We develop two pricing formulae for European options in the SABR model with beta= 1 case by means of Malliavin Calculus. We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call \"Dyson series on the return's idiosyncratic noise\" yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor's (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion.The second approach, which we call \"Dyson series on the common noise\" necessitates the calculation of only a one-dimensional integral, but the formula is more complex.
","abstract_html":"<p>We develop two pricing formulae for European options in the SABR model with beta= 1 case by means of Malliavin Calculus. We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call "Dyson series on the return's idiosyncratic noise" yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor's (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion.The second approach, which we call "Dyson series on the common noise" necessitates the calculation of only a one-dimensional integral, but the formula is more complex.</p>","abstract_has_math":false,"creators":["Guo, Zhengji"],"institution":null,"degree_name":"Mathematics, PhD","degree_level":"Open Access Dissertation","degree_discipline":"Institute of Mathematical Sciences","degree_department":null,"school":null,"contributors":["John Angus","Qidi Peng"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-01-01T08:00:00Z","date_published":"2019-01-01T08:00:00Z","updated_at":"2026-07-24T01:39:59Z","subjects":["Exponential formula","Malliavin Calculus","option pricing","SABR model","stochastic volatility"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarship.claremont.edu/cgu_etd/510","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["John Angus","Qidi Peng"]},{"key":"dc:creator","label":"Author","values":["Guo, Zhengji"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2023-03-22T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Institute of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics, PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Exponential formula","Malliavin Calculus","option pricing","SABR model","stochastic volatility"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarship.claremont.edu/cgu_etd/510"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We develop two pricing formulae for European options in the SABR model with beta= 1 case by means of Malliavin Calculus. We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call \"Dyson series on the return's idiosyncratic noise\" yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor's (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion.The second approach, which we call \"Dyson series on the common noise\" necessitates the calculation of only a one-dimensional integral, but the formula is more complex.
"]},{"key":"dc:title","label":"Title","values":["A full asymptotic series of European call option prices in the SABR model with beta=1"]}]}],"canonical_facts":{"dc:contributor":["John Angus","Qidi Peng"],"dc:creator":["Guo, Zhengji"],"dc:date.available":["2023-03-22T07:00:00Z"],"dc:description.abstract":["We develop two pricing formulae for European options in the SABR model with beta= 1 case by means of Malliavin Calculus. We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call \"Dyson series on the return's idiosyncratic noise\" yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor's (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion.The second approach, which we call \"Dyson series on the common noise\" necessitates the calculation of only a one-dimensional integral, but the formula is more complex.
"],"dc:identifier":["https://scholarship.claremont.edu/cgu_etd/510"],"dc:subject":["Exponential formula","Malliavin Calculus","option pricing","SABR model","stochastic volatility"],"dc:title":["A full asymptotic series of European call option prices in the SABR model with beta=1"],"thesis:degree_discipline":["Institute of Mathematical Sciences"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Mathematics, PhD"]},"updated_at":"2026-07-24T01:39:59Z"}