lecture 12 sandor zoltan n´emeth -...
TRANSCRIPT
![Page 1: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/1.jpg)
Mathematical Finance
Lecture 12
Sandor Zoltan Nemeth
University of Birmingham
Autumn
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 1 / 12
![Page 2: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/2.jpg)
Outline
MATERIAL COVERED: Lecture Notes, page 58
Subection 5.5.3
Exercise Sheet 5, Question 2
1 Perpetual Options
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 2 / 12
![Page 3: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/3.jpg)
Outline
MATERIAL COVERED: Lecture Notes, page 58
Subection 5.5.3
Exercise Sheet 5, Question 2
1 Perpetual Options
2 American Vanilla Call Option with a Constant Dividend Yield:Equations (reminder)
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 2 / 12
![Page 4: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/4.jpg)
Outline
MATERIAL COVERED: Lecture Notes, page 58
Subection 5.5.3
Exercise Sheet 5, Question 2
1 Perpetual Options
2 American Vanilla Call Option with a Constant Dividend Yield:Equations (reminder)
3 Solving the American Perpetual Vanilla Call
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 2 / 12
![Page 5: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/5.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 6: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/6.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 7: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/7.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:1 Impossible to predict change of r , σ, D0 in time
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 8: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/8.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:1 Impossible to predict change of r , σ, D0 in time2 Risk: Writer or underlying company go out of business before option
exercised
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 9: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/9.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:1 Impossible to predict change of r , σ, D0 in time2 Risk: Writer or underlying company go out of business before option
exercised
3 Analytical Solutions tested against the corresponding Americanoption with a finite expiry date
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 10: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/10.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:1 Impossible to predict change of r , σ, D0 in time2 Risk: Writer or underlying company go out of business before option
exercised
3 Analytical Solutions tested against the corresponding Americanoption with a finite expiry date
4 Independent of time: If underlying same value today as yesterday,then it should have same value today as yesterday ←− infinite time
available for excercise
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 11: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/11.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:1 Impossible to predict change of r , σ, D0 in time2 Risk: Writer or underlying company go out of business before option
exercised
3 Analytical Solutions tested against the corresponding Americanoption with a finite expiry date
4 Independent of time: If underlying same value today as yesterday,then it should have same value today as yesterday ←− infinite time
available for excercise
5 Steady state solutions for corresponding American option withfinite time to expiry
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 12: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/12.jpg)
Perpetual American options
1 Same as regular American Options but T = ∞
2 Not traded because:1 Impossible to predict change of r , σ, D0 in time2 Risk: Writer or underlying company go out of business before option
exercised
3 Analytical Solutions tested against the corresponding Americanoption with a finite expiry date
4 Independent of time: If underlying same value today as yesterday,then it should have same value today as yesterday ←− infinite time
available for excercise
5 Steady state solutions for corresponding American option withfinite time to expiry
6 All ∂
∂tterms are zero and Sf (t) is replaced by Sf = constant
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 3 / 12
![Page 13: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/13.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations
1 For S > Sf (t):C = S − E
and
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 4 / 12
![Page 14: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/14.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations
1 For S > Sf (t):C = S − E
and
1
2σ2S2 ∂2C
∂S2+
∂C
∂t− rC + (r −D0)S
∂C
∂S< 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 4 / 12
