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Competitively-Issued Convertible Bank Notes in a Theory of Finance: Earl Thompson Meets Fischer Black

Joshua R. Hendrickson

Abstract

In this paper, I show the validity of and the relationship between two previously unrelated claims in monetary theory. The first claim, made by Earl Thompson, is that privately-issued bank notes pay a positive rate of return in a competitive equilibrium. The second claim, made by Fischer Black, is that it is possible to have a gold standard in which the gold reserves of the central bank are near zero. I show that both of these claims are correct under the assumption of complete markets and perfect commitment. The link between these claims is the Black-Scholes equation applied to convertible bank notes. In commodity-based monetary systems, bank notes are perpetual American options. I extend the model to consider the implications of a lack of commitment on the part of the bank and incomplete markets. I show that both arguments break down when banks lack commitment to redemption or markets are incomplete. I conclude with implications for macroeconomic theory.

JEL Classification: E42; E50

Corresponding author: Joshua R. Hendrickson, Department of Economics, University of Mississippi, 306 Odom Hall, University, MS, 38677, USA, E-mail:

Appendix

A.1 Proof of Proposition 1

Imagine that a bank issues nn bank notes denominated in N dollars. The bank’s assets consist of B dollars of risk-less bonds that earn a risk-free rate of return, r. It follows that dB rBdt. The bank also holds nc ounces of the commodity. It follows that the net worth, W, of the issuing bank can be written as

W=pnc+Bnnv(p)

Assuming that the quantity of the commodity and the quantity of bank notes are held constant, it follows that

dW=ncdp+dBnndv

or

dW=(ncμp+rBnnμvv)dt+(ncσpnnσvv)dz

The bank can perfectly hedge its risk by choosing an optimal reserve ratio. Specifically, the bank can choose its reserve ratio such that

ncσpnnσvv=0

Solving for the reserve ratio yields:

ncpnnv=σvσ

When the bank chooses this reserve ratio, the net worth of the bank is given as

dW=(ncμp+rBnnμvv)dt

In other words, the net worth of the bank grows at certain rate. If this rate is lower than the risk-free rate, r, then it wouldn’t make sense for the bank to operate. If this rate is above the risk-free rate, r, then the banks could earn a positive profit from arbitrage without any risk. However, in a competitive banking system, banks would enter until the rate of return on the net worth of the bank is equal to the risk-free rate. Thus, in any equilibrium in which banks exist, the net worth of the bank evolves according to

dW=rWdt

Given the portfolio above, this implies that

dW=r(pnc+Bnnv)dt

Thus, it must be true that

ncμp+rBnnμvv=rpnc+rBrnnv

Or,

ncμpnnμvv=rpncrnnv

Dividing both sides by nnv yields

ncpnnvμμv=ncpnnvrrμvr=ncpnnv(μr)

Plugging in the reserve ratio that perfectly hedges risk yields

μvrσv=μrσ

Plugging in μv and σv from Equations (3) and (4), respectively, yields

1v[μpv(p)+12σ2p2v(p)]r1vσpv(p)=μrσ

Or,

μpv(p)+12σ2p2v(p)rv(p)pv(p)=μr

Re-arranging yields:

12σ2p2v(p)+rpv(p)rv(p)=0

This is the Black-Scholes (1973) equation for a perpetual American option. Given this equation, we need to verify parts (a), (b), and (c) of the proposition.

To do so, let’s conjecture that

v(p)=ApB

Plugging this function and the appropriate derivatives into the differential equation demonstrates that this is a solution if B is a solution to the following quadratic equation:

(5)12σ2B2+(r12σ2)Br=0

This equation has one positive solution and one negative solution. The solution to the differential equation can be written as

v(p)=A1pB1+A2pB2

where A1 and A2 are positive constants and B1 is the positive solution and B2 is the negative solution to Equation (5).

This can be simplified further by using economic theory. For example, in the hypothetical scenario in which the price of the commodity goes to zero, the option to purchase the commodity at a fixed price becomes worthless because the commodity becomes worthless and isn’t traded. The option to buy a worthless commodity is therefore also worthless. In other words, the ability to go to the bank and exchange one ounce of the commodity for K dollars is unlikely to be honored as the commodity loses all of its market value. This implies that

limp0v(p)=0

This requires that A= 0. As a result, the solution for v(p) is

(6)v(p)=A1pB1

Also, at the precise price at which the option to buy the commodity is exercised, the individual is indifferent to holding the bank note and buying the commodity. Let p* denote the price at which the option to purchase the commodity is exercised. It follows that

v(p*)=(p*K)NK

where N is the denomination of the note and K is the strike price, or official price of the commodity. The conjectured solution implies that

A1(p*)B1=(p*K)NKA1=(p*)B1[p*K]NK

Substituting this expression for A1 into Equation (6) provides a solution to the value as a function of the current price, the price when the option is exercised, the denomination, and the strike price:

v(p)=(pp*)B1[p*K]NK

To this point, I have referred to p* as the price of the commodity when the option is exercised. However, I have not yet determined this threshold for the price. The threshold can be determined through economic reasoning. A higher threshold increases the value of exercising at the exercise date. However, a higher threshold also reduces the present value of the option. The note holder wants to optimally balance this trade-off and can do so by choosing the threshold, p*, that maximizes v(p). The threshold is given as

(7)p*=(B1B11)K

Now recall that B1 is the positive solution to Equation (5). Let’s write the solution in the form:

(B+λ1)(B+λ2)=0

It follows that

λ1λ2=2rσ2
λ1+λ2=2rσ21

It is straightforward to solve for λ1 = −1 and λ2=2rσ2 and therefore B1 = 1. This is an important result because it will prove parts (a) and (b) of the proposition.

Consider first the implications for part (a) of the proposition. From the threshold for p shown in Equation (7), it follows that

limB11p*=

This means that, in the absence of risk-free arbitrage, the option to redeem the bank note for the commodity is never exercised. This proves part (a).

For part (b) of the proposition, recall that

v(p)=(pp*)B1[p*K]NK

So, when B1 = 1, it follows that

v(p)=(pp*)[p*K]NK=(1Kp*)pNK

and therefore

v(p)=pNK

This proves part (b) of the proposition.[5]

Part (c) says that both the expected rate of return on the bank note and the expected rate of return on the commodity are equal to the real risk-free rate, r. Let’s start with the assumption that the expected rate of return on the bank note is r. Formally, this implies that

r=1dtEdvv

Or,

rv(p)=1dtE(v(p)dp+12v(p)(dp)2)

Plugging in Equation (1) yields

(8)12σ2p2v(p)+μpv(p)=rv(p)

I have already shown that the solution for v(p) is v(p)=pNK. It follows that

v(p)=NK
v(p)=0

Plugging these into Equation (8) yields:

μ=r

Thus, both the expected rate of return on the bank note and the underlying commodity are equal to the risk-free rate. This proves part (c) of the proposition and completes the proof.

A.2 Proof of Proposition 2

Recall that the value of the bank note is given as v(p). Given the probability of bankruptcy, it follows that

dv=v(p)dp+12v(p)(dp)2+ρ[0v(p)]dt

Plugging in Equation (1) yields

dv=[μpv(p)+12σ2p2v(p)ρv(p)]dt+σpv(p)dz

or

dvv=μˆvdt+σvdz

where

μˆv=1v[μpv(p)+12σ2p2v(p)ρv(p)]

and σv is defined as in (4).

From Proposition 1, I know that in the absence of risk-free arbitrage it must be the case that

μˆvrσv=μrσ

Plugging in the values for μˆv and σv yields

1v[μpv(p)+12σ2p2v(p)ρv]r1vσpv(p)=μrσ

Simplifying yields

12σ2p2v(p)+rpv(p)(ρ+r)v=0

This establishes point number (1) in Proposition 2.

From the proof of Proposition 1, I know that the solution to this differential equation is of the form

v(p)=α1pβ1

where β1 is the positive root of

12σ2β2+(r12σ2)β(ρ+r)=0

Or,

β2+(2rσ21)β2(ρ+r)σ2=0

Let λ1 and λ2 be the roots of this equation. It follows that

λ1+λ2=2rσ21
λ1λ2=2(ρ+r)σ2

Note that unlike in the proof to Proposition 1, λ1 ≠ −1.

The positive root is given as

β1=(2rσ21)+(2rσ21)2+8(ρ+r)σ22

It is easy to verify that β1 > 1.

Using the derivation from the proof of Proposition 1, the threshold for the price of the commodity is

p*=(β1β11)K

This establishes point (2) of Proposition 2.

Given p*, it follows that

v(p)=(pp*)β1[1β11]N

where

β1=(2rσ21)+(2rσ21)2+8(ρ+r)σ22

Thus, the threshold for the price of the commodity is finite and, since β1 > 1, the value of the bank note is a convex function of the price of the underlying commodity. This establishes point (3). QED.

A.3 A Sketch of a Proof of Claim 1

With jumps, the price of the commodity is

dpp=μdt+σdzδdq

Let v(p) again denote the value of the bank note. It follows that

dv=v(p)[μdt+σdzδdq]+12v(p)σ2p2dt+{v[(1δ)p]v(p)}dq

Or,

dvv=μvdt+σvdz+1v{v[(1δ)p]v(p)}dq

Suppose that a bank constructs the following portfolio:

W=pnc+Bnnv(p)

It follows that dW is

dW=[ncμp+rBnnμvv]dt+[ncσp+nnσvv]dz+[nn{v[(1δ)p]v(p)}ncδp]dq

Suppose that as in Proposition 1 that the bank chooses its reserve ratio such that

ncpnnv=σvσ

It follows that

dW=[nn(σvvμσμvv)+rB]dt+nn[{v[(1δ)p]v(p)}σvδvσ]dq

The risk from the “jump” that is due to discoveries of the commodity cannot be eliminated. I say that this is a sketch of a proof because this does not prove that the risk cannot be perfectly hedged. As mentioned in the text, some argue that jumps should be treated as idiosyncratic risk that can be diversified away.

A.4 Proof of Result 1

Suppose that the bank note is valued as though it has an expected rate of return equal to the risk-free rate. It follows that

Edvv=rdt

Using Ito’s Lemma, this implies that

12σ2p2v(p)+μpv(p)+λ{v[(1δ)p]v(p)}=rv(p)

Or,

12σ2p2v(p)+μpv(p)(r+λ)v(p)+λv[(1δ)p]=0

Let’s conjecture that

v(p)=Γpγ

Plugging this into the differential equation above yields:

12σ2γ2+(μ12σ2)γ(r+λ)+λ(1δ)γ=0

A positive solution for γ can be obtained via numerical methods. Given that the conjectured solution is a solution, it follows from the proof to Proposition 1 that the value of the bank note can be written as

v(p)=(pp*)γ[p*K]NK

and

p*=(γγ1)K

QED.

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Received: 2020-02-10
Accepted: 2021-02-24
Published Online: 2021-04-02

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