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AmericanPayoffAtExpiry drops the (H/S)^(2 mu) N(D2) term whenever N(D2) underflows to zero. At low volatility the power it multiplies is large enough that the product is not negligible: at vol 0.1%, r -2% and q 3%, a down-and-in put on a barrier at 95 paying 10 at expiry priced at 1.0002 against 1.0176. Where N(D2) underflows or the power overflows, the product is now formed in logs, with the asymptotic tail of log N. Over 1152 options with volatility from 10% down to 1e-6, master had 4 values more than 1e-6 away from the closed form in 50-digit arithmetic, up to 1.7%; with this change the largest difference is 1.6e-8.
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coverage: 78.252% (+0.001%) from 78.251% — mkzung:fix/digital-american-at-expiry-small-variance into lballabio:master |
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AmericanPayoffAtExpiry, whichAnalyticDigitalAmericanEngineuses for a payoff at expiry, setsY_to zero whenever N(D2) underflows. At low volatility the power (H/S)^(2 mu) that multiplies N(D2) is large enough that the product is not negligible. At vol 0.1%, r -2% and q 3%, a down-and-in put on a barrier at 95 paying 10 at expiry in one year prices at 1.0002, where the same formula in 50-digit arithmetic gives 1.0176.Where N(D2) underflows or the power overflows, the product is now formed in logs, with the asymptotic tail of log N where N underflows. Below the threshold at which the variance counts as zero nothing changes.
Over 1152 options with volatility from 10% down to 1e-6, master has 4 values more than 1e-6 away from the 50-digit closed form, the worst 1.7% off. With this change the largest relative difference is 1.6e-8.
testCashAtExpiryOrNothingAmericanLowVolatilitychecks four such cases and fails on master.DigitalOptionTestsand the full suite pass.The at-hit calculator has a related problem, which a separate pull request covers; the two touch different files.