Fair Strikes, Pricing, and Payoffs of Variance and Volatility Swaps
Summary
The document explains how a fair strike relates to the pricing and payoff of volatility and variance swaps. The fair strike is presented as the level at which expected realized variance or volatility matches the contract strike, making the expected payoff neutral under the stated assumptions. For variance swaps, it describes a theoretical fair value based on a replicating portfolio of out-of-the-money options weighted by inverse squared strike, while noting that trading constraints create bid and offer prices around that value.
The payoff illustration gives a long variance position’s profit or loss as variance notional multiplied by realized variance minus strike and by time to maturity. If realized variance is zero, the resulting loss is the strike multiplied by notional and time. The answers also mention initial and maintenance margin, risk premia, hedging costs, and market supply and demand. These are introductory explanations rather than a complete pricing framework; contract terms, conventions, and the distinctions between variance and volatility payoffs require more detail in practice.
Key ideas
- A fair strike is the contract level where expected realized volatility or variance matches the agreed strike under the pricing assumptions.
- A variance swap’s theoretical fair strike can be derived from a replicating portfolio of out-of-the-money options.
- Liquidity, transaction costs, hedging constraints, and risk premia can produce tradable prices away from theoretical fair value.
- A long variance swap’s payoff depends on realized variance relative to strike, scaled by notional and time.
- Swap contracts can require margin even when their main cash flow settles at maturity.
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Full text
# What is the Fair Strike in a Var/Vol Swap and how does it relate to its price? # What is the Fair Strike in a Var/Vol Swap and how does it relate to its price? I am a student trying to price volatility and variance swaps. People who price those two products usually try to get the "fair strike", and don't seem to care about the price. However, I have a hard time understanding what fair strike means. Also, if people just want to get this value, how do they get a price out of it ? I couldn't find any good resources about my question, so I decided to ask here. Edit: Are swaps free? Do the two entities just decide to take the swap at the "fair strike" and then the loser pays to the winner at the end? ## Answer by Hans-Peter Schrei (score 1) https://quant.stackexchange.com/a/74972 The comments of nbbo2 and AKdemy and the answer by Newquant are correct. In the following, I am trying to expand on their comments and give an explanation which might clarify some concepts for a beginner in the field. Fair Strike In the context of volatility and variance swaps, the "fair strike" refers to the strike level at which the swap is considered fairly priced, given the current market conditions and expectations of future volatility. It's the level at which both parties (the buyer and the seller of the swap) believe that the expected future realized volatility or variance is equal to the strike level. This means that both parties should have no advantage over each other in terms of expected returns. Price of Volatility and Variance Swaps Swaps are not free, and they do have a price associated with them. The fair strike is used as a starting point for pricing the swap, but there are additional factors that come into play. The price of a swap typically consists of the fair strike adjusted for factors like the risk premium (the amount one party demands for taking on the risk of the swap) and the cost of hedging the swap (which includes transaction costs, bid-ask spreads, and other market frictions). The pricing of a volatility or variance swap can be more complex than simply looking at the fair strike because market participants may have different expectations about future volatility and risk. The ultimate price of the swap will be determined by supply and demand, and the willingness of the parties involved to accept the risk associated with the swap. In a swap, the two parties agree to exchange cash flows based on the difference between the realized volatility or variance and the agreed-upon strike level. This exchange occurs at the end or at agreed-upon times during the lifetime of the swap, and the party that has a negative value (i.e., the "loser") pays the other party the absolute value of the difference. The net result is a transfer of risk from one party to another, with one party benefiting from increased volatility and the other benefiting from decreased volatility. ## Answer by Newquant (score 1) https://quant.stackexchange.com/a/74973 Vol and Var swaps are less 'swap' and more 'forwards'. There's no intermediate transfer of interest before maturity. Contracts will specify how much margin to be posted initially, as well as the required transfer of maintenance margin when required. The price of a variance swap is the strike. From a purely theoretical standpoint, the price of a variance swap is the price of it's replicating portfolio (OTM options weighted at 1/k^2). In reality because of liquidity and replication constraints there's a bid and offer around the fair value. The P/L of a long position is given by: variance notional * (realised variance - strike) * T If realised variance was 0, then the P/L would be -strike * variance notional * T. I.E the max loss or the price of the swap at inception.
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