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JSFH Exchange, Basket and other multi-asset options JSFH Igor Hlivka MUFG Securities International, LONDON JSFH This application demonstration reviews several classes of multi-asset options and as such extends the concept on multi-variability in Finance presented / discussed in other applications. Multi-variability is important concept in financial engineering as many non-standard structured products in the market are exposed to multiple source of randomness. Multi-variability is not trivial in terms of handling multiple dependencies, however suitable change of martingale measure and dimension-reduction techniques can help simplifying the multi-variable process into more manageable routines. Although multi-dependency in many instances requires numerical processing, we will show that with Maple we can do better. Our aim is to devise an analytical solution to this problem and will show how Maple's symbolic engine can efficiently cope with this task

Exchange Options are special class of bi-variate options that give the buyer the right to exchange at option maturity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5the asset LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. This means that the option holder will exercise his (her) right to exchange the asset if one has greater value than the other. In this respect, we can define the option's payoff as: 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 A key to valuation of such options is to assess the behavior of assets under two sources of randomness: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ Both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGPg==in isolation are independent Wiener processes with 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 with following properties: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

In case of two source of randomness, we define the SDE for two stochastic processes as: 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From here we can establish: 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the correlation between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVGLi1GIzYlLUY7NiRRIjJGJ0Y+RjJGNUZARj4= To value this option analytically, we first change the option's payoff by performing the "change of measure" The standard way of valuing option is to apply the risk-neutral measure, such that the asset processes when normalized by appropriate numeraire become martingale: 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 where B is money-market account We can therefore express the Exchange option payoff with new measure as follows: 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 To see why the change is measured is required, we examine two stock process in detail. Under standard risk-neutral measure P we get: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== 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

KiYmSSNTMEc2IjYjIiIiRictSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmKiYsJkkickdGJUYnKiYjRiciIiNGJykmSSZzaWdtYUdGJUYmRjRGJyEiIkYnSSJ0R0YlRidGJyomRjZGJyZJIldHRiVGJkYnRidGJw==

KiYmSSNTMEc2IjYjIiIjIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJiomLCZJInJHRiVGKComI0YoRidGKCkmSSZzaWdtYUdGJUYmRidGKCEiIkYoSSJ0R0YlRihGKComRjZGKCZJIldHRiVGJkYoRihGKA==

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

KigmSSNTMEc2IjYjIiIjIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsKiooI0YoRidGKEkidEdGJUYoKSZJJnNpZ21hR0YlRiZGJ0YoISIiKiZGNEYoJkkiV0dGJUYmRihGKCooRjFGKEYyRigpJkY1NiNGKEYnRihGKComRjxGKCZGOUY9RihGNkYoJkYkRj1GNg==

We observe that the volatility of the ratio of two assets is equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRJyYjOTYzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RIl5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGTi9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lJ2FjY2VudEdGUkY1 = 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 for the ratio process to be martingale, the following condition has to to satisfied: 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 However, under the risk-neutral measure P, this is not the case as: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRictRiM2JS1GOzYkUSIyRidGNUY+RkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkkvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRl1vLUYsNihGLkZCRkJGR0ZKRkxGNQ== \342\211\240 (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 because in bi-variate setting 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 Let's build the new volatility and transformed measure vectors under new measure Q 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjU1ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWFjdGlvbkdGJDYlLUkobWZlbmNlZEdGJDYoLUYjNiYtSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYmLUkkbXRkR0YkNigtRiM2Ji1GIzYmLUYsNiVRJyYjOTYxO0YnL0YwRj1GOS1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmZvLUklbXN1YkdGJDYlLUYsNiVRJyYjOTYzO0YnRmFvRjktRiM2Ji1JI21uR0YkNiRRIjJGJ0Y5LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0Y9RjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y5LUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GYHEtRmlvNiVGW3AtRiM2Ji1GYXA2JFEiMUYnRjlGZHBGZ3BGOUZpcEY5LyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zbci8lK2dyb3VwYWxpZ25HRltyLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRmJyRmlxRlxyRl5yLUZlbjYmLUZobjYoLUYjNiZGaG9GYm8tSSZtc3FydEdGJDYjLUYjNiZGZnFGXHEtRiM2JC1JJW1zdXBHRiQ2JUZeb0ZgcC8lMXN1cGVyc2NyaXB0c2hpZnRHRltxRjlGOUY5RmlxRlxyRl5yRmByRmNyRmlxRlxyRl5yLyUmYWxpZ25HUSVheGlzRicvRmpxUSliYXNlbGluZUYnL0ZdclEnY2VudGVyRicvRl9yUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGMS8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmR0LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRl91LyUmZnJhbWVHRl91LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGPS8lLWVxdWFsY29sdW1uc0dGPS8lLWRpc3BsYXlzdHlsZUdGPS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZcdUZkcEZncEY5RjkvSSttc2VtYW50aWNzR0YkUSpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zidi8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRicvJSlydGFibGVpZEdRKTU0MDMxNzQ4RidGZHBGZ3BGOQ==

