4.6 A First Look at Option Pricing
43
p = K e
−ρt N (−d 2 ) − S 0 N (−d 1 )
= K e
−ρt [1 − N (d 2 )] − S 0 [1 − N (d 1 )]
(4.44)
= K e
−ρt
− S 0 + S 0 N (d 1 ) − K e
−ρt N (d 2 )
= K e
−ρt
− S 0 + c ,
which can be rewritten as
c + K e
−ρt
= p + S 0 .
(4.45)
The two sides of this equation can be interpreted as the values of two portfolios at the
time t = 0 the options are sold. The portfolio on the right hand side of equation (4.45)
consists of a put option and shares S 0 . The one on the left hand side consists of a
call option c and a cash value K e
−ρt invested with rate ρ. But why should the bank
give you rate ρ and what is a fair price for the call option c and put option p? For
simplicity we assume that we deposit some amount K e
−r f t at the bank at the risk-free
rate r f . In the next chapter we will see that there is an intricate relation between this
assumption and fair pricing of options.
We can now ask what the values of the two portfolios is at the strike time t = T.
First consider the portfolio on the right-hand side. If the stock price exceeds the
strike price we forfeit the put option and keep the stock which has a value S T . If
the stock price is below the strike price we exercise the put option and make a profit
of K − S T and the value of the stock is S T such that the value of this portfolio
is K − S T + S T = K . We now compare this outcome with that of the portfolio
described by the left-hand side in (4.45). If the stock value S T at time T is above
the strike price S T > K we exercise the option which constitutes a value of S T − K .
Jointly with the bank deposit which has grown to value K the total value of this
portfolio is thus S T − K + K = S T . If the stock value is below the strike price
S T < K we forfeit the call option but still have the value K in the bank which
constitutes the total value of our portfolio.
Summarizing, we find that both portfolios have the value S T if S T > K and K
in case S T < K . Thus the value of both portfolios is equal in all cases, just as the
(4.45) indicates. We note, however, that there is a deeper link between the risk-free
rate r f at which we deposited a sum of cash and the fair prices for the options. In the
preceding chapter we always assumed that the growth rate of the stock was given by
some rate ρ, possibly derived from CAPM. So the valuation of the options in this
chapter is not necessarily fair, but depends on guessing a reasonable value for ρ.
In the next chapter we will resolve this puzzle by determining a differential equation for the temporal evolution of option prices and building portfolios that are (quasi)
risk-free, and imply fair prices for the options. But before doing so that let us briefly
digress on the calculation of expectation values.
43
p = K e
−ρt N (−d 2 ) − S 0 N (−d 1 )
= K e
−ρt [1 − N (d 2 )] − S 0 [1 − N (d 1 )]
(4.44)
= K e
−ρt
− S 0 + S 0 N (d 1 ) − K e
−ρt N (d 2 )
= K e
−ρt
− S 0 + c ,
which can be rewritten as
c + K e
−ρt
= p + S 0 .
(4.45)
The two sides of this equation can be interpreted as the values of two portfolios at the
time t = 0 the options are sold. The portfolio on the right hand side of equation (4.45)
consists of a put option and shares S 0 . The one on the left hand side consists of a
call option c and a cash value K e
−ρt invested with rate ρ. But why should the bank
give you rate ρ and what is a fair price for the call option c and put option p? For
simplicity we assume that we deposit some amount K e
−r f t at the bank at the risk-free
rate r f . In the next chapter we will see that there is an intricate relation between this
assumption and fair pricing of options.
We can now ask what the values of the two portfolios is at the strike time t = T.
First consider the portfolio on the right-hand side. If the stock price exceeds the
strike price we forfeit the put option and keep the stock which has a value S T . If
the stock price is below the strike price we exercise the put option and make a profit
of K − S T and the value of the stock is S T such that the value of this portfolio
is K − S T + S T = K . We now compare this outcome with that of the portfolio
described by the left-hand side in (4.45). If the stock value S T at time T is above
the strike price S T > K we exercise the option which constitutes a value of S T − K .
Jointly with the bank deposit which has grown to value K the total value of this
portfolio is thus S T − K + K = S T . If the stock value is below the strike price
S T < K we forfeit the call option but still have the value K in the bank which
constitutes the total value of our portfolio.
Summarizing, we find that both portfolios have the value S T if S T > K and K
in case S T < K . Thus the value of both portfolios is equal in all cases, just as the
(4.45) indicates. We note, however, that there is a deeper link between the risk-free
rate r f at which we deposited a sum of cash and the fair prices for the options. In the
preceding chapter we always assumed that the growth rate of the stock was given by
some rate ρ, possibly derived from CAPM. So the valuation of the options in this
chapter is not necessarily fair, but depends on guessing a reasonable value for ρ.
In the next chapter we will resolve this puzzle by determining a differential equation for the temporal evolution of option prices and building portfolios that are (quasi)
risk-free, and imply fair prices for the options. But before doing so that let us briefly
digress on the calculation of expectation values.