PRICING FORMULA FOR COMMODITY-LINKED BONDS.pdf

8/11/2019 PRICING FORMULA FOR COMMODITY-LINKED BONDS.pdf

1/30

Asia-Pacific Financial Markets 5: 129158, 1998.

1998Kluwer Academic Publishers. Printed in the Netherlands. 129

The Pricing Formula for Commodity-LinkedBonds with Stochastic Convenience Yields

and Default Risk

RYOZO MIURA1 and HIROAKI YAMAUCHI2

1Faculty of Commerce, Hitotsubashi University, Kunitachi, Tokyo, Japan2Student in Doctoral program at Graduate School of Commerce, Hitotsubashi University,

Kunitachi, Tokyo, Japan

Abstract. At the maturity, the owner of a commodity-linked bond has the right to receive the face

value of the bond and the excess amount of spot market value of the reference commodity bundle over

the prespecified exercise price. This payoff structure is an important characteristic of the commodity-

linked bonds.

In this paper, we derive closed pricing formulae for the commodity-linked bonds. We assume that

the reference commodity price and the value of the firm (bonds issuer) follow geometric Brownian

motions and that the net marginal convenience yield and interest rate follow OrnsteinUhlenbech

processes. In the appendix, we derive pricing formulae for bonds which are the same as the above

commodity-linked bonds, except that the reference commodity price in the definition of the payoff

at the maturity is replaced by the value of a special asset which depends on the convenience yield.

Key words:bond pricing, commodity-linked bond, convenience yield, default probability, PDE.

1. Introduction

1.1. REVIEWS

Schwartz (1982) introduced a general framework for pricing commodity-linkedbonds where (1) the reference commodity price follows a geometric Brownianmotion and the interest rate is constant. He also covered in his framework the threeother cases where (2) the commodity price and the bond price (i.e., the interest rateis stochastic) follow geometric Brownian motions, (3) the commodity price and thevalue of the firm (bonds issuer) follow geometric Brownian motions and the inter-est rate is constant, and (4) the interest rate behaves stochastically as an extension

to the case. There he obtained the closed pricing formulae of commodity-linkedbonds for the first three cases (1), (2) and (3). Defaults at the time of the maturityof the contingent claim (or, the bonds) of the issuing firms were considered in(3), where a pricing formula was derived. But he did not derive any closed pricingformula for the case (4). In his paper there was no discussion about the convenienceyields.


2/30

130 RYOZO MIURA AND HIROAKI YAMAUCHI

Carr (1987) derived a closed pricing formula for the commodity-linked bondsfor an extended case of the above (3), where the bond prices follow a third geo-metric Brownian motion without referring to the interest rate process, which is verysimilar to, but a little different from the above case (4). Carrs pricing formula takescare of the default of the issuing firms. The convenience yields were not consideredin his paper either.

Gibson and Schwartz (1990) is the first to consider the stochastic convenienceyields for the bond pricing model. They derived the partial differential equation forthe price functions of the assets defined as functions of spot commodity price andthe net marginal convenience yield. They estimated parameters for the behaviorof the net marginal convenience yield from market data, and calculated numer-ically the futures prices of the commodity.1 Bjerksund (1991) derived a closedpricing formula for the commodity contingent claims where the commodity pricefollows a geometric Brownian motion, the net marginal convenience yield followsan OrnsteinUhlenbech process, and the interest rate is constant. He did not con-

sider the default of the issuing firms at the maturity of the commodity contingentclaims. Gibson and Schwartz (1993) utilized Bejerksunds (also two other parties)pricing formula and Blacks (1976) formula to fit the market prices of the crudeoil futures options. Since our concerns in the present paper are the mathematicalpricing formulae for the commodity contingent claims, we do not further refer theirfitting results. They were able to calculate numerically the present prices of thecommodity-linked bonds, but did not derive a closed analytical pricing formula.

1.2. CONVENIENCE YIELD

The owner of the commodity has the rights (this is an option) to decide how he/she

will treat the commodity; sell, lend, or store it, or even consume it. As for theconsumption-use commodities such as crude oil or copper, the owner may con-sume it for his/her own manufacturing activities, or he/she may also store it forhis/her future consumption or future sell-out. The owner of the futures contracts orthe other contingent claims, however, does not have this rights because of lack ofstorage until the maturity.

The commodities prices are also seen to change with regards to the storage levelof the participants in the market. Since all the participants make their own decisiontaking account of their own current and future perspective of inventory levels andtime intervals, the market prices will change as aggregated results of each activityconducted by them. As Duffie (1989) discusses, the convenience yield is seen as

the value of the option to sell out of storage. We will thus assume that the yield willchange in relation to the scarcity of the commodity in the market. A low inventorylevel in the market, that is, scarcity of storage, leads to be backwardation of amarket where the futures prices of distant contract months are lower than those ofthe nearby (see, for example, Edwards and Ma, 1992). This means that backwar-dation occurs when there is a shortage of the available physical commodity. This


3/30

PRICING FORMULA FOR COMMODITY-LINKED BONDS 131

shortage implies the following attitude of the holder; the holder of the physicalcommodity are unwilling to part with it, even for short period of time (Edwardsand Ma, 1992) and thus generates the convenience yields. In this respect, we mayassume that there is an inverse relationship between the changes of the convenienceyield and the changes of current inventory level in the market. Kaldor (1939) andWorking (1948) examined and affirmed this hypothesis.2

A statistical analysis for the net marginal convenience yield can be done usingspot and futures prices. Brennan (1991) squeezed out the net marginal convenienceyield from futures prices of gold, silver, platinum, copper, No. 2 heating oil, lumber,and plywood. By analyzing those data, he showed their mean-reverting movements.Gibson and Schwartz (1990, 1993) used the relations between futures prices withdifferent contract months to estimate parameter values in the models for the netmarginal convenience yields movements and utilized the estimated values for theirnumerical pricing of the contingent claims.

1.3. OUR RESULTS

In this paper, we take the approach of Gibson and Schwartz (1990, 1993), Bjerk-sund (1991) to express the price change of the reference commodity in relationto its convenience yield. In Appendix C, we drive a pricing formula for a specialderivative. The underlying asset of this derivative itself is a derivative security,which is seen in Bjerksund (1991), that consists of a commodity and the continu-ously reinvested net marginal convenience yield. These two appraoches reflect twoways of treatments, that we could take, for the pricing of the commodity contingentclaims. The latter one uses the value of the ownership of the commodity as itsunderlying variables which receive the total expected return derived from its price

changes and net marginal convenience yield. Then, we see that the resulting pricingformula does not explicitly depend on the parameters related to the movements ofthe convenience yield. On the other hand, the former one uses the market price ofthe commodity where the owner of the commodity contingent claim cannot receivethe convenience yield deriving from the ownership of the commodity, but receivesthe total expected return from its price changes.

