Gamma for Call/Put options

[Sandor Kelemen]

Hi there,

In the assignment x3/3.1 the part of the question is to determine generally the sign of Gamma for the call/put options. I thought that this is generally indifferent. I see that for both options the gamma is positive in the B-S framework.

If Gamma is positive for the call option then every additional small increase or decrease in the share price has a smaller impact on the option price. Hence I feel here a hidden risk-aversion assumption. Therefore in a market with risk-seeking investors gamma might be actually negative.

Can you reinforce/refute my ideas above please?

Thx.

 

[mugono]

Hi there,

In the assignment x3/3.1 the part of the question is to determine generally the sign of Gamma for the call/put options. I thought that this is generally indifferent. I see that for both options the gamma is positive in the B-S framework.

If Gamma is positive for the call option then every additional small increase or decrease in the share price has a smaller impact on the option price. Hence I feel here a hidden risk-aversion assumption. Therefore in a market with risk-seeking investors gamma might be actually negative.

Can you reinforce/refute my ideas above please?

Thx.

I suspect you're over thinking it.

The sign of the gamma will likely hold regardless of the model paradigm chosen.*

Buying an option limits your risk but has a cost - the option premium.

The cost of option is time, which decays as you approach expiration. Options that have negative theta have a positive gamma. Negative gamma arises where you are short (not long) an option and receive a credit for putting on a limited reward / unlimited risk position.

The gamma of an option represents a sensitivity. It tells you how the option changes following changes in the option's delta. In other words it describes the acceleration of an option as the underlying price changes.

Black scholes is one model used to price options and derive option sensitivities; but it isn't the only one.

*If, as you suggest it were possible to buy an option with negative gamma, this would imply you could purchase it for a credit (and not at a cost to you). You should immediately see that this would represent an arbitrage opportunity as you would be guaranteed a non-zero return regardless of how the underlying performed.

 

[Sandor Kelemen]

Hi Mugono,

We have an approximative equality as follows:
c_new ~ c_old + Delta*(S_new - S_old) + 0.5*Gamma*(S_new - S_old)^2

S_old - old current share price
S_new - hypothetical new current share price to test the sensitivity of the derivative
c_old & c_new - call prices respectively
Delta = first partial derivative of c with respect to S_t
Gamma = second partial derivative of c with respect to (S_t)^2

So Gamma is allowed to be negative and still creating positive c_new price. For example (hypothetical - I do not know whether it is really possible to reach this....):
S_old = 100, S_new = 99
Delta = 0.01, Gamma = -0.01
c_old = 2

Then we would have c_new = 2 - 0.01 -0.005 = 1.985 still positive.

What do you think?

With the approximation above I see it as follows:

c_new ~ c_old + Delta*(S_new - S_old)
This part gives a linear relationship in the current option price vs. current share price plain. The sign of the Gamma then decides what curvature will c_new have against the current share price. And that, the curvature, is basically driven by the risk profile of the investors. Therefore I think, as I wrote above, that theoretically for the market with risk-seeking investors Gamma for some options might be negative for some rare cases (with Delta still positive).

...after exam days... it was very tough...

 

[mugono]

Hi Mugono,

We have an approximative equality as follows:
c_new ~ c_old + Delta*(S_new - S_old) + 0.5*Gamma*(S_new - S_old)^2

S_old - old current share price
S_new - hypothetical new current share price to test the sensitivity of the derivative
c_old & c_new - call prices respectively
Delta = first partial derivative of c with respect to S_t
Gamma = second partial derivative of c with respect to (S_t)^2

So Gamma is allowed to be negative and still creating positive c_new price. For example (hypothetical - I do not know whether it is really possible to reach this....):
S_old = 100, S_new = 99
Delta = 0.01, Gamma = -0.01
c_old = 2

Then we would have c_new = 2 - 0.01 -0.005 = 1.985 still positive.

What do you think?

With the approximation above I see it as follows:

c_new ~ c_old + Delta*(S_new - S_old)
This part gives a linear relationship in the current option price vs. current share price plain. The sign of the Gamma then decides what curvature will c_new have against the current share price. And that, the curvature, is basically driven by the risk profile of the investors. Therefore I think, as I wrote above, that theoretically for the market with risk-seeking investors Gamma for some options might be negative for some rare cases (with Delta still positive).

...after exam days... it was very tough...

The risk aversion of an investor / market will be relevant in how an asset is valued; ie lower / higher the more risk averse / risk seeking investors are.

Derivatives are priced off the underlying and apply no arbitrage principles. Knowledge of investors’ risk preferences is not needed to price the derivative.

As previously explained, a long option with a negative gamma (if such a thing existed - it doesn’t) would likely present an arbitrage opportunity as it would be possible to replicate the payoff that eliminated risk and earned a higher than risk free return.

Good luck with the exam!

 

[Sandor Kelemen]

The risk aversion of an investor / market will be relevant in how an asset is valued; ie lower / higher the more risk averse / risk seeking investors are.

Derivatives are priced off the underlying and apply no arbitrage principles. Knowledge of investors’ risk preferences is not needed to price the derivative.

As previously explained, a long option with a negative gamma (if such a thing existed - it doesn’t) would likely present an arbitrage opportunity as it would be possible to replicate the payoff that eliminated risk and earned a higher than risk free return.

Good luck with the exam!

Hi Mugono,

I think I am getting the intuition that under reasonable constructions (like the lognormal share price model) the delta must increase with the share price (as the probability of falling below the call strike decreases). So I am OK with the positivity of Gamma.

However, can you imagine (i.e. let's forget all our assumptions on Black Scholes) a market where investors would suspect that calls on shares with very high prices are suspicious, hence they would not buy such an option. This would theoretically stop or even revert the increasing character of Delta.

 

[mugono]

Hi Mugono,

I think I am getting the intuition that under reasonable constructions (like the lognormal share price model) the delta must increase with the share price (as the probability of falling below the call strike decreases). So I am OK with the positivity of Gamma.

However, can you imagine (i.e. let's forget all our assumptions on Black Scholes) a market where investors would suspect that calls on shares with very high prices are suspicious, hence they would not buy such an option. This would theoretically stop or even revert the increasing character of Delta.

Sandor,

Read my posts again - they are not constrained by the BS assumptions.

You may find reading around the subject fruitful.