Adapted downhill simplex method for pricing convertible bonds
The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and minimax optimization. Stock prices (underlying asset) were modeled under the as...
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irk-123456789-45172009-11-25T12:00:30Z Adapted downhill simplex method for pricing convertible bonds Mishchenko, K. Mishchenko, V. Malyarenko, A. The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and minimax optimization. Stock prices (underlying asset) were modeled under the assumption of the geometric Brownian motion of their values. The Monte Carlo method was used for calculating the real payoff which is the objective function. The minimax optimization problem was solved using the derivative-free Downhill Simplex method. The performed numerical experiments allowed to formulate recommendations for the choice of appropriate size of the initial simplex in the Downhill Simplex Method, the number of generated trajectories of underlying asset, the size of the problem and initial trajectories of the behavior of investors and issuers. 2007 Article Adapted downhill simplex method for pricing convertible bonds / K. Mishchenko, V. Mishchenko, A. Malyarenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 130–147. — Бібліогр.: 5 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4517 en Інститут математики НАН України |
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The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and minimax optimization.
Stock prices (underlying asset) were modeled under the assumption of the geometric Brownian motion of their values. The Monte Carlo method was used for calculating the real payoff which is the objective function. The minimax optimization problem was solved using the derivative-free Downhill Simplex method.
The performed numerical experiments allowed to formulate recommendations for the choice of appropriate size of the initial simplex in the Downhill Simplex Method, the number of generated trajectories of underlying asset, the size of the problem and initial trajectories of the behavior of investors and issuers. |
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Mishchenko, K. Mishchenko, V. Malyarenko, A. |
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Mishchenko, K. Mishchenko, V. Malyarenko, A. Adapted downhill simplex method for pricing convertible bonds |
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Mishchenko, K. Mishchenko, V. Malyarenko, A. |
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Mishchenko, K. |
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Adapted downhill simplex method for pricing convertible bonds |
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Adapted downhill simplex method for pricing convertible bonds |
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Adapted downhill simplex method for pricing convertible bonds |
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Adapted downhill simplex method for pricing convertible bonds |
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Adapted downhill simplex method for pricing convertible bonds |
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adapted downhill simplex method for pricing convertible bonds |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4517 |
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Adapted downhill simplex method for pricing convertible bonds / K. Mishchenko, V. Mishchenko, A. Malyarenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 130–147. — Бібліогр.: 5 назв.— англ. |
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AT mishchenkok adapteddownhillsimplexmethodforpricingconvertiblebonds AT mishchenkov adapteddownhillsimplexmethodforpricingconvertiblebonds AT malyarenkoa adapteddownhillsimplexmethodforpricingconvertiblebonds |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.130–147
KATERYNA MISHCHENKO, VOLODYMYR MISHCHENKO AND
ANATOLIY MALYARENKO
ADAPTED DOWNHILL SIMPLEX METHOD FOR
PRICING CONVERTIBLE BONDS
The paper is devoted to modeling optimal exercise strategies of the
behavior of investors and issuers working with convertible bonds.
This implies solution of the problems of stock price modeling, payoff
computation and minimax optimization.
Stock prices (underlying asset) were modeled under the assumption
of the geometric Brownian motion of their values. The Monte Carlo
method was used for calculating the real payoff which is the objective
function. The minimax optimization problem was solved using the
derivative-free Downhill Simplex method.
The performed numerical experiments allowed to formulate recom-
mendations for the choice of appropriate size of the initial simplex in
the Downhill Simplex Method, the number of generated trajectories
of underlying asset, the size of the problem and initial trajectories of
the behavior of investors and issuers.
1. Introduction and Problem Formulation
Convertible Bonds One type of securities at modern financial market
is a convertible bond. They belong to most popular securities of modern
financial market. This type of securities is of interest for both small devel-
oping companies for attracting investments and for investors, since for the
latter such bonds are highly profitable.
Strategies of the behavior of investors and issuers for early exercise de-
cision must be chosen in such a way that the issuers’ payoff will be minimal
while the investors’ profit will be maximal.