The pricing formula in the former case includes the parameters related to theconvenience yield. By using this formula, we draw several graphs of the bondprices and the default probabilities. The default occurs when the total payoff tothe bond holder exceeds the value of the issuer at the maturity. The figures forthe default probabilities provide us useful information to the bond issuer/holder inregard to the risk management.

1.4. ORGANIZATION OF THIS PAPER

This paper is organized as follows. In Section 2, we define the random variables,namely the commodity price, the value of the issuer, and the net marginal con-


4/30


venience yield, and the models for the behavior of these random variables. Wealso give some market conditions. Then we briefly derive the partial differentialequation (in short, PDE) for the price function of the commodity-linked bonds andobtain a closed pricing formula of the bonds by solving the PDE analytically. InSection 3, we show several figures for the bond prices and default probabilitiesas functions of parameter values. In Section 4, we show our results, rather indetail, which is the extended case of Section 2 where the instantaneous interestrate behaves stochastically, following another OrnsteinUhlenbech processes. Insubsection 4.1 we set some additional assumptions. In subsection 4.2, we derivethe PDE by using the standard no-arbitrage argument. In subsection 4.3, we derivethe closed pricing formula for the commodity-linked bond B(St, Vt, t, rt, ) byapplying FeynmanKac Theorem. Some related analytical details are presented inAppendix A and B. Appendix C shows a brief mathematical derivation for thepricing function of the bond linked to the special derivative. In Appendix D, wepresent Mathematicas program list to calculate prices of the commodity-linked

bondB(St, Vt, t, ).

2. Closed Pricing Formula for the Commodity-Linked BondsB(St, Vt, t, )

In this section, we derive the pricing functions of the commodity-linked bonds,B(St, Vt, t, ). To start with, we define our stochastic variables and derive thePDE. Then we obtain the closed pricing formula for the commodity-linked bondsthat satisfies the derived PDE with its payoff at the maturity as the boundarycondition.

LetS, V, and be stochastic processes. Stis the spot price of the commodity,Vtdenotes the value of the issuer (or the value of the firm), and trepresents the

instantaneous net marginal convenience yield rate. We assume thatS

, V

, and

satisfy following stochastic differential equations (in short, SDE):

dS

S=S dt+S dWS (1)

dV

V=V dt+ V dWV (2)

d =k ( )dt+ dW , (3)

whereWS,WV, andW are the standard Wiener processes and their correlation aresuch that dWS dWV = SVdt, dWS dW = S dt, and dWV dW = V dt.

We postulate that the parameters S,V, , , S, V,, SV,S , andV areconstants. We assume in the above that d follows OrnsteinUhlenbeck processthat can take negative values. This is not a problem for the convenience yield, be-cause our definition (3) is for the net marginal convenience yield. The net marginalconvenience yield is defined by the differences that the gross convenience yieldsubtracted by the cost of carry, thus it sometimes takes negative values.


5/30


We also postulate that there is a risk free interest rate rand that this is a constantduring the time interval fromtto T. The length of this time interval is denoted by .In this paper, we assume that assets are infinitely divisible and that a short positionis allowed. We also assume that there is no-arbitrage opportunity in the market.

Commodity-linked bonds have the payoff at the majority such that the ownerof the bonds has right to receive, in the case of no default, in addition to the facevalue, the excess amount of the spot market price of the reference commodity overthe prespecified exercise price. In the case where the default is considered, thepayoff at the maturity is the minimum of either the payoff in the case of no defaultor the value of the issuer at the maturity. The total amount of the payment to thebond owner at the maturity is

min[VT, F +max{ST K, 0}]. (4)

F andK are constants, Fis the face value of the bond and K is the prespecified

exercise price of the reference commodity.Also we assume that there are N (N 3) different assets in the market with

price functionsBi (St, Vt, t, )fori-th asset, wherei =1, 2, , N, that have thesame reference commodity. This is not an unrealistic assumption. Moreover, wepostulate that for any choice of the three assets, the three vectors each of whichconsists of iS,

iV,

i in the following equation (6) for Bi (St, Vt, t, )are linearly

independent to each other: that is, the following matrices are non-singular for anychoice of the three derivative assets.

iS

iV

i

j

S j

V j

kS kV

k

, (5)

wherei, j , k =1, , N, andi =j,i =k , andj =k .Next, we derive the PDE for the pricing function of the commodity-linked bond,

B(St, Vt, t, ). By using Itos lemma, we obtain the following equation.

dBi

Bi=B,i dt+

iS dWS+

iV dWV +

i dW , (6)

where

B,i =

BiS

S S+ Bi

VV V +

Bi

( ) Bi

+ 12

2Bi

S2S22S +

12

2Bi

V2V22V +

12

2Bi

22

+ 2Bi

SVSV SVSV +

2BiS

SSS + 2Bi V

V VV

Bi,

iS=Bi

S

SS

Bi, iV =

Bi

V

V V

Bi, i =

Bi

Bi.


6/30


We construct a portfolio W such that the portfolio consists of three differentderivative assets and the commodity. We denote the weights of each assets in thisportfolio asxi (i =1, , 4) and the sum of these portfolio weights is equal to 1,i.e.,

4i=1

xi =1 .

Then the rate of return of the portfolio Wis given by

dW

W=x1

dB1

B1+x2

dB2

B2+x3

dB3

B3+x4

dS

S+t dt

. (7)

This equation utilizes the property that the total rate of return of the owner of thereference commodity is the sum of the price changes of the commodity and itsconvenience yield.

By using the standard no-arbitrage argument, we obtain the following equa-tions:

B,1r

B,2r

B,3r

S+ tr

=1

1S 2S 3SS

+2

1V 2V 3V0

+3

1 2 30

. (8)

Note that 1,2, and3are the market prices of risk for the commodity price, thevalue of the issuer, and the net marginal convenience yield, respectively. They are,logically at now, time dependent and vary with regard to the choice of the assetsin the portfolio W. We show in the Appendix A that these do not depend on the

choice of the assets. We assume that theses are constants in solving the followingPDE (9). Note also that first, second, and third row of (8) are the equations for anyderivative assets and the fourth row of (8) is the equation for the commodity itself.From (8) we have the following PDE.

B

SS(r ) +

B

VV (V 2V)+

B

{( ) 3}

B

r B +

1

2

2B

S2S22S +

1

2

2B

V2V22V +

1

2

2B

22 +

+ 2B

SVSV SVSV +

2B

SSSS +

2B

V V VV =0 .

