Sunday, October 11, 2009
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Saturday, September 12, 2009
APPLICATION OF STOCHASTIC METHODS IN INTEGRATION FOR EVALUATING OPTION PRICE
ABSTRACT
In evaluating the value of integration which can not be found exactly, we can implement numerical approach. The numerical method is categorized into deterministic and stochastic. Application of stochastic method in integration can be used for evaluating price of an European call option and European put option which approach the result of Black-Scholes formula.
European call option and European put option price is based on assumption that stock price follows a Geometric Brownian Motion with parameters r, , S, Sp, and T. Starting from a basic formula, the price of an European call option P= exp(-rt) E(max(ST –Sp,0)) and European put option P=exp(-rt)E( max(Sp - ST, 0)) are modified into multivariable integral form. Then, using random numbers generated in MATLAB and implementing stochastic method , the integral can be evaluated to approach option price.
The graph shows, the higher the amount of random number used in stochastic method in integration then error is smaller. Convergence of percentage error is proportional to convergence of 1/N.
Key Words : Integral, Stokastik, Option, Metode Stokastik Dalam Integral
In evaluating the value of integration which can not be found exactly, we can implement numerical approach. The numerical method is categorized into deterministic and stochastic. Application of stochastic method in integration can be used for evaluating price of an European call option and European put option which approach the result of Black-Scholes formula.
European call option and European put option price is based on assumption that stock price follows a Geometric Brownian Motion with parameters r, , S, Sp, and T. Starting from a basic formula, the price of an European call option P= exp(-rt) E(max(ST –Sp,0)) and European put option P=exp(-rt)E( max(Sp - ST, 0)) are modified into multivariable integral form. Then, using random numbers generated in MATLAB and implementing stochastic method , the integral can be evaluated to approach option price.
The graph shows, the higher the amount of random number used in stochastic method in integration then error is smaller. Convergence of percentage error is proportional to convergence of 1/N.
Key Words : Integral, Stokastik, Option, Metode Stokastik Dalam Integral
COMPUTATION OF STATISTICS FORMULA WITHOUT DUMMY USING SPREADSHEET PACKAGE SOFTWARE
By Made Wijana
wijanaimade@yahoo.com
ABSTRACT
Generally, statistics formula involving data summation of a random variable, product of a random variable, and summation of product several random variables and function of random variable. In implementing statistics formula in a worksheet of spreadsheet package software like Microsoft Excel we often need a dummy which can be a cell or collection of cells (range).
This paper discuss computation of statistics formula in a worksheet of spreadsheet package software like Microsoft Excel without a dummy which can be a cell or collection of cells (range). Computation of statistics formula in a worksheet of spreadsheet package software like Microsoft Excel can be done without a dummy by utilization function SUMPRODUCT.
Computation of statistics formula in without a dummy doesn’t need more cell or range as well as the size of file is smaller. However, the computation need a long formula and often it is quite complicated with functions which is possible that a function nest in other function.
.
Key Word: Statistics formula computation, Computation without a dummy, Computation in a spreadsheet
wijanaimade@yahoo.com
ABSTRACT
Generally, statistics formula involving data summation of a random variable, product of a random variable, and summation of product several random variables and function of random variable. In implementing statistics formula in a worksheet of spreadsheet package software like Microsoft Excel we often need a dummy which can be a cell or collection of cells (range).
This paper discuss computation of statistics formula in a worksheet of spreadsheet package software like Microsoft Excel without a dummy which can be a cell or collection of cells (range). Computation of statistics formula in a worksheet of spreadsheet package software like Microsoft Excel can be done without a dummy by utilization function SUMPRODUCT.
Computation of statistics formula in without a dummy doesn’t need more cell or range as well as the size of file is smaller. However, the computation need a long formula and often it is quite complicated with functions which is possible that a function nest in other function.
.
Key Word: Statistics formula computation, Computation without a dummy, Computation in a spreadsheet
COMPARISON OF SAVINGS WITH INTEREST BASED ON THE LOWEST BALANCE, AVERAGE BALANVE, AND DAILY BALANCE
By Made Wijana
Wijanaimade@yahoo.com
ABSTRACT
Generally, interest of money saved in a financial institution which is computed every month depends on balance of transactions and interest rate. In computing the interest, nowadays we recognize three interest computation methods those are: interest based on the lowest balance, average balance and daily balance.
