Certified Valuation of Arithmetic Asian Options

I. Introduction

Computing a model price is one question, and proving how closely the discrete model behind the computation approximates the continuous model is another; numerical valuation of derivatives involves both. With arithmetic Asian options on fixed observation dates, the positive-part payoff, stochastic variance and the projection of the discrete variance all enter together. To be of practical use, a conclusion should state an error in price units at the specified step size, with payoff conversion, transform evaluation, infinite tails and rounding all accounted for in a single error budget.

We organize the theory around that requirement, studying the risk-neutral Heston model together with a precisely specified scheme, which applies a positive-part projection to the variance Euler update and uses the current variance in the log-price update. We want to control the difference between expectations of the actual arithmetic-average payoff. So we begin at the level of the payoff: we identify a Gaussian random factor shared by all observed prices, integrate it exactly, and bound the error using the structure of the remaining variables. The quantity that then governs the argument is not a negative moment of the variance at an individual time, but

where $A$ is the arithmetic average, $G$ is the geometric average, and $I$ is the integrated variance over the first observation interval.

The main result of this paper puts a weighted payoff remainder together with complete transform error control for the original projected Euler kernel. Our contributions are the following.

1. A weighted payoff remainder. Lemma 3.1 gives an exact linear-reference decomposition of the arithmetic payoff whose nonnegative remainder is controlled by a weighted second moment. Section 5 verifies the common Gaussian structure for a deterministic-variance class and for the specified Heston model and projected kernel. That pair has stochastic variance and a boundary atom induced by projection, so the theory strictly extends the deterministic-variance class.

2. A complete certificate for the original kernel. Theorem 4.1 gives a signed enclosure of the difference between the projected Euler and continuous prices. Ninety-one real loadings control the nonlinear term and thirteen complex loadings determine the linear term. At the parameters (4.3) and $h=1/768$, Corollary 4.2 gives the complete monetary bound .011025$.011025$ and encloses the continuous-model price in an interval of width below .010498$.010498$. Section 8 adds certificates at two further step sizes and at a second initial variance.

3. An analysis of the certificate width. Section 4.1 derives the optimal common geometric scale for the exact nonlinear contribution and attributes about 94.5 percent of the interval width at the main point to payoff conversion. The step-grid study in Section 8 identifies a remainder-dominated width plateau. Proposition I.1 gives a direct coupled-remainder inequality, which brings the nonlinear width in the deterministic example of Appendix I down to about 0.573 percent of the separate-law width.

4. Two further results. For $\xi=0$, Theorem 6.1 gives a common first-order expansion with explicit remainder bounds for nine puts and the Asian spread. Theorem 7.1 transfers price differences to posterior quantiles at every level, and Section 7 uses it to bound the displacement of posterior target quantiles on the restricted prior (7.5) with synthetic quotes.

Conditioning and geometric reference variables are well established in Asian option pricing [1, 2, 3, 4], and the framework of Fusai and Kyriakou in particular covers the Heston model and includes error upper bounds based on conditional variances [4]. The transform machinery comes from the affine structure of Heston's model [5], and discrete geometric Asian options admit recursive transform methods [6] and publicly available implementations [7]. The next subsection compares our results with those of Fusai and Kyriakou directly.

The logical structure of the paper is

Here $\mathcal S$ denotes conditional Gaussian smoothing and rigorous transform error accounting; $\mathsf M$ consists of the payoff decomposition with explicit constants and transform error bounds; and $\mathsf B$ is a rigorous enclosure of the actual price difference. Section 2 makes these objects precise and Sections 3–5 prove the main chain of implications. The weak expansion follows in Section 6 and the posterior transfer in Section 7. Section 8 presents the numerical certificates and their financial use, and Section 9 sets out the domain of validity.