(9)

Equation (9) is the PDE which every B (St, Vt, t, )must satisfy.Next, we derive the closed form of the pricing function for the commodity-

linked bonds B(St, Vt, t, ) under the payoff function (4) at the bond maturity.This is done by applying FeynmanKac Theorem (see Friedman, 1975, Chapter 6,Theorem 5.3). To calculate the expected value of the payoff function, where we


7/30


write [St, Vt, t] for the corresponding stochastic processes, we need to obtain thejoint distribution function of [St, Vt, t] based on PDE (9). The detailed derivationis shown in subsection 4.3.

From (9) we have the following SDE.

d

St

Vt

t

=

St(r t)

Vt(V 2V)

( t)3

dt+Gd

Z1,t

Z2,t

Z3,t

, (10)

where the Z1,t, Z2,t, Z3,t are another set of independent standard Wiener processes.A choice ofG is given by

G= StS 0 0

Vt

V

SV

V V

c 0

S e f

,

where c =

12SV , e = V SVS

1 2SV

, f =

1

(V SVS )2

12SV2S.

The derivation ofGis shown in Appendix B.Because one of the drift term of SDE (10) is the product of St and t, SDE

(10) is non-linear. However, we make the following change of variables in orderto transform this non-linearSDE to a linearSDE. Then let Pt = log St and Jt =log Vt. By applying Itos lemma, we have

d

Pt

Jt

t

=

t+(r 12

2s)

V 2V 12

2V

( t)3

dt+

+

S 0 0

VSV V c 0

S e f

d

Z1,t

Z2,t

Z3,t

.

(11)

Now by using Theorem 8.2.2 in Arnold (1973, p. 129), we can solve the SDE(11) and derive the joint distribution of[ Pt, Jt, t]for the time interval[t0, T]. Thesolution of (11) is given by

PtJt

t

=

()()

()

+

XtYt

Zt

,


8/30


where =t t0,[(),(), ()]are deterministic functions such that

()()()

= Pt0 + t0

e 1

+ r 2s

2 +( 3)1e

2

Jt0 + (V 2V 12

2V)

t0 e +( 3)

1e

,

and [Xt, Yt, Zt] are jointly normally distributed. Their means are zero and thevariance-covariance matrix

is given by

=

Var(Xt) XY XZ

XY Var(Yt) Y Z

XZ Y Z Var(Zt)

,

where

Var(Xt)=

2s 2

SS

+

2

2

+2(1e )

SS

2

2

3

+

+ (1e2 )2

23

Var(Yt)= 2

V

Var(Zt)= 2(1e

2 )

2

XY =SVSV VV

+VV (1e

)

2

XZ =SS (1e

)

2(1e )

2 +

2(1e2 )

22

Y Z =VV (1e

)

The joint density function of[ Xt, Yt, Zt]is given by

f (xt, yt, zt)= 1

(2 )3/2 detexp{ 1

2vT

1

v},

where

v=

xtyt

zt

.


9/30


Then we calculate the present value of the expected value of the payoff function atthe maturity (see Figure 1 for the payoff chart at the maturity). The solution of thePDE (9) with the boundary condition (i.e., payoff) (4) is given by

B(St, Vt, t, t) =E

min{VT, F+max(ST K, 0)} er

St = StVt = Vtt = t

=E

min{e()+yT, F+max(e()+xT K, 0)} er

St = StVt = Vtt = t

=F exp{r}

H

R

fXY(xt, yt)dxtdyt+

+ exp{()r }

H

R

exp{yt}fXY(xt, yt)dxtdyt+

+ exp{()r }

Q

R

exp{yt}fXY(xt, yt)dxtdyt+

+ (F K )exp{r}

Q

R

fXY(xt, yt)dxtdyt+

+ exp{()r}

Q

R

exp{xt}fXY(xt, yt)dxtdyt

(12)

where fXY(xt, yt) is the marginal density function of xt and yt and {H|yT =log F ()}, {R|xT = log K ()}, and {Q|yT = log{F +exp{()+xT} K} ()} .

3. Bond Prices and Default Probabilities

In this section we show several figures for the bond prices and the default proba-bilities as functions of parameters. We list a Mathematicas program for computingthe above pricing function (12) in Appendix D. The default probabilities are givenby the following formula, that is, we calculated the probability that the total payoffto the bond holder is equal to VT, i.e., the light gray area in the (ST, VT)domainof Figure 1.

Default probability = H

R

f (xt, yt)dxtdyt+

+

Q

R

f (xt, yt)dxtdyt.(13)

When we use these formulae, we have to estimate each of paramters of theprocesses for St, Vt, t and a constant r. For the commodity price St, and the


10/30


Figure 1. Payoff chart at the maturity.

interest rater , they are quoted in the market, thus we can observe directly each ofthem and estimate the parametersSand the constantr . But it is difficult to observethe value of the issuer Vt,and the net marginal convenience yield t,because theyare not quoted or reported in the market. Therefore we have to estimate each ofVt andtor to use some proxies instead. For the value of the issuer, one idea toestimate each of parameters V andV ofVt is that: we can treat stock price ofthis issuer firm as a call option on the value of the issuer, then we squeeze out thevalue of the issuer firm from its stock price. Unfortunately to accomplish this idea

is not an easy task because it is difficult to know the whole cash flow of the issuer.In this section, the initial value ofVtand the parameters for the behavior of Vtare set rather subjectively. For the net marginal convenience yield, we recommenda simplified estimation procedure by Gibson and Schwartz (1990) or its revisedscheme by Yamauchi (1998). Their methods use futures prices of the commoditywith various delivery months. By isolating the differences between futures priceswith neighboring delivery months and also by excluding interest rates effects forthe futures prices, they approximately estimate one month or two months net mar-ginal FORWARD convenience yield rate and might regard the AGGREGATED netmarginal forward convenience yield rate as the net marginal covenience yield rate.Then they can be used to estimate each of parameters oft.