A standard convertible bond is a bond that gives the holder (investor)
the right to exchange (convert) it into a predetermined number of stock
during a certain, predetermined period of time [1].
2000 Mathematics Subject Classifications. 62P05, 65K10, 91B28, 90C47.
Key words and phrases. Convertible bonds, Monte Carlo simulation, optimal strate-
gies, Downhill Simplex method, minimax optimization problem
130
ADAPTED DOWNHILL SIMPLEX METHOD 131
Convertible bonds are characterized by the following options:
The Issuers Options
1. Call price Kt
This option allows the issuer to call back the convertible bond at the
time t with the payment Kt to the investor.
2. Call Notice Period timenotice
Before calling back the convertible bond the issuer announces his
intend and can call convertible bond only after the call notice pe-
riod timenotice. During this period the investor may convert the bond
(”force conversion”).
The Investors Options
1. Number of stocks n
The investor may convert the bond into n stocks at any time during
the predetermined period.
2. Put price Pt
This option allows the investor to sell the convertible bond at the
price Pt during the predetermined period.
Common Options of Convertible Bonds and Stocks
1. Maturity Time T
The expiry time of a convertible bond.
2. Face Value of Convertible Bond N
The predetermined price of a convertible bond which the issuer will
pay to the investor at maturity time.
3. Redemption Ratio κ
This is a preset percentage of the face value of a convertible bond
which increases the price of the face value. So, the κN instead of N
may be paid by the issuer to the investor. Usually κ is equal to 1.
4. Price of Coupon Bond Bt
This is the premium paid by the issuer to the investor at some fixed
time moments during the preset period.
5. Value of continuation Vt
This is the price of a convertible bond at every time moment during
the period when this convertible bond is alive. Vt is valued by the
amount of money which the owner may get by converting the bond
into stocks.
132 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
6. Current Stock price St
This is the current price of stocks owned by the issuer.
7. One Option
The issuer or investor can perform only one action with the convert-
ible bond, i.e. if the investor converts the bond the latter expires, as
well as if the issuer wants to call the bond this cannot be stopped.
Finally, the initial conditions at the time t = 0 are:
1. Kt > N
The call price of a convertible bond is greater than its face value.
2. Pt < N
The put price of a convertible is less than its face value.
3. n = N/(S0 · η), where η > 1.
n is the number of stocks obtained by conversion of a bond. This
value is calculated under assumption that the stock price at the initial
moment is greater than the real one.
Problem Formulation Objective function is as a payoff gained by the
investor. This means that the payoff is a nonnegative number. It is clear
that the investor wants to maximize the payoff, while the the issuer wants
to minimize it. This payoff is based on the behaviors of the investor and
the issuer and on current stock price at the time t.
The problem under consideration is to find such strategies for the investor
Convt and the issuer Callt which maximize the investor’s payoff and min-
imize the payoff payed by the issuer to investor, simultaneously. In other
words, we have a minimax optimization problem for computing the issuer’s
and investor’s strategies.
max
Convt
min
Callt
payoff(Convt, Callt) (1)
The natural choice of the boundary conditions for the investor’s and issuer’s
strategies is the following:
Convt ≥ N (2)
Kt ≥ Callt ≥ N (3)
We consider (3) only under the condition T − t > timenotice because
otherwise the investor will not have enough time to realize his right to call
back a convertible bond.
ADAPTED DOWNHILL SIMPLEX METHOD 133
Additionally, we introduce some initial settings, where we assume that we
have a zero coupon convertible bond (no coupon payment during the preset
period is done: Bt = 0).
Also, we use the following preset parameters:
N = constant
κ = 1
Kt = constant, Kt > N
S0 = constant. (S0 · n < N)
η > 1, say η = 1.1
n = N/(S0 · η)
Initial Guess: Convt = Kt + ε; Callt = N + ε
timenotice = const
The maximization problem (1) - (4) will be solved by the global Down-
hill Simplex method, see [4] and stock prices are modeled by Monte Carlo
simulations presented in [2].
The fact that such a problem can be solved numerically using this ap-
proach was shown in [3].