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

KigmSSNTMEc2IjYjIiIjIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsLiooI0YoRidGKEkidEdGJUYoKSZJJnNpZ21hR0YlRiZGJ0YoISIiKihGNEYoKSwmRihGKCokKUkkcmhvR0YlRidGKEY2RjFGKCZJIldHRiVGJkYoRigqKEYxRihGMkYoKSZGNTYjRihGJ0YoRjYqKEY8RihGNEYoJkY+RkJGKEYoKipGPEYoRjRGKEZBRihGMkYoRigqJkZBRihGREYoRjZGKCZGJEZCRjY=

As we can see, under this measure transformation, the ratio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRIlNGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYkUSIyRidGOC9GNlEldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYjNiUtRi82JS1GMjYlRjRGQUZDLUYjNiUtRj42JFEiMUYnRjhGQUZDRkVGQUZDLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZYLyUpYmV2ZWxsZWRHRjdGOA== can be expressed as: 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. What does this imply? Under the transformed measure Q, the bi-variate process 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can be reduced to the univariate counterpart with : 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 We use this result for the option valuation: We will first define the new martingale process X Decide the limits of integration for which the option payoff holds 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 Integrate the payoff similar to Black-Scholes with 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KiQpLCgqJCkmSSZzaWdtYUc2IjYjIiIiIiIjRitGKyoqRixGK0kkcmhvR0YpRitGJ0YrJkYoNiNGLEYrISIiKiQpRi9GLEYrRisjRitGLEYr

KigmSSNTMEc2IjYjIiIjIiIiJkYkNiNGKCEiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYqKCNGKEYnRigsKCokKSZJJnNpZ21hR0YlRipGJ0YoRigqKkYnRihJJHJob0dGJUYoRjhGKCZGOUYmRihGKyokKUY8RidGKEYoRihJInRHRiVGKEYrKigpRjVGNEYoKUY/RjRGKEkieUdGJUYoRihGKA==

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

LCQqKiMiIiIiIiNGJSwqKiZJInRHNiJGJSkmSSZzaWdtYUdGKjYjRiVGJkYlRiUqLEYmRiVJJHJob0dGKkYlJkYtNiNGJkYlRixGJUYpRiUhIiIqJkYpRiUpRjFGJkYlRiUqJkYmRiUtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYjKiYmSSNTMEdGKkYuRiUmRj9GMkYzRiVGJUYlKSwoKiRGK0YlRiUqKkYmRiVGMEYlRixGJUYxRiVGMyokRjVGJUYlRiRGMylGKUYkRjNGJQ==

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

JCIrcW09WjUhIzU=

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Maple calculates the sensitivities with ease. Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ is a denominator in the exchange option formula, the Delta (first derivative w.r.t LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+) for this factor is negative. 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

