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Generally with swaps, one distinguishes pricing from valuation:

Variance swaps have a theoretical replication. A vanilla option trader following a delta-hedging strategy is essentially replicating the payoff of a weighted variance swap where the daily squared returns are weighted by the option’s dollar gamma. Taking this argument one step further, a fair variance swap can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes. These weights are being inversely proportional to squared strikes, an application of the BlackScholes closed-form formula for gamma.

Due due to practical difficulties in replicating the actual log payout across strikes, the market for equity index varswaps usually trades at a basis to the replicating portfolio.

For Vol Swaps, things are a bit messier. Simplified, a Volswap is a varswap - convexity adjustment and the convexity adjustment can be replicated with a portfolio of options on var. So you essentially have 2 replicating portfolios.

There are two documents from JP Morgan Variance Swaps and Just what you need to know about Variance Swaps with the latter being more concise. On a side remark, the way delta and gamma are defined here is a simplification. It will only work intraday. Ideally, the Greeks are directly derived from the replicating portfolio. However, such a full decomposition is not something (many) vendors offer and mainly tier 1 banks have implemented.

 Towards a Theory of Volatility Trading by Peter Carr et al. is probably the best paper to read.

Generally, with swaps, one distinguishes pricing from valuation:

Variance swaps have a theoretical replication. A vanilla option trader following a delta-hedging strategy is essentially replicating the payoff of a weighted variance swap where the daily squared returns are weighted by the option’s dollar gamma. Taking this argument one step further, a fair variance swap can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes. These weights are being inversely proportional to squared strikes, an application of the Black Scholes closed-form formula for gamma.

Due to practical difficulties in replicating the actual log payout across strikes, the market for equity index Var swaps usually trades at a basis to the replicating portfolio.

For Vol Swaps, things are a bit messier. Simplified, a Vol swap is a Var swap - convexity adjusted and the convexity adjustment can be replicated with a portfolio of options on var. So, you essentially have 2 replicating portfolios.

There are two documents from JP Morgan Variance Swaps and Just what you need to know about Variance Swaps. On a side remark, the way delta and gamma are defined here is a simplification. It will only work intraday. Ideally, the Greeks are directly derived from the replicating portfolio. However, such a full decomposition is not something (many) vendors offer, and mainly tier 1 banks have implemented.  Towards a Theory of Volatility Trading by Peter Carr et al. is probably the best paper to read.

An intuitive graphical representation of a variance swap computation can be found here.

For Variance Swaps (and Vol swaps with some caveats), the Black Scholes model is the main tool used for pricing. It is just less obvious.

For Variance Swaps (and Vol swaps with some caveats), the Black Scholes model is the main tool used for pricing. It is just less obvious.

There is no assumption on pricing configuration or even numeraire. Therefore, you cannot simply use Black Scholes. It depends largely on the underlying how you price this. Is it equity, an index, a commodity (mostly futures), an exchange rate, a bond? Or whatever other underlying you can think of. Does the stock pay dividends? In the case of FX, what currency is your notional and premium in (Garman Kohlhagen assumes notional in ccy1 and premium in ccy2, everything else requires (simple) adjustments to the formula).

There is no assumption on pricing configuration or even numeraire. Therefore, you cannot simply use Black Scholes. It depends largely on the underlying how you price this. Is it equity, an index, a commodity (mostly futures), an exchange rate, a bond? Or whatever other underlying you can think of. Does the stock pay dividends? In the case of FX, what currency is your notional and premium in (Garman Kohlhagen assumes notional in ccy1 and premium in ccy2, everything else requires (simple) adjustments to the formula).