Section 2 of this paper is devoted to the description of the methods for
stock price generating, payoff computation, approximations of the strategy
of the behaviors of the investor and the issuer, as well as the description of
the Downhill Simplex method.
In Section 3 we present the results of the numerical experiments with
model under consideration. In these experiments we try to determine the
best values for such input parameters as size of the simplex in Downhill
Simplex method, the best choice for the initial trajectories, the optimal
number of trajectories used for the stock price generation and the appro-
priate number of the points for trajectories approximations.
We finalize our work by making conclusions and giving some guidelines
for further investigation in Section 4.
2. Numerical Issues
Stock Price Generating The first step in solving the problem (1) - (3)
is to generate stock prices. This can be done by different methods, and
we base our computation on the method producing the Brownian motion
134 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
type trajectories (see e.g. [2]). This method generates trajectories without
jumps.
The initial stock price S0 is given. The formula for generating the stock
price without jumps at t + 1 time moment is:
St+1 = St · e(r−δ−σ2/2)·�+σ·√�·Zt+1 (4)
where St is the stock price at the current time moment t; r is the interest
rate; δ is the dividend yield of the issuer stocks (underlying asset); σ is the
volatility; � = 1
250
is the time step at 250 working days a year and {Zt+1, t ≥
0} is a sequence of independent standard normal random variables.
Computation of the Payoff We use the Monte Carlo method for mod-
eling a real payoff. For this purpose we generate M, a large number of
trajectories, of the issuer stock prices according to (4), and then compute
payoffi for each of them according to Algorithm 1.
The put price Pt is less than the face value N . In the case of maximiza-
tion of the investor’s payoff we will not take into account Pt and suppose
that the investor does not use the possibility of choosing the put option.
Finally, the real payoff is:
payoff =
1
M
·
M∑
i=1
payoffi (5)
In this study we simplify the problem by setting r = const; δ = const
and σ = const.
Optimization procedure by Downhill Simplex Method As we solve
a minimax optimization problem (1), the optimization procedure is to be
applied twice at each iteration: once as a maximizer of the objective w.r.t.
the investor’s strategy and then as a minimizer w.r.t. the issuer’s strategy.
In other words, we find a pair of trajectories (Conv∗
t , Callt∗) which
satisfy the conditions:
payoff(Conv∗
t , Callt) = max
Convt
payoff(Convt, Callt) (6)
payoff(Conv∗
t , Callt∗) = min
Callt
payoff(Conv∗
t , Callt) (7)
The optimization problem under consideration is very computational
expensive due to the Monte Carlo simulations used for evaluation of the
objective function, thus we do not consider the optimization methods based
on derivatives computation.
To perform the optimization procedure in the most efficient way we use
a derivative-free Downhill Simplex method [4], which is also easy imple-
mentable.
ADAPTED DOWNHILL SIMPLEX METHOD 135
Algorithm 1 Payoff (objective function) Computation
payoff = 0
flagnotice = 0
timecheck = 0
for t = 0 : T (T = maturity · 250) do
if St > Convt then
payoff = St (conversion or force conversion during the notice pe-
riod)
break
end if
if flagnotice = 1 then
if timecheck = timenotice then
payoff = Kt (call)
break
else
timecheck = timecheck + 1
end if
else
if (St > Callt) and ((T − t) > timenotice) then
flagnotice = 1 (start call notice period)
end if
end if
if t = T then
payoff = N (the face value)
end if
end for
136 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
Below we give a description of this algorithm for minimizing some func-
tion f(x).
Firstly we need to specify the initial m + 1 dimensional simplex by
taking m points around the initial guess x1. The point with the highest
function value fmax is called xmax.
The main idea of the Downhill Simplex method is to substitute a point
with the coordinates xmax by another point with better function value (e.g.
get lower fmax in case of minimization problem). This is done by means of
Reflection, Expansion, Contraction and Multiple Contraction. Let consider
these functions shortly.