![Page 15: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/15.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations
1 For S > Sf (t):C = S − E
and
1
2σ2S2 ∂2C
∂S2+
∂C
∂t− rC + (r −D0)S
∂C
∂S< 0
2 For S < Sf (t):C ≥ max(S − E , 0)
and
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 4 / 12
![Page 16: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/16.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations
1 For S > Sf (t):C = S − E
and
1
2σ2S2 ∂2C
∂S2+
∂C
∂t− rC + (r −D0)S
∂C
∂S< 0
2 For S < Sf (t):C ≥ max(S − E , 0)
and
1
2σ2S2 ∂2C
∂S2+
∂C
∂t− rC + (r −D0)S
∂C
∂S= 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 4 / 12
![Page 17: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/17.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 18: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/18.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
∂C
∂S= 1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 19: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/19.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
∂C
∂S= 1
4 At S = 0:C = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 20: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/20.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
∂C
∂S= 1
4 At S = 0:C = 0
5 At t = T :C = max(S − E , 0)
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 21: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/21.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
∂C
∂S= 1
4 At S = 0:C = 0
5 At t = T :C = max(S − E , 0)
6 Here D0, r , σ,E ,T = constants
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 22: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/22.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
∂C
∂S= 1
4 At S = 0:C = 0
5 At t = T :C = max(S − E , 0)
6 Here D0, r , σ,E ,T = constants
7 It is necessary to find C (S , t) for 0 ≤ S ≤ Sf (t) and for−∞ < t ≤ T
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 23: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/23.jpg)
American Vanilla Call Option with a Constant Dividend
Yield: Equations 2
3 At S = Sf (t):C = Sf (t)− E
and
∂C
∂S= 1
4 At S = 0:C = 0
5 At t = T :C = max(S − E , 0)
6 Here D0, r , σ,E ,T = constants
7 It is necessary to find C (S , t) for 0 ≤ S ≤ Sf (t) and for−∞ < t ≤ T
8 It is also necessary to find Sf (t) for −∞ < t ≤ T
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 5 / 12
![Page 24: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/24.jpg)
Solving the American Perpetual Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
1 For S > Sf :C = S − E
and
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 6 / 12
![Page 25: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/25.jpg)
Solving the American Perpetual Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
1 For S > Sf :C = S − E
and
1
2σ2S2d
2C
dS2− rC + (r −D0)S
dC
dS< 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 6 / 12
![Page 26: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/26.jpg)
Solving the American Perpetual Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
1 For S > Sf :C = S − E
and
1
2σ2S2d
2C
dS2− rC + (r −D0)S
dC
dS< 0
2 For S < Sf :C ≥ max(S − E , 0)
and
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 6 / 12
![Page 27: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/27.jpg)
Solving the American Perpetual Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
1 For S > Sf :C = S − E
and
1
2σ2S2d
2C
dS2− rC + (r −D0)S
dC
dS< 0
2 For S < Sf :C ≥ max(S − E , 0)
and
1
2σ2S2d
2C
dS2− rC + (r −D0)S
dC
dS= 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 6 / 12
![Page 28: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/28.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
3 At S = Sf :C = Sf − E
and
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 7 / 12
![Page 29: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/29.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
3 At S = Sf :C = Sf − E
and
dC
dS= 1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 7 / 12
![Page 30: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/30.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
3 At S = Sf :C = Sf − E
and
dC
dS= 1
4 At S = 0:C = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 7 / 12
![Page 31: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/31.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
3 At S = Sf :C = Sf − E
and
dC
dS= 1
4 At S = 0:C = 0
5 Here D0, r , σ,E = constants
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 7 / 12
![Page 32: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/32.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
3 At S = Sf :C = Sf − E
and
dC
dS= 1
4 At S = 0:C = 0
5 Here D0, r , σ,E = constants
6 It is necessary to find C (S) for 0 ≤ S ≤ Sf
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 7 / 12
