Certified Valuation of Arithmetic Asian Options

I. Introduction

Most options written on oil, metals, freight and corporate currency flows pay on an average. A refiner buys crude every day of the month and an airline burns fuel every day of the quarter, so their physical contracts are priced on monthly averages and their hedges follow. The NYMEX average price option on crude oil settles on the mean of the daily futures settlements in a calendar month [10]. The London Metal Exchange lists options on the monthly average of its official settlement prices, the price at which physical metal changes hands [25]. Mexico's treasury insures its oil revenue with puts on the average price over a year [35]; the program for 2009 covered 330 million barrels at 70 dollars and paid out 5.085 billion dollars [4].

The payoff of such a contract depends on the arithmetic average $A$ of the fixings, whose law has no closed form in any standard model. The geometric average $G$ does, and practice rests on that fact. Kemna and Vorst use the geometric option as a control variate [21]. Curran, and Rogers and Shi, condition on $G$ and obtain a lower bound that is accurate to several digits in Black's model [12, 31]. Fusai and Kyriakou carry the same bound to stochastic volatility and jumps by transform methods [16]. Under stochastic volatility the working alternative is simulation on a time grid, which adds a discretization bias to the sampling error.

Two numbers in this workflow usually come without a proof. One is the distance between the conditioning lower bound and the price. The other is the distance between the price under the discretized dynamics and the price under the model that was calibrated. The classical estimate of the first is of first order in the conditional standard deviation of $A$ given $G$, which makes it loose exactly where the lower bound is good. The second is known to be of first order in the step for smooth payoffs [28], with an unknown constant. A valuation is certified when both distances are replaced by inequalities whose every term has been computed with a rigorous error. This paper does so, and it starts from a description of the first distance that makes its size predictable.

A. The idea

Suppose the holder of an arithmetic call must decide exercise by looking at a proxy $B$ instead of $A$: she receives $A-K$ when $B\gt K$ and nothing otherwise. She can only lose by this, and she loses on two events. If the proxy is below the strike while $A$ is above, she forgoes $A-K$. If the proxy is above while $A$ is below, she pays $K-A$. In both cases the strike lies between the proxy and the truth, so the loss is at most $|A-B|$ and it occurs with a probability of order $|A-B|$ times the density of the proxy at the strike. The cost is of second order in the basis $A-B$. Figure 1 shows the two events for the contract of Section 5.

The second-order statement becomes an inequality when the density at the strike can be bounded, and here a structural fact helps. The return realized before the first fixing moves every fixing by the same factor. If part of that return is Gaussian given everything else, with volatility $\sigma$, then conditionally the option is a Black–Scholes call on that factor, and the density at the strike is at most $1/(\sigma\sqrt{2\pi})$ in log-strike. With $B=cG$ the basis depends only on the shape of the path, since $A/G$ is unchanged when all fixings are scaled together. The cost of the kink is therefore controlled by a ratio: the dispersion of the fixings about their own average, divided by the level risk that precedes them.

B. Results

Section 3 proves the general statements. Proposition 3.1 identifies the cost of the kink with the Bregman divergence generated by the call payoff and shows that its integral across strikes is exact: with weights $dK$ it equals $E(A-B)^2/2$, and with the weights $dK/K^2$ that replicate a log contract it equals $E\,d(A/B)$, where $d(x)=x-1-\log x$. Theorem 3.3 adds the common factor. For every strike,

and the same bound holds with $E[(A-B)^2/\sigma]/(2K\sqrt{2\pi})$ on the right. Each bound is a total mass from Proposition 3.1 multiplied by the largest value a Gaussian density of width $\sigma$ can take. A matching lower bound shows that (1.1) is attained for at-the-money options whose dispersion is small relative to $\sigma$. For $B=cG$ the best constant is the mean of $A/G$ weighted by $1/\sigma$, and the bound at that scale is $K/\sqrt{2\pi}$ times $E[1/\sigma]$ times the mean log deviation of $A/G$ (Proposition 3.4). Because the Curran, Rogers–Shi and optimized Fusai–Kyriakou lower bounds dominate the proxy value, (1.1) also bounds their errors.