Firstly, we introduce the point xcenter (the center point of the simplex
for current iteration):
xcenter =
1
m + 1
m+1∑
i=1
xi (8)
1. Reflection
The point xmax is reflected into x∗
max so that it lies on the opposite
(to the point xmax) side of the line containing xmax and xcenter. The
distance between xcenter and x∗
max depends on a positive constant α -
reflection coefficient and is computed as
x∗
max = (1 + α) · xcenter − α · xmax (9)
2. Expansion
The expansion process is a prolongation of Reflection, and the new
point x∗∗
max is found as:
x∗∗
max = γ · x∗
max + (1 − γ) · xcenter (10)
where γ(γ > 1) is the expansion coefficient.
3. Contraction
The contraction process puts the next point between xmax and xcenter,
and the new point x∗
max can be found as:
x∗
max = β · xmax + (1 − β) · xcenter (11)
where 0 < β < 1 is the contraction coefficient ;
4. Multiple Contraction
Here all the points are shifted according to the following rule:
xi = (xi + xmin)/2, i = 1, ..m + 1 (12)
where i is the number of points in the simplex and xmin corresponds
to the point with the minimal function value fmin.
ADAPTED DOWNHILL SIMPLEX METHOD 137
In Algorithm 2 we implement the Downhill Simplex method described
in [5].
We denote as xi the whole strategy on ith iteration consisting of all j
points and xj is the j component of the strategy x.
So, for solving the maximization problem (6) we use the Algorithm 2,
where we minimize the objective function −f . The problem (7) is solved
by Algorithm 2 directly.
Dimension of the Problem and Investor(Issuer) Trajectory Ap-
proximation The trajectories Convt and Callt are the vectors of length
250 ·Maturity and 250 ·Maturity−CallNoticePeriod, respectively. Since
the lengthes of these vectors are the dimension of the optimization problem,
it is clear that such a problem cannot be solved efficiently by any optimiza-
tion method. In order to reduce the dimension of the problem, solved by
the Downhill Simplex Method we shall consider some approximations of the
trajectories instead of the original extremely costly computable trajectories.
So, instead of considering whole trajectories Convt and Callt in opti-
mization procedure described above, we use a set of threshold points xi- the
critical dates from the first possible exercise date till last exercise date at
maturity T . The trajectories are approximated by piecewise linear functions
with nodes being the threshold points xi.
We will use 5−15 points approximation of the trajectories which means
that the optimization problem will be 5 − 15 dimensional, too.
Since we have a Brownian type modeling, the deviations of the prices
from the initial state will increase while approaching the maturity. This is
also natural for any market that the most interesting and important actions
take place at the end. So, the distributions of the points should meet this
requirement. According to [3] we consider the following distribution of the
threshold points:
xi+1 =
xm − xi
2
+ xi, i = 1, ..m − 2 (13)
where x1 = 0 and xm = T or xm = T − timenotice − 1 for the investor’s
and issuer’s trajectories accordingly. Also m is the number of points in
approximation.
Note that for the investor’s trajectory the function value at the point xm =
T is always equal to N .
Short Description of the Main Algorithm As a termination criterion
we use the value of the gap. The gap is the difference between the investors
and issuers payoff. For each iteration it is defined as:
gap = payoff(Conv∗
t , Callt) − payoff(Conv∗
t , Callt∗) (14)
138 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
Algorithm 2 Downhill Simplex Method
Choose the initial guess x1
Choose the size of initial simplex k
Choose the maximum number of iterations maxiter
Choose the termination criterion ε
for i = 2 : m + 1 do
xi = x1
xi
i = xi
i−1 + k
f(i) ≡ f(xi)
end for
MAIN LOOP
for i = 1 : maxiter do
Find fmax, fmin, and fnearestmax
if ((i == maxiter) or ((fmax − fmin) < ε)) then
break
end if
Reflection xmax to x∗
max, and find f ∗
max
if f ∗
max < fmin then
Expansion x∗
max to x∗∗
max and find the new function value f ∗∗
max
if f ∗∗
max < fmax∗ then
f ∗
max = f ∗∗
max, x∗
max = x∗∗
max
end if
else
if f ∗
max > fnearestmax then
Contraction xmax to x∗
max, and find f ∗
max
end if
end if
if f ∗
max > fmax then
Multiple Contraction
else
xmax = x∗
max, fmax = f ∗
max
end if
end for
ADAPTED DOWNHILL SIMPLEX METHOD 139
where payoff(Conv∗
t , Callt) and payoff(Conv∗
t , Callt∗) are the optimizers
for the problems (6)- (7) respectively.
