CC BY
38
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.
Статья посвящена численному изучению двух модельных задач финансовой математики при помощи метода Ньютона.
Key words: differential equation, the Newton’s method, financial modeling problems.
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).
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
only the asset volatility s cannot be observed directly, s > 0 the volatility is a constant parameter that determines the strength of the random fluctuations.
© Б. Сер-Од
Монголия, Улан-Батор, Монгольский (государственный) университет науки и технологии
ЧИСЛЕННОЕ ИССЛЕДОВАНИЕ ДВУХ МОДЕЛЬНЫХ ФИНАНСОВЫХ ЗАДАЧ С ПРИМЕНЕНИЕМ МЕТОДА НЬЮТОНА1
Introduction
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
- rV = О .
(1)
The solution is
C (S, t) = S • N (d1) - E ■ є~ r (T-t) N (d2)
(2)
and
P(S, t) = E ■ e-r (T-t) N (-d2) - S ■ N (-d1). (3)
The Black-Scholes call and put values depend on S, E, r, T-t and s2. Of these five quantities,
(3)
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 s? One approach is to extract the volatility from the ob-
1 Работа выполнена при финансовой поддержке РФФИ (О9-О1-9О2О3-Монг-а).
101
served market data - given a quoted option value, and knowing S, t, E, r and T, find the s that leads to this value. Having found s , we may use the Black-Scholes formula to value other options on the same asset. A s computed this way is known as an implied volatility. The name indicates that s is implied by option value data in the market. We focus here on the case of extracting s 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 s* that solves C(s) = 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 se [0, ¥ are of interest. Let us look at C(s) in the case of large or small volatility.
d2C SyfT-t -k2 d,d2 d,d2 dC
77 = n— e 2 = —^. (4)
asz V2p s s ds
Where
log(S / E) + (r + -S )(T -1) log(S / E) + (r - -S )(T -1) _____
d =-----------------, 2-----------, d2 =----------------------------------, 2- , d2 = dl -svT-1.
sv T -1 аы T -1
9C
It follows from (4) that-------is maximum over [0, ¥ at s = (J, where
ds
а =
log S / E + r (T -1)
T -1
(5)
Э 2C ds2
Moreover, -—- may be written in the form:
dC = T-l s4 -s4 dC (6)
ds2 4s3 ( )ds' ()
The identity (6) shows us that C( s) is convex for s <(7 and concave for s > (J. This will allow us to get a globally convergent Newton iteration by suitably choosing the starting value.
We will write our nonlinear equation for s * in the form F(s) = 0, where F( s ):=C( s) - C .
Newton's method takes the form
Where F1 (s) is given by F1 (s) = dC / ds. Moreover, we can write
*
0 < Sn+1 S < 1 for all n > 0. (10)
s„-s
n
Therefore, the error decreases monotonically as n increases. In a similar manner, it can be shown that (10) holds in the case where s > s*. Overall, we conclude that with the choice s0 = s the error will always decrease monotonically as n increases. Hence, s0 =<y 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.
2
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:
UBB
File
Edit View Text Debug Breakpoints Web Window Help
h Computes implied volatility for a European call
%%%%%%%%%%%%%%%%%%%% parameters %%%%%%%%%%%%%%%%%%%%
r=0.03; % interest rate
S=200; % asset price
E=200; % exercise price
T=3; % expiry date
tau=T;
sigmatrue=0.3; % starting value
[Ctrue,Cdelta,P,Pdelta]=ch08(S,E,r,sigmatrue,tau);
sigmahat=sqrt[2*ai>s [ [log[S/E)+r*T) /T)); %^%%%%%%%%%%%%%%%% Newton's me tod %%%%%%%%%%%%%%%%%% tol=le-8; sigma=sigmahat; sigmadiff=1; k=l;
kmax=100;
while (sigmadiff>=tol & k<kmax)
[C,Cdelta,Cvega,P,Pdelta,Pvega]=chl0(S,E,r,sigma, increment=(Ctrue-C)/Cvega; sigma=sigma-increment; k=k+l;
sigmadiff=abs(increment);
end
sigma
File Edit View Web Window Help
□ | & 1 £ 1 9 | Currer .
