LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Maple for Commodity Finance Igor Hlivka MUFJ Securities International, LONDON In this application we show how Maple can handle financial options models that have become popular in commodity finance. The financial trading of commodities has dramatically increased over past years as finance community started to look for non-standard instruments uncorrelated with the traditional financial products such as stocks, bonds, rates and currencies. Commodities, unlike their financial counterparts, require different approach to the process modeling: (i) commodities exhibit seasonality effects, (ii) commodity futures are exposed to many deformation modes, (iii) futures volatility is driven by the "Samuelson" effect that causes its drop as the expiry time shortens. To accommodate these features into the option pricing we propose a two-factor Markov model with deterministic volatility. A reader familiar with financial products will recognise its patterns as two-factor Heath-Jarrow-Morton [HJM] model adapted to commodities. Similar to HJM setting, the stochastic object here is the commodity future rather than the spot price that allows us ignoring the "convenience yield" that is a systemic feature of commodities pricing. This approach has become quite popular amongst energy traders and the model has been recently implemented by Murex - a leading provider of financial trading systems. Multi factor models, where two-factor is a particular case, are not easily handled due to complex dependencies. Maple with its powerful symbolic engine can handle this task with ease and the analytical approach presented below provides an interesting insight into the underlying stochastic process.

In line with the introduction statement we are interested in two factors affecting the futures volatility. We opt to model the futures prices directly and assume that under the suitable martingale measure the futures prices follow the two-factor dynamics: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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TLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZFdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYnt 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 and for the volatilities we select the following specification: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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 Here 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 are volatilities of the short and long part of the curve, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82I1EhRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEifkYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURjU=is the correlation between the spot prices and long-end of the futures curve, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is a speed of mean-reversion and 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 are the seasonality adjustments that oscillate around 0. They will help achieving the correct seasonality effect in commodities. The "Samuelson" effect is handled through the 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 term and is present in each factor of volatility. As seen from the model specification, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWUY1drives the short-end part of the futures curve and will diminish as \316\272 increases. On the other hand, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWUY1contains the same term, but also a linear factor with long-term volatility LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW9HRiQ2LVEoJmluZmluO0YnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSVtc3VwR0YkNiUtRjs2LVEvJkV4cG9uZW50aWFsRTtGJ0Y1L0Y/USZ1bnNldEYnL0ZBRmluL0ZDRmluL0ZFRmluL0ZHRmluL0ZJRmluL0ZLRmluRkxGTy1GIzYmLUYvNiVRJyYjOTQ1O0YnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYvNiVRIlRGJ0ZRRlNGUUZTRjVGUUZTLyUxc3VwZXJzY3JpcHRzaGlmdEdGVy1GOzYtUSJ+RidGNUY+RkBGQkZERkZGSEZKRkxGT0Y1which controls the long-term part of the curve. As seen from above, this is the "derived" version of HJM model, however the model itself is assymetric due to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 term.

We will progress in several steps: Define the model variables Derive the processes Find the moments Match them with the standard distributional patterns Value options

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QyY+Jkkmc2lnbWFHNiI2IyIiIiwmKigsJiomSSRyaG9HRiZGKCZGJTYjIiIhRihGKCZGJTYjSSlpbmZpbml0eUclKnByb3RlY3RlZEchIiJGKC1JJGV4cEdGJjYjLCZJImRHRiZGKEkmYWxwaGFHRiZGKEYoLUY3NiMsJComSSZrYXBwYUdGJkYoSSJ0R0YmRihGNUYoRigqJkYxRigtRjc2I0Y7RihGKEYoPiZGJTYjIiIjKipGLkYoLUklc3FydEdGJjYjLCZGKEYoKiRGLUZIRjVGKEY2RihGPEYoRig=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqKCwmKiZJJHJob0c2IiIiIiZJJnNpZ21hR0YnNiMiIiFGKEYoJkYqNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyEiIkYoLUkkZXhwRzYkRjBJKF9zeXNsaWJHRic2IywmSSJkR0YnRihJJmFscGhhR0YnRihGKC1GMzYjLCQqJkkma2FwcGFHRidGKEkidEdGJ0YoRjFGKEYoKiZGLUYoLUYzNiNGOUYoRig=

KiomSSZzaWdtYUc2IjYjIiIhIiIiLCZGKEYoKiRJJHJob0dGJSIiIyEiIiNGKEYsLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJkkiZEdGJUYoSSZhbHBoYUdGJUYoRigtRjA2IywkKiZJJmthcHBhR0YlRihJInRHRiVGKEYtRig=