The methods are commonly implemented in financial institutions especially in banks. They have different properties as well as give different results. Most debtors do not understand interest computation of their savings in financial institutions. They do not know interest rate and interest computation method implemented by financial institutions.
This paper discusses interest computation methods of savings especially comparing their properties and results. Spreadsheet package program was used to find the results.
In conclusions, the value of interest based on the lowest balance(ILB) is less sensitive to the value of row balances than the value of interest based on average balance(IAB) and interest based on daily balance(IDB). The value of IAB is the most sensitive to the fluctuation of interest rates. Computation of ILB is the simplest as it does not depend on all balances. Computation of IAB depends on balances of all rows however it does not cover interest rate fluctuations. Meanwhile, computation of IDB covers all row balances and the fluctuation of interest rates as well as able to implement multilevel interest rate although it is complicated. If the same interest rate is implemented to all methods, ILD gives the lowest result than IAB and IDB. The results of IAB and IDB will be the same if the interest rate is not fluctuated, conversely their results will be different.
Keywords: Saving, Interest computation, Lowest balance, Average balance, Daily balance
Wijanaimade@yahoo.com
ABSTRACT
Generally, interest of money saved in a financial institution which is computed every month depends on balance of transactions and interest rate. In computing the interest, nowadays we recognize three interest computation methods those are: interest based on the lowest balance, average balance and daily balance.
The methods are commonly implemented in financial institutions especially in banks. They have different properties as well as give different results. Most debtors do not understand interest computation of their savings in financial institutions. They do not know interest rate and interest computation method implemented by financial institutions.
This paper discusses interest computation methods of savings especially comparing their properties and results. Spreadsheet package program was used to find the results.
In conclusions, the value of interest based on the lowest balance(ILB) is less sensitive to the value of row balances than the value of interest based on average balance(IAB) and interest based on daily balance(IDB). The value of IAB is the most sensitive to the fluctuation of interest rates. Computation of ILB is the simplest as it does not depend on all balances. Computation of IAB depends on balances of all rows however it does not cover interest rate fluctuations. Meanwhile, computation of IDB covers all row balances and the fluctuation of interest rates as well as able to implement multilevel interest rate although it is complicated. If the same interest rate is implemented to all methods, ILD gives the lowest result than IAB and IDB. The results of IAB and IDB will be the same if the interest rate is not fluctuated, conversely their results will be different.
Keywords: Saving, Interest computation, Lowest balance, Average balance, Daily balance
Trigonometry For High School
Trigonometry (from the Greek trigonon = three angles and metro = measure) is a part of elementary mathematics dealing with angles, triangles and trigonometric functions such as sine (abbreviated sin), cosine (abbreviated cos) and tangent (abbreviated tan). It has some connection to geometry, although there is disagreement on exactly what that connection is; for some, trigonometry is just a section of geometry.
Overview and definitions in Trigonometry
A standard right triangle.
Trigonometry uses a large amount of specific words to describe parts of a triangle. Some of the definitions in trigonometry are:
Right triangle - A right triangle is a triangle that has one angle that is equal to 90 degrees. (A triangle can not have more than one right angle.) The standard trigonometric ratios can only be used on right triangles.
Hypotenuse - The hypotenuse of a triangle is the longest side, and the side that is opposite the right angle. For example, for the triangle on the right, the hypotenuse is side c.
Opposite of an angle - The opposite side of an angle is the side that does not intersect with the vertex of the angle. For example, side a is the opposite of angle A in the triangle to the right.
Adjacent of an angle - The adjacent side of an angle is the side that intersects the vertex of the angle but is not the hypotenuse. For example, side b is adjacent to angle A in the triangle to the right.
