General Practical Example

The complexity really reaches its peak when one investigates exotic products. 

For most exotic products the BS framework is not adequate for pricing such structures. 

For some exotic options, adjusting volatility (finding the exotic IV) to match the price is not even possible. 

This is because no volatility can be found such that the model produces a price that is consistent with the market price. 


Let us go through an example that explains the concept of vega-gamma-theta hedging. 

Suppose we have a ptf with the following Greek representation: 

- Delta = 300 000

- Gamma = 2500

- Theta = -500 000

- Vega = 750 000


Let us assume the following market parameters: 

- Stock price = 1500

- IR r = 2.5%

- Dividend yield q = 0%

- Volatility = 25%


Is it possible to design a ptf that would offset all the Greeks simultaneously? 


Well, we can obviously eliminate Delta without making a difference to any other Greeks by selling stocks. 

To hedge out the other Greeks, we need to use instruments that have non-zero Greeks --> option universe. 


Let us assume we have 3 options available: 


Strike 1200 1500 1800
Time to Maturity 1 1.5 0.5
Premium 27.85 207.98 25.95
Delta -0.13 0.61 0.19
Gamma 0.0006 0.0008 0.001
Theta -0.09 -0.21 -0.22
Vega 3.2 7.05 2.90


If we construct a ptf with the following weights: 

  OTM Puts ATM Calls OTM Calls Stocks
Volume -542,270 -1,157,791 3,673,006 227,891
Position Short Short Long Long


How did we obtain these weights? 

Is it always possible to find those kinds of positions such that we get back what we want? 


If we define the weights: w1, w2, w3, w4 respectively. 

We are then trying to obtain a ptf with ptf Greeks equal to the ones we are after. 

Mathematically speaking: 

Set equations


From basic linear algebra, this set of equations has a solution if its determinant is different from zero: 

Determinant matrix


As it turns out, if we try to use 3 options with identical maturities, this determinant gets to be very close to zero. 

The reason for this is of course that all options are similar derivatives on the same instrument and the relationship between the Greeks makes the difference between using one option or three options very small. 

In other words, these options are not independent enough to build up an arbitrary ptf. 

A good mix of strikes and maturities solves this problem. 


In the above example, we were looking for the hedging ptf. 

If we put on this hedge, it is typically still a dynamic hedge that will need to be adjusted as market moves. 

But because we minimised more Greeks, the hedge is more stable and rebalancing won't have to happen as often. 

It becomes much more of a semi-static hedge. 

If we have more options available, we can minimise more Greeks and make it an even better hedge. 


One can prove that if we are trying to hedge an exotic derivative whose payout is only determined by the terminal value ST, then there exists a static hedge, provided we can use all the different strikes. 


In practice, one can build a ptf with features that the trader finds desirable --> ex: long vega, short gamma. 

This ptf exercise is the foundation for vega, gamma and theta trading. 






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