6'-%+AXESLABELSG6%Q)sigma[1]6"Q$rho6"Q!6"-%*AXESSTYLEG6#%'NORMALG-%%GRIDG6&;$""#!"#$""#!"";$!"&!""$""&!""X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm":":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10B9BFB3FB75983155B85543FB6CA0FA355676F3FB636EE4603E6B03FB59FD3E1D6C9A03FB5046AC5EE2BB23FB46450752687123FB3BF12CCCB027A3FB3142C4C34F14F3FB262FF23CD7CA8-%&COLORG6#%%ZHUEG-%&TITLEG6$Q9Exchange~Option~Premium~6"-I%FONTG6"6%I&TIMESG6"I%BOLDG6""#7-%,ORIENTATIONG6$$!%g9!""$"$q&!""

As the chart above shows, the Exchange Option is quite sensitive to correlation and volatility. Negative correlation between two assets will make the the ratio volatility LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRJyYjOTYzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RIl5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGTi9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lJ2FjY2VudEdGUi1GOzYtUSJ+RidGNUY+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRmVuRjU=higher and hence lead to a higher premium.

Results for the Exchange options easily extends into the valuation of option contract that pays the "best" or "worst" of two assets:

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 The value of the Best of two option is then: 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

LCQqKCMiIiIiIiNGJSwsKiZGJkYlKSZJI1MwRzYiNiNGJUYmRiUhIiIqJiZGKzYjRiZGJS1JJGVyZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLDYjLCQqLCNGJSIiJUYlKUYmRiRGJSwqKiZGJkYlLUkjbG5HRjQ2IyomRipGJUYwRi5GJUYlKiZJInRHRixGJSkmSSZzaWdtYUdGLEYtRiZGJUYuKixGJkYlSSRyaG9HRixGJSZGR0YxRiVGRkYlRkRGJUYlKiZGREYlKUZKRiZGJUYuRiUpLCgqJEZFRiVGJSoqRiZGJUZJRiVGRkYlRkpGJUYuKiRGTEYlRiVGJEYuKUZERiRGLkYlRiVGJSomLUYzNiMsJCosRjpGJSwqRkNGJUZIRi5GS0YlRj5GJUYlRk1GLkZSRi5GPEYlRiVGJUYqRiVGLkYwRi5GKkYlRiVGKkYuRi4=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnUGF5b2ZmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JVEkbWluRicvRjNRJmZhbHNlRicvRjZRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Znbi1GSjYtUSJ+RidGP0ZNRk9GUUZTRlVGV0ZZL0ZmblEmMC4wZW1GJy9GaW5GXm9Gam5GOi1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiUtRi82JVEjU1RGJ0YyRjUtRiM2Ji1JI21uR0YkNiRRIjFGJ0Y/RkFGQ0Y/RkYtRko2LVEiLEYnRj9GTS9GUEY0RlFGU0ZVRldGWUZdby9GaW5RLDAuMzMzMzMzM2VtRictRiw2JUZnby1GIzYmLUZdcDYkUSIyRidGP0ZBRkNGP0ZGRkFGQ0Y/Rj9Gam5Gam5GSUZqbkZmcC1GSjYtUSomdW1pbnVzMDtGJ0Y/Rk1GT0ZRRlNGVUZXRlkvRmZuUSwwLjIyMjIyMjJlbUYnL0ZpbkZhcS1JJW1zdXBHRiQ2JS1GYW82JC1GIzYoRmZwLUZKNi1RKCZtaW51cztGJ0Y/Rk1GT0ZRRlNGVUZXRllGYHFGYnFGZW9GQUZDRj9GPy1GIzYmLUZKNi1RIitGJ0Y/Rk1GT0ZRRlNGVUZXRllGYHFGYnFGQUZDRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R0ZIRkFGQ0Y/ The value of the Best of two option is then: 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

