Breeden and Litzenberger (1978) show that one can replicate Arrow-Debreu prices using the concept of butterfly spread on European call options. This spread entails selling two call options at exercise price , buying one call option at and another at , where is the stepsize between the adjacent call strikes. These four options constitute a butterfly spread centered on . If the terminal underlying asset value is equal to then the payoff of of such butterfly spreads is defined as:
where
As tends to zero, this security becomes an Arrow-Debreu security paying
if and zero in other states. As it is assumed that has a continuous
distribution function on
, the probability of any given level of
is zero and thus, in this case, the price of an Arrow-Debreu security is zero. However,
dividing one more time by , one obtains the price of
and as tends to 0 this price tends
to
for . Indeed,
This can be proved by setting the payoff of this new
security
in (8.1) and letting tend to 0. Indeed, one
should remark that:
If one can construct these financial instruments on a continuum of states (strike prices) then at infinitely small a complete state pricing function can be defined.
Moreover, as tends to zero, this price will tend to the second derivative of
the call pricing function with respect to the strike price evaluated at :
Equating (8.3) and (8.4) across all states yields:
Here denotes the drift and the volatility assumed to be constant.
The analytical formula for the price in of a call option with a terminal date
, a strike price , an underlying price , a risk-free rate
, a continuous dividend yield
, and a volatility ,
is:
As a consequence of the assumptions on the underlying asset price
process the Black-Scholes SPD is a log-normal density with mean
and
variance
for
:
The Black-Scholes SPD can be calculated in XploRe using the following quantlet:
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The arguments are the strike prices , underlying price , risk-free interest rate , dividend yields , implied volatility of the option , and the time to maturity . The output consist of the Black-Scholes SPD (bsspd.fbs), (bsspd.delta), and the (bsspd.gamma) of the call options. Please note that spdbs can be applied to put options by using the Put-Call parity.
However, it is widely known that the Black-Scholes call option formula is not valid empirically. For more details, please refer to Chapter 6. Since the Black-Scholes model contains empirical irregularities, its SPD will not be consistent with the data. Consequently, some other techniques for estimating the SPD without any assumptions on the underlying diffusion process have been developed in the last years.