On the Valuation of Discrete Asian Options in High Volatility Environments

Figures & data

Table 1. Limit values of option prices when volatility parameters $|\mathrm{\Sigma }|\to \mathrm{\infty }$ tend to infinity.

	Arithmetic Asian		Geometric Asian		Averaged
Option price for $\left|\mathrm{\Sigma }\right|\to \mathrm{\infty }$	Call	Put	Call	Put	Call	Put
	$s_0$	K	0	K	$s_0$	K

Table 2. Simulated option prices under the Black-Scholes model using different control variates and volatilities for strike 1.

σ	Control variate
Call	None	Kemna-Vorst	Modified Kemna-Vorst	Averaged
$25\mathrm{\%}$	$0.2570\pm 0.0006$	$0.2567\pm 0.0001$	$0.2566\pm 0.0000$	$0.2566\pm 0.0000$
$100\mathrm{\%}$	$0.8279\pm 0.0374$	$0.8214\pm 0.0332$	$0.7990\pm 0.0002$	$0.7991\pm 0.0001$
$200\mathrm{\%}$	$0.3134\pm 0.0514$	$0.3250\pm 0.0501$	$0.9853\pm 0.0001$	$0.9853\pm 0.0001$
$500\mathrm{\%}$	$0.0000\pm 0.0000$	$0.0000\pm 0.0000$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$
Put	None	Kemna-Vorst	Modified Kemna-Vorst	Averaged
$25\mathrm{\%}$	$0.2566\pm 0.0003$	$0.2566\pm 0.0000$	$0.2567\pm 0.0001$	$0.2566\pm 0.0000$
$100\mathrm{\%}$	$0.7989\pm 0.0003$	$0.7990\pm 0.0002$	$0.8214\pm 0.0332$	$0.7991\pm 0.0001$
$200\mathrm{\%}$	$0.9853\pm 0.0001$	$0.9853\pm 0.0001$	$0.3250\pm 0.0501$	$0.9853\pm 0.0001$
$500\mathrm{\%}$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$	$0.0000\pm 0.0000$	$1.0000\pm 0.0000$

Table 3. Simulated option prices under the Heston model using different control variates and growing initial variances for strike 1.

${\nu }_0$	Control variate
Call	None	Kemna-Vorst	Modified Kemna-Vorst	Averaged
$50\mathrm{\%}$	$0.6213\pm 0.0056$	$0.6256\pm 0.0037$	$0.6324\pm 0.0001$	$0.6323\pm 0.0001$
$100\mathrm{\%}$	$0.6320\pm 0.0059$	$0.6365\pm 0.0038$	$0.6442\pm 0.0001$	$0.6441\pm 0.0001$
$10000\mathrm{\%}$	$0.1272\pm 0.0504$	$0.6623\pm 0.0157$	$0.9971\pm 0.0000$	$0.9971\pm 0.0000$
$1000000\mathrm{\%}$	$0.0000\pm 0.0000$	$0.6261\pm 0.0000$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$
Put	None	Kemna-Vorst	Modified Kemna-Vorst	Averaged
$50\mathrm{\%}$	$0.6325\pm 0.0004$	$0.6324\pm 0.0001$	$0.6256\pm 0.0037$	$0.6323\pm 0.0001$
$100\mathrm{\%}$	$0.6442\pm 0.0004$	$0.6442\pm 0.0001$	$0.6365\pm 0.0038$	$0.6441\pm 0.0001$
$10000\mathrm{\%}$	$0.9971\pm 0.0000$	$0.9971\pm 0.0000$	$0.6623\pm 0.0157$	$0.9971\pm 0.0000$
$1000000\mathrm{\%}$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$	$0.6261\pm 0.0000$	$1.0000\pm 0.0000$

Table 4. Simulated option prices under the Heston model using different control variates and growing long-term variances for strike 1.

θ	Control variate
Call	None	Kemna-Vorst	Modified Kemna-Vorst	Averaged
$50\mathrm{\%}$	$0.6213\pm 0.0056$	$0.6256\pm 0.0037$	$0.6324\pm 0.0001$	$0.6323\pm 0.0001$
$100\mathrm{\%}$	$0.7252\pm 0.0189$	$0.7358\pm 0.0156$	$0.7862\pm 0.0002$	$0.7861\pm 0.0001$
$10000\mathrm{\%}$	$0.0000\pm 0.0000$	$0.0000\pm 0.0000$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$
$1000000\mathrm{\%}$	$0.0000\pm 0.0000$	$0.0000\pm 0.0000$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$
Put	None	Kemna-Vorst	Modified Kemna-Vorst	Averaged
$50\mathrm{\%}$	$0.6325\pm 0.0004$	$0.6324\pm 0.0001$	$0.6256\pm 0.0037$	$0.6323\pm 0.0001$
$100\mathrm{\%}$	$0.7863\pm 0.0003$	$0.7862\pm 0.0002$	$0.7358\pm 0.0156$	$0.7861\pm 0.0001$
$10000\mathrm{\%}$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$	$0.0000\pm 0.0000$	$1.0000\pm 0.0000$
$1000000\mathrm{\%}$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$	$0.0000\pm 0.0000$	$1.0000\pm 0.0000$