The Carr-Madan replication formula ensures that knowledge of the prices of options at every strike is equivalent to knowing the entire pricing distribution . In many situations, the available market data is insufficient to determine this distribution precisely,  and the question arises: what are the bounds for the option price at a specified strike, given the market-implied constraints? Applying techniques from the analysis of quantum systems, operator algebra methods are here used to generate an upper bound for the price of a basket option , depending only on a covariance matrix generated from the constituent assets in the basket . The result is then used to create converging families of  bounds for vanilla options, interpolate the volatility smile, and analyse the relationships between swaption and caplet prices  .
Option pricing is the most elementary challenge of mathematical finance. Knowledge of the prices of options at every strike is equivalent to knowing the entire pricing distribution for a security, as derivatives contingent on the security can be replicated using options. The available data may be insufficient to determine this distribution precisely, however, and the question arises: What are the bounds for the option price at a specified strike, given the market-implied constraints? Positivity of the price map imposed by the principle of no-arbitrage is here utilised, via the Gelfand-Naimark-Segal construction, to transform the problem into the domain of operator algebras. Optimisation in this larger context is essentially geometric, and the outcome is simultaneously super-optimal for all commutative subalgebras. This generates an upper bound for the price of a basket option . With innovative decomposition of the assets in the basket , the result is used to create converging families of price bounds for vanilla options, interpolate the volatility smile, and analyse the relationships between swaption and caplet prices . Beyond these practical applications, the approach is liberated from the requirement that the assets are commutative, accommodating an extension to the framework for mathematical finance that is equally applicable in noncommutative geometries .
R
The Carr-Madan replication formula ensures that knowledge of
Option pricing is the most elementary challenge of mathematical finance. Knowledge of
R
. In many situations, the available market data is
for a security, as derivatives contingent on the security can be replicated using options. The available data may be
A
however,
R
what
What
R
Applying techniques from the analysis of quantum systems, operator algebra methods are here used to generate
Positivity of the price map imposed by the principle of no-arbitrage is here utilised, via the Gelfand-Naimark-Segal construction, to transform the problem into the domain of operator algebras. Optimisation in this larger context is essentially geometric, and the outcome is simultaneously super-optimal for all commutative subalgebras. This generates
R
, depending only on a covariance matrix generated from the constituent
. With innovative decomposition of the
R
. The result is then
, the result is
A
price
A
. Beyond these practical applications, the approach is liberated from the requirement that the assets are commutative, accommodating an extension to the framework for mathematical finance that is equally applicable in noncommutative geometries