There are three main trigonometric ratios for right triangles, and three reciprocals of those ratios. There are 6 total ratios. They are:
Sine (sin) - The sine of an angle is equal to the opposite/hypotenuse
Cosine (cos) - The cosine of an angle is equal to the adjacent/hypotenuse
Tangent (tan) - The tangent of an angle is equal to the opposite/adjacent
The reciprocals of these ratios are:
Cosecant (csc) - The cosecant of an angle is equal to the hypotenuse/opposite or 1/sin
Secant (sec) - The secant of an angle is equal to the or hypotenuse/adjacent
Cotangent (cot) - The cotangent of an angle is equal to the opposite/adjacent or 1/tangent
The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, such as SOH-CAH-TOA:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
Diposkan oleh School Maths di 19:00 0 komentar
Geometry For Elementary School
Geometry
Geometry is a kind of mathematics used to work with shapes.
Examples of Shapes
There are flat shapes and solid shapes in geometry. Squares, circles and triangles are some of the simplest shapes in flat geometry. Cubes, cylinders, cones and spheres are simple shapes in solid geometry.
Measuring in Geometry
Geometry can be used to measure a flat shape's area and perimeter.It can also be used to measure a solid shape's volume and surface area.
Many things have the shapes found in geometry. Geometry can be used to measure many things by seeing them as made of geometrical shapes. For example, geometry can help people find:
the surface area of a house, so they can buy the right amount of paint
the volume of a box, to see if it is big enough to hold a litre of food
the area of a farm, so it can be divided into equal parts
the distance around the edge of a pond, to know how much fencing to buy.
Some Simple Ideas in Geometry
In mathematics, geometry starts with a few simple ideas:
A point is shown on paper by touching it with a pencil or pen, without making any sideways movement. We know where the point is, but it has no size.
A straight line is the shortest distance between two points. For example, Sophie pulls a piece of string from one point to another point. A straight line between the two points will follow the path of the tight string.
A plane is flat surface that does not stop in any direction. A ball placed any place on this flat surface will not move if gravity on the surface is constant.
Triangle
A triangle is a shape. It has three straight sides and three points. The three angles of a triangle add to 180 degrees. It is the polygon with the least possible number of sides.
Triangles can be grouped according to how long their sides are
In an equilateral triangle all three sides have the same length
In an isosceles triangle two sides have the same length
In a scalene triangle all sides have different lengths
Triangles can also be grouped by their angles.
A right triangle has one angle that is 90 degrees (a right angle). The side opposite the right angle is the hypotenuse.
An obtuse triangle has one angle that is larger than 90 degrees (an obtuse angle)
An acute triangle has angles that are all less than 90 degrees (acute angles)
Perimeter
In geometry, perimeter is the distance around a flat object. For example, all four sides of a square rhombus have the same length, so a rhombus with side length 2 inches would have a perimeter of 8 inches (2+2+2+2=8).
Real-life objects have perimeters as well. A football field, including the end zones, is 360 feet long and 160 feet wide. So the perimeter of the field is 360+160+360+160=1040 feet.
The perimeter of a circle is usually called the circumference. It may be calculated by multiplying the diameter times "Pi". Pi is a constant which is equal to 3.14159; however, the places to the right of the decimal are endless. The number of places used depend on the accuracy required for the result.
A right triangle, (also called a right-angled triangle), has one angle that is 90 degrees. The other two angles always add up to 90 degrees but can be different sizes. The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
Examples of shapes
2D shapes
circles
squares
triangles
ovals
kites
These are two dimensional shapes or flat plane geometry shapes. They can look like anything and can have any number of sides. The sides can be straight or curved. Triangles and squares are polygons.
The sides of these shapes are lines.
3D shapes
spheres
cubes
cones
pyramids
These are three dimensional shapes. The sides of these shapes are surfaces. Again, the sides can be straight or curved
Overview and definitions in Trigonometry
A standard right triangle.
Trigonometry uses a large amount of specific words to describe parts of a triangle. Some of the definitions in trigonometry are:
Right triangle - A right triangle is a triangle that has one angle that is equal to 90 degrees. (A triangle can not have more than one right angle.) The standard trigonometric ratios can only be used on right triangles.
Hypotenuse - The hypotenuse of a triangle is the longest side, and the side that is opposite the right angle. For example, for the triangle on the right, the hypotenuse is side c.
Opposite of an angle - The opposite side of an angle is the side that does not intersect with the vertex of the angle. For example, side a is the opposite of angle A in the triangle to the right.
Adjacent of an angle - The adjacent side of an angle is the side that intersects the vertex of the angle but is not the hypotenuse. For example, side b is adjacent to angle A in the triangle to the right.
There are three main trigonometric ratios for right triangles, and three reciprocals of those ratios. There are 6 total ratios. They are:
Sine (sin) - The sine of an angle is equal to the opposite/hypotenuse
Cosine (cos) - The cosine of an angle is equal to the adjacent/hypotenuse
Tangent (tan) - The tangent of an angle is equal to the opposite/adjacent
The reciprocals of these ratios are:
Cosecant (csc) - The cosecant of an angle is equal to the hypotenuse/opposite or 1/sin
Secant (sec) - The secant of an angle is equal to the or hypotenuse/adjacent
Cotangent (cot) - The cotangent of an angle is equal to the opposite/adjacent or 1/tangent
The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, such as SOH-CAH-TOA:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
Diposkan oleh School Maths di 19:00 0 komentar
Geometry For Elementary School
Geometry
Geometry is a kind of mathematics used to work with shapes.
Examples of Shapes
There are flat shapes and solid shapes in geometry. Squares, circles and triangles are some of the simplest shapes in flat geometry. Cubes, cylinders, cones and spheres are simple shapes in solid geometry.
Measuring in Geometry
Geometry can be used to measure a flat shape's area and perimeter.It can also be used to measure a solid shape's volume and surface area.
Many things have the shapes found in geometry. Geometry can be used to measure many things by seeing them as made of geometrical shapes. For example, geometry can help people find:
the surface area of a house, so they can buy the right amount of paint
the volume of a box, to see if it is big enough to hold a litre of food
the area of a farm, so it can be divided into equal parts
the distance around the edge of a pond, to know how much fencing to buy.
Some Simple Ideas in Geometry
In mathematics, geometry starts with a few simple ideas:
A point is shown on paper by touching it with a pencil or pen, without making any sideways movement. We know where the point is, but it has no size.
A straight line is the shortest distance between two points. For example, Sophie pulls a piece of string from one point to another point. A straight line between the two points will follow the path of the tight string.
A plane is flat surface that does not stop in any direction. A ball placed any place on this flat surface will not move if gravity on the surface is constant.
Triangle
A triangle is a shape. It has three straight sides and three points. The three angles of a triangle add to 180 degrees. It is the polygon with the least possible number of sides.
Triangles can be grouped according to how long their sides are
In an equilateral triangle all three sides have the same length
In an isosceles triangle two sides have the same length
In a scalene triangle all sides have different lengths
Triangles can also be grouped by their angles.
A right triangle has one angle that is 90 degrees (a right angle). The side opposite the right angle is the hypotenuse.
An obtuse triangle has one angle that is larger than 90 degrees (an obtuse angle)
An acute triangle has angles that are all less than 90 degrees (acute angles)
Perimeter
In geometry, perimeter is the distance around a flat object. For example, all four sides of a square rhombus have the same length, so a rhombus with side length 2 inches would have a perimeter of 8 inches (2+2+2+2=8).
Real-life objects have perimeters as well. A football field, including the end zones, is 360 feet long and 160 feet wide. So the perimeter of the field is 360+160+360+160=1040 feet.
The perimeter of a circle is usually called the circumference. It may be calculated by multiplying the diameter times "Pi". Pi is a constant which is equal to 3.14159; however, the places to the right of the decimal are endless. The number of places used depend on the accuracy required for the result.
A right triangle, (also called a right-angled triangle), has one angle that is 90 degrees. The other two angles always add up to 90 degrees but can be different sizes. The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
Examples of shapes
2D shapes
circles
squares
triangles
ovals
kites
These are two dimensional shapes or flat plane geometry shapes. They can look like anything and can have any number of sides. The sides can be straight or curved. Triangles and squares are polygons.
The sides of these shapes are lines.
3D shapes
spheres
cubes
cones
pyramids
These are three dimensional shapes. The sides of these shapes are surfaces. Again, the sides can be straight or curved
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