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

A.1 Proof of Proposition 1

Imagine that a bank issues n _n 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 n _c ounces of the commodity. It follows that the net worth, W, of the issuing bank can be written as

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

or

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

Solving for the reserve ratio yields:

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

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

Given the portfolio above, this implies that

Thus, it must be true that

Or,

Dividing both sides by n _n v yields

Plugging in the reserve ratio that perfectly hedges risk yields

Plugging in μ _v and σ _v from Equations (3) and (4), respectively, yields

Or,

Re-arranging yields:

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

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:

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

where A ₁ and A ₂ are positive constants and B ₁ is the positive solution and B ₂ 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

This requires that A ₂ = 0. As a result, the solution for v(p) is

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

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

Substituting this expression for A ₁ 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:

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

Now recall that B ₁ is the positive solution to Equation (5). Let’s write the solution in the form:

It follows that

It is straightforward to solve for λ ₁ = −1 and λ 2 = 2 r σ 2 and therefore B ₁ = 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

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

So, when B ₁ = 1, it follows that

and therefore

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

Or,

Plugging in Equation (1) yields

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

Plugging these into Equation (8) yields:

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

Plugging in Equation (1) yields

or

where

and σ _v is defined as in (4).

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

Plugging in the values for μ ˆ v and σ _v yields

Simplifying yields

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

where β ₁ is the positive root of

Or,

Let λ ₁ and λ ₂ be the roots of this equation. It follows that

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

The positive root is given as

It is easy to verify that β ₁ > 1.

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

This establishes point (2) of Proposition 2.

Given p ^*, it follows that

where

Thus, the threshold for the price of the commodity is finite and, since β ₁ > 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

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

Or,

Suppose that a bank constructs the following portfolio:

It follows that dW is

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

It follows that

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

Using Ito’s Lemma, this implies that

Or,

Let’s conjecture that

Plugging this into the differential equation above yields:

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

and

QED.