The initial values required for the calculations of the price functions and defaultprobabilities are set in the following way. The issuer of the commodity-linked bondhas the business of producing and selling the commodity which is the underlyingasset of that commodity-linked bond. The issuer wants to issue this bond withface value F = 100 and the maturity of 5 years. The strike price is set equalto the price of commodity at the time of issuance. The current interest rate is r


11/30


is 4% per year. The initial value of this issuer Vt is 200 which consists of thiscommodity-linked bond and the equity. Its expected growth ratio V is 2% peryear and its volatility Vis 30% annually. These parameters of the value of theissuing firm are set so that the probability ofV

T < Fis approximately 16% at the

maturity. The current prices of commodity St is 20 and the price volatility s is39.2% per year. This volatility parameter is estimated from WTI crude oil pricesdata in NYMEX for the period from September 4, 1990 to June 20, 1994. To setthe initial values of the parameters of the process of the net marginal convenienceyields, we refer the detail to Yamauchi (1998). In his paper, he first calculated, fromdaily futures prices of different maturity, the daily values of 3 months net marginalconvenience yield rates in a similar way to the one by Gibson and Schwartz (1990).Then, based on these daily values, he estimated the parameters of the process . Thewhole estimation period is from September 4, 1990 to June 22, 1993. He dividedthis estimation period into two periods; from September 4, 1990 to June 22, 1991and from June 23, 1991 to June 22, 1993. From the former period, the estimated

parameters were such that: = 19.122, = 0.324, and = 1.3050. From thelatter period, =4.547, =0.021, and =0.2673. The estimated parameterssuggest that the convenience yields of crude oil at the former period showed avery wild movements. During the latter period, the convenience yields seem to berelatively stable. In this paper, we set two situations, namely situation A and B.We use the estimated parameters from the former period for situation A and thelatter ones for situation B. We set the current level of the convenience yield rate tat 0.25 for both situations. When the spot price of this commodity moves up, theconvenience yield rate tends to move up in the effect according to their correlation.Their correlationS is set at 0.75 for situation A and at 0.50 for situation B. Thesecorrelation parameters are estimated from WTI crude oil prices and convenience

yield rates. Since we suppose this issuing firm sells this commodity to the market,the value of this issuer is positively correlated to the changes of this commodityprices and the convenience yields. Thus the commodity prices and the value of theissuer is set to behave with correlation SV =0.50 for both situations. Also we setthe correlation parametersV between the value of the issuer and the convenienceyield at 0.50 for situation A and at 0.33 for situation B.

F =100 K =20 =5 r =0.04 St =20 s =0.392

Vt =200 V =0.02 V =0.3 2 =0.067 t =0.25 SV =0.5

Situation A Situation B

=19 =0.32 =1.31 =4.5 =0.02 =0.273 =0.214 S =0.75 V =0.5 3 =0.074 S =0.5 V =0.33

Figure 2 shows that the graph of the commodity-linked bond prices as a functionof the speed of adjustment . To calculate the bond prices for Figure 2 and Figure 3,


12/30


Figure 2.

Figure 3.

we set = 0.1, 3 = 0.12, = 0.5, S = 0.6, and V = 0.4 apart fromsituation A and B, we draw these two figures to see overall responses of bond pricesand default probabilities to the values of . Figure 2 suggests that a smaller level of makes the bond prices higher than that of a larger level of . This means that thepremium portion of bond prices decrease as become large, that is, movementsof the convenience yield become more stable rather than that of smaller level ofwhen other parameters are kept fixed. Figure 3 describes the default probabilities ofthe commodity-linked bond. This figure shows that the default probability becomehigh as is at a smaller level. This result makes sense that the high premium,which means in part that the expected value of the payoff at the maturity is large,corresponds to the high default probability of this bond at the maturity.

Figure 4 suggests that the bond prices will increase as increases for situationB, while the bond prices do not seem to be affected by the changes ofin situation

A. This is because in a large level of , the convenience yield rate returns to itslong term mean quickly even if is at a large level. Consequently, the premiumportion changes little as become large. Figure 5 shows a graph of the defaultprobabilities as a function of.


13/30


Figure 4. Thin line : Situation A. Thick line : Situation B.




14/30






15/30




Figure 6 shows that the higher the commodity prices Stare, the more expensivethe bond prices are. This is very natural. The strike price K is equal to 20, the

premium increases as St moves across K from out of the money to in the money.Figure 7 is a graph of the default probabilities of the commodity-linked bond as afunction ofSt. This figure also shows that the default probabilities become high asStbecomes large which is the same as Figure 6.

Figure 8 through 11 are the graphs in relation to the value of the issuer. Figure 8shows that the higher the value of the issuer is, the more expensive the bond priceis. Figure 9 suggests that the default probability decreases as the value of the issuerVtincreases. This is very natural. IfVtis very small, the default probability of thisbond at the maturity is anticipated to be high. As for Figure 10 and 11, we see thatthe larger the volatility of the value of the issuer Vis, the lower the bond price isand, at the same time, the default probability is high. These are also natural.


16/30


4. Closed Pricing Formula for the Commodity-Linked Bonds

B(St, Vt, t, rt, )

In this section, we describe a straight extension of Section 2 where the instanta-

neous interest rate changes stochastically, following another OrnsteinUhlenbechprocess.

4.1. ASSUMPTIONS FOR THE PRICING FORMULA OF COMMODITY-LINKEDBONDSB (St, Vt, t, rt, )

Assume the same situation as we postulated in the Section 2 except that the interestrate behaves stochastically

drt =g (r rt)dt+r dWr , (14)

whereWr is another standard Wiener process and correlations are such that

dWs dWr =Sr dt, dWV dWr =V r dt, and dW dWr = r dt .

This assumption for the behavior of the interest rate rtmight be a little problematic,because there may occur negative interest rate. However, this probability is small.In our paper, for the ease of the derivation of the pricing formula, we assumeOrnsteinUhlenbech process forrt.

We also postulate that the following parameters are constants:

S, V, , , g, r , S, V, , r , SV, S , Sr , V , V r , r .

We assume that there are N (N 4) different assets with price functions

Bi (St, Vt, t, rt, ), fori =1, 2, , N,that have the same reference commodityin the market. For any choice of the four assets, we assume that there exists theinverse of the following matrices:

iS iV

i

ir

j

S j

V j

jr

kS kV

k

kr

lS lV

l

lr

, (15)

where i, j , k , l = 1, , N, andi = j, i = k,i = l,j = k, j = l, and k = l .

These quantities are defined later in the following equality (16).


17/30


4.2. PARTIAL DIFFERENTIAL EQUATION

In this subsection, we derive the PDE for the pricing function of the commodity-linked bond Bi (St, Vt, t, rt, ). By using Itos lemma, we obtain the following

equation for thei-th commodity-linked bond(i =1, 2, , N );

dBi

Bi=B,i dt+

iS dWS+

iV dWV +

i dW +

ir dWr , (16)

where

B,i =

BiS

S S+ Bi

VV V +

Bi

( ) + Bi

rg(r r )

Bi

+ 12

2Bi

S2S22S+

12

2Bi

V2V22V +

12

2Bi

22 +

12

2Bi

r 22r

+ 2Bi

SVSV SVSV +

2BiS

SSS + 2BiSr

SSr Sr

+ 2Bi V V VV +

2Bi V r V Vr V r +

2Bir r r

Bi

,

iS=Bi

S

SS

Bi, iV =

Bi

V

V V

Bi,

i =Bi

Bi, ir =

Bi

r

r

Bi.

