Basically what I am trying to do is adjust the Beta of a security to the beta of the option using that options delta.

I believe that the equation is:  Beta of Underlying / (Delta / Option Price Per BS) = Beta of Option

Where:
Beta of Underlying = coefficient to % change in market in relation to the underlying security
Delta = change of option in dollars to $1 change in the underlying

I tested it and it looks like it works, but I need someone to confirm it for me.  I'm not really looking for an opinion as to whether or not this is the best way to do this.  I realize that there is a way to do this using historical options prices, but I do not have access to this data.

Delta is computed as follows:



Black Schols Options Price Formula:

Where:

Please help if you can.  There is nothing published in current financial literature on this.

See eqn (15) in orig. Black-Scholes formula article.

Found it:  https://www.cs.princeton.edu/courses/ar ... oles73.pdf

Further analysis reveals... Unfortunately, this does not solve the problem in the manner that I was looking for.

For ATM options the delta is very stable with respect to any volatility you input, for somewhat OTM options very sensitive to input vol.  Naturally again affect the Beta of option, OTM betas sensitive to vol.  Should one use implied vol to calculate the Beta or some other vol estimate (for atm dose not matter) ? Depends on market, how risk limits are measured (minimizing market to market?) etc, and for what use etc.

see also my know your weapon article, a little about option beta also there.  Look at DdeltaDVol, one could have similar DBetadVol: sensitivity in option Beta for change in implied vol.

For ATM options the delta is very stable with respect to any volatility you input, for somewhat OTM options very sensitive to input vol.  Naturally again affect the Beta of option, OTM betas sensitive to vol.  Should one use implied vol to calculate the Beta or some other vol estimate (for atm dose not matter) ? Depends on market, how risk limits are measured (minimizing market to market?) etc, and for what use etc.

see also my know your weapon article, a little about option beta also there.  Look at DdeltaDVol, one could have similar DBetadVol: sensitivity in option Beta for change in implied vol.

I'm assuming three things, and I am smart enough to know when and when not to use it.
(1)  The options are fairly liquid
(2)  Closest to in the money
(3) no historical data is available to gage sensitivity

Where can I find your article?

For ATM options the delta is very stable with respect to any volatility you input, for somewhat OTM options very sensitive to input vol.  Naturally again affect the Beta of option, OTM betas sensitive to vol.  Should one use implied vol to calculate the Beta or some other vol estimate (for atm dose not matter) ? Depends on market, how risk limits are measured (minimizing market to market?) etc, and for what use etc.

see also my know your weapon article, a little about option beta also there.  Look at DdeltaDVol, one could have similar DBetadVol: sensitivity in option Beta for change in implied vol.

I'm assuming two things, and I am smart enough to know when and when not to use it.
(1)  The options are fairly liquid
(2)  Closest to in the money
(3) no historical data is available to gage sensitivity

Where can I find your article?

It should be on this website, but also here Know Your Weapon

 Closest to in the money

also can be sensitive to vol input, calculate dDeltadVol for your specifics. in liquid market many would use implied vol possibly

So the equation that you propose is Beta Option =  (Strike / Call) * Change in Call?  Am I defining your variables properly?

In theory I could I take (Delta * (Spot/Call Price) * Beta) since delta is the change in change of option in relation to change of underlying?  Or would this be correct alone: Beta/(Delta / Current Call Price)? ...you get two very different sets of numbers.  One is much higher than the other.

So the equation that you propose is Beta Option =  (Strike / Call) * Change in Call?  Am I defining your variables properly?

In theory I could I take (Delta * (Spot/Call Price) * Beta) since delta is the change in change of option in relation to change of underlying?  Or would this be correct alone: Beta/(Delta / Current Call Price)? ...you get two very different sets of numbers.  One is much higher than the other.

yes just   Delta*Spot/CallPrice *BetaStock. Not strike /call. That is Elasticity * BetaStock, first mention by Black-Scholes 73 paper as Alan refer to. The DDeltaDVol in my paper could be interesting if you are interested in the sensitivity of Beta in relation to vol you input to calculate option beta.

So the equation that you propose is Beta Option =  (Strike / Call) * Change in Call?  Am I defining your variables properly?

In theory I could I take (Delta * (Spot/Call Price) * Beta) since delta is the change in change of option in relation to change of underlying?  Or would this be correct alone: Beta/(Delta / Current Call Price)? ...you get two very different sets of numbers.  One is much higher than the other.

yes just   Delta*Spot/CallPrice *BetaStock (Not strike /call). That is simply Elasticity * BetaStock, first mention by Black-Scholes 73 paper as Alan refer to. The DDeltaDVol in my paper could be interesting if you are interested in the sensitivity of Beta in relation to vol you input to calculate option beta.

I think it makes sense to write all corresponding equations related to betas, deltas and the problem is to resolve the system with respect to a variable you need. Otherwise it looks like informal exchange of ideas.

So the equation that you propose is Beta Option =  (Strike / Call) * Change in Call?  Am I defining your variables properly?

In theory I could I take (Delta * (Spot/Call Price) * Beta) since delta is the change in change of option in relation to change of underlying?  Or would this be correct alone: Beta/(Delta / Current Call Price)? ...you get two very different sets of numbers.