Warrants (Finance)

Warrants (=stock options) are a method that companies use to compensate employees (in addition to salary.) They can be priced using the BSM formula.

Situation §

At time $t=0$, company issues warrants each with strike price $K$ and expiration $T$.

• $N_{out}$: number of outstanding shares
• $N_w$: number of warrants issued
• $S_t$: stock value at time $t$
• $V(0)=N_{out}{S}_0+N_{w}W(0)$: Company value (=MCAP)
  • We currently don’t know $W(0)$ At time $t=T$, suppose all warrant holders exercise their warrants:
• ${\text{MCAP}}_{\text{before}}(T)=N_{out}{S}_{T,\text{before}}$ before the warrants exercised
• ${\text{MCAP}}_{\text{after}}(T)=(N_{out}+N_w)S_{T,\text{after}}$ after the warrants exercised We don’t know $S_{T,\text{after}}=\frac{{\text{MCAP}}_{\text{before}}(T)+N_{w}K}{N_{out}+N_w}=\frac{{\text{MCAP}}_{\text{after}}}{N_{out}+N_w}$

Pricing §

Consider two portfolios $A$ and $B$.

• $A$ is just a simple warrant that expires at time $T$ with strike $K$
  • time $t=0$, $V_A(0)=W(0)$
  • time $t=T$, $V_A(T)=\max \left\{\frac{{\text{MCAP}}_{\text{after}}}{N_{out}+n_w}-K,0\right\}$
• $B$ is a long of $\frac{N_{out}}{N_{out}+N_w}$ units of call option on the stock.
  • time $t=0$, $V_B(0)=\frac{N_{out}}{N_{out}+N_w}{C}_0^E$
  • time $t=T$, $V_B(T)=\frac{N_{out}}{N_{out}+N_w}\max \left\{\frac{{\text{MCAP}}_{\text{before}}}{N_{out}}-K,0\right\}$ Now,

$$\begin{array}{rl}{V}_A(T)& =\max \left\{\frac{{\text{MCAP}}_{\text{after}}}{N_{out}+N_w}-K,0\right\}\\ & =\max \left\{\frac{{\text{MCAP}}_{\text{before}}+\cancel{N_{w}K}}{N_{out}+N_w}-\frac{K(N_{out}+\cancel{N_w})}{N_{out}+N_w},0\right\}\\ & =\max \left\{\frac{{\text{MCAP}}_{\text{before}}}{N_{out}+N_w}-\frac{K{N}_{out}}{N_{out}+N_w},0\right\}\\ & =\frac{N_{out}}{N_{out}+N_w}\max \left\{\frac{{\text{MCAP}}_{\text{before}}}{N_{out}}-K,0\right\}\\ & \\ & =V_B(T)\end{array}$$

By the Law of One Price $V_A(0)=V_B(0)$, meaning that:

$$W(0)=\frac{N_{out}}{N_{out}+N_w}{C}_0^E$$

■