whether the bias of the actual projected scheme can be enclosed at a fixed step size, in price units, while accounting for every finite computation used to obtain the bound.

The continuous model is the risk-neutral Heston stochastic-volatility model. With independent Brownian motions W and B, the variance and log-price satisfy

\begin{aligned} dV_t&=\kappa(\bar v-V_t)\,dt+\xi\sqrt{V_t}\,dW_t\\ dZ_t&=(r-V_t/2)\,dt+\sqrt{V_t}\bigl(\rho\,dW_t+\sqrt{1-\rho^2}\,dB_t\bigr)\\ S_t&=S_0e^{Z_t},\qquad Z_0=0,\quad V_0=v_0 \end{aligned}

The discrete implementation projects each variance update onto the nonnegative half-line and uses the current variance in the log-price update. With independent standard normal innovations G and H, it is

\begin{aligned} V_{j+1}^h&=\left[(1-\kappa h)V_j^h+\kappa\bar v h+\xi\sqrt{hV_j^h}\,G_j\right]^+\\ Z_{j+1}^h&=Z_j^h+(r-V_j^h/2)h\\ &\quad+\sqrt{hV_j^h}\bigl(\rho G_j+\sqrt{1-\rho^2}\,H_j\bigr) \end{aligned}

The same innovation G drives both updates. Certification concerns this original transition kernel, including its projection and current-variance convention.

The target contract is a capped arithmetic Asian call spread. For n fixed observation dates, let A and G denote the arithmetic and geometric averages, respectively. The payoff and the sign convention for the pricing error are

\begin{gathered} A=\frac1n\sum_{i=1}^n S_{t_i},\qquad G=\left(\prod_{i=1}^n S_{t_i}\right)^{1/n}\\ \psi(A)=(A-K_1)^+-(A-K_2)^+\\ p_c=e^{-rT}E_P\psi(A),\qquad p_h=e^{-rT}E_{Q_h}\psi(A)\\ e_h=p_h-p_c \end{gathered}

Conditioning on a geometric reference is established in Asian option valuation, including Curran (1994) and Fusai and Kyriakou (2016). The contribution here is the payoff-level weighted remainder argument together with complete, computable error accounting for the specified kernel.

2. Conditional smoothing and the core bound

The structural hypothesis is that every observed price contains a common Gaussian factor. The positive coefficients and conditional standard deviation are measurable with respect to a conditioning sigma-algebra:

S_{t_i}=e^Ua_i,\qquad U\mid\mathcal G\sim N(0,\sigma^2)

For a fixed positive reference multiplier c, assume integrability of A and G and finiteness of the weighted moment

D_\nu(c)=E_\nu\!\left[\frac{(A-cG)^2}{\sigma}\right]

Conditional smoothing lemma. For a positive strike K, the remainder left after expanding the call payoff around cG satisfies

\begin{aligned} R_{K,\nu}=E_\nu\bigl[&(A-K)^+-(cG-K)^+\\ &-(A-cG)\mathbf1_{\{cG>K\}}\bigr]\\ 0\le R_{K,\nu}&\le\frac{D_\nu(c)}{2K\sqrt{2\pi}} \end{aligned}

Proof idea. Convexity gives the lower bound pathwise. Conditional integration over U makes the call a twice-differentiable function of its positive reference value. Taylor's integral remainder supplies a quadratic bound. The exponential factor in that curvature bound also appears in the conditional second moment of A−cG, so taking expectations yields the stated weighted remainder. The full proof is given in Lemma 3.1 of the manuscript.

The spread payoff decomposes into a transform-computable linear part and two nonnegative call remainders. When each model supplies upper bounds for its weighted moment and a signed enclosure for the difference of the linear parts, the main theorem combines those intervals into a signed enclosure of the true Euler-minus-continuous price difference.

For Heston, the shared Gaussian contribution comes from the independent stock Brownian motion over the first observation interval. Its conditional variance is

\sigma^2=(1-\rho^2)I,\qquad I=\int_0^{t_1}V_t\,dt

The key moment is thus weighted by the inverse square root of first-period integrated variance. The projected scheme admits the corresponding construction with its discrete integrated variance.

3. The fixed-step Heston certificate

The worked instance uses twelve monthly observations over one year, initial stock price 100, interest rate 0.01, strikes 95 and 110, and step size 1/768. The model parameters and geometric scaling are

\begin{gathered} (\kappa,\bar v,\xi,\rho,v_0)=(3,\;0.045,\;0.23,\;-0.55,\;0.045)\\ c=\frac{1254433}{1250000} \end{gathered}

The quadratic payoff remainder is bounded using 91 real transform loadings and a nonnegative Laplace integral. The linear term uses thirteen complex loadings. The complete budget includes bounds for the original projection kernel, Laplace quadrature and infinite tails, Fourier frequency truncation and periodization, complex-logarithm branches, and outward rounding.

Quantity	Certified enclosure or bound
Euler-minus-continuous price difference	[−0.011024692273, 0.010642371599]
Continuous-model price	[6.508371733, 6.518868974]
Projected Euler price	[6.507844281, 6.519014106]
Absolute discretization bias	Less than 0.011025 price units

These intervals are deterministic arithmetic enclosures. The source repository records a 256-bit Arb implementation and a full recomputation of both principal modules through a second implementation at 384-bit precision. This overview reports those manuscript results; the proofs, exact endpoints, verification inputs, and reproduction commands are provided in the linked manuscript and repository.