From the optimization point of view, due to the definition (14) the gap
is always nonnegative.
The main algorithm for solution the problem (1) - (4) is sketched in the
Algorithm 3.
Algorithm 3 Body of the main loop
gap = gapold = 10
while ((gap > ε) and (gapold > ε)) do
gapold = gap
compute payoff(Conv∗
t , Callt) from (6) by one step of Algorithm 2
compute payoff(Conv∗
t , Callt∗) from (7) by one step of Algorithm 2
gap = payoff(Conv∗
t , Callt) − payoff(Conv∗
t , Callt∗)
end while
where ε is a small constant, say ε = 0.0001.
3. Numerical Results
The numerical model described above has a quite complicate structure
and is sensitive to the choice of the input parameters. In this section we
investigate the dependence of the performance of the method on some of
parameters. This analysis is used to validate the numerical model and
choose the parameters which give a reasonable and fast solution.
In each experiment we compute and compare the optimal strategies of
the investor and the issuer. Moreover, we present the history of the behavior
of the optimal value of the objective, i.e. the investor’s payoff.
While comparing the results of the numerical experiments with different
initial parameters it is necessary to assume that due to the specificity of the
given optimization problem the following conditions are to be fulfilled:
• The strategies of the issuer and the investor must change their behav-
ior near the maturity time, see Section 2;
• The objective function must produce dumping oscillations due to the
nature of the minimax problem.
Below we present the experiments where 4 parameters (initial condi-
tions) of the optimization problem are varied. These parameters are: the
number of generated trajectories, the size of the simplex, the initial guess
and the number of points for approximation strategies of the investor and
the issuer (the size of the problem).
140 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
In each experiment the optimization problem is solved 3 times for 3
experimental values for each of the four parameters, three other parameters
being fixed (are from the basic set).
The basic set of the parameters is the following:
number of generated trajectories M = 525 (15)
size of simplex k = 3 (16)
size of the problem m = 10 (17)
initial guess ε = 5 (18)
The rest of the parameters of the problem are constant values, see Table 1.
Initial Stock Price $ 98
Convertible Bond Price (Face value) $ 100
Call Price $ 110
Call Notice Period 10 days
Interest Rate 0.05 (5%)
Dividend yield 0.1
Volatility 0.2 (20%)
Maturity 2 (two years)
Table 1: Constant parameters used in the numerical experiments
Experiment 1: Different number of trajectories for Stock Price
generating Presented here are the results for 3 different sets of generated
trajectories: M = 50, 525 and 1000. The rest of the parameters are (16) -
(18).
Analyzing the above figures, one can see that the solution corresponding
to the case with 50 trajectories cannot be considered to be proper as the
number of trajectories is insufficient. Firstly, subplot 1 in Figure 1 shows
that the behavior of the strategies close to the maturity does not change.
This means that the amount of generated strategies has no real affect on
the strategies of the investor and the issuer at the end of the bond lifetime.
Secondly, Figure 2 shows that the method terminates rather fast, which is
not appropriate for this minimax problem.
The behavior of the investor’s and the issuer’s optimal strategies as well
as the objective function history are similar in the cases with 525 and 1000
trajectories (see Figure 2 and subplots 2-3, Figure 1). Thus, these numbers
of generated stock price trajectories can be accepted for future experiments.
In the basic set (15) - (18) we consider 525 trajectories since the time needed
for function evaluation for this case is much shorter than the one for the
case with 1000 trajectories, but the solution is acceptable thereat.
ADAPTED DOWNHILL SIMPLEX METHOD 141
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
P
ric
e
in
$
Results of Global Optimization
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
Life time of bond
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
Figure 1: Strategies of Investor and Issuer. Different number of trajectories
0 2 4 6 8 10 12 14 16 18
107.2
107.4
107.6
107.8
108
108.2
108.4
108.6
108.8
Optimal value of the objective function
Number of iterations
P
ay
of
f
50 traject.
