Numerical Analysis of the Optimal Exercise Boundary of an American Put Option

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UPPSALA UNIVERSITET UPPSALA UNIVERSITY Inst. f¨or informationsteknologi Information Technology Avd. f¨or teknisk databehandling Dept. of Scientific Computing

Numerical Analysis of the Optimal Exercise Boundary of an American Put Option

by

Per Lindberg Gunnar Marcusson Gustav Nordman Internal Report No. 2002:7

May 2002

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(11)

0 2 4 6 8 10 12 14 16 18 20

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Discrete Stockprice ∆s

s a s

s b i

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3µ

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(12)

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(13)

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(14)

· Æ Å ² ± ² ¾ ²Læ

5 ßuà#áÌ ú Ð[ÄÅÆ'Å2À ÐS[¿ÁZ¾-Æ2À ÐS[Ä·'ÐÅ² Ð[¿·Á2ÅÆ Ð[± Å ± Ð Ð[± æ ± ²

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lW$'G&RkuP$' *!$'u )d)(K

8

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50 45 40 35 30 25 20 15 10 5 0

10 12 14 16 18 20 22 24 26 28

Time to maturity

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c = 3 c = 9 Perpetual put

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(15)

0 5 10 15 20 25 30 35 40 45 50

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time to maturity

e 1 e 2 e 3

³6!.'G&*ÆX·ÑUTÒwVK&*PxloK=K $E /eNIhS&*PK!)(ÒlW$G&(!K)BJ *T T

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3[¤yTKuP$'/@EG*!."%$'&O/0-G&* *!K)n!&*.&#TYcX']k'&*)K4T

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²

(16)

50 45 40 35 30 25 20 15 10 5 0 10

12 14 16 18 20 22 24 26 28 30

time to maturity

Stock price

c = 3 c = 18 Perpetual put

³6!.'G&*À[·6UT^ZG/@&* PX- GJ$"*Td^/@&* P@EG$'E*!$'p"&g"%&*$/ /e8G&( =k

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0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

16 18 20 22 24 26 28 30

Time to maturity

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σ = 0.2 σ = 0.3 σ = 0.4 σ = 0.5

×

³6!.G&*%æX·gè=/@EP3J$'T&RH /e-*G&*w=ke)($'!GS!$u$"|X-&Rki!.NTX$':8! =k[

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(17)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10

⁻⁴

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

Time to maturity

f

_num

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f

_num

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10

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Time to maturity

f

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Time to maturity

f

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Time to maturity

f

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(18)

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24 25 26 27 28 29 30

Time to maturity

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(19)

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

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Time to maturity

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(20)

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Numerical Analysis of the Optimal Exercise Boundary of an American Put Option

UPPSALA UNIVERSITET UPPSALA UNIVERSITY Inst. f¨or informationsteknologi Information Technology Avd. f¨or teknisk databehandling Dept. of Scientific Computing