Section 4 evaluates the bound where contracts trade. In Black's model with volatility $v$, a lead time $t_1$ before the first fixing and a window $\tau$, the bound per unit of strike is

to leading order under continuous monitoring, with an explicit correction for $n$ fixings (Proposition 4.1). For a calendar-month crude oil option one to twelve months ahead, ball arithmetic gives price intervals 0.0023 to 0.0081 cents per barrel wide, and simulation puts the true cost at 65 to 94 percent of the bound. The classical first-order bound for the same contracts is 2.6 cents. Under Heston dynamics with a volatility of volatility of 0.7 the cost is about 0.005 to 0.008 cents and the bound 0.006 to 0.013. For an annual-average put on 330 million barrels the bound is 0.91 cents per barrel in Black's model, three million dollars on a premium of 569 million.

Section 5 turns the bound into a certificate for a contract on which it is not small: a 95–110 call spread on twelve monthly fixings under Heston with $S_0=100$. The proxy value is computed by validated transform inversion and the weighted second moment by validated quadrature, with projection, tails, periodization and rounding inside the budget. The continuous price lies in

the price under absorbed Euler with step $h=1/768$ lies in $[6.507844281,\;6.519014106]$, and their difference in $[-0.011024692273,\;0.010642371599]$.

Section 6 explains the Euler error that these intervals enclose. One Euler step of the stochastic differential equation acts on exponential-affine functions as one explicit Euler step of the Riccati equation (Proposition 6.1). The finite sums already enclosed by the certificate then display a first-order bias in the proxy value: −0.000406686$-0.000406686$, −0.000202992$-0.000202992$ and −0.000101409$-0.000101409$ at 384, 768 and 1536 steps, with leading coefficient −0.156$-0.156$. Simulation puts the change in the cost of the kink between the two laws below 10⁻⁶$10^{-6}$. The Euler price at 768 steps is therefore about 0.0002 below the continuous price, well inside the proved bound of 0.011025. On the deterministic-variance face the leading error is a vega multiplied by the quadrature error of the Euler variance (Theorem 6.2), and a coupling of the two laws narrows the allowance for the cost of the kink by a factor of 175.

Section 7 carries the fixed-parameter bounds through a calibration and bounds the displacement of every posterior quantile of the Asian price. Section 8 describes two uses: an acceptance test for pricing engines and a numerical component of a valuation reserve.

C. Related work

Geometric conditioning starts with Curran [12] and Rogers and Shi [31]. Rogers and Shi bound the error of their lower bound by half the expected conditional standard deviation of the average, an estimate of first order. Thompson [33], Nielsen and Sandmann [29] and Lord [26] sharpen it in the lognormal model; in Thompson's example the Rogers–Shi upper bound is 9.039 against a lower bound of 8.8275 and his own upper bound of 8.8333. Fusai and Kyriakou [16] optimize the exercise threshold, treat stochastic volatility and jump models through the joint transform of the price and its geometric average, and state a conditional-variance error bound of the Rogers–Shi type (their Theorem 4), which they evaluate numerically. Moment matching [34, 24] and the Laplace transform of Geman and Yor [17] are the other classical routes in the lognormal model.

Relative to this literature, Theorem 3.3 is an a priori error bound of second order that holds beyond the lognormal model and needs only moments of the fixings weighted by $1/\sigma$. It covers every lower bound of the conditioning family at once (Corollary 3.5), and the cubic law of Section 4 gives its size before any computation.

The decomposition of a convex payoff into calls is due to Breeden and Litzenberger [6] and Carr and Madan [9], and the weights $dK/K^2$ are those of the variance swap [13]. In forecast evaluation the same pathwise quantity is the elementary score of Ehm, Gneiting, Jordan and Krüger [14], who read it as the regret of an investor acting on a point forecast and prove the mixture representation of Bregman scores; Proposition 3.1 is that representation with exercise in the role of investment.

