Pricing of Warrants in a Firm

1
Pricing of Warrants in a Firm

• P.W.A.Dayananda
• Dept. of Mathematics
• University of St.Thomas
• St.Paul, MN 55105

2
What is a financial warrant?

• This is a financial instrument that allows the
  holder to acquire a share of stock by paying K
  at time
• Normally issued with bonds as a sweetner.
• In many cases they trade seperatly from the
  associated bond issue.
• Exercise is normally 5-15 yrs time.

3
What are the main charcteritics?

• Normally they can not be traded.
• Similar to exchange trade Call option
• Similar to executive stock options.
• Warrants may be called prior to exercise and even
  the exercise price could be changed.

4
Growth of financial warrants

• Market capitalization of warrants grew at the
  rate of 26 per annum for the last decade.
• In US Chrysler gave the federal government stock
  warrants in exchange for loans.
• In some countries in Asia warrants account for
  about 0.4 of total market value.

5
How do the warrants affect the Firm?

• When they are exercised the firm needs to issue
  new shares.
• Firms equity gets diluted when warrants are
  exercised.
• Warrants holders on exercise have voting rights
  and other privileges when they are exercised as
  they become common share-holders.

6
Assumptions for modeling

• Stock pays no dividends.
• Firm has N common shares and n warrants only.
  Share price at time t is S(t).
• Let total equity value at time t be V(t)
  v(t)V(t)/N.
• Stock price dynamics
  (1)
• Equity per share dynamics
• 
  (2)

7
Preliminaries-1

• Payoff of a warrant at exercise time
• 
  (3)
• The present value of a warrant at time t0
• is
  (4)
• and
  (5)

8
Preliminaries-2

• Note W(t) is a random variable
• Value is deterministic
• We can not use Black-Scholes formula to value
  as the process is not observable and it does not
  have known volatility.
• However, text books use Black-Scholes to value
  warrants-which we will discuss later.

9
Preliminaries-3

• One can show that satisfies the pdf
• (6)
• Also Total equitytotal market value of shares
  total discounted value of warrants. This gives
• Nv(t)NS(t)nW(t).
• Hence
  (7)

10
Preliminaries-4

• Take expectation of (7) with respect to risk
  neutral measure giving
• (8)
• where

11
Previous method used for warrant pricing

• Authors assumed that the process v(t) has
  geometric Brownian motion with constant
  volatility and used BS formula
• (9)
• together with
  (8)
• Iteratively to find

12
Volatility of equity compared to that of stock
price volatility -1

• We use Itos formula for relation in (7)
  giving (10)
• Thus the Elasticity (E) of the warrant price
• (11)

13
Volatility of equity compared to that of stock
price volatility-2

• But , where (12)
• Note Egt1

14
Volatility of equity compared to that of stock
price volatility -3

• Thus, equating the volatility terms on both sides
  of (10) and using (11) gives
• (13)
• Hence (14)
• So when . (15)
• Empirical work of Galai (1991) supports it.

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Pricing Warrants-1

• We write , and .
• Then from (13) we have
• (16)
• Differentiating (16) with respect to v, we get
• (17)

16
Pricing warrants-2

• But we had from (6) that
• (6)
• Compare (6) with (17)
• (17)
• This gives
• (18)

17
Relations for warrant price-

• From first two terms in (18), we have
• 
  (19)
• Hence
• (20)

18
Plot of warrant price against alpha
19
Conditions for gt0

• Using the numerator of the expression (20) for
  warrant price as a quadratic in
• we have the relation that must hold.

20
Ordinary differential equation for alpha ( )

• The first and third terms in (18) gives the pdf
  (21)
• Substituting for from (20), we get the ODE
• (22)

21
Boundary conditions to solve (22)

• One can show that at exercise time
• (23)
• This gives the relation for the boundary value
  for
• (24)
• (Explain why sign is used)

22
Solving the ODE in alpha then warrant price

• Our prime task now is to solve the equation (24)
  which is of the form
• (25)
• And then substitute for in (20)
• (20)
• to obtain warrant price - at time t0.

23
Approximation- to second order in alpha

• We neglect third higher order terms in
• (25) giving
• (26)
• Numerical illustrations are provided in the
• tables when r0.05,K55,s(0)50 ,
• and different values.

24
Table -1

25
Table -2
26
Table-3