A signed certificate can also transfer an existing discrete-model price enclosure into a continuous-model enclosure:

\begin{gathered} p_h\in[q_L,q_U],\qquad e_h\in[\ell,u]\\ \Longrightarrow\quad p_c\in[q_L-u,\;q_U-\ell] \end{gathered}

This provides a concrete numerical risk budget: sampling error, discretization bias, and other valuation errors can be allocated separately and checked against a tolerance in price units.

4. A common weak expansion for ten payoffs

A second analysis sets volatility of volatility to zero. The variance path becomes deterministic but remains nonconstant when the initial and long-run variances differ. The twelve Gaussian log-increment densities are differentiated with respect to their variances; one density perturbation then gives a common leading term for nine European puts and the Asian spread.

\begin{gathered} p_h(g)-p_c(g)=h\beta_g+R_h(g)\\ \beta_g=e^{-r\tau}E_s\!\left[g(Y)\sum_j d_j\mathcal S_j(Y)\right]\\ |R_h(g)|\le e^{-r\tau}\frac L2\,C_2h^2 \end{gathered}

Here L is the length of the payoff range, the score terms differentiate the Gaussian density, and the constants retain explicit lower bounds on increment variance. The nine puts have strikes 90, 100, and 110 and maturities 1/4, 1/2, and 1. On the specified family with mean-reversion rate 3, long-run variance 0.045, initial variance in [0.03, 0.06], and fixing-aligned step size at most 1/768, the Asian second-order remainder is bounded by 0.000165217. A separate finite-step density comparison bounds its absolute bias by 0.004990924 at step size 1/768.

This expansion applies to the deterministic-variance family. The nine put coefficients are nonzero when the initial and long-run variances differ. The Asian coefficient has an exact score representation and an effective absolute bound; its nonvanishing is not established. A corresponding first-order expansion at positive volatility of volatility requires additional regularity and remainder analysis.

5. Posterior price-quantile transfer

Numerical approximation changes both the calibration likelihood and the target price. A posterior comparison must therefore retain each model's own likelihood, normalizing constant, and target. The study couples the marginal posteriors of initial variance, then extends that coupling to the full parameter posteriors.

Suppose the coupled initial variances differ by at most $\delta_u$ , the continuous reference price has Lipschitz constant $L_0$ , the reference price difference is bounded by $\varepsilon_0$ , and the true targets lie within radii $e_{P,J}$ and $e_{Q,J}$ of their reference targets. Every corresponding left generalized price quantile then satisfies

\begin{gathered} |q_Q(p)-q_P(p)|\le\varepsilon_0+e_{P,J}+e_{Q,J}+L_0\delta_u\\ 0<p<1 \end{gathered}

The numerical examples fix the mean-reversion rate at 3 and the long-run variance at 0.045. They use independent uniform priors for initial variance in [0.03, 0.06], volatility of volatility in [10⁻⁷, 10⁻⁶], and correlation in [−0.8, −0.3]. Calibration uses synthetic put quotes with observation covariance

\Sigma_d=0.75^2\bigl(0.75I_d+0.25\mathbf1\mathbf1^T\bigr)

Here d is one for the single-quote case and nine for the full calibration vector. Likelihoods are enclosed over 4,096 entire initial-variance cells, with normalizing mass retained when constructing the posterior CDFs.

Calibration case	Maximum displacement of any price quantile
One annual at-the-money put quote	0.008018821658 price units
Nine put quotes across three maturities and three strikes	0.012716404217 price units

These are displacement bounds between two posterior target distributions. They do not locate the absolute value of an individual quantile. The prior's small positive volatility-of-volatility range is distinct from the main Heston certificate's value of 0.23.

6. Scope and contribution

The fixed-step certificate applies to the specified contract, parameters, observation dates, and projected kernel. Extending it uniformly across a parameter region requires moment, transform, projection, and tail bounds valid throughout that region. The weak expansion is established at zero volatility of volatility; the posterior certificates concern the stated three-dimensional prior and synthetic quotes.

The work combines established conditioning, affine transforms, Gaussian density expansions, Bayesian approximation, and validated arithmetic into executable numerical-finance certificates. Its specific contribution is to make the mathematical transfer steps effective for the original scheme and retain all constants needed for the reported price-unit conclusions. Application to market calibration additionally requires evidence about quote quality and model fit.

Selected references

Curran, M. (1994). Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price. Management Science, 40(12), 1705–1711. Article.

Fusai, G., and Kyriakou, I. (2016). General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options. Mathematics of Operations Research, 41(2), 531–559. Article.

Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327–343. Article.

Talay, D., and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Analysis and Applications, 8(4), 483–509. Article.

Cotter, S. L., Dashti, M., and Stuart, A. M. (2010). Approximation of Bayesian Inverse Problems for PDEs. SIAM Journal on Numerical Analysis, 48(1), 322–345. Article.

Johansson, F. (2017). Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic. IEEE Transactions on Computers, 66(8), 1281–1292. Article.

Manuscript and implementation

The full research manuscript contains the theorem statements, complete proofs, appendices, numerical budgets, and reproducibility details underlying this overview.

For more information, visit the GitHub repository.