Moment Matching for FX Asian Options with a Harmonic Payoff
Summary
The document asks whether domestic–foreign symmetry can price a foreign-paying fixed-strike Asian option when its arithmetic average is approximated by a lognormal distribution. The proposed change of measure uses the average itself as a numeraire, but the response does not establish that this is a valid measure change or resolve whether the symmetry argument works.
Instead, the answer describes pricing the reciprocal-average payoff directly. If the average is modeled with a lognormal or shifted-lognormal density, its moments can be inferred from the component spot rates under the relevant probability measure, and the payoff expectation can then be integrated over that density. This offers a route to valuation without relying on the proposed symmetry. The response is brief: it gives no derivation of the moment formulas, numerical example, comparison of approximations, or assessment of model error. Its usefulness therefore lies mainly in identifying a direct distribution-based approach, whose accuracy depends on the assumed approximation and chosen moments.
Key ideas
- The proposed change of measure based on the arithmetic average is posed as a question, not validated by the response.
- A direct alternative is to approximate the average's distribution and integrate the reciprocal-payoff expectation.
- The response allows lognormal or shifted-lognormal approximations using moments inferred from component spot rates.
- The document provides no derivation or accuracy analysis for the approximation.
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Full text
# FX Asian Option Moment-matching in Harmonic case
# FX Asian Option Moment-matching in Harmonic case
I need to price a "foreign-paying" fixed-strike Asian (i.e., average) option. Thus, the payoff is:
$$\left(\frac{A_T - K}{A_T}\right)^{+} = \left(1 - \frac{K}{A_T}\right)^{+} = K \left(\frac{1}{K}-\frac{1}{A_T}\right)^{+}$$
where $A_T$ is the arithmetic average, at expiry $T$, of some spot FX rates. E.g, $A_T = \sum_{i=1}^N w_i S_{t_i}$ for spot FX rates $S_{t_i}$ and weights $w_i$. In the reduced case $N = 1$, we simply have a European vanilla option, which we can price via domestic-foreign symmetry, as in https://quant.stackexchange.com/a/44541/43050.
My question is: Can we use the same domestic-foreign symmetry for the Asian option, e.g. if we approximate the distribution of $A_T$ as lognormal via moment-matching? Following the logic for the vanilla case, we would set $\frac{dP^f}{dP^d}\big|_t = \frac{A_t B^f_t}{A_0 B^d_t}$, where $B^d_t$ and $B^f_t$ are the respective domestic and foreign money-market account values at time $t$, and assuming we can prove that the expectation is $1$ so it's an actual Radon-Nikodym derivative. Then we'd have:
\begin{align*} E^f\bigg( \frac{1}{B^f_T A_T} (A_T-K)^+\bigg) &= E^d\bigg(\frac{A_T B^f_T}{A_0 B^d_T} \frac{1}{B^f_T A_T} (A_T-K)^+\bigg)\\ &= \frac{1}{A_0}E^d\bigg(\frac{1}{B^d_T} (A_T-K)^+\bigg), \end{align*}
thus reducing to the domestic-payout case, which we know how to price (via moment-matching). Does this make any sense, and, if not, why not? If the symmetry approach does not work, can we use moment-matching on the harmonic average directly? I have only seen moment-matching discussed in the arithmetic case. Many thanks in advance.
## Answer by ir7 (score 1)
https://quant.stackexchange.com/a/65864
Regarding the second question, if we assume that $A$ is lognormal (shifted lognormal) and imply its two (three) moments from respective spot rate components under the appropriate probability measure, then we have its probability density function $f_A$ which allows us to compute the expectation of the payoff the usual way:
$$ E\left[\left(\frac{1}{K}-\frac{1}{A}\right)^+ \right] =\int_K^\infty \left(\frac{1}{K}-\frac{1}{x}\right)f_A(x) dx. $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.