A. Comparison with conditional Asian bounds

Fusai and Kyriakou [4] obtain optimized conditioning bounds for Heston Asian options (Sections 2.2–3), conditional-variance error bounds (Theorem 4, equations (51)–(54)), and experiments at several monitoring frequencies (Section 5.1, Table 5). The reviewed source reports no certificate for kernel (2.2), no outward-rounded error budget and no executable signed Euler-minus-continuous enclosure. We provide all three at the specified parameters and grid. Our principal result is the finite error enclosure for the original kernel (2.2), which we build from a payoff remainder weighted by first-period integrated variance and a validated computation of every stated error contribution. Table 1 sets out the mathematical and computational objects on each side.

Page references in Table 1 use the internal numbering of the accepted manuscript linked in [4], after its repository cover, rather than the journal pagination.

Table 1. Comparison with the conditional Asian bounds of Fusai and Kyriakou.

Relative to the reviewed source, this paper adds the weighted remainder and its combination with error control for the original projected kernel. Section 4 analyzes the width of the certificate that results.

B. Three distinct domains of validity

Every monetary Asian conclusion concerns the same contract, the arithmetic call spread in (2.3), whose payoff is bounded by 15. The three results hold on different domains, which Table 2 keeps apart. In particular, the posterior example concerns a near-deterministic perturbation regime, not calibration stability at the main Heston point.

Table 2. Domains of the three certified results.

II. Models and structural conditions

A. The continuous model, discrete kernel, and payoff

Let $W,B$ be independent standard Brownian motions, and let $\theta=(\kappa,\bar v,\xi,\rho,v_0)$. Assume $S_0,\kappa,\bar v,v_0\gt 0$, $\xi\ge0$, $|\rho|\lt 1$, and $r\in\mathbb R$. The discrete model uses $h\gt 0$, with each $t_i/h$ a positive integer. The continuous model is

On the aligned grid with step size $h$, define the original discrete kernel $Q_h$ by

where $G_j,H_j$ are independent standard normal variables, the two updates at the same step share $G_j$, and the initial conditions agree. In particular, the price update uses $V_j^h$.

Fix $0\lt t_1\lt \cdots\lt t_n=T$, and set

where $0\lt K_1\lt K_2$. Write

We use this sign convention for the error throughout, and when one step size is understood we write $Q=Q_h$. The numerical example uses $n=12,t_i=i/12,T=1$, and

The calibration payoffs are the nine European puts with maturities $1/4,1/2,1$ and strikes 90,100,110$90,100,110$.

B. The common Gaussian structure $\mathsf H_s$

A pricing law $\nu$ is said to satisfy $\mathsf H_s(c)$ if there exist a sub-$\sigma$-algebra $\mathcal G$, positive $\mathcal G$-measurable random variables $a_1,\ldots,a_n,\sigma$, and a random variable $U$ such that

Set $a=n^{-1}\sum_i a_i$ and $g=(\prod_i a_i)^{1/n}$. We require

Here $c\gt 0$ is a constant chosen in advance. Condition (2.6) describes the stochastic structure of the model, and (2.7) is an integrability condition that one can check through moments or Laplace transforms, so neither condition already contains the price-difference conclusion we want to prove.

C. The finite verification condition $\mathsf H_v$

The structural condition still has to be connected to finite computation, and that is the job of the condition $\mathsf H_v$, which consists of the following inputs:

1. A computable upper bound for $D_\nu(c)$, obtained from enclosures at finitely many Laplace nodes, provable quadrature enclosures, and a bound on the infinite tail.

2. Outward enclosures for the finite set of transform loadings required in Section 3. For the discrete kernel, each loading also has a computable bound on the difference between the true kernel and its algebraic recursion.

3. Complete bounds on the infinite-frequency tail and periodization error in Fourier inversion, together with branch control for the continuous complex transform.

We define these inputs through specific transforms or moments, and it is the theorems below that connect them to the target price. We write $\mathsf H=\mathsf H_s+\mathsf H_v$. Appendices A–C specify the finite catalogs, precision, tail-bound constants and kernel error formulas.

