Discrete Variance Swap Valuation and Forward-Start Replication

Summary

The document addresses valuation of a variance swap observed from a current date when realized variance is computed from daily closing-price returns. It points to several replication-based approximations for discrete variance swaps, including continuous replication, Derman’s method, trapezoidal or Simpson approximations, and an optimal quadratic hedge. It does not provide the full pricing formulas for those methods.

For a forward-starting period, it rewrites average squared returns between two future dates as a weighted difference of averages from the valuation date to each endpoint. This leads to a payoff representation as a calendar spread of two spot-starting variance swap payoffs. The weights depend on the relative lengths of the time intervals and sum to a unit difference. The response also points to a broader reference, but gives no numerical valuation, market inputs, or comparison of approximation accuracy; choosing a method requires more detail about the contract and model assumptions.

Key ideas

• Discrete variance swaps can be approached with replication-based approximations, including continuous, quadrature, and quadratic-hedging methods.
• A forward-period average of squared returns can be expressed as a weighted difference of two spot-starting averages.
• The forward-starting variance payoff therefore has a calendar-spread representation using two spot-starting swaps.
• The response does not give a complete pricing formula or assess the accuracy of the listed approximations.

Tags

# Valuation of Variance Swap


# Valuation of Variance Swap












Let say I have a `Variance Swap` contract which is based on daily closing prices (not the continuous variance calculation) and will last between the day interval $T_1$ and $T_2$ against a strike with $K^2$.

Standing at time $T_0$, I need to value this contract.

Is there any analytical Valuation formula to achieve this?

Any pointer will be highly appreciated.

## Answer by ir7 (score 1, accepted)

https://quant.stackexchange.com/a/57840

This resource surveys the main available replication-based approximations of discrete variance swap pricing:

- continuous method

- Derman's method

- Trapezoidal/Simpson methods

- Optimal Quadratic Hedge (Leung and Lorig)

Edit:

We have:

$$ A_{m,n}:=(t_n-t_m)^{-1}\sum_{i=m+1}^n R^2_i = (t_n-t_m)^{-1}\left(\sum_{i=1}^n R^2_i -\sum_{i=1}^m R^2_i\right) $$

$$ = w_1 (t_n-t_0)^{-1} \sum_{i=1}^n R^2_i - w_2 (t_m-t_0)^{-1} \sum_{i=1}^m R^2_i $$ $$ = w_1A_{0,n} - w_2A_{0,m}$$

with $w_1-w_2 =1$, $w_1 = (t_n-t_m)^{-1}(t_n-t_0) $, $w_2 = (t_n-t_m)^{-1}(t_m-t_0) $.

Forward-starting variance swap payoff is then a calendar spread of two spot-starting variance swap payoffs:

$$ A_{m,n} - K^2 = w_1 (A_{0,n} - K^2) - w_2 (A_{0,m}- K^2). $$

Edit 2:

Bossu et al. paper 'Everything you need to know about variance swaps' has, well, everything, including a term sheet sample.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.