## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 13, Number 3 (2003), 890-913.

### Volatility time and properties of option prices

#### Abstract

We use a notion of stochastic time, here called *volatility time*, to show convexity
of option prices in the underlying asset if the contract function is convex as well as continuity and monotonicity of the option price in the
volatility. The volatility time is obtained as the almost surely unique stopping time
solution to a random ordinary differential equation related to
volatility. This enables us to write price processes, or processes
modeled by local martingales, as Brownian motions with respect
to volatility time. The results are shown under very weak assumptions and are
of independent interest in the study of stochastic differential equations.
Options on several underlying assets are also studied and we prove
that if the volatility matrix is independent of time,
then the option prices decay with time if the contract function is convex.
However, the option prices are no longer necessarily convex in the
underlying assets and the option prices do
not necessarily decay with time, if a time-dependent volatility is allowed.

#### Article information

**Source**

Ann. Appl. Probab., Volume 13, Number 3 (2003), 890-913.

**Dates**

First available in Project Euclid: 6 August 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1060202830

**Digital Object Identifier**

doi:10.1214/aoap/1060202830

**Mathematical Reviews number (MathSciNet)**

MR1994040

**Zentralblatt MATH identifier**

1061.91028

**Subjects**

Primary: 91B28 60H30: Applications of stochastic analysis (to PDE, etc.) 35K15: Initial value problems for second-order parabolic equations

**Keywords**

Volatility stochastic time contingent claim convexity time decay

#### Citation

Janson, Svante; Tysk, Johan. Volatility time and properties of option prices. Ann. Appl. Probab. 13 (2003), no. 3, 890--913. doi:10.1214/aoap/1060202830. https://projecteuclid.org/euclid.aoap/1060202830