525 traject.
1000 traject.
Figure 2: Objective function history. Different number of trajectories
142 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
Experiment 2: Different sizes of simplex This experiments concerns
the proper choice of the parameter k, which is the distance between the
neighbor nodes in the initial simplex used in Downhill Simplex method.
This experiment we run for 3 sizes of the initial simplex: k = 1, 3 and 5.
The rest of the parameters are (15) and (17) - (18).
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
P
ric
e
in
$
Results of Global Optimization
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
Life time of bond
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
Figure 3: Strategies of Investor and Issuer. Different sizes of initial simplex
Figure 4 shows that for the minimal size of simplex k = 1 we have os-
cillation with small amplitude. So, the behavior of strategies of the investor
and the issuer do not change essentially w.r.t the initial guess (see subplot
1, Figure 3).
On the contrary, for maximal size of simplex k = 5, the first solution
of the minimization has a dominating effect. In other words, the first step
down (the solution of the minimization problem) has a big magnitude, which
does not allow to produce sufficiently big second step up (the solution of
the maximization problem). This effect manifests itself in subplot3, Figure
3, where the functions corresponding to the strategies of the investor and
the issuer are flat near the maturity time.
So, we choose the size of simplex k = 3, which produces reasonable steps
for both strategies (see subplot 2, Figure 3).
Experiment 3: Different initial trajectories For any global opti-
mization procedure the initial guess, which is the initial trajectories are of
ADAPTED DOWNHILL SIMPLEX METHOD 143
0 2 4 6 8 10 12 14 16 18
107.3
107.4
107.5
107.6
107.7
107.8
107.9
Optimal value of the objective function
Number of iterations
P
ay
of
f
size of simplex = 1
size of simplex = 3
size of simplex = 5
Figure 4: Objective function history. Different sizes of initial simplex
importance. We analyze 3 values of the initial trajectories ε = 1, 5 and 9.
The rest of the parameters are (15)-(17).
Figure 6 shows that all three experiments terminate with the same ob-
jective function value, but give different points (strategies), see Figure 5.
The case with ε = 1 requires the maximal number of iterations (almost
twice as many as the case for ε = 5 and three times as many as for ε = 10).
The case with ε = 9 seems to be not very informative, since almost
nothing happens close to the maturity (see Figure 5, subplot 3).
So, the most interesting cases are ε = 1 and ε = 5, but we choose ε = 5
in the basic set because in this case the number of iterations is twice lower
in comparison with the case ε = 1, and it produces an acceptable result.
Experiment 4: Different sizes of the problem Instead of using the
whole trajectories for issuers and investors, we used the approximated tra-
jectories. The amount of points in (13) which gives a reasonable solution to
the problem is the subject of investigation in this experiment.
We solved our problem for different sizes of the problem: m = 5, 10 and
15. The rest of the parameters are (15)-(16) and (18).
Figure 8 shows that the objective function history for all the cases is
almost similar.
Nevertheless, 5-point approximation of the strategies is not sufficient.
Since the most interesting part of the strategy is the second, it is not enough
144 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
120
P
ric
e
in
$
Results of Global Optimization
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
Life time of bond
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
Figure 5: Strategies of Investor and Issuer. Different initial trajectories.
0 5 10 15 20 25 30 35
106.5
107
107.5
108
108.5
109
109.5
Optimal value of the objective function
Number of iterations
P
ay
of
f
delta = 1
delta = 5
delta = 9
Figure 6: Objective function history. Different initial trajectories
ADAPTED DOWNHILL SIMPLEX METHOD 145
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
120
P
ric
e
in
$
Results of Global Optimization
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
0 50 100 150 200 250 300 350 400 450 500
80
90
100
110
120
Life time of bond
P
ric
e
in
$
Trajectory of Investor
Trajectory of Issuer
Call Price
Price of Convertible Bond
Figure 7: Strategies of Investor and Issuer. Different problem sizes
0 2 4 6 8 10 12 14 16 18 20
107.3
107.4
107.5
107.6
107.7
107.8
107.9
108
108.1
Optimal value of the objective function
Number of iterations
P
ay
of
f
5 point approx.
