# Local Volatility Model

Exotic equity derivatives usually require a more sophisticated model than the BS model.

The most popular alternative model is a local volatility model (LocVol), which is the only complete consitent volatility model.

LocVol models try to stay close to the BS model by introducing more flexibility into the volatility.

A LocVol model describes the instantaneous volatility of a stock, whereas BS is the average of the instantaneous volatilities between spot and strike.

LocVol is instantaneous vol of underlying

Instantaneous volatility is the volatility of an underlying at any given local point, which we shall call the local volatility. We shall assume the LocVol is fixed and has a normal negative skew. There are many paths from spot to strike and, depending on which path is taken, they will determine how volatile the underlying is during the life of the option. BS vol is average of local volatilities

It is possible to calculate the LocVol surface from the BS IV surface. This is possible as the BS IV of an option is the average of all the paths between spot and the maturity and strike of the option. A reasonable approximation is the average of all local volatilities on a direct straight-line path between spot and strike. For a normal relatively flat skew, this is simply the average of two values, the ATM LocVol and the strike LocVol.

BS skew is half LocVol skew as it is the average

If the LocVol surface has a 22% implied at the 90% strike, and 20% implied at the ATM strike, then the BS IV for the 90% strike is 21% (average of 22% and 20%). As ATM implieds are identical for both local and BS IV, this means that 90%-100% skew is 2% for LocVol but 1% for BS. LocVol skew is therefore twice the BS skew.

ATM vol is the same for both BS and LocVol

For ATM implieds, the LocVol at the strike is equal to ATM, hence the average of the two identical numbers is simply equal to the ATM implied. For this reason, BS ATM implied is equal to LocVol ATM implied.

LocVol is the only complete consistent model

Complete --> it allows hedging based only on the underlying

Consistent --> it does not contain a contradiction

It is often used to calculate exotic option IVs to ensure the prices for these exotics are consistent with the values of observed vanilla options and hence prevent arbitrage. A constant BS volatility model (constant IV for all strikes and expiries) can be considered to be a special case of a static LocVol model (where the LocVols are fixed and constant for all strikes and expiries).

In the LocVol model the only stochastic behaviour introduced into the volatility function is a result of it being a function of the underlying asset price (if rt and qt are deterministic). So there is still just one source of stochasticity, ensuring the completeness of the BS model is perserved. Completeness is important, because it guarantees unique prices. This is the stated reason to develop the local volatility model in Dupire’s original paper.

Since we can look in the vanilla market and find prices or equivalently IVs for vanilla options at any strike and expiry, can we find the LocVol σlocal(St,t) function so that if spot follows the below process, then the fair values of vanilla options exactly match the market?

dS/S = µ dt + σlocal(St,t) dW          --> ameliorer format formule

The answer is yes and gives us a powerful tool to price exotic options in a model that is consistent with the vanilla market.

So knowing the market prices of vanilla options, we can derive the LocVol function using the following formula:

....

We can complete the LocVol formulation by deriving the PDE to use for pricing. The derivation is identical to that of the BS equation and the result is as follows:

....

The resulting LocVol surface is fully non-parametric.

This LocVol model allows a full fitting of an arbitrage-free IV surface.

From LocVol to Implied Vol

Given the LocVol model, the computation of the IV is only approximate. There exists several approximation, one of which is the most likely path. This method gives intuition as to how the IV is built from local volatility.

To maintain intuition on this calculation, we present the Monte Carlo simulation approach. We simulate many paths and keep only the one that finishes around the strike. We obtain a stream of trajectories that start at the initial spot and finish around the strike. We average on each date all these paths and obtain the most likely path. We can also extract the variance around this path. We obtain the IV estimation from it (thanks to the most likely path and the width around it).

The construction of the LocVol from the IV is a simpler problem theoretically but constitutes a difficult numerical problem in practice.

See Gatheral's paper for more details.

From Implied Vol to LocVol     --> formula above     --> mixer tout!

Known problems of the LocVol model:

- Forward skew is smaller than it should be

- Vol of vol is small

- Numerical problems in implementation

Despite all this, polls within IBs show that LocVol model is used by more than 90% of production systems in day-to-day risk management.

It is important to note that traders adjust their prices an their greeks if ever the LocVol model is not the adequate pricing model --> for Cliquets options or options on variance for example.

Strengths of the LocVol model:

The strength of the LocVol model lies in its signature to the product.

Only the Vega KT map can give precise sensitivies for all vanilla options (K,T).

The LocVol model allows a more precise projection of the global vega. It provides a powerful means to find the right strikes and maturities whereby to project the total vega

It is also a P&L explanation for the varied moves in the IV surface. This tool is used every day dozens of times to explain the impact on the book movements in the vol surface.