D. The intermediate property and target conclusion

The property $\mathsf M$ has two components: an exact decomposition of the actual payoff into a linear term and a remainder, and signed enclosures for each remainder and each linear-transform error. The target conclusion $\mathsf B$ is a definite finite interval $[\underline e,\overline e]$ such that

Each endpoint is obtained by finitely many rational operations and validated enclosures of transcendental functions.

III. The core argument: from a common Gaussian factor to a weighted remainder

A. A conditional smoothing lemma

Lemma 3.1. Suppose that $\mathsf H_s(c)$ holds. For $K\gt 0$, define

Then

Proof. Convexity of the positive-part function gives pathwise nonnegativity. Conditional on $\mathcal G$, let

Since $\sigma\gt 0$, this function is twice differentiable, with

where $\varphi$ is the standard normal density. The final bound follows by maximizing at $\log(x/K)=-2\sigma^2$.

Applying Taylor's formula with integral remainder between the positive arguments $a,cg$ gives

The conditional derivative satisfies

Since

taking expectations proves (3.2). The call and linear terms are integrable because $E(A+G)\lt \infty$. The conditional second-moment identity is used in the sense of nonnegative conditional expectations, and the weighted remainder is controlled by $D_\nu(c)\lt \infty$. Q.E.D.

The proof works because the two factors $e^{2\sigma^2}$ match exactly, which lets us write the curvature contribution from conditional smoothing directly as a weighted second moment under the true model. The residual $A-cG$ may have either sign.

B. The payoff decomposition

Define

The conditional normal variance is strictly positive, so $G$ has no atom at any positive threshold. By (3.1),

The identity keeps the original arithmetic payoff as the object of interest and splits off a linear part that a small number of transforms can compute.

Let $Y=\log(G/S_0)$ and $y_K=\log(K/(cS_0))$, and define the distribution functions of two positive measures by

Expanding (3.4) on each of its intervals gives

The corresponding transforms are

Thus the linear part requires thirteen loadings when $n=12$.

IV. The main theorem and the complete price certificate

Theorem 4.1 (A signed enclosure of the price difference between two models). Suppose that $P,Q_h$ each satisfy $\mathsf H_s(c)$, and that computable constants satisfy $D_P(c)\le d_P,D_Q(c)\le d_Q$. If $\mathsf H_v$ provides

then, with $\gamma=e^{-rT}/(2\sqrt{2\pi})$,

Proof. Apply (3.5) to each law and subtract. The lower-end contributions are $-R_{K_2,Q}-R_{K_1,P}$, and the upper-end contributions are $R_{K_1,Q}+R_{K_2,P}$. Lemma 3.1 and (4.1) give the result. Q.E.D.

The last step of the main theorem only combines intervals, so the mathematical substance lies elsewhere, in Lemma 3.1 and in the rigorous verification of $\mathsf H_v$. The interval that comes out is generally not symmetric about zero, and the computation retains its center.

Corollary 4.2 (The specified Heston parameter point). For (2.1)–(2.5), take

Then

Furthermore,

Proof. Section 5 establishes the common Gaussian structure. Appendices A–C specify all requirements for the validated arithmetic computation, which yields

Here $d_\nu=W_\nu/\sqrt{1-\rho^2}$. The center of the finite linear difference is approximately −.000202992309237006$-.000202992309237006$, and its complete error radius is at most .000594238060$.000594238060$. Combining the exact rational endpoints outward with (4.2) gives (4.4). Applying the corresponding error contributions to (3.5) separately for each law gives (4.5). The source code, exact endpoints, and the recomputation of the entire catalog by a second implementation together constitute the executable evidence specified in the appendices. Q.E.D.

A. Optimizing the weighted remainder and its width limitation

Since the nonlinear conversion bound depends on the common scale $c$, we ask which scale makes it smallest. Assume, in addition to the structural condition, that

are finite for $\nu=P,Q$. Put

The nonlinear contribution to the width in (4.2), using exact moment inputs, is

Consequently its unique minimizer on $c\gt 0$ is