Heston's affine structure supplies the continuous transforms [19], and recursive transforms for discrete geometric averages are available [22, 30]. The scheme studied here is the absorption fix in the classification of Lord, Koekkoek and van Dijk [27]; Broadie and Kaya [8] and Andersen [1] give exact and moment-matched alternatives. Mickel and Neuenkirch prove weak order one for the absorbed scheme and smooth payoffs when the Feller index exceeds one [28]; Talay and Tubaro [32] and Bally and Talay [2] are the classical references for weak expansions. Error control for transform inversion follows Lee [23], and all enclosures use the ball arithmetic of Arb [20].

II. Average-price contracts and their two averages

A. The contract

Fix dates $0\lt t_1\lt \cdots\lt t_n=T$ and let $S_{t_1},\ldots,S_{t_n}$ be the fixings of a positive price. Put

An average price call pays $(A-K)^+$ at $T$ and a put pays $(K-A)^+$. We call $t_1$ the lead time and $\tau=t_n-t_1$ the window. For the exchange-traded contracts of the introduction the window is a calendar month of daily fixings and the lead time is whatever remains until that month begins. For Mexico's hedge the window is a year.

Two elementary reductions are used throughout. Put–call parity holds for the average, $(A-K)^+-(K-A)^+=A-K$, so a statement about calls is a statement about puts. And a fixing that has been observed is a constant: if $m$ fixings with mean $\bar a$ are known, then

where $A'$ is the mean of the remaining fixings. An option inside its window is an option on the fixings still to come with a shifted strike, or a forward if $K'\le0$. The lead time is therefore always the time to the next unknown fixing.

B. The gap between the averages is dispersion

Write $z_i=\log S_{t_i}$, $\bar z$ for their mean, $u_i=z_i-\bar z$, and

Then $G=e^{\bar z}$ and $X=n^{-1}\sum_ie^{u_i}$. Since the $u_i$ sum to zero and $e^u-1-u$ lies between $u^2e^{-|u|}/2$ and $u^2e^{|u|}/2$,

The arithmetic average is the geometric average marked up by half the dispersion of the log fixings about their own mean. The markup does not change when every fixing is multiplied by the same number: $X$ is a function of the shape of the path only.

In financial terms the position long $A$ and short $G$ is a claim on the realized variance of the fixing series, with no exposure to the level at first order. In Black's model with volatility $v$ and continuous monitoring over a window $\tau$, $s^2$ has mean $v^2\tau/6$ and variance $v^4\tau^2/45$ (Section 4), so $X$ is about $1+v^2\tau/12$ and its randomness is of order $v^2\tau$ times 0.07$0.07$. Under stochastic volatility the dispersion inherits the volatility risk, and with a negative correlation between price and variance it is larger on paths that end low. That asymmetry is visible in the numerical profile of Section 5.

III. Exercising on a proxy

A. The cost of the kink

Let $A$ and $B$ be positive random variables with finite means; $B$ is the proxy. For $K\gt 0$ put

$L_B(K)$ is the value of the call when exercise is decided by $B$; we call it the proxy value and $R_B(K)$ the cost of the kink. For $B=cG$ we write $L_c$ and $R_c$. Prices are discounted at the end.

Proposition 3.1 (The cost of the kink). For all positive numbers $A,B,K$,

The put payoff $(K-A)^+$ and its proxy version $(K-A)\mathbf 1_{\{B\le K\}}$ differ by the same quantity. Consequently $R_B(K)\ge0$ is the expected distance from $A$ to the strike on the event that the strike separates $A$ from $B$, for calls and puts alike. If $\varphi$ is convex on $(0,\infty)$ with a locally absolutely continuous derivative, then

both sides being finite or infinite together. In particular

Proof. Identity (3.2) is a check of the four cases given by the positions of $A$ and $B$ relative to $K$; the put follows from parity applied to both payoffs. For (3.3), fix $A$ and $B$ and integrate the right side of (3.2) against $\varphi''$. If $B\lt A$, integration by parts gives

and the case $A\lt B$ gives the same expression. Taking expectations with Tonelli's theorem proves (3.3). The choices $\varphi(K)=K^2/2$ and $\varphi(K)=-\log K$ give (3.4). Q.E.D.

