Black and Scholes option pricing model

by syoyo

[En]

I’m getting interest to financial engineering because there are a lot of
similarity against (MC-based) global illumination.
For example, massive monte carlo simulation, probability theory, statistics and
low discrepancy sequences.

Thus I’m scrating the surface of financial enginnering field by
starting reading 2 famos papers in classic finantial engineering,
i.e. the paper of Black-Scholes option pricing theorem.

Black, F. and M. Scholes,
“The Pricing of Options and Corporate Liabilities,”
Journal of Political Economy, Vol. 81, pp. 637-654. 1973.

Merton. R. C.,
“Theory of Rational Option Pricing,”
Bell Journal of Economics and Management Science, Vol. 4, pp. 141-183. 1973.

Though I’m a newbie of the financial engineering, I couldn’t understand these yet.
B&S 1973 was written simply(Black said so) but needs to read a lot of citation to understand the background, while Merton 1973 is too complexed!(a lot of equations).

I can say B&S 1973 is similar to Kajiya’s the rendering equation paper and
Merton 1973 is like Veach’s Ph.D thesis.

And more, I couldn’t find the Name of Kiyoshi Ito in the References,
who established the stocastic calclus and his Ito’s lennma heavily affects Black
to derive the Black-Scholes equation(its story is well known in many FE books).

http://www.siam.org/pdf/news/1059.pdf
http://www.nikkei.co.jp/topic3/kinyu/20001222a63cm001_22.html
http://en.wikipedia.org/wiki/Kiyoshi_Ito

Option pricing theorem, including this Black-Scholes equation, would be
applicable to (theoretical) global illumination problem.
For example optimal stopping in adaptive sampling,
but I have no concrete idea of appliyng financial engineering technique to
computer graphics for now … ๐Ÿ™‚

[Ja]

้‡‘่žๅทฅๅญฆใฎ่ซ–ๆ–‡ใงๆœ€ใ‚‚่‘—ๅใชใ‚‚ใฎใจใ„ใ†ใ“ใจใงใ€ไธŠ่จ˜ไบŒใคใฎ
ใƒ–ใƒฉใƒƒใ‚ฏโ€ขใ‚ทใƒงใƒผใƒซใ‚บๅผใฎๅ…ƒ่ซ–ๆ–‡ใ‚’่ปฝใ่ชญใ‚“ใงใฟใพใ—ใŸใ€‚

้‡‘่žๅทฅๅญฆ็ณปใฎ่ซ–ๆ–‡ใฏใ€้‡ใฏ CG ็ณปใ‚ˆใ‚Šใ‚‚ๅคšใ„ใ‚ˆใ†ใชใฎใงใ™ใŒใ€
ๆœ€่ฟ‘ใฎใ‚‚ใฎใงใ‚‚ใ‚ใพใ‚Š web ใงใฏๅ…ฌ้–‹ใ•ใ‚Œใฆใ„ใชใ„ใ‚ˆใ†ใงใ™ใ€‚
(non random walker ใจใ—ใฆ็Ÿฅใ‚‰ใ‚Œใ‚‹? MIT ใฎ Andrew Lo ๆ•™ๆŽˆใจใ‹ใฏ
ๅ…ฌ้–‹ใ—ใฆใ„ใ‚‹ใ‚ˆใ†ใงใ™ใŒ)
ใพใ‚่ซ–ๆ–‡่‡ชไฝ“ๅคใ„ใ“ใจใ‚‚ใ‚ใ‚Š web ใงใฏไธŠ่จ˜่ซ–ๆ–‡ใฏๅ…ฅๆ‰‹ใงใใชใ‹ใฃใŸใฎใงใ€
ๅ›ณๆ›ธ้คจใซ่กŒใฃใฆ่ค‡ๅ†™ใ—ใฆใใ‚‹ใ“ใจใซใชใ‚Šใพใ—ใŸใ€‚

Black & Scholes 1973 ใฏใ€ใ‚ชใƒชใ‚ธใƒŠใƒซใฎใƒ–ใƒฉใƒƒใ‚ฏโ€ขใ‚ทใƒงใƒผใƒซใ‚บๆ–น็จ‹ๅผใซใคใ„ใฆใ€
Merton 1973 ใฏใใฎๅผใฎๆ•ฐๅญฆ็š„ใช่จผๆ˜Žใ‚’ไธŽใˆใ‚‹ใ€ใจใ„ใ†ๆ„Ÿใ˜ใงใ™ใ€‚