LCQqKCMiIiIiIiNGJSwsKihGJkYlJkkjUzBHNiI2I0YmRiUmRio2I0YlRiVGJSomRilGJS1JJGVyZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYjLCQqLCNGJSIiJUYlKUYmRiRGJSwqKiZGJkYlLUkjbG5HRjI2IyomRi1GJUYpISIiRiVGJSomSSJ0R0YrRiUpJkkmc2lnbWFHRitGLkYmRiVGQSosRiZGJUkkcmhvR0YrRiUmRkZGLEYlRkVGJUZDRiVGJSomRkNGJSlGSUYmRiVGQUYlKSwoKiRGREYlRiUqKkYmRiVGSEYlRkVGJUZJRiVGQSokRktGJUYlRiRGQSlGQ0YkRkFGJUYlRiUqJi1GMTYjLCQqLEY4RiUsKkZCRiVGR0ZBRkpGJUY8RiVGJUZMRkFGUUZBRjpGJUYlRiVGLUYlRkFGKUZBRi1GJUYlRi1GQUYl

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It is worth mentioning that "Best of" / "Worst of" options are relatively expensive as their premium is close to the underlying asset price.

JSFH Basket options are special case of multi-asset options where several underlying assets are bundled together and this basket performance then determines the option's payoff. By virtue of construction, baskets options are essentially a particular case of a broader family of portfolio options where the stochastic variable is the baskets (portfolio) of assets. As such, basket options can be utilized to monetize the expectation of the future behavior of combined assets pool. Similar to the univariate case, basket options can be structured as Puts or Calls. Puts provide a positive payoff if the weighted-average of basket assets drops below a strike level. Calls give rise to a positive payoff if the weighted-averarge at maturity is greater than the strike. Formally: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn = 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiKkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZDLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYlLUZKNiVRImlGJ0ZNRlBGTUZQLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGLw== where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is i-th asset and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is its weight in the basket with 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 = 1 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 A key to valuation of basket options analytically is to determine the parameters of joint distribution. We observe that each asset return in the basket individually (under the appropriate measure) follows a normal distribute: 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 and return dependency structure is characterized by the correlation matrix \316\243. This allows us deriving basket's variance as a product of individual weights and variance-covariance matrix. Once basket variance has been determined, we can proceed with valuation as in univariate case - just replace the asset with the weighted-average of basket assets and volatility with basket's volatility. Let's examine this is practice and valued 5 stocks Call Basket Option: 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Here we have derived the basket's variance-covariance matrix. We will obtain the basket's volatility and weighted average of assets through simple factor/matrix manipulation: 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

KiQpLEAqLiIiIyIiIiZJIndHNiI2I0YnRicmRik2IyIiJUYnJkkmc2lnbWFHRipGK0YnJkYwRi1GJyZJJHJob0dGKjYkRidGLkYnRicqLkYmRidGKEYnJkYpNiMiIiZGJ0YvRicmRjBGN0YnJkYzNiRGJ0Y4RidGJyouRiZGJ0YoRicmRik2I0YmRidGL0YnJkYwRj5GJyZGMzYkRidGJkYnRicqLkYmRidGKEYnJkYpNiMiIiRGJ0YvRicmRjBGREYnJkYzNiRGJ0ZFRidGJyomKUYoRiZGJylGL0YmRidGJyomKUYsRiZGJylGMUYmRidGJyouRiZGJ0ZDRidGLEYnRkZGJ0YxRicmRjM2JEZFRi5GJ0YnKi5GJkYnRkNGJ0Y2RidGRkYnRjlGJyZGMzYkRkVGOEYnRicqJilGQ0YmRicpRkZGJkYnRicqLkYmRidGPUYnRixGJ0Y/RidGMUYnJkYzNiRGJkYuRidGJyouRiZGJ0Y9RidGNkYnRj9GJ0Y5RicmRjM2JEYmRjhGJ0YnKi5GJkYnRj1GJ0ZDRidGP0YnRkZGJyZGMzYkRiZGRUYnRicqJilGPUYmRicpRj9GJkYnRicqJilGNkYmRicpRjlGJkYnRicqLkYmRidGLEYnRjZGJ0YxRidGOUYnJkYzNiRGLkY4RidGJyNGJ0YmRic=

