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


At time , company issues warrants each with strike price and expiration .

  • : number of outstanding shares
  • : number of warrants issued
  • : stock value at time
  • : Company value (=MCAP)
    • We currently don’t know At time , suppose all warrant holders exercise their warrants:
  • before the warrants exercised
  • after the warrants exercised We don’t know


Consider two portfolios and .

  • is just a simple warrant that expires at time with strike
    • time ,
    • time ,
  • is a long of units of call option on the stock.
    • time ,
    • time , Now,

By the Law of One Price , meaning that: