Worst of Put

Payoff for this structure is Max( 100 – Min(Stock A, Stock B),0). Here’s a sample payoff graph attached with 20% Correlation between stock A & stock B.

1.      Both move away from 100 into higher prices, this gives a value of 0 for the structure

2.      Both move away from 100 into lower prices this gives a positive value to the structure

So in all with 50% probability structure gives a positive payout when correlation is positive. With negative correlation it gives a maximum payout, hence the structure as a whole is majorly Short Correlation. If I calculate prices for 20% Correlation and 90% Correlation then as per my analysis 20% Correlation Price – 90% Correlation price should be negative (Short Correlation)

Plot

So we are right most of the time, but at high stock levels the structure is not strictly short correlation.

Let’s analyze other Greeks, for example Delta. If StockA = 110 then obviously worst performing stock is B and it becomes a normal vanilla put (as shown in delta profile Blue Line)

If StockA = 80, then as StockB crosses 80, worst performing stock changes to A, hence delta due to B is 0 (Red Line)

Note that delta becomes 0 sooner when Stock A = 80 compared to Stock A = 110 (as worst performing stock change happens sooner in first case)

The verification can easily be done with Delta A (next picture) which is active as soon as B loses worst performance signature.

First case when Stock A = 110 will be flat, coz even when it becomes worst performing price will be 0 as it's Out of the Money.

When Stock A = 80, and Stock B > 80, the position of worst performing stock changes to Stock A. Since put on stock A is In The Money we see constant delta of -1.

Similar profiles are visible for other values of Stock A



Consider the gamma profile when both stocks are moving in the same direction 99% Correlation, Vol A = 10%, Vol B = 20%



In all the above cases initially we see gamma due to Stock B (worst performing), as it moves towards Stock A it replaces it and Stock A becomes worst performing stock.

There's a slight negative Gamma (negative kinks in above graphs) for prices between 70 to 90 (Purple Line). When Stock B overtakes stock A due to higher volatility it still has some probability of becoming the worst performing stock. So we see the slight gamma sign change possibility.

When A becomes worst performing Stock, then Gamma for Stock B should definitely fall, if both are moving in same direction

Now let's throw in correlation scenario, and see what happens! If correlation is -99%, Both stocks can move towards each other or away from each other .

  1. Move away [ First Stock Increases, Second Stock Decreases]

  2. Move towards [ First Stock Decreases, Second Stock Increases]


In both cases, correlation will be negative 99%. (Drawing the same picture for easy comparison)

 

When Stock A = 80 (Red Line), For Stock B < 80 it is the worst performing stock hence so Gamma A should be 0 till B is 80. But as in the previous case, there are two different things

  1. Volatility is higher for Stock B

  2. Correlation is negative

So if they Stock B is moving forward, then Stock A is moving backward (Due to negative correlation), hence it becomes the worst performing stock sooner, hence the Gamma shows up even before 80. It will actually show up much before when correlation was 99%. From the above two graphs the Red Line in first one rises above 0 for stock B < 50 (-99%), and for 99% correlation it rises above 0 for Stock B > 50. This is just the repelling effect.

Similar understanding is there when asset move towards each other (Stock B decreases, Stock A increases). So correlation does effect gamma, and intuitive understanding is much better rather than mathematical.

CrossGamma Effect

This is a mixed derivative term, and it can mean two things

1.      Change in Delta (with respect to stock A) due to change in Stock B

2.      Change in Delta (with respect to stock B) due to change in Stock A

Finding customized baskets of these stocks (namely Stock A, and Stock B) won’t yield any results. If for example the two stocks are Tata Steel and IFCI, then both of them are part of larger basket called NIFTY Index. So hedging cross gamma means taking positions in NIFTY Index options. As explained before Worst of Put is short correlation, so we need a long correlation basket option to hedge this exposure. A call on basket is long correlation, so a call on NIFTY Index will hedge the cross gamma for this product.

Theoretically it is easy, but in practical sense, by hedging with Index options we introduce another risk to our portfolio. When markets trend downwards people buy protection using Puts on NIFTY Index. The stocks might not go down by that percentage but our hedge will be drastically affected by Index movements. This is mainly due to non-perfect correlation hedge.