With the same standard no-arbitrage argument used in Section 2, we obtain

B

SSt(rtt)+

B

VVt(V 2V)+

B

{( t)3} +

+B

r{g(r rt)4r } +

1

2

2B

S2S2t

2S +

1

2

2B

V2V2t

2V+

+1

2

2B

22 +

1

2

2B

r 22r +

2B

SVStVtSVSV+

+ 2B

SStSS +

2B

SrSts r Sr +

2B

V VtVV +

+ 2B

V rVtVr V r +

2B

rr r

B

rt B =0 ,

(17)

where 2, 3,and 4 are the market prices of risk for the value of the issuer, thenet marginal convenience yield, and the interest rate, respectively. Equation (17) isthe PDE which everyB(St, Vt, t, rt, )must satisfy.


18/30


4.3. CLOSED PRICING FORMULA

In this subsection, we derive the closed pricing formula of the commodity-linkedbondB (St, Vt, t, rt, )by applying the FeynmanKac Theorem. In the calcula-

tion of the expected value of the payoff function (4), we need the joint distributionfunction of[St, Vt, t, rt]based on PDE (17). From (17), we have the followingSDE:

d

St

Vt

t

rt

=

St(rtt)

Vt(V 2V)

( t)3

g(r rt)4r

dt+

+

StS 0 0 0

Vt

V

SV

Vt

V e 0 0

S f h 0

r Sr r g r i r j

dZ1,t

Z2,t

Z3,t

Z4,t

,

(18)

where

e =

1 2SV , f =

V SV S

e, g =

V r SVSr

e

h=

1p2S f

2 , i =r S Sr f g

h, j =

1 2Sr g

2 i2

and Z1,t, Z2,t, Z3,t, Z4,t,are independent standard Wiener processes.Next, to transform the non-linearSDE (18) to linear one, let P

t = log S

t and

Jt = log Vt. Then we have the following SDEs for Pt and Jtby applying Itoslemma.

d

t

t

t

rt

=

t+rt 12

2S

V 2V 12

2V

( t)3

g(r rt)4r

dt+

+

S 0 0 0

VSV V e 0 0

S f h 0

r Sr r g r i r j

d

Z1,t

Z2,t

Z3,t

Z4,t

.

(19)


19/30


By solving the stochastic differential Equations (19), we derive the joint proba-bility density function of[ Pt, Jt, t, rt]for the time interval [t0, T]using theorem8.2.2 in Arnold (1973, p. 129). By theorem 8.2.2, the SDE

dQt =(A(t) Qt+a(t))dt+B(t)dZt

has the solution

Qt =t(Qt0 +

tt0

1s a(s) ds +

tt0

1s B(s) dZs ) (20)

with the initial valueQt0 ,where

Qt =

Pt

Jt

trt

, Qt0 = Pt0Jt0t0rt0

,

A(t)= A =

0 0 1 10 0 0 00 0 00 0 0 g

,

a(t)= a =

1

22S

V 2V 12

2V 3

gr 4r

,

B(t)= B =

S 0 0 0VSV V e 0 0S f h 0r Sr r g r i r j

, and Zt =

Z1,t

Z2,t

Z3,t

Z4,t

.

In this solution (20), tstands for the fundamental matrix of.

Qt= A(t) Qt tand its inverse are given by

t =eA(tt0) =

1 0 1

(e(tt0) 1) 1

g(1eg(tt0))

0 1 0 0

0 0 e(tt0) 0

0 0 0 eg(tt0)

, (21)


20/30


1t =

1 0

1

(e(tt0) 1)

1

g(1eg(tt0))

0 1 0 0

0 0 e(tt0) 00 0 0 eg(tt0)

, (22)

By substituting (21) and (22) into (20), we have

Pt

Jt

trt

=

()

()

()

()

+

xtytztwt

,

where

()

()

()

()

=

Pt0 + t0e 1

+rt01eg

g

2s

2

+( 3)1e

2 (gr 4r )

1eg g

g2

Jt0 +

V 2V

12

2V

t0 e +( 3)

1e

rt0 eg +(gr 4r )

1eg

,

xtytztwt

=

s

S

+ r Srg

tt0

dZ1,s + S

tt0

e(s t )dZ1,s

r Sr

g t

t0eg(s t )dZ1,s +

rg

g

f

t

t0dZ2,s

+f

tt0

e(s t )dZ2,s rg

g

tt0

eg(s t )dZ2,s

+

r ig

h

tt0

dZ3,s + h

tt0

e(s t )dZ3,s

r ig

tt0

eg(s t )dZ3,s + r j

g

tt0

(1eg(s t ))dZ4,s

VSVt

t0dZ1,s + Ve

tt0

dZ2,s

St

t0e(s t )dZ1,s+ f

tt0

e(s t )dZ2,s

+ht

t0e(s t )dZ3,s

r Sr

tt0 e

g(s t )

dZ1,s + rgt

t0 eg(s t )

dZ2,s

+r it

t0eg(s t )dZ3,s+ r j

tt0

eg(s t )dZ4,s

,

and =tt0. Note that the stochastic integrals

{}dZi,s(i =1, 2, 3) are normallydistributed.


21/30


Then, the pricing formula for the commodity-linked bond B (St, Vt, t, rt, )isobtained by calculating the expected present value of the bond with payoff function(4).

B(St, Vt, t, rt, t)

=E

min{VT, F +max(ST K, 0)} exp

Tt

rudu

St =StVt =Vtt =trt =rt

=E

min{e()+yT , F +max(e()+xT K, 0)}

exp

Tt

((u t )+wu)du

St =StVt =Vtt =trt =rt

=exp{ ( )}

H

R

exp{yt+rt}f (xt, yt, r

t)dxtdytdr

t +

+ F exp{()}

H

R

exp{r t}f (xt, yt, rt)dxtdytdr

t +

+exp{ ( )}

Q

R

exp{yt+rt}f (xt, yt, r

t)dxtdytdr

t +

+ (FK)exp{()}

Q

R

exp{rt}f (xt, yt, rt)dxtdytdr

t +

+exp{( )}

Q

R

exp{xt+rt}f (xt, yt, r

t)dxtdytdr

t ,

(23)

where

{H |yT =log F ()} , {R |xT =log K ()} ,

{Q| yT =log{F +exp{()+xT} K } ()} ,

and

() ( )

( )

=

gr 4r

g + gr 4r

g rt 1eg

g

Pt 2S

2 3

1e

t

1e

Jt+

V 2V 12

2V

gr 4r

g

1e

g

g

rt

1eg

g

.