Will look at the volatility graphs in deterministic case with some arbitrary values: QyQtSSdwbG90M2RHNiI2Jy1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQ8JCZJJnNpZ21hR0YlNiMiIiImRi02IyIiIzcnL0kiZEdGJSQhIighIiMvSSZhbHBoYUdGJSQiI1ghIiQvSSJ0R0YlIiIkL0kma2FwcGFHRiUkIiNhRj0vSSRyaG9HRiUkIiIoISIiLyZGLTYjIiIhOyQiIiZGOCQiI05GOC8mRi02I0kpaW5maW5pdHlHRilGTi9JJWF4ZXNHRiVJJmJveGVkR0YlL0kmdGl0bGVHRiU3JFFCU2hvdH5hbmR+TG9uZ35Wb2xhdGlsaXR5fnN1cmZhY2V+RiUvSSVmb250R0YlNyVJJlRJTUVTR0YlSSVCT0xER0YlIiM4Ri8=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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The Futures process is a standard Ito process with two factors of randomness for which the terminal value is well known: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRjBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkxGOQ== = initial value of the stochastic process LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjRlRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn = terminal value of the stochastic process LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYy= futures process drift LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5= time to maturity (in years) QyQ+SSNGVEc2IiomSSNGMEdGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCgqJiwoSSNtdUdGJUYoKiQmSSZzaWdtYUdGJTYjRigiIiMjISIiRjcqJCZGNTYjRjdGN0Y4RihJInRHRiVGKEYoKihGNEYoLUklc3FydEdGKzYjRj1GKEkjUjFHRiVGKEYoKihGO0YoRj9GKEkjUjJHRiVGKEYoRihGKA==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

JSFH

We take the Mean and Standard deviation of the Futures process LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRlRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

QyY+SSNNbkc2Ii1JKXNpbXBsaWZ5R0YlNiMtSSVNZWFuR0YlNiNJI0ZUR0YlIiIiPkkkU3RkR0YlLUYnNiMtSTJTdGFuZGFyZERldmlhdGlvbkdGJUYrRi0=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJ0R0YoIiIiSSNtdUdGKEYsRixJI0YwR0YoRiw=

KiQsJCosLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkidEdGKiIiIkkjbXVHRipGLiIiI0kjRjBHRipGMCwmKiotRiY2IyosRi1GLiZJJnNpZ21hR0YqNiMiIiFGMC1GJjYjSSJkR0YqRjAtRiY2I0kmYWxwaGFHRipGMC1GJjYjKiZJJmthcHBhR0YqRi5GLUYuISIjRi4tRiY2IyosRi1GLkY7RjBGPkYwRkFGRSZGODYjSSlpbmZpbml0eUdGKEYwRi4tRiY2IyowRi1GLkY7Ri5GPkYwRkEhIiJJJHJob0dGKkYuRjdGLkZJRi5GMC1GJjYjKihGLUYuRklGMEY+RjBGLkZPKiYtRiY2IyowRi1GLkY7RjBGPkYwRkFGRUZQRi5GN0YuRklGLkYwLUYmNiMqLEYtRi5GO0YuRj5GMEZBRk9GSUYwRjBGLkYuRlVGRUZYRkVGTyNGLkYw

We now look at the distributional patterns of the futures process by assuming certain arbitrary parameters: Qyg+SShudW1kaXN0RzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVldmFsR0YoNiRJI0ZUR0YlNysvSSNGMEdGJSIjNS9JI211R0YlIiIhLyZJJnNpZ21hR0YlNiNGNCQiI0khIiMvJkY3NiNJKWluZmluaXR5R0YoJCIjTkY7L0kidEdGJSIiJC9JJmFscGhhR0YlJCIjWCEiJC9JImRHRiUkISIoRjsvSSZrYXBwYUdGJSQhIiZGOy9JJHJob0dGJSQiIikhIiJGVi1JKnJhbmRvbWl6ZUdGJUYlIiIiPkkiQUdGJS1JJ1NhbXBsZUdGJTYkRiQtSSJeR0YoNiRGMSIiJUZZ

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IishZXJIRSI=

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Take the histogram of this random sample: QyQtSSpIaXN0b2dyYW1HNiI2I0kiQUdGJSIiIg==

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%;

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We now want to match the moments of the Log-Normal distribution with the empirical distribution followed by the Futures QyQ+SSxEaXN0TW9tZW50c0c2IjckLUknTW9tZW50R0YlNiQtSSpMb2dOb3JtYWxHRiU2JEkodXBzaWxvbkdGJUkmb21lZ2FHRiUiIiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtRig2JEYqIiIjRi8=

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print(); # input placeholder

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print(); # input placeholder