10 point approx.
15 point approx.
Figure 8: Objective function history. Different problem sizes
146 K.MISHCHENKO, V.MISHCHENKO AND A.MALYARENKO
to have only 3 points for approximation of the second part of the strategy.
As seen from subplot 1, Figure 7, the strategies are not smooth enough in
the vicinity of the maturity time.
15-point approximation gives very interesting results, but the size of the
problem becomes too high as well. So, the best choice for the basic set is
10-points approximation of the strategies, which is the tradeoff between two
other approximations.
4. Conclusions and Suggestions for Further Investigation
In this study we considered a method for computing the strategies of the
investor and the issuer dealing with convertible bonds. This method consists
of two main stages: stock price generating and solution of the minimax
optimization problem. For stock price generating we used a Monte-Carlo
method based on the formula (4), and applied the Downhill Simplex method
for the solution of the global nonlinear optimization problem. The results
of our investigation allow to draw the following conclusions.
1. The proposed method is sensitive to the number of generated trajecto-
ries of Stock Price. It means that for some small number of generated
trajectories the method does not produce any reasonable solution. We
suggest 500 trajectories as the minimal number required for achieving
a reasonable solution;
2. The Simplex Downhill method is sensitive to the size of the initial
simplex. It is very important to choose the initial simplex of a proper
size, otherwise there exists a risk to get non-acceptable solution. We
recommend the size of simplex k = 3;
3. The Downhill Simplex is also sensitive to the choice of a good initial
guess. The best choice in our experiments was ε = 5;
4. The dimensions (size of the problem) is important in our experiments,
too. Very large size of the problem requires too much computational
time, but for a small size we get non-acceptable solution. We took 10
points (solved 10-dimensional problem).
For further investigation the following is to be taken into account :
All the results presented in this study were obtained for predetermined
constant values such as initial stock price, call price, face value of convertible
bond, etc. So, it is very interesting to run experiments with other values of
the economic parameters. Also, such parameters as volatility, interest rate,
dividend yield may vary during the lifetime of the bond. For example, they
may be recalculated every day, which will make stock price generation more
complicated. Finally, the problem may be solved for nonzero coupon bond.
ADAPTED DOWNHILL SIMPLEX METHOD 147
We used Brownian type stock price generation without jumps (see Sec-
tion 2) which is one of the options. It is possible to consider some other
stock price generation algorithms which have another nature (with jumps)
and may give an interesting effect on the results.
From the viewpoint of the optimization it would be extremely useful
to consider another global solver, since Downhill Simplex is so sensitive to
the choice of the initial point and the size of initial simplex. On the other
hand, some local optimization methods may be useful, since the problem is
constrained and the feasibility area is quit narrow.
For more efficient strategies approximation it may be very helpful to
consider other distribution of points (e.g. equidistant distribution) and
other ways of approximation, e.g. cubic splines.
References
1. Amman, M., Kind, A., Wilde, C., Simulation-Based Pricing of Convertible
Bonds, Journal of Empirical Finance, (2007).
2. Garcia, D., Convergence and Biases of Monte Carlo estimates of Amer-
ican option prices using a parametric exercise rule, Journal of Economic
Dynamics & Control, 27, (2003), 1855–1879.
3. Isaksson, C., Pricing Convertible Bonds with Monte Carlo simulations,
Mälardalen University Master Thesis, 2006.
4. Nelder, J. A., Mead, R., A simplex method for function minimization, The
Computer Journal, 7, (1964), 308–313.
5. Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Nu-
merical Recipes in C. The Art of Scientific Computing, Second Edition,
Cambridge University Press, (1997).
Department of Mathematics, Mälardalen University, Box 883,
SE-72123, Väster̊as, Sweden
E-mail: kateryna.mishchenko@mdh.se, anatoliy.malyarenko@mdh.se
Master student graduated from Royal Institute of Technology,
Stockholm, Sweden
E-mail: vladimir mishchenko@yahoo.com
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