22/30


f (xt, yt, rt) is the joint density function such that

f (xt, yt, rt) =

1

(2 )3/2 detexp

1

2vT

1

v

,

wherev =

xtyt

rt

and is its variance-covariance matrix which is given by

=

Var(xt) xy xr xy Var(yt) yr

xr yr Var(rt)

,

where

Var(xt) =

2S +

2

2 +

2r

g2 2

SS

+2

Sr Sr

g2

r r

g

+

+2

2 3(1e2)+

2r

2g3(1e2g) +

+ 2(1e)

2

SS

+

r r

g

2(1eg)r

g2

SSr +

r

g

r

2 r r

g(+ g)(1e(+g))

Var(yt) =2V

Var(zt) =

2r

g2 2

2r

g3(1eg

)+

2r

2g3 (1e2g

)

xy =

SVSV

VV

+

Vr V r

g

+

+V V

2 (1e)

Vr V r

g2 (1eg)

xr =

Sr Sr

g

2r

g2 +

r r

g

+

+ r r

g(+ g)(1e(+g))

2r

2g3(1e2g)

r r

g1e

+

1eg

g

+

+Sr Sr

g2 (1eg)+2

2r

g3(1eg)

yr = Vr V r

g+

Vr V r

g2 (1eg)


23/30


The default probability is given by

Default probability =

H

R

f (xt, yt, rt)dxtdytdr

t

+

Q

R

f (xt, yt, rt)dxtdytdr

t .

(24)

5. Concluding Remarks

In this paper, we have derived pricing formulae for the commodity-linked bondsin three cases. In Section 2, the first pricing function B(St, Vt, t, ) had threevariables beside the time to maturity, namely the commodity price, the value ofthe issuer, and the net marginal convenience yield. We have shown several fig-

ures of the bond prices and of the default probabilities as functions of parametervalues in Section 3. These figures provided us certain implication for prices ofthe commodity-linked bond and the default probabilities. In Section 4, the secondpricing formula B(St, Vt, t, rt, ) was an extended version of the first one. Thisformula was the pricing function when we allow the stochastic changes for theinterest rate. In Appendix C, the pricing function, B(Ct, Vt, t, ), was differentfrom the first and second ones with respect to its underlying asset. The underlyingasset of B(Ct, Vt, t, ) was a special asset which is a self-financing portfolioconsisting of a unit of commodity and the continuously reinvested net marginalconvenience yield. We note that the closed-form pricing function of this bonddid not include any parameter associated with the convenience yield. In these

derivation procedures, we used the standard no-arbitrage argument to derive PDEsand took the change of the variables which transformed the non-linearSDEs tothe linearones to find the joint probability distribution for calculating the pricingformulae.

As is well known, commodities and their derivative securities are one categoryof the risky assets for the corporate finance. We have to consider various efficientways for their hedging schemes in regard to financial risk management. We hopethat the bond pricing formulae that we derived here will provide some help topractitioners who want to hedge the risk of price fluctuations of commodities.

Acknowledgements

This paper is an improved version of our working paper, Miura and Yamauchi(1996). We are grateful to the anonymous referees for their valuable comments onour earlier draft.

The first author would like to thank the Commodity Futures Association ofJapan for its financial support to this research work.


24/30


Appendix A: Proof for the Independence of1,2, and3on the Choice of

the Assets

By the standard no-arbitrage argument, we derived constants 1, 2, and 3 for

the choice of the three assets, namely, 1st, 2nd, and 3rd bonds in Section 2. Theyare tentatively dependent on the choice of three assets. In this appendix, we proveits independence on the choice of the assets. First, we exchange 3rd bonds to 4thbonds. The same no-arbitrage argument is valid for this new portfolio and we obtaina new set of constants 1,

2, and

3with the following equations:

B,1r

B,2r

B,4r

S+ tr

=1

1S 2S 4SS

+2

1V 2V 4V0

+3

1 2 40

. (A.1)

From equationsD = 1 A+ 2B+3 Cand (A.1) and (5) (the assumption ofnon-singularity), we obtain

123

=

12

3

=

1S 1V 1 2S 2V 2

S 0 0

1

B,1 rB,2 r

S+ tr

.

This procedure can be iterated. The iteration is done for exchanging 2nd bondto 5th bond, and 1st bond to 6th bond. Then we have a unique time dependentconstant set,1,2, and3. Thus, these constants are independent of the choice ofassets. Each of these is called the market price of risk.

Of course, we can imply the same result to the set of constants 1, 2,3, and4in section 4.2 and also

1,

2, and

3in Appendix C.

Appendix B: Decomposition of G GT

Gis a matrix such that

GGT =

S2t2S StVtSVSV StSSStVtSVSV V2t 2V VtVV

StSS VtVV 2

.

To getG, the following decomposition helps:

GGT =

StS 0 0

0 VtV 00 0

1 SV SSV 1 V S V 1

StS 0 0

0 VtV 00 0

.

Next, we decompose the second matrix of the RHS above 1 SV SSV 1 V

S V 1

=

a 0 0b c 0

d e f

a b d0 c e

0 0 f

T

,


25/30


where

a2 =1, b2 +c2 =1, d2 + e2 +f2 =1

ab = SV, ad =S , bd+ ce = V .

When we assigna =1, we have

b= SV, d=S and c2 =12SV .

Now we suppose c and fbe positive (of course, negative values are feasible. Butwe select positive values forc andfas one of the choices),

c=

12SV,e =

V SVS1 2SV

, f =

1

(V SVS )2

1 2SV 2S .

Then we can writeG as follows:

G =

StS 0 00 VtV 0

0 0

a 0 0b c 0

d e f

=

StS 0 0VtVSV VtV c 0

S e f

.

This argument can be used to obtain a choice ofG for the SDE (18) in relationto the PDE (17) in subsection 4.3 and the same thing applies to the SDE (C.5) inAppendix C.

