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Численное исследование двух модельных финансовых задач с применением метода Ньютона.

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B. Ser-Od. Numerical investigation of two financial modeling problems with help of the Newton’s method
УДК 517.977
ББК 22.19
© B. Ser-Od
Mongolia, Ulaanbaatar, Mongolian University of Science and Technology
NUMERICAL INVESTIGATION
OF TWO FINANCIAL MODELING PROBLEMS
WITH HELP OF THE NEWTON’S METHOD
The article is devoted to numerical investigation of two financial modeling problems via the Newton’s
method.
Key words: differential equation, the Newton’s method, financial modeling problems.
© Б. Сер-Од
Монголия, Улан-Батор, Монгольский (государственный) университет науки и технологии
ЧИСЛЕННОЕ ИССЛЕДОВАНИЕ
ДВУХ МОДЕЛЬНЫХ ФИНАНСОВЫХ ЗАДАЧ
С ПРИМЕНЕНИЕМ МЕТОДА НЬЮТОНА1
Статья посвящена численному изучению двух модельных задач финансовой математики при
помощи метода Ньютона.
Key words: differential equation, the Newton’s method, financial modeling problems.
Introduction
Most introductory numerical analysis texts have a problem on solving nonlinear equations. An
excellent and up-to-date specialist treatment that includes MATLAB codes is (Kelley, 1995). The
convergence analysis for Newton's method is based on the article (Manaster and Koehler, 1982).
It is also mentioned in (Kwok,1998).
1. First problem: Option value as a function of volatility
Since the formulation by Black and Scholes the idea of using stochastic calculus for modeling
prices of risky assets in financial investment has been generally accepted.
A call option gives its holder the right to purchase from the writer a prescribed asset for a prescribed price at a prescribed time in the future. We say the value of the call option at the expiry
date, denoted by C, is C = max( E − S (T ), 0).
The direct opposite of a call option is a put option. The value of the put option at the expiry
date, denoted by P, is P = max( S (T ) − E , 0) , where E – exercise price, S(t) – asset price, T-t –
expiry date, C – value of call, P – value of put, r – the risk free interest rate, V(S, t) – function
gives option value for any asset price S > 0 at any time 0< t < T.
The Black-Scholes partial differential equation is
∂V 1 2 2 ∂ 2V
∂V
+ σ S
+ rS
− rV = 0 .
(1)
2
∂t 2
∂S
∂S
The solution is
C ( S , t ) = S ⋅ N (d1 ) − E ⋅ e− r (T −t ) N (d 2 )
(2)
and
P ( S , t ) = E ⋅ e − r (T −t ) N (− d 2 ) − S ⋅ N (− d1 ) .
(3)
The Black-Scholes call and put values depend on S, E, r, T-t and σ 2 . Of these five quantities,
only the asset volatility σ cannot be observed directly, σ ≥ 0 the volatility is a constant parameter that determines the strength of the random fluctuations.
We now put a Newton's method to work on the problem of computing the implied volatility.
How do we find a suitable value for σ ? One approach is to extract the volatility from the ob1
Работа выполнена при финансовой поддержке РФФИ (09-01-90203-Монг-а).
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ВЕСТНИК БУРЯТСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2009/9
served market data - given a quoted option value, and knowing S, t, E, r and T, find the σ that
leads to this value. Having found σ , we may use the Black-Scholes formula to value other options on the same asset. A σ computed this way is known as an implied volatility. The name indicates that σ is implied by option value data in the market. We focus here on the case of extracting σ from a European call option quote. An analogous treatment can be given for a put, or, alternatively, the put quote could be converted into a call quote via put-call parity.
We assume that the parameters E, r and T and the asset price S and time t are known. Given a
quoted value C*, our task is to find the implied volatility σ * that solves C( σ ) = C*. Computing
the implied volatility requires the solution of a nonlinear equation and we may use the bisection
method or Newton’s method. We will find that it is possible to exploit the special form of the nonlinear equation arising in this context.
Since volatility is non-negative, only values σ ∈ [ 0, ∞[ are of interest. Let us look at C( σ ) in
the case of large or small volatility.
∂ 2C S T − t − 12 d12 d1d 2 d1d 2 ∂C
=
e
=
.
(4)
∂σ 2
σ
σ ∂σ
2π
Where
1
1
log( S / E ) + (r + σ 2 )(T − t )
log( S / E ) + (r − σ 2 )(T − t )
2
2
d1 =
, d2 =
, d 2 = d1 − σ T − t .
σ T −t
σ T −t
It follows from (4) that
∂C
is maximum over [ 0, ∞[ at σ = σˆ , where
∂σ
σˆ := 2
log S / E + r (T − t )
.
T −t
(5)
∂ 2C
may be written in the form:
∂σ 2
∂ 2C T − t 4
∂C
=
.
(6)
(σˆ − σ 4 )
2
3
∂σ
4σ
∂σ
The identity (6) shows us that C( σ ) is convex for σ < σˆ and concave for σ > σˆ . This will allow us to get a globally convergent Newton iteration by suitably choosing the starting value.
We will write our nonlinear equation for σ * in the form F( σ ) = 0, where F( σ ):=C( σ ) – C*.
Newton's method takes the form
F (σ )
σ n+1 = σ n − / n .
(7)
F (σ n )
Moreover,
Where F / (σ ) is given by F / (σ ) = ∂C / ∂σ .
Moreover, we can write
σ −σ *
0 < n +1 * < 1 for all n > 0.
(10)
σn −σ
Therefore, the error decreases monotonically as n increases. In a similar manner, it can be
shown that (10) holds in the case where σˆ > σ * . Overall, we conclude that with the choice
σ 0 = σˆ the error will always decrease monotonically as n increases. Hence, σ 0 = σˆ is a proof
starting value for Newton's method on this particular nonlinear equation. This is therefore our method of choice for computing the implied volatility.
102
B. Ser-Od. Numerical investigation of two financial modeling problems with help of the Newton’s method
Consider we know exercise price, asset price, expiry date and interest rate. We shall to calculate volatility for Newton’s method by Matlab programs.
Let r = 0.03, S = 200, E = 200 and T = 3.
So, we can be found that a volatility is 64.3817 for our choice, following:
The widely reported phenomenon that the implied volatility is not constant as other parameters
are varied does, of course, imply that the Black-Scholes formulas fail to describe perfectly the option values that arise in the marketplace. This should be no surprise, given that the theory is based
on a number of simplifying assumptions.
2. Second problem: Financial investment
Modern finance theory is based on the principles of compound interest. With compound interest the amount of interest earned is set at the same percentage for the term of the loan, but the
amount of the interest for the second and subsequent time periods is calculated on the original deposit plus the accumulated interest to date. We will calculate the sum that needs to be invested
now (PV) in order to accumulate to a required future amount (FV). The relationship previously
used was
FV = PV ⋅ (1 + i ) n .
Rearrange to give:
FV
PV =
.
(1 + i ) n
The interest rate is sometimes knows as the cost of capital and can be considered to be the cost
borrowing money by the business or the rate of return that an investor may obtain if the money is
invested with security.
The cash flow formula for net present value which is
CFn
CF1
CF2
+
+ ... +
,
NPV = −CF0 +
1
2
(1 + i ) (1 + i )
(1 + i ) n
103
ВЕСТНИК БУРЯТСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2009/9
where NPV – net present value, CF0 – the cost of the investment outlay at commencement, being
a negative cash flow (cost of initial investment), CFk – cash flow at the end of period k, 1 ≤ k ≤ n ,
n – the last cash flow, usually including the reversion value, i – the discount rate.
Discount rate are the same as interest rates but the expression ‘discount rate’ is used where the
problem involves discounting rather than compounding. The internal rate of return is the periodic
average rate of return from the financial investment. The IIR is defined that equates the present
value of the expected future cash flows to the initial capital invested.
The IIR is the discount rate that results in a net present value of zero.
How calculate the IRR?
n
CFk
. Considering a new function F, which is the following
Hence, NPV = −CF0 + ∑
k
k =1 (1 + i )
n
CFk
F (i ) = −CF0 + ∑
− NPV .
k
k =1 (1 + i )
The equation F(i) = 0 is a nonlinear equation for i value.
CFk
then function F(i) is,
If we choose ak (i ) :=
(1 + i ) k
n
F (i ) = −CF0 + ∑ ak (i ) − NPV .
k =1
1
1
and ak" (i ) = k (k + 1) ⋅ CFk
>0.
k +1
(1 + i )
(1 + i )k + 2
Therefore, ak (i ) is convex for all k (sum convex functions F(i) is a convex),
It is easy to see that ak' (i ) = − k ⋅ CFk
n
F (i ) = −CF0 + ∑ ak (i ) − NPV = 0 .
k =1
So, we can be found a minimum value for i.
We have a data for efficiency of investment (NPV) and cash flow of Mongolian National company APU. We consider that cash flow is constant since 2007 years. This company invested
8275.2 million tugrugs in 2003.
Year
2004
2005
2006
2007
2008
2009
2010
2011
2012
CF
339.3 million
526 million
1520.7 million
2967 million
2967 million
2967 million
2967 million
2967 million
2967 million
NPV
-7980.16
-7582.42
-6582.54
-4884.15
-3411.02
-2128.31
-1012.9
-42.986
800.4205
If known a cash flow in 2004 – 2012 years and interest rate (r = 0.15), then NPV is,
9
339.3
526.0
1520.7
2967.0
NPV = −8275.2 +
+
+
+
= 800.4205
∑
2
3
1 + 0.15 (1 + 0.15) (1 + 0.15) k = 4 (1 + 0.15)k
n
CFk
million tugrugs, where CF0 is 8275.2. So, F (i ) = −8275.2 + ∑
− 800.4205 = 0 .
k
k =1 (1 + i )
We shall find a minimum percentage of the profit. If in is converges to a solution i* then
F (i )
in +1 = in − / n .
F (in )
104
B. Ser-Od. Numerical investigation of two financial modeling problems with help of the Newton’s method
Let be starting value is 15%.
Apply Newton’s method to previous real data which is the following:
The solution is minimum percentage of profit, so that 0.1281. Other side this solution is internal rate of return.
References
1. D.J. Higman. An introduction to financial option valuation Cambridge University press. –
2004.– Vol. 13. – P.123-139.
2. T. Mikosch. Elementary stochastic calculus. – University Copenhagen Denmark press, 1998. –
P. 167-181.
105
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