CG ใฎไธ–็•Œใง่จ€ใˆใฐใ€Black & Scholes 1973 ใฏ Kajiya ใฎ the rendering equation,
Merton 1973 ใฏ Veach ใฎ Ph.D thesis, ใจใ„ใ†ไฝ็ฝฎใฅใ‘ใชๆ„Ÿใ˜ใงใ™ใ€‚

Black & Scholes 1973 ใฏ Black ๅ…ˆ็”Ÿ่‡ช่บซใ‚‚่จ€ใ†ใ‚ˆใ†ใซใ€
่ชญใฟๆ‰‹ใŒ็†่งฃใ—ใ‚„ใ™ใ„ใ‚ˆใ†ใซ็ฐกๆฝ”ใซๆ›ธใ‹ใ‚Œใฆใ‚ใ‚‹ใจใฎใ“ใจใงใ—ใŸใ€‚
ใŸใ—ใ‹ใซ็ฐกๆฝ”ใซๆ›ธใ‹ใ‚Œใฆใ‚ใ‚‹ใ“ใจใฏใ‚ˆใใ‚ใ‹ใ‚‹ใฎใงใ™ใŒใ€
ๅˆ†้‡ŽใŒ้•ใ†ใ‹ใ‚‰ใจใ„ใ†ใฎใ‚‚ใ‚ใ‚Šใ€ใใ‚Œใ ใ‘่ชญใ‚“ใงใ‚‚ใ„ใพใ„ใกใ‚ˆใใ‚ใ‹ใ‚Šใพใ›ใ‚“ใ€‚

ใจใ„ใ†ใ‹ๅคšใใŒ … ใซใคใ„ใฆใฏ … ใ‚’ๅ‚็…งใ€ใชใ‚“ใฆๆ›ธใ‹ใ‚Œใฆใ‚ใ‚‹ใฎใงใ€
cite ใ•ใ‚Œใฆใ„ใ‚‹่ซ–ๆ–‡ใ‚‚่ชญใฟ่พผใ‚“ใงใ„ใ‹ใชใ„ใจใชใœใ“ใ“ใงใ“ใ‚“ใช็†่ซ–ใ‚„ๅผใซใชใ‚‹ใฎ๏ผŸ
ใจใ„ใ†ใฎใŒใพใฃใŸใ่ฆ‹ใˆใชใ„ใ‚ใ‘ใงใ™ใ€‚ใพใ‚ใ“ใ‚Œใฏ the rendering equation ใซใ‚‚่จ€ใˆใ‚‹ใ“ใจใงใ™ใญใ€‚

Merton 1973 ใฏใ€ใ“ใ‚Œใฏใ‚‚ใ†ๆ•ฐๅผใฎใ‚ชใƒณใƒ‘ใƒฌใƒผใƒ‰ใงใ™ใ€‚
็งใฎใ‚ˆใ†ใช็•ฐๅˆ†้‡Žใฎไบบ้–“ใŒใ“ใ‚Œใ‚’ใ„ใใชใ‚Šใ‚ˆใ‚“ใงใ‚‚ใ€ใพใฃใŸใใกใ‚“ใทใ‚“ใ‹ใ‚“ใทใ‚“ใงใ—ใŸใ€‚
(ใƒ™ใƒซ็ ”ใ‚ธใƒฃใƒผใƒŠใƒซใฏใฟใ‚“ใชใ“ใ‚“ใชใซใ™ใ”ใ„ใฎใ ใ‚ใ†ใ‹?…)
ไธ‹ๆ‰‹ใช็†่ซ– GI ่ซ–ๆ–‡ใ‚ˆใ‚Šใ‚‚ใพใฃใจใ†ใซๆ•ฐๅญฆ็š„ใงใ™ใ€‚้‡‘่žๅทฅๅญฆใชใ‚“ใฆๆ‰€่ฉฎ
้‡‘ๅ„ฒใ‘ใฎใŸใ‚ใฎๅฎŸๅ‹™็š„ใชๅญฆๅ•ใ ใจๆ€ใฃใฆใ„ใŸ็งใŒๆต…ใฏใ‹ใงใ—ใŸใ€‚ใ”ใ‚ใ‚“ใชใ•ใ„ใ€‚
Merton ๅ…ˆ็”Ÿใซใฏใฒใ‚Œไผใ—ใพใ™ใ€‚
(ไปŠใฏ GI ใฎใปใ†ใŒ้‡‘่žๅทฅๅญฆใ‚ˆใ‚Šใ‚‚ๆต…ใฏใ‹ใ ใชใใจใ„ใ†ๆ„Ÿใ˜ใงใ™)