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Maple returns two solutions. We know that the standard deviation of a random variable must be positive, so will check which solution set suits our needs: QyY+SSVTb2wxRzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVldmFsR0YoNiQmJkklc29sbkdGJTYjIiIiRjA3Ky9JI0YwR0YlIiNEL0kjbXVHRiUiIiEvJkkmc2lnbWFHRiU2I0Y4JCIjTiEiIy8mRjs2I0kpaW5maW5pdHlHRihGPS9JInRHRiVGMS9JJmFscGhhR0YlJCIiI0Y/L0kiZEdGJSQhIihGPy9JJmthcHBhR0YlJCEiJkY/L0kkcmhvR0YlJCIiKCEiIkYxPkklU29sMkdGJS1GJzYjLUYrNiQmJkYvNiNGSUYwRjJGMQ==

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print(); # input placeholder

L0kmb21lZ2FHNiIkIiszQyYpXE4hIzU=

L0kmb21lZ2FHNiIkISszQyYpXE4hIzU=

Since the second set standard deviation is negative, we pick the fist set as the "correct" solution of the matching problem.

Having determined the matching parameters, we will now derive the options formulas for both Call and Put We will do it in two steps: a) Define random variable b) Determine the conditional expectation Qyg+SSJXRzYiLUkvUmFuZG9tVmFyaWFibGVHRiU2Iy1JKkxvZ05vcm1hbEdGJTYkSSJhR0YlSSJiR0YlISIiPkkoQ3BheW91dEdGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JC1JIz49R0YzNiRGJEkiS0dGJSwmRiQiIiJGOEYuRjo+SShQcGF5b3V0R0YlLUYyNiQyRiRGOCwmRjhGOkYkRi5GOg==

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print(); # input placeholder

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print(); # input placeholder

KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiZJInJHRigiIiJJInRHRihGLSEiIkYtLCoqJi1GJDYjLCZJImFHRihGLSokSSJiR0YoIiIjI0YtRjhGLS1JJGVyZkdGJTYjLCQqKEY3Ri9GOEY5LCgtSSNsbkdGJTYjSSJLR0YoRi1GNUYvRjZGL0YtRjlGLSNGL0Y4KiZGQ0YtLUY7NiMsJCooRjhGOSwmRkBGLUY1Ri9GLUY3Ri9GOUYtRjlGMkY5RkNGREYt

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These are European Call and Put formulas in terms of arbitrary parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY5 We will replace them with out computed matched parameters QyY+SSVDYWxsRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRik2JDwkL0kiYUdGJS1JJHJoc0dGKTYjJiZJJXNvbG5HRiU2IyIiIjYjIiIjL0kiYkdGJS1GMzYjJkY2RjhJKEFyYkNhbGxHRiVGOT5JJFB1dEdGJS1GJzYjLUYtNiRGL0knQXJiUHV0R0YlISIi

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print(); # input placeholder

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

This is the desired Call Option formula with factored volatility function. We can now value it with some numerical parameters. Let's price 6 months European Call and Put options on a hypothetical commodity with current price = 25 and Call strike = 27 [Put strike = 23] and the set of volatility parameters shown below: JSFH

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JSFH QyY+SShudW1jYWxsRzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVldmFsR0YoNiRJJUNhbGxHRiU3LS9JI0YwR0YlIiNEL0kiS0dGJSIjRi9JI211R0YlIiIhL0kickdGJSRGMSEiJC8mSSZzaWdtYUdGJTYjRjckIiM8ISIjLyZGPjYjSSlpbmZpbml0eUdGKCQiI05GQi9JInRHRiUkIiImISIiL0kmYWxwaGFHRiUkIiIjRkIvSSJkR0YlJCEiKEZCL0kma2FwcGFHRiUkRkxGQi9JJHJob0dGJSQiIihGTSIiIj5JJ251bXB1dEdGJS1GJzYjLUYrNiRJJFB1dEdGJTctRi8vRjMiI0JGNUY4RjxGQ0ZJRk5GUkZWRllGZ24=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrdkooPWQmISM1

JCIrYiY+eHQlISM1

JSFH

Analytical result derived above allows us examining options' sensitivities w.r.t various input parameters. For example, correlation parameter sensitivity on the Call option price can be examined graphically: 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Joint impact due to correlation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and speed of mean reversion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy 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Analytical formula allows us computing derivatives easily: For example: Put option delta - first derivative w.r.t the Futures price: 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print(); # input placeholder

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%;

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

Put option cross- Vega - mixed derivative w.r.t both volatility parameters 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print(); # input placeholder

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

This is quite complex formula, however Maple handles it with ease We can investigate the properties of this cross-vega graphically: 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JSFH JSFH

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We have demonstrated that Maple is well-positioned to handle more complex financial commodity products exposed to dual sources of randomness. While two-factor models stand high on the complexity scale of handling, symbolic manipulation of expressions coupled with moment matching technique enable us deriving close-from solution in straightforward manner.

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