The right side of (3.3) is the Bregman divergence of $\varphi$ between $A$ and $B$ [7], and the integrand in (3.2) is that divergence for the call payoff itself. Formula (3.3) is the spanning of a convex payoff by calls [6, 9], applied to the pair of payoffs. The two cases in (3.4) are conservation laws: however the cost of the kink is spread across strikes, its total is a moment of the basis. The weights $dK/K^2$ are those that replicate a log contract, so a strip of proxy errors weighted like a variance swap sums to the expected log-gap $E\,d(A/B)$. For $B=cG$ that gap involves only the shape variable $X$.

Proposition 3.1 needs no model. It also shows what a model must supply for a bound at a single strike: a limit on how much of the total mass can concentrate there.

B. A common factor

Assumption (CF). There are a $\sigma$-field $\mathcal G$, positive $\mathcal G$-measurable random variables $a_1,\ldots,a_n$ and $\sigma$, and a random variable $U$ such that

Put $a=n^{-1}\sum_ia_i$ and $g=(\prod_ia_i)^{1/n}$, so that $A=ae^U$, $G=ge^U$ and $X=a/g$. Given $\mathcal G$ the shape of the path is known and its level is lognormal. The assumption holds in the following cases.

1. Black's model with volatility $v$: $U=vW_{t_1}$, $\sigma=v\sqrt{t_1}$, and $\mathcal G$ is generated by the Brownian increments after $t_1$.

2. Stochastic volatility, $dS_t/S_t=r\,dt+\sqrt{V_t}\,(\rho\,dW_t+\sqrt{1-\rho^2}\,dB_t)$ with $V$ adapted to $W$ and $B$ independent of $W$. Heston, 3/2, Stein–Stein and rough volatility models are of this form.

3. The same with jumps in price or variance that are independent of $B$: the jump path is placed in $\mathcal G$.

4. Time-stepping schemes with Gaussian price innovations, such as the Euler kernel of Section 5.

In the second case $\mathcal G$ is generated by $W$ and by the increments of $B$ after $t_1$, and

in the fourth the integrals are replaced by their sums over the grid. The assumption is a scaling property: the Gaussian part of the return before $t_1$ multiplies every fixing by the same factor and leaves the later volatility path as it was. Dynamics whose drift and volatility are free of the price level have it, and so does an average over several futures contracts that move in parallel.

Let $\Phi$ and $\phi$ be the standard normal distribution function and density, $\phi_\sigma(u)=\phi(u/\sigma)/\sigma$, and for $x,s,K\gt 0$

$F_K(x;s)=E(xe^{sZ}-K)^+$ for a standard normal $Z$ is the undiscounted Black–Scholes value of a call on a lognormal factor with total volatility $s$, and $\Gamma_K=\partial_x^2F_K$ is its gamma.

Proposition 3.2 (Conditional price and conditional cost). Assume (CF) and let $B=be^U$ with $b\gt 0$ $\mathcal G$-measurable. Then

and the conditional cost of the kink is

In the log-strike $k=\log K$,

where $\pi_{a,b}$ is the probability density proportional to $|ae^{-t}-1|$ on the interval between $\log b$ and $\log a$.

Proof. The first identity in (3.7) is the definition of $F_K$. Completing the square in $e^{sz}\phi(z)$ gives $E[e^{sZ}\mathbf 1_{\{Z\gt q\}}]=e^{s^2/2}\Phi(s-q)$, hence $\partial_xF_K(b;\sigma)=e^{\sigma^2/2}\Phi(\log(b/K)/\sigma+\sigma)$ and

Adding the two gives the second identity. Since $\sigma\gt 0$, $F_K(\cdot;\sigma)$ is twice continuously differentiable, and (3.8) is Taylor's formula with integral remainder between $b$ and $a$. For (3.9) substitute $x=e^t$: then $\Gamma_K(e^t;\sigma)e^t=Ke^{-t}\phi_\sigma(k-t)$ and $(a-e^t)e^{-t}=ae^{-t}-1$, whose integral between $\log b$ and $\log a$ is $a/b-1-\log(a/b)$. Q.E.D.