GI ใฎใŸใ‚ใซ็ขบ็Ž‡่ซ–ใ‚„็ตฑ่จˆใ‚’ๅญฆใฐใ‚“ใจใ™ใ‚‹ใชใ‚‰ใฐใ€ๅฎŸ้š›ใฎ้‡‘่žๅธ‚ๅ ดใงๅฎŸ่ทตใฎใงใใ‚‹
้‡‘่žๅทฅๅญฆใ‹ใ‚‰ๅ…ฅใ‚‹ใฎใŒใ‚ˆใ•ใใ†ใงใ™ใ€‚
(่บซ้Šญใ‚’ๅˆ‡ใฃใฆๅฎŸ่ทตใ™ใ‚Œใฐใ€ใชใ‹ใชใ‹ๆœฌๆฐ—ใซๅ‹‰ๅผทใงใใ‚‹ใงใ—ใ‚‡ใ†ใ—)

ใ•ใฆใ€ไบŒใคใฎ่ซ–ๆ–‡ใงใ‚‚ใฃใจใ‚‚ๆฐ—ใซใชใฃใŸใฎใฏใ€
References ใซๆˆ‘ใ‚‰ใŒๆ†งใ‚ŒใฎไผŠ่—คๅ…ˆ็”Ÿใฎๆ–‡ๅญ—ใŒใฉใกใ‚‰ใฎ่ซ–ๆ–‡ใซใ‚‚็พใ‚Œใฆใ„ใชใ„ใ“ใจใ€‚
(ใคใ„ใงใซใ„ใ†ใจ Bachelier 1900 ใ‚‚)
ใˆใฃใจใ€B&S equation ใ‚’ๅฐŽๅ‡บใ™ใ‚‹ใฎใซไผŠ่—คใฎ lenma ใŒๅคšๅคงใชใ‚‹่ฒข็Œฎใ‚’
ใ—ใŸใ‚“ใ˜ใ‚ƒใชใ‹ใฃใŸใ‚“ใงใ—ใ‚‡ใ†ใ‹๏ผŸใใ‚Œใ‚’ cite ใ—ใชใ„ใจใฏใชใซใ”ใจใž?
ใใ‚Œใจใ‚‚ cite ใ•ใ‚Œใฆใ„ใ‚‹่ซ–ๆ–‡ใซใฏๅซใพใ‚Œใฆใ„ใ‚‹ใฎใ‹ใช๏ผŸ

…ใ—ใฐใ‚‰ใใฏ่ซ–ๆ–‡ใฎๅญซๅผ•ใใฎใŸใ‚ใซใ€ๅ›ณๆ›ธ้คจใธ้€šใ†ๆ—ฅใ€…ใซใชใ‚Šใใ†ใงใ™ใ€‚

ใกใชใฟใซใ€ใƒ–ใƒฉใƒƒใ‚ฏโ€ขใ‚ทใƒงใƒผใƒซใ‚บๆ–น็จ‹ๅผใ‚’ๅซใ‚ใ€
ใ‚ชใƒ—ใ‚ทใƒงใƒณ็†่ซ–ใ‚„้‡‘่žใฎ็†่ซ–ใฏ CG ใธใฎๅฟœ็”จใŒๅคšใใ‚ใ‚Šใใ†ใงใ™ใ€‚
ใŸใจใˆใฐ้ฉๅฟœ็š„ใ‚ตใƒณใƒ—ใƒชใƒณใ‚ฐใฎๆœ€้ฉใชๆ‰“ใกๅˆ‡ใ‚Šๅ›žๆ•ฐใชใฉใงใ™ใ€‚

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