Martingale pricing

Overview

Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options, futures, interest rate derivatives, credit derivatives, etc. In contrast to the PDE approach to pricing, martingale pricing formulae are in the form of expectations, which can be efficiently solved numerically using a Monte Carlo approach. As such, martingale pricing is preferred when valuing high-dimensional contracts, such as a basket of options. On the other hand, valuing American-style contracts is troublesome and requires discretizing the problem (making it like a Bermudan option) and only in 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing American options.

Measure theory representation

Suppose the state of the market can be represented by the filtered probability space,

( Ω , (

F

t

)

t ∈ [ 0 , T ]

,

P

~

)

{\displaystyle (\Omega ,({\mathcal {F}}_{t})_{t\in [0,T]},{\tilde {\mathbb {P} }})}

. Let

{ S ( t )

}

t ∈ [ 0 , T ]

{\displaystyle \{S(t)\}_{t\in [0,T]}}

be a stochastic price process on this space. One may price a derivative security,

V ( t , S ( t ) )

{\displaystyle V(t,S(t))}

under the philosophy of no arbitrage as,

See also

Brownian model of financial markets Martingale (probability theory)