Appendix C: Derivation for the Closed Pricing Formula of

Commodity-Linked BondsB(Ct, Vt, t, ): One of the Underlying Asset is a

Self-Financing Portfolio

In this appendix, we derive the closed pricing formula for commodity-linked bondsB(Ct, Vt, t, )with payoff at the maturity such that

min[VT, F +max{CT K, 0}] . (C.1)

We assume thatC(St, t, t)(in short,Ct) is the value of a self-financing portfo-lio, treated in Bjerksund (1991), has a unit of commodity S, and reinvests the net

marginal convenience yield continuously. This has its initial value C(St, t, t) =S(t). This portfolio was treated in Bjerksund (1991), where the evolution of itsvalue is described by the following SDE:

dC

C=(S+ t)dt+S dWS. (C.2)


26/30


This equation (C.2) means that the total expected return of this special portfoliocomes from both the expected price changes of the commodity Sand the netmarginal convenience yield t. SDEs for Vt andtare the same as in Section 2.Their correlations between each other are also the same as in Section 2.

We assume here N (N 4) different B in the market and the followingmatrices are non-singular for any choice of three assetsB i, B

j, andB

k .

iS iV ijS jV jkS

kV

k

, (C.3)

wherei, j , k =1, . . . , N andi =j,i =k , andj =k .Note thatiS,

iV, and

i are defined as follows:

iS=Bi

C

CS

Bi, iV =

Bi

V

V V

Bi, i =

Bi

Bi.

Other assumptions are the same as in Section 2.By using the standard no-arbitrage argument as we did this in subsection 4.2,

we have the following PDE for the pricing function of this bond:

B

CC r +

B

VV (V

2V)+

B

{( )

3}

B

r B +

+1

2

2B

C2 S22S +

1

2

2B

V2V22V +

1

2

2B

2 2 +

+ 2B

CVCV SVSV +

2B

CCSS +

2B

V V VV =0 ,

(C.4)

where

2 and

3 are the market prices of risk for the value of the issuer andthe net marginal convenience yield, respectively. They are dependent on time andindependent on the choice of portfolio assets (see Appendix A).

This (C.4) is the PDE which every B(Ct, Vt, t, )must satisfy. This (C.4) isdifferent from (9) in regard to the coefficients ofB/S (and B/C), namely,the coefficient of the first partial derivative in this PDE is C r and this does notinclude .

Next, we derive the present value of the expected value of the payoff function(C.1) at the maturity by applying FeynmanKac theorem. From (C.4), we have thefollowing SDE:

d Ct

Vtt

=

Ct rVt(V

2V)

( t)3

dt+

+

CtS 0 0VtVSV VtV c 0

S e f

d

Z1,tZ2,t

Z3,t

,

(C.5)


27/30


where

c=

1

2

SV, e =

V SVS1 2SV

, f =

1

(V SV S )2

12SV

2

S .

Note that the drift terms in this SDE (C.5) are linear in the variables. This makes iteasier to solve the PDE (C.4) than to solve (9). By solving the equation (C.5), the

joint distribution of[Ct, Vt,t]for the time interval [t0, T]can be derived by thetheorem 8.5.2 in Arnold (1973, p. 141). The solutions are given by

Ct

Vt

t

=

Ct0 exp{() +Xt}

Vt0 exp{()+Yt}

() + Zt

,

where

()

r

1

22s

()

V 2V

12

2V

() t0 e +(1e )

3

,

=t t0, and[Xt,

Yt,

Zt]are jointly normal distributed with zero means and thevariance-covariance matrix

is given by

=

2S SVSV SS(1e )

SVSV 2

V VV (1e )

SS(1e )

VV

(1e )

2 (1e2 )

.

The density functionm( Xt, Yt, Zt)of[ Xt, Yt, Zt]is below.

m(Xt, Yt, Zt)=1

(2 )3/2

det exp

1

2vT1

v

, where v=

XtYt

Zt

.


28/30


Finally, we derive the present value of the expected value of the payoff functionat the maturity. The solution of the PDE (C.4) with initial condition (C.1) is asfollows:

B(Ct, Vt, t, t) =

=E

min{VT, F+max(CT K, 0)} er

Ct =CtVt =Vtt =t

=E

min{Vte()+yT , F +max(Cte()+xT K, 0)} er

Ct =CtVt =Vtt =t

=Vt er()

H

R

exp{yt} mXY(xt, yt)dxtdyt+

+F er

H

R

mXY(xt, yt)dxtdyt+

+Vt er()

Q

R

exp{yt} mXY(xt, yt)dxtdyt+

+(FK)er

Q

R

mXY(xt, yt)dxtdyt+

+Ct er()

Q

R

exp{xt} mXY(xt, yt)dxtdyt,

(C.6)

where mXY(xt, yt) is the marginal density function ofxt and yt and {H | yT =

log F ()}, {R | xT = log K ()}, and {Q | yT = log{F +exp{() +xT} K} ()}.

We note that the derived pricing function (C.6) does not include t and theparameters related to movements oft.

Appendix D: Pricing Program on Mathematica for B(St, Vt, t, )

CLB[St_,sp_,k_,a_,ld_,sd_,DE_,Vt_,muv_,sv_,lv_,Rhosd_,Rhosv_,Rhovd_,tau_,K_,F_,r_]:=With[

{

sx = Round[Sqrt[((sp^2)-(2*sp*sd*Rhosd)/k+(sd^2)/(k 2))*tau+2*(1-Exp[-k*tau])*((sp*sd*Rhosd)/(k^2)-

(sd^2)/(k^3))+(1-Exp[-2*k*tau])*((sd^2)/(2*(k^3)))]*10000000]/10000000,

sy = Round[Sqrt[(sv^2)*tau]*10000000]/10000000,

cov = Round[(sp*sv*Rhosv*tau-(sv*sd*Rhovd*tau)/k+(sv*sd*Rhovd*(1-Exp[-k*tau]))/(k^2))*10000000]

/10000000,

rho = Round[(((sp*sv*Rhosv*tau-(sv*sd*Rhovd*tau)/k+(sv*sd*Rhovd*(1-Exp[-k*tau]))/(k^2)))/Sqrt[(((sp^

2)-(2*sp*sd*Rhosd)/k+(sd^2)/(k^2))*tau+2*(1-Exp[-k*tau])*((sp*sd*Rhosd)/(k^2)-(sd^2)/(k^3))+

(1-Exp[-2*k*tau])*((sd^2)/(2*(k^3))))*((sv^2)*tau)])*10000000]/10000000,

alpha = Round[(N[Log[St]]+(DE*(Exp[-k*tau]-1))/k+(r-(sp 2)/2)*tau+(k*a-ld*sd)*((1-Exp[-k*tau]

-k*tau)/(k^2)))*10000000]/10000000,

beta = Round[(N[Log[Vt]]+(muv-lv*sv-(sv^2)/2)*tau)*10000000]/10000000,

R = Round[(N[log[K]]-N[Log[St]]+DE*(Exp[-k*tau]-1)/k+(r-(sp^2)/2)*tau+(k*a-ld*sd)*((1-Exp[-k*tau]

