In their second article, Johan Beumée and Paul Wilmott look at how to construct a pricing model for warrants.
To construct our pricing model we focus on the capital structure of the company rather than the share price as issuing warrants alters the asset and liability balance, and warrant exercise might affect the share price.
Imagine that a company issues M fixed strike (K) warrants with price W(0) thereby adding an amount of MW(0) cash to the company's assets. At the time of issuance this cash offsets the newly created liability exactly since the warrant holders presumably paid the fair price for their warrants. For the sake of simplicity we assume the asset value of the company in terms of machines, buildings, quality of its management, etc. plus the cash received from the warrants can be represented by an independent variable A(t).
The total equity value of the company at time t is given by A(t)-MW(t) (assets plus cash minus liabilities) and the N shares currently in existence have an equal claim on this value. Hence at any time t after or including issuance it must be true that NS(t)+MW(t)=A(t), where S(t) is the value of a share. The warrants may be European style or American style and once they are exercised the company loses the corresponding liability and gains an amount of cash K.
By design the asset value of the company A(t) remains invariant under rational exercise and a Black-Scholes type equation of motion can now be constructed under the assumption the warrant premium can be expressed as a function of the total asset value of the company. Hence, we write W(t)=W(A(t),t). It is assumed that A(t) satisfies the usual log-normal stochastic differential equation with volatility Vol(A). Neither the mean nor the volatility depends on the share price nor the warrant premium so A(t) remains invariant under warrant exercise.
An important point arises when the company receives cash as a result of a warrant exercise. The actual asset value of the company upon reception of the strike jumps by an amount of cash K and so a choice must be made as to how this cash must be managed. It could be invested in bonds, for instance, or the company could decide to acquire more machinery or use the cash to merge with another company. As the asset mean and volatility reflect the overall corporate policy of this company it seems unreasonable to assume the charecteristics of the company change as a result of the arrival of a cash payment. In the remainder it will be assumed that the asset value incorporates the arrived cash payment and will otherwise continue to evolve according to some log-normal equation. This has a bearing on the optimal exercise policy discussed below.
By Black-Scholes analogy, the equation of motion for the warrant price is now obtained by creating a risk free portfolio H(t) containing one option W(t)=W(A(t),t) hedged by a short share position. The appropriate delta depends on the absolute number of shares and warrants and the warrant sensitivities vis-à-vis the asset value A(t). Incorporating a realistic dividend payment scheme as a continuous yield q the equations of motion becomes (see equation 1) where r is the risk free interest rate. This non-linear diffusion equation is the equivalent of the Black-Scholes differential equation expressing the price of the warrant as a function of the asset value rather than the share price.
Using this representation the effect of warrant exercise can be quantified. Assume that a warrant is exercised at time t so the company issues a share which is priced at S(t) and receives the strike K. If S*(t) is the share value immediately after exercise then it can be shown that S*(t)=S(t)+g(W(t)-S(t)+K, where 1/g=(N+1)(1+df). Here d is the warrant delta (with respect to the underlying share price) which necessarily must be between zero and one and f is approximately proportional to the fraction M/N. Exercising a warrant when W(t) >= S(t)-K will add value to the shares of the remaining shareholders and can therefore not be an optimal exercise policy. A warrant holder will wait until W(t)=S(t)-K before exercising a warrant at which point it is not optimal to hold the hedged warrant portfolio any longer. Notice that at this exercise boundary the share price does not change under exercise hence the warrant premium of the remaining warrants must also remain constant.
To obtain the premium for an American style warrant the equation must be solved under the boundary condition W(t)<S(t)-K which can be rewritten as W(t)=(A(t)-NK)/(N+M). In the case of a European style warrant we search for a solution with the final boundary (payoff) W(T)=max(S(T)-K,0), which upon substitution is seen to be equivalent to (see equation 2) with T being option expiry.
Though the warrant equation is very similar to the Black-Scholes equation the term containing the dividend makes the solution to this equation more complicated. The dividend term, in particular, is not linear so a simple closed form solution does not present itself even for European style warrants. The non-linear term is important since the premium of the warrant is very sensitive with respect to the dividend policy of the company.
To find a first approximation to a solution of this equation for European style warrants the non-linear term is simply ignored which is equivalent to assuming that the dilution effect is small. The equation then reduces to the standard Black-Scholes equation which has an analytical answer in the case of European warrants. In fact, solving under the appropriate final conditions we find the implicit warrent price (see equation 3) where BS(S+cW(0),NK,r,q,vol(A) is the standard Black-Scholes solution with S+cW(0),NK,r,q,vol(A) being the spot, strike, risk-free rate, continuous dividend yield and volatility of the company asset value respectively and c=M/N, S=S(0) is the spot share price. This is the dilution suggested by Hull.
The warrant premium can be obtained implicitly from this equation or can be approximated using the deltas and gammas of the original Black-Scholes expression via a Taylor expansion. In fact, some manipulation reveals that approximately W(0) =BS(S,K,r,q,vol(A))/(1+c-cd) where d is the delta option. This shows that a European style warrant correction typically increases if the option moves further into the money and decreases if the potential dilution c becomes larger.
