DIFFUSIONS AND STOCHASTIC CALCULUS
It is interesting to observe that the solution does not involve the parameter μ (in fact, no μ
appears in the PDE). This seems quite mysterious at first thought: for example, one would
think that a call option would be more valuable for a stock that is expected to rise a great
deal than for one that is expected to fall! This is a point well worth pondering, and we will
discuss this further later.
This lack of dependence on μ is fortunate. To use the B-S formula, we do not need
to know or even try to estimate the μ that appeared in our original stochastic model of
the stock price behavior. The parameter σ does enter into the formula, however. But σ is
easier to estimate than μ.
Some final B-S comments: the Black-Scholes formula is really a sort of test of consistency:
the prices of the stock, the option, and the interest rate should be consistent with
each other. In particular, the theory underlying the formula is not an equilibrium theory;
we do not have to worry about people’s utility functions and other imponderables. That
may be a key reason why the formula is so successful in practice as compared with many
other results in economics: it does not require us to pretend that we know many things
that we cannot know. Briefly, the reason we can get away with this is that the option is in
fact “redundant” given the stock and bond—a portfolio can be formed using just the stock
and the bond that duplicates the cash flows from the option. Thus, if we observe the stock
price process and the interest rate, then the option price is determined by a no-arbitrage
condition.
hope that may help sombody,got the whole paper if anybody wants it,
It is interesting to observe that the solution does not involve the parameter μ (in fact, no μ
appears in the PDE). This seems quite mysterious at first thought: for example, one would
think that a call option would be more valuable for a stock that is expected to rise a great
deal than for one that is expected to fall! This is a point well worth pondering, and we will
discuss this further later.
This lack of dependence on μ is fortunate. To use the B-S formula, we do not need
to know or even try to estimate the μ that appeared in our original stochastic model of
the stock price behavior. The parameter σ does enter into the formula, however. But σ is
easier to estimate than μ.
Some final B-S comments: the Black-Scholes formula is really a sort of test of consistency:
the prices of the stock, the option, and the interest rate should be consistent with
each other. In particular, the theory underlying the formula is not an equilibrium theory;
we do not have to worry about people’s utility functions and other imponderables. That
may be a key reason why the formula is so successful in practice as compared with many
other results in economics: it does not require us to pretend that we know many things
that we cannot know. Briefly, the reason we can get away with this is that the option is in
fact “redundant” given the stock and bond—a portfolio can be formed using just the stock
and the bond that duplicates the cash flows from the option. Thus, if we observe the stock
price process and the interest rate, then the option price is determined by a no-arbitrage
condition.
hope that may help sombody,got the whole paper if anybody wants it,