-k*tau)/(k^2))))*100000]/100000,


29/30


P = Round[(N[Log[F]]-(N[Log[Vt]]+(muv-lv*sv-(sv^2)/2)*tau))*100000]/100000

},

Q = N[Log[GF-K+Exp[alpha+x]]]-beta;

N[1/2*Pi*sx*sy*Sqrt[1-rho^2])*0.00001*

(Exp[beta-r*tau]*NSum[NIntegrate[Exp[y]*Exp[(((x^2)/(sx^2))+((y^2)/(sy^2))-((2*rho*x*y)/(sx*sy)))/

((-2)*(1-rho^2))],{y,Round[-6*sy*100]/100,P}],{x,Round[-6*sx*100]/100,R,0.00001},Method->Integrate]

+F*Exp[-r*tau]*NSum[NIntegrate[Exp[(((x^2)/(sx^2))+((y^2)/(sy^2))-((2*rho*x*y)/(sx*sy)))/((-2)*

(1-rho^2))],{y,P,Round[6*sy*100]/100}],{x,Round[-6*sx*100]/100,R,0.00001},Method->Integrate]

+Exp[beta-r*tau]*NSum[NIntegrate[Exp[y]*Exp[(((x^2)/(sx^2))+((y^2)/(sy^2))-((2*rho*x*y)/(sx*sy)))/

((-2)*(1-rho^2))],{y,Round[-6*sy*100]/100,(Round[Q*100000]/100000)}],{x,R,Round[6*sx*100]/100,0.00001}

,Method->Integrate]

+(F-K)*Exp[-r*tau]*NSum[NIntegrate[Exp[(((x^2)/(sx^2))+((y^2)/(sy^2))-((2*rho*x*y)/(sx*sy)))/((-2)*

(1-rho^2))],{y,(Round[Q*100000]/100000),Round[6*sy*100]/100}],{x,R,Round[6*sx*100]/100,0.00001}

,Method->Integrate]

+Exp[alpha-r*tau]*NSum[NIntegrate[Exp[x]*Exp[(((x^2)/(sx^2))+((y^2)/(sy^2))-((2*rho*x*y)/(sx*sy)))/

((-2)*(1-rho^2))],{y,(Round[Q*100000]/100000),Round[6*sy*100]/100}],{x,R,Round[6*sx*100]/100,0,00001}

,Method->Integrate]

)

]

]

Note in Proof

After the final revision of this paper toward this publication, we came to know the work of K.R.

Miltersen and E.S. Schwartz (1997) Pricing of Options on Commodity Futures with Stochastic Term

Structure of Convenience Yields and Interest Rates. Publications from Department of Management,

School of Business and Economics, Odense University. Their paper develops a model for pricing

options on commodity futures in the presence of stochastic rates as well as stochastic convenience

yields.

Notes

1. The spot (or sheer) price of some commodities such as crude oil are not observed in the market.We have a forward contract of the crude oil, since it takes a few weeks to deliver the oil. A

device to calculate the convenience yield made by Gibson and Schwartz (1991) works without

using spot prices.

2. For example, the production and transportation of crude oil cannot match with the demand of

the market simultaneously, thus the shortage of crude oil costs much to the buyers of crude oil

at the time being. So, if the shortage occurs in the crude oil market, then the owner of crude

oil can enjoy his privilege, and the convenience yield will be recognized in the crude oil market.

Moreover, in equilibrium, the convenience yield on the marginal unit of storage will be equalized

among all of the potential and current owners of the commodities throughout their competition.

ReferencesArnold, Ludwig (1973), Stochastic Differential Equations: Theory and Applications, Wiley-

Interscience.

Bjerksund, Petter (1991), Contingent claims evaluation when the convenience yield is stochastic:Analytical results, Working Paper No. 1, Norges Handelshyskole, Norway.

Black, Fisher (1976), The pricing of commodity contracts, J. Financial Econ. 3, 167179.


30/30


Brennan, Michael J. (1991), The Price of convenience and the valuation of commodity contingent

claims, in D. Lund and B. ksendal (eds), Stochastic Models and Option Values: Application to

Resources, Environment and Investment Problems, North Holland.

Carr, Peter (1987), A note on the pricing of commodity-linked bonds, J. Finance42 (4), 10711076.

Duffie, Darrell (1989), Futures Markets, Prentice Hall.Edwards, F.R. and Ma, C.W. (1992), Futures & Options, McGraw-Hill.

Friedman, Avner (1975),Stochastic Differential Equations and Applications: Vol. 1, Academic Press.

Gibson, R. and Schwartz, E.S. (1990), Stochastic convenience yield and the pricing of oil contingent

claims,J. Finance45 (3), 959976.

Gibson, R. and Schwartz, E.S. (1991), Valuation of long term oil-linked assets, in D. Lund and B.

ksendal (eds), Stochastic Models and Option Values: Application to Resources, Environment

and Investment Problems, North Holland.

Gibson, R. and Schwartz, E.S. (1993), The pricing of crude oil futures options contracts, Advances

in Futures and Options Research6, 291311.

Kaldor, Nicholas (1939), Speculation and economic stability, Rev. Econ. Stud.7 (3), 127.

Miura, Ryozo (1989),Modern Portfolio no Kiso (in Japanese), Dobunkan.

Miura, R. and Yamauchi, H. (1996), The pricing formula for commodity-linked bonds with stochastic

convenience yields and default risk, Working Paper Series No. 8, Department of Commerce,

Hitotsubashi University, Japan.Schwartz, Eduardo S. (1982), The pricing of commodity-linked bonds, J. Finance37 (2), 525541.

Working, Holbrook (1948), The theory of the price of storage, Am. Econ. Rev.39, 12541262.

Yamauchi, Hiroaki (1995), The commodity-linked bond the pricing theory and the data analysis

(in Japanese), Master Degree Thesis of Graduate School of Commerce, Hitotsubashi University,unpublished.

Yamauchi, Hiroaki (1998), Syouhin-kakakukinrikonbiniensu-irudo ga fukakujitsu ni hendou su-

rubaai no syouhin-sakimono-kakakushiki to gen-yu konbiniensu-irudo no data bunseki (in

Japanese),Keiei-Zaimu-Kenkyu-Sosyo, forthcoming.

PRICING FORMULA FOR COMMODITY-LINKED BONDS.pdf

Documents

Transcript of PRICING FORMULA FOR COMMODITY-LINKED BONDS.pdf