Volatility swap

Overview

In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. Volatility swaps allow investors to trade the volatility of an asset directly, much as they would trade a price index. Its payoff at expiration is equal to

(

σ

realised

−

K

vol

)

N

vol

{\displaystyle (\sigma _{\text{realised}}-K_{\text{vol}})N_{\text{vol}}}

where:

σ

realised

{\displaystyle \sigma _{\text{realised}}}

is the annualised realised volatility,

K

vol

{\displaystyle K_{\text{vol}}}

is the volatility strike, and

Definition of the realized volatility

Definition of the annualized realized volatility depends on traders viewpoint on the underlying price observation, which could be either discretely or continuously in time. For the former one, with the analogous construction to that of the variance swap, if there are

n + 1

{\displaystyle n+1}

sampling points of the observed underlying prices, says,

S

t

0

,

S

t

1

, . . . ,

S

t

n

{\displaystyle S_{t_{0}},S_{t_{1}},...,S_{t_{n}}}

where

0 ≤

t

i − 1

Pricing and valuation

In general, for a specified underlying asset, the main aim of pricing swaps is to find a fair strike price since there is no cost to enter the contract. One of the most popular approaches to such fairness is exploiting the Martingale pricing method, which is the method to find the expected present value of given derivative security with respect to some risk-neutral probability measure (or Martingale measure). And how such a measure is chosen depends on the model used to describe the price evolution. Mathematically speaking, if we suppose that the price process

S = (

S

t

)

0 ≤ t ≤ T

{\displaystyle S=(S_{t})_{0\leq t\leq T}}

follows the Black-Scholes model under the martingale measure

Q

{\displaystyle \mathbb {Q} }

, then it solves the following SDE:

d

S

t

S

t

= r ( t ) d t + σ ( t ) d

W

t

,

Pricing volatility swap with continuous-sampling

Regarding the argument of Carr and Lee (2009), in the case of the continuous- sampling realized volatility if we assumes that the contract begins at time

t

0

= 0

{\displaystyle t_{0}=0}

,

r ( t )

{\displaystyle r(t)}

is deterministic and

σ ( t )

{\displaystyle \sigma (t)}

is arbitrary (deterministic or a stochastic process) but independent of the price's movement i.e. there is no correlation between

σ ( t )

{\displaystyle \sigma (t)}

and

S

t

{\displaystyle S_{t}}

, and denotes by

C

See also

Variance swap Volatility (finance) Forward volatility Option on realized variance

External links

Prepackaged Volatility Plays