Gamma-Theta: Always Flirting

There is no free lunch in owning an option. 

- you will make free gamma money

- you will lose precious theta money

 

There is a balance to be found. 

 

A trader that bought an option paid the intrinsic value (if any) and the time value. 

He knows the time value will be lost during the lifetime of the option. 

The only thing he does not know is how big the future daily losses/gains will be. 

However, the sum of all these will have to make up for the extrinsic premium paid.

He will have to work for his money by actively delta hedging and locking in the gamma profits. 

 

Every single day the trader who owns the option tries to fight the theta by:

- rebalancing his ptf at appropriate times. 

- leaving the position open if he expects ths stock to keep on moving in the same direction. 

 

By waiting, he gets to lock even more profit in one single rebalance but he is taking a risk. 

 

Within the BS setup, we can derive an expression that exactly specifies this relation between these 2 greeks: 

Thetagammaa

 

This relation is interesting because it is telling us how all the different Greeks lead to the price. 

We know that the cost of any derivative is determined by the cost of hedging. 

Let's first consider a ptf of an option that we will delta hedge. 

 

Cash part --> premium V and the loan with value = value of the stock holdings in the ptf --> ∆*S. 

The interest accumulated over shirt time interval ∆t on this net amount --> r*(V - S∆) *∆t. 

That is the deterministic cost of the construction. 

 

On the dynamic hedging procedure, we know we will lose the amount of Thetaa

 

The Gamma copieterm requires slightly more attention to explain. 

 

The Gamma copieprofit would be given by the cash gamma term: 1/2Gamma copie(∆S)2

Since the volatility is relative, we need to multiply it with current level of the stock to get an absolute number. 

In a time interval ∆t, the typical mvmt would be ∆S = Daily typical mvmt

 

So the combination of theta and gamma effect over ∆t is given by: Thetagam    --> leads to 1st equation above. 

This relation reveals the core of the delta-hedging procedure. 

Gamma and theta constantly need to be balanced in order to make up for the premium and the cost of hedging. 

Understanding this balance is key to understand how to manage a book of options. 

 

If we rewrite the 1st equation by using the explicit partial derivatives: 

Bs equ

 

This equation is a PDE known as the BS equation. 

Its solution is unique when the boundary and initial conditions are set. 

The boundary condition is given by the payout profile at maturity. 

The initial conditions are set by the stock price at time t0

This PDE is closely related to the heat equatin in Physics (link between the two is simple transformation). 

 

 

We have often ignored in the previous discussions the fact that the time value can be negative. 

Now it is time to tie up our loose ends. 

The price of a put option can go below the IV. We discussed this in the first section (Basic Instruments/IV and TV.). 

 

So the theta can be positive for put options. 

What does that mean for the theta-gamma balance that we just unveiled throughout this chapter? 

This means that a trader long the option will gain from both the gamma and the theta, so there is no longer a trade-off, but money is rolling in from everywhere. 

 

Where is the catch? 

 

Well, the premium for these far ITM put options is quite high. 

In order to buy this, we need to borrow an amount of cash equal to the premium. 

This loan needs to be paid back as well; this cost is high and the payments are due every day in the BS model. 

 

Why this argument would not hold for a call option as well? 

 

When we own a put, we also need to buy the stock to hedge ourselves --> substantial loan we had becomes even bigger. 

For a call, they would cancel each other out to some extent wheras for the put the effect gets reinforced. 

 

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