42
4 Stochastic Processes
This caveat notwithstanding, we can determine the price of the European put
option p in the same way we used for the call option above by integrating over the
shaded region on the left-hand side in Fig. 4.3. This leads to
p = e
−ρt
K
−∞
(K − S))(S, t)d S ,
(4.41)
where K –S is the pay-off function of the put option for S < K . The integral, in the
same way as before, weighs the pay-off with probability (S, t)d S and then backpropagates this expectation value to the initial time with the discount factor e
−ρt
.
Evaluating the integrals leads to the well-known form
p = K e
−ρt N
ln(K /S 0 ) − ρt + σ
2 t/2
σ
√
t
−S 0 N
ln(K /S 0 ) − ρt − σ
2 t/2
σ
√
t
= K e
−ρt N
−
ln(S 0 /K ) + (ρ − σ
2
/2)t
σ
√
t
(4.42)
−S 0 N
−
ln(S 0 /K ) + (ρ + σ
2
/2)t
σ
√
t
,
where, in the second form, the cumulative distribution function N has the same
arguments as in (4.40). These arguments are commonly used in the financial literature
and are denoted by
d 1 =
ln(S 0 /K ) + (ρ + σ
2
/2)t
σ
√
t
d 2 =
ln(S 0 /K ) + (ρ − σ
2
/2)t
σ
√
t
= d 1 − σ
√
t .
(4.43)
Note that the derivation did not use any assumptions of risk-balancing a portfolio. All
we did was calculating the expectation value of the expected pay-off if the average
value grows with rate ρ and the distributions of stocks, shares or options are lognormal.
The prices of put and call options with the same strike price K are linked by the
so-called put-call parity, which follows directly from the pricing formulas for the
respective options. To prove this we write the definition of the put option and use the
property of the cumulative distribution function, namely that N (−z) = 1 − N (z).
We find
4 Stochastic Processes
This caveat notwithstanding, we can determine the price of the European put
option p in the same way we used for the call option above by integrating over the
shaded region on the left-hand side in Fig. 4.3. This leads to
p = e
−ρt
K
−∞
(K − S))(S, t)d S ,
(4.41)
where K –S is the pay-off function of the put option for S < K . The integral, in the
same way as before, weighs the pay-off with probability (S, t)d S and then backpropagates this expectation value to the initial time with the discount factor e
−ρt
.
Evaluating the integrals leads to the well-known form
p = K e
−ρt N
ln(K /S 0 ) − ρt + σ
2 t/2
σ
√
t
−S 0 N
ln(K /S 0 ) − ρt − σ
2 t/2
σ
√
t
= K e
−ρt N
−
ln(S 0 /K ) + (ρ − σ
2
/2)t
σ
√
t
(4.42)
−S 0 N
−
ln(S 0 /K ) + (ρ + σ
2
/2)t
σ
√
t
,
where, in the second form, the cumulative distribution function N has the same
arguments as in (4.40). These arguments are commonly used in the financial literature
and are denoted by
d 1 =
ln(S 0 /K ) + (ρ + σ
2
/2)t
σ
√
t
d 2 =
ln(S 0 /K ) + (ρ − σ
2
/2)t
σ
√
t
= d 1 − σ
√
t .
(4.43)
Note that the derivation did not use any assumptions of risk-balancing a portfolio. All
we did was calculating the expectation value of the expected pay-off if the average
value grows with rate ρ and the distributions of stocks, shares or options are lognormal.
The prices of put and call options with the same strike price K are linked by the
so-called put-call parity, which follows directly from the pricing formulas for the
respective options. To prove this we write the definition of the put option and use the
property of the cumulative distribution function, namely that N (−z) = 1 − N (z).
We find