![Page 33: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/33.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
3 At S = Sf :C = Sf − E
and
dC
dS= 1
4 At S = 0:C = 0
5 Here D0, r , σ,E = constants
6 It is necessary to find C (S) for 0 ≤ S ≤ Sf
8 It is also necessary to find Sf
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 7 / 12
![Page 34: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/34.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Insert C ∼ Sλ into
1
2σ2S2 d
2C
dS2+ (r −D0)S
dC
dS− rC = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 8 / 12
![Page 35: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/35.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Insert C ∼ Sλ into
1
2σ2S2 d
2C
dS2+ (r −D0)S
dC
dS− rC = 0
=⇒ 1
2σ2S2
d2CdS2
︷ ︸︸ ︷
λ(λ− 1)Sλ−2+(r −D0)S
dC
dS︷ ︸︸ ︷
λSλ−1−r
S︷︸︸︷
Sλ = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 8 / 12
![Page 36: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/36.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Insert C ∼ Sλ into
1
2σ2S2 d
2C
dS2+ (r −D0)S
dC
dS− rC = 0
=⇒ 1
2σ2S2
d2CdS2
︷ ︸︸ ︷
λ(λ− 1)Sλ−2+(r −D0)S
dC
dS︷ ︸︸ ︷
λSλ−1−r
S︷︸︸︷
Sλ = 0
=⇒ 1
2σ2λ(λ− 1)Sλ + (r −D0)λS
λ − rSλ = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 8 / 12
![Page 37: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/37.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Insert C ∼ Sλ into
1
2σ2S2 d
2C
dS2+ (r −D0)S
dC
dS− rC = 0
=⇒ 1
2σ2S2
d2CdS2
︷ ︸︸ ︷
λ(λ− 1)Sλ−2+(r −D0)S
dC
dS︷ ︸︸ ︷
λSλ−1−r
S︷︸︸︷
Sλ = 0
=⇒ 1
2σ2λ(λ− 1)Sλ + (r −D0)λS
λ − rSλ = 0
=⇒ 1
2σ2λ(λ− 1) + (r −D0)λ− r = 0
Divide by (1/2)σ2:
=⇒ λ(λ− 1) + (k − kD)λ− k = 0,
k = 2rσ2 and kD = 2D0
σ2
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 8 / 12
![Page 38: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/38.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 39: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/39.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 40: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/40.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
∆ = (k − kD − 1)2 + 4k > 0 =⇒ distinct real roots
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 41: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/41.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
∆ = (k − kD − 1)2 + 4k > 0 =⇒ distinct real roots
λ1 > λ2 roots
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 42: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/42.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
∆ = (k − kD − 1)2 + 4k > 0 =⇒ distinct real roots
λ1 > λ2 roots
=⇒ (λ− λ1)(λ− λ2) = λ2 + (k − kD − 1)λ− k
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 43: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/43.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
∆ = (k − kD − 1)2 + 4k > 0 =⇒ distinct real roots
λ1 > λ2 roots
=⇒ (λ− λ1)(λ− λ2) = λ2 + (k − kD − 1)λ− k
=⇒ λ2 − (λ1 + λ2)λ + λ1λ2 = λ2 + (k − kD − 1)λ− k
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 44: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/44.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
∆ = (k − kD − 1)2 + 4k > 0 =⇒ distinct real roots
λ1 > λ2 roots
=⇒ (λ− λ1)(λ− λ2) = λ2 + (k − kD − 1)λ− k
=⇒ λ2 − (λ1 + λ2)λ + λ1λ2 = λ2 + (k − kD − 1)λ− k
=⇒ λ1λ2 = −k < 0 =⇒ λ1 > 0 > λ2
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 45: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/45.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
λ(λ− 1) + (k − kD)λ− k = 0
=⇒ λ2 + (k − kD − 1)λ− k = 0
∆ = (k − kD − 1)2 + 4k > 0 =⇒ distinct real roots
λ1 > λ2 roots
=⇒ (λ− λ1)(λ− λ2) = λ2 + (k − kD − 1)λ− k
=⇒ λ2 − (λ1 + λ2)λ + λ1λ2 = λ2 + (k − kD − 1)λ− k
=⇒ λ1λ2 = −k < 0 =⇒ λ1 > 0 > λ2
=⇒
λ1 = 12
(
1− k + kD +√
(k − kD − 1)2 + 4k)
> 0
λ2 = 12
(
1− k + kD −√
(k − kD − 1)2 + 4k)
< 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 9 / 12
![Page 46: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/46.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 47: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/47.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
C = 0 at S = 0
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 48: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/48.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
C = 0 at S = 0
=⇒ 0 = limS→0 a1Sλ1 + a2S
λ2 = a1 × 0+ a2 ×∞
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 49: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/49.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
C = 0 at S = 0
=⇒ 0 = limS→0 a1Sλ1 + a2S
λ2 = a1 × 0+ a2 ×∞
=⇒ 0 =
0︷ ︸︸ ︷
a1 × 0+a2 ×∞ = a2 ×∞ =⇒ a2 = 0 =⇒ C = a1Sλ1C = a1Sλ1C = a1Sλ1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 50: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/50.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
C = 0 at S = 0
=⇒ 0 = limS→0 a1Sλ1 + a2S
λ2 = a1 × 0+ a2 ×∞
=⇒ 0 =
0︷ ︸︸ ︷
a1 × 0+a2 ×∞ = a2 ×∞ =⇒ a2 = 0 =⇒ C = a1Sλ1C = a1Sλ1C = a1Sλ1
C = Sf − E at S = Sf =⇒ a1Sλ1f
a1Sλ1f
a1Sλ1f
= C (Sf ) = Sf − E= Sf − E= Sf − E
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 51: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/51.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
C = 0 at S = 0
=⇒ 0 = limS→0 a1Sλ1 + a2S
λ2 = a1 × 0+ a2 ×∞
=⇒ 0 =
0︷ ︸︸ ︷
a1 × 0+a2 ×∞ = a2 ×∞ =⇒ a2 = 0 =⇒ C = a1Sλ1C = a1Sλ1C = a1Sλ1
C = Sf − E at S = Sf =⇒ a1Sλ1f
a1Sλ1f
a1Sλ1f
= C (Sf ) = Sf − E= Sf − E= Sf − E
dC
dS= 1 at S = Sf =⇒ λ1a1S
λ1−1|S=Sf= 1 =⇒ λ1a1S
λ1−1f
= 1λ1a1Sλ1−1f
= 1λ1a1Sλ1−1f
= 1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 52: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/52.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
General Solution: C = a1Sλ1 + a2S
λ2
C = 0 at S = 0
=⇒ 0 = limS→0 a1Sλ1 + a2S
λ2 = a1 × 0+ a2 ×∞
=⇒ 0 =
0︷ ︸︸ ︷
a1 × 0+a2 ×∞ = a2 ×∞ =⇒ a2 = 0 =⇒ C = a1Sλ1C = a1Sλ1C = a1Sλ1
C = Sf − E at S = Sf =⇒ a1Sλ1f
a1Sλ1f
a1Sλ1f
= C (Sf ) = Sf − E= Sf − E= Sf − E
dC
dS= 1 at S = Sf =⇒ λ1a1S
λ1−1|S=Sf= 1 =⇒ λ1a1S
λ1−1f
= 1λ1a1Sλ1−1f
= 1λ1a1Sλ1−1f
= 1
{
a1Sλ1f
= Sf − E
λ1a1Sλ1−1f
= 1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 10 / 12
![Page 53: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/53.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
{
a1Sλ1f
= Sf − E
λ1a1Sλ1−1f
= 1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 11 / 12
![Page 54: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/54.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
{
a1Sλ1f
= Sf − E
λ1a1Sλ1−1f
= 1
Multiply the first equation by λ1 and the second equation by Sf
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 11 / 12
![Page 55: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/55.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
{
a1Sλ1f
= Sf − E
λ1a1Sλ1−1f
= 1
Multiply the first equation by λ1 and the second equation by Sf
=⇒{
λ1a1Sλ1f
= λ1(Sf − E )
λ1a1Sλ1f
= Sf
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 11 / 12
![Page 56: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/56.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
{
a1Sλ1f
= Sf − E
λ1a1Sλ1−1f
= 1
Multiply the first equation by λ1 and the second equation by Sf
=⇒{
λ1a1Sλ1f
= λ1(Sf − E )
λ1a1Sλ1f
= Sf
λ1(Sf − E ) = Sf =⇒ λ1Sf − λ1E = Sf =⇒ λ1Sf − Sf = λ1E
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 11 / 12
![Page 57: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/57.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
{
a1Sλ1f
= Sf − E
λ1a1Sλ1−1f
= 1
Multiply the first equation by λ1 and the second equation by Sf
=⇒{
λ1a1Sλ1f
= λ1(Sf − E )
λ1a1Sλ1f
= Sf
λ1(Sf − E ) = Sf =⇒ λ1Sf − λ1E = Sf =⇒ λ1Sf − Sf = λ1E
=⇒ (λ1 − 1)Sf = λ1E =⇒ Sf =λ1E
λ1 − 1, a1 =
Sf − E
Sλ1f
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 11 / 12
![Page 58: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/58.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Thus if S < Sf , then C =
a1︷ ︸︸ ︷
Sf − E
Sλ1f
Sλ1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 12 / 12
![Page 59: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/59.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Thus if S < Sf , then C =
a1︷ ︸︸ ︷
Sf − E
Sλ1f
Sλ1
If S ≥ Sf , then C = S − E
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 12 / 12
![Page 60: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/60.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Thus if S < Sf , then C =
a1︷ ︸︸ ︷
Sf − E
Sλ1f
Sλ1
If S ≥ Sf , then C = S − E
Note: If D0 = 0, then kD = 2D0σ2 = 0 which implies
λ1 = 12
(
1− k + kD +√
(k − kD − 1)2 + 4k)
=
12
(
1− k +√
(k − 1)2 + 4k)
= 12
(
1− k +√
(k2− 2k + 1) + 4k)
=
12
(
1− k +√k2 + 2k + 1
)
= 12
(
1− k +√
(k + 1)2)
=
12 (1− k + k + 1) = 1
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 12 / 12
![Page 61: Lecture 12 Sandor Zoltan N´emeth - web.mat.bham.ac.ukweb.mat.bham.ac.uk/S.Z.Nemeth/3g11/presentation12-3g11.pdf · PerpetualAmericanoptions 1 Same as regular American Options but](https://reader034.vdocuments.us/reader034/viewer/2022050312/5f741f6c9fdac2015450955e/html5/thumbnails/61.jpg)
Solving the Perpetual American Vanilla Call with a
Constant Dividend Yield: Exercise Sheet 5, Question 2
Thus if S < Sf , then C =
a1︷ ︸︸ ︷
Sf − E
Sλ1f
Sλ1
If S ≥ Sf , then C = S − E
Note: If D0 = 0, then kD = 2D0σ2 = 0 which implies
λ1 = 12
(
1− k + kD +√
(k − kD − 1)2 + 4k)
=
12
(
1− k +√
(k − 1)2 + 4k)
= 12
(
1− k +√
(k2− 2k + 1) + 4k)
=
12
(
1− k +√k2 + 2k + 1
)
= 12
(
1− k +√
(k + 1)2)
=
12 (1− k + k + 1) = 1
Thus Sf = λ1E
λ1−1 = λ1E
0 = ∞
American Perpetual Vanilla Call Without Dividends −→ NO early exercise
S Z Nemeth (University of Birmingham) Mathematical Finance Autumn 12 / 12