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

LCwqJiZJIlNHNiI2IyIiIkYoJkkid0dGJkYnRihGKComJkYlNiMiIiNGKCZGKkYtRihGKComJkYlNiMiIiRGKCZGKkYyRihGKComJkYlNiMiIiVGKCZGKkY3RihGKComJkYlNiMiIiZGKCZGKkY8RihGKA==

The valuation is now straightforward -we have reduced the multivariate dependency to a univariate case which we know how to value. Assuming joint lognormality of each assets we set: 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 where 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRJyYjOTYzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RIl5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGTi9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lJ2FjY2VudEdGUi1GOzYtUSJ+RidGNUY+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRmVuRjU== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIi5GJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVy1GLzYlUSZDb1ZhckYnRjJGNUZARi5GRA== 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

This is the "full-form" valuation, and despite the long string of functional parameters Maple handles it well. We can now compute the option premium with parameters substitution: 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

JCIqVnBqcyIhIzU=

On the other hand, we could have taken a "shortcut" - computing the premium symbolically for the set of theoretical parameters and then making the substitution for volatility and mean. 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JCIqV3BqcyIhIzU=

JSFH As the above analysis indicates, the basket options can be quite complex, particularly when the number of underlying assets becomes large. In this respect, basket options are also more difficult to manage as multi-dependency and multi-variability creates complex web of sensitivities - including multiple deltas, gammas, vegas etc.

This application example shows that multi-asset options can become quite complex when number of underlying assets increases. Computational efficiency greatly increases when CAS is employed to simplify processes symbolically. Maple, as this example again demonstrates, handles these tasks very well.

Legal Notice: The copyright for this application is owned by the authors. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities. TTdSMApJNVJUQUJMRV9TQVZFLzUwNDc5NTM2WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJiImJiUiU0c2IyIiIiZGKDYjIiIjJkYoNiMiIiQmRig2IyIiJSZGKDYjIiImNiI=TTdSMApJNVJUQUJMRV9TQVZFLzUxMDI1Mjc2WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJiImJiUmc2lnbWFHNiMiIiImRig2IyIiIyZGKDYjIiIkJkYoNiMiIiUmRig2IyIiJjYiTTdSMApJNVJUQUJMRV9TQVZFLzUxMDMzMTI0WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJiImJiUid0c2IyIiIiZGKDYjIiIjJkYoNiMiIiQmRig2IyIiJSZGKDYjIiImNiI=TTdSMApJNVJUQUJMRV9TQVZFLzUxMDM5MjMyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjOiImIiYiIiImJSRyaG9HNiRGJyIiIyZGKTYkRiciIiQmRik2JEYnIiIlJkYpNiRGJyIiJkYoRicmRik2JEYrRi4mRik2JEYrRjEmRik2JEYrRjRGLEY1RicmRik2JEYuRjEmRik2JEYuRjRGL0Y3RjtGJyZGKTYkRjFGNEYyRjlGPUY/Ric2Ig==TTdSMApJNVJUQUJMRV9TQVZFLzUxMDYzNzQ4WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjOiImIiYqJCYlJnNpZ21hRzYjIiIiIiIjKihGKEYrJkYpNiNGLEYrJiUkcmhvRzYkRitGLEYrKihGKEYrJkYpNiMiIiRGKyZGMTYkRitGNkYrKihGKEYrJkYpNiMiIiVGKyZGMTYkRitGPEYrKihGKEYrJkYpNiMiIiZGKyZGMTYkRitGQkYrRi0qJEYuRiwqKEYuRitGNEYrJkYxNiRGLEY2RisqKEYuRitGOkYrJkYxNiRGLEY8RisqKEYuRitGQEYrJkYxNiRGLEZCRitGM0ZGKiRGNEYsKihGNEYrRjpGKyZGMTYkRjZGPEYrKihGNEYrRkBGKyZGMTYkRjZGQkYrRjlGSUZQKiRGOkYsKihGOkYrRkBGKyZGMTYkRjxGQkYrRj9GTEZTRlcqJEZARiw2Ig==