Warrants

A warrant is an option-like security on equity, but it is issued by the same company which has issued the equity, and when a warrant is exercised, a new stock is issued. This new stock is issued at a the warrant strike price, which is lower than the current stock price (If it wasn't the warrant would not be exercised.) Since the new stock is a a fractional right to all cashflows, this stock issue waters out, or dilutes, the equity in a company. The degree of dilution is a function of how many warrants are issued.

Warrant value in terms of assets

Let $K$ be the strike price, $n$ the number of shares outstanding and $m$ the number of warrants issues. Assume each warrant is for 1 new share, and let $A_t$ be the current asset value of firm. Suppose all warrants are exercised simultaneously. Then the assets of the firm increasy by the number of warrants times the strike price of the warrant.

$$A_t+ m K,$$

but this new asset value is spread over more shares, since each exercised warrant is now an equity. The assets of the firm is spread over all shares, hence each new share is worth:

$$\frac{A_t+ m K}{m+n}$$

making each exercised warrant worth:

$$\frac{A_t+ m K}{m+n} - K = \frac{n}{m+n}\left(\frac{A_t}{n}-K\right)$$

If we knew the current value of assets in the company, we could value the warrant in two steps:

1. Value the option using the Black Scholes formula and $\frac{A_t}{n}$ as the current stock price.
2. Multiply the resulting call price with $\frac{n}{m+n}$.

If we let $W_t$ be the warrant value, the above arguments are summarized as:

$$W_t = \frac{n}{n+m} C_{BS} \left(\frac{A}{n},K,\sigma,r,(T-t)\right),$$

where $C_{BS}(\cdot)$ is the Black Scholes formula.

Valuing warrants when observing the stock value

However, one does not necessarily observe the asset value of the firm. Typically one only observes the equity value of the firm. If we let $S_t$ be the current stock price, the asset value is really:

$$A_t = n S_t + m W_t$$

Using the stock price, one would value the warrant as

$$W_t = \frac{n}{n+m} C_{BS} \left(\frac{nS_t+mW_t}{n},K,\sigma,r,(T-t)\right)$$

or

$$W_t = \frac{n}{n+m} C_{BS} \left(S_t +\frac{m}{n}W_t,K,\sigma,r,(T-t)\right)$$

Note that this gives the value of $W_t$ as a function of $W_t$. One need to solve this equation numerically to find $W_t$.

The numerical solution for $W_t$ is done using the Newton-Rhapson method. Let

$$g(W_t) = W_t - \frac{n}{n+m}C_{BS}\left(S_t + \frac{m}{n}W_t,K,\sigma,r,(T-t) \right)$$

Starting with an initial guess for the warrant value $W^o_t$, the Newton-Rhapson method is that one iterates as follows

$$W^{i}_t= W^{i-1}_t - \frac{g(W^{i-1}_t)}{g^\prime(W^{i-1}_t)},$$

where $i$ signifies iteration $i$, until the criterion function $g(W^{i-1}_t)$ is below some given accuracy $\epsilon$. In this case

$$g^\prime(W_t) = 1-\frac{m}{m+n} N(d_1)$$

where

$$d_1 = \frac{\ln\left(\frac{S_t+\frac{m}{n}W_t}{K}\right)+(r+\frac{1}{2}\sigma^2)(T-t)} {\sigma\sqrt{T-t}}$$

An obvious starting value is to set calculate the Black Scholes value using the current stock price, and multiplying it with $\frac{m}{m+n}$.

Code 7.1 implements this calculation.

Readings

McDonald (2002) and Hull (2003) are general references. A problem with warrants is that exercise of all warrants simultaneously is not necessarily optimal.

Press et al. (1992) discusses the Newton-Rhapson method for root finding.