# From Options to Variance Swaps

To create a ptf of options with constant exposure to variance --> ensure  constant for moves both in S and t.

The cost of this portfolio represents the price of exposure to realised variance.

When considering how to construct such an exposure, there are three possible approaches to take:

1. Use a single vanilla option, but buy/sell additional amounts of it to keep  constant over time.

Advantages: use only a single option strike

- position could end up with enormous amounts of option as gamma decreases

2. Re-strike the option to maintain a constant gamma.

Start with ATM option and on each re-hedging step, sell/hold option and buy new ATM to achieve constant gamma.

Better than 1. but still requires dynamic trading of options

3. Construct ptf of options so that  remains constant over both moves in S and t.

- to some extent independent of the volatility process driving the underlying

Disadvantage: require a strip of options across a continuum of strikes

In theory --> 3. is used.

In practice --> more like a combination of 2. and 3.

The third approach is to some extent independent of the volatility process driving the underlying.

The first two approaches would require continually calculating the gamma over the course of the trade, and this gamma will be highly (and dangerously) dependent on the assumed IV (and volatility process) at that time.

What kind of ptf is needed to achieve a constant dollar gamma across strikes?

Fig. 102:

- peak dollar gamma increases linearly with strike.

- contribution of low-strike options is small compared to high-strike options.

- we need to increase weights of low-strike options and decrease weights of high-strike options.

Fig 103.

- Naively, it may be thought that weighting by 1/K will achieve constant dollar gamma.

- It has the property that each option in the ptf has an equal peak dollar gamma.

Fig. 104:

- dollar gammas of higher strike options "spread out" more.

Fig. 105:

- summing 1/K - weighted options across all strikes leads to  exposure increasing linearly with S (see obs F. 104).

- weighting each option by 1/K2 will achieve constant

Why a ptf of options weighted by 1/K2 gives a constant exposure to volatility?

--> only  prevents the direct exposure to variance.

We have to find a ptf with gamma proportional to 1/S2 to have a constant exposure.

Gamma = second derivative of ptf's value w.r.t underlying --> this second derivative must be proportional to 1/S2

The negative natural log of S represents such a payoff:

By integrating 1/S2 twice, any function of S of the form  − a ln S + bS + c will have a constant dollar gamma (of a).

The action of delta-hedging this combined portfolio will recover the RV of the underlying.

Note that the bS + c terms represent a static positions in bonds (c) and forwards (b) with no volatility exposure.

Unfortunately, log contracts are not traded in the market.

But we can replicate such a contract with vanilla options using an infinite sum of calls and puts across the continuum of strikes, each weighted by the inverse square of strike.

Integrating the value of this ptf at expiry demonstrates that the non-linear part of the payoff is a negative log contract:

To summarise:

1. A portfolio of calls and puts, weighted as 1/strike-squared, has constant dollar-gamma;

2. Delta-hedging this portfolio provides constant exposure to the difference between implied and realised variance regardless of where the volatility is delivered;

3. Hence the p/l from delta-hedging this portfolio is proportional to difference between realised and implied variance.

A variance swap can therefore be created by replicating a log contract with options which are then delta-hedged.

Although this argument is easy and intuitive, it does depend on certain strong (Black-Scholes) assumptions on the underlying.

The power of the more general replication argument outlined below lies in the fact that it applies under much less restrictive conditions.

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