» C =
0.0691
Cdelta =
0.2586
P =
0.4953
Pdelta =
-0.7414
sigma =
64.3817
» ’i
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
CF CF CF
NPV = -CF0 +-CF- + -CF^ +... + - C n
(1 +1)1 (1 +1)2 (1 + i)n
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 CF
Hence, NPV = -CF0 + V---------kr . Considering a new function F, which is the following
k=1 (1 +i)
n CF
F (0=-CF0+V jrFt -NPV. k=1 (1 + i)
The equation F(i) = 0 is a nonlinear equation for i value.
CF
If we choose ak (i) := ——kk then function F(i) is,
(1+o*
F(i) = -CFo + £ a„ (i) - NPV.
k=1
It is easy to see that ak(i) = -k • CFk (1 ■)k+1 and ak(i) = k(k + 1) ' CFk (1 •)k+2 > °.
(1 + i) (1 + i)
Therefore, ak (i) is convex for all k (sum convex functions F(i) is a convex),
n
F (i) = -CFa + ^ a„ (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 CF NPV
2004 339.3 million -7980.16
2005 526 million -7582.42
2006 1520.7 million -6582.54
2007 2967 million -4884.15
2008 2967 million -3411.02
2009 2967 million -2128.31
2010 2967 million -1012.9
2011 2967 million -42.986
2012 2967 million 800.4205
If known a cash flow in 2004 - 2012 years and interest rate (r = 0.15), then NPV is,
„nc „ 339.3 526.0 1520.7 ^ 2967.0
NPV = -8275.2 +-----------------+------------ +------------ + V-------------- = 800.4205
1 + 0.15 (1 + 0.15)2 (1 + 0.15)3 k=4(1 + 0.15)k
n CF
million tugrugs, where CF0 is 8275.2. So, F (i) = -8275.2 + V---------------- 800.4205 = 0.
k=1 (1 +i)
We shall find a minimum percentage of the profit. If in is converges to a solution i* then
i = i - Fern.
n+1 n j—i / / • \
F (in )
Let be starting value is 15%.
Apply Newton’s method to previous real data which is the following:
C: \WI NDOWS \Desktop\NEWT 0 N 2. M '
File Edit View Web Window Help
le Edit View Text Debug Breakpoints Web Window Help 3 G? B I £ % fe «■> <-* I sllll /.I © © I B1 ® Hf |№ ^ ° 01 I % I 7 I Current CH
1
2
3
4
5
6
7
8 9
10
11
12
13
14
15
16
17
18
19
20 21 22
23
24
25
26 27
method, to NPV=-CF (0) +sum (CF (k) / (1+i) % initial value
first payment second payment thirth payment cost of initial inves
other payment
%Apply Newton i(1)=0.16; idiff=1; k=l;
kmax=8;
tol=le-4;
CF(1)=339.3; %
CF[2)=526; %
CF (3)=1520.7; %
№7=800.4205; % CF(0)=8275.2
for j=4:kmax
CF[ j)=2967; %
end
%%%%%%%%%%%%%%%%%% Newton's method while (idiff>=tol & k<kmax);
Fval=CF(l)/(l+i(k)J-8275.2;
Fprime=-2*CF(1)/[l+i(k))A(-2); for j =2:kmax
Fval=Fval+CF(j)/((1+i(k))Aj);
Fprime=Fprime-(j+1)*CF(j)/((1+i(k))A(j+1));
end
increment=(Fval-NPV)/Ferine;
%i(k+1)=i(k)+increment ; i(k+1)=i(k)-increment; k=k+l;
% idiff=ahs(increment); idiff=ahs(i(k)-i(k-1)); end
i(k) % discount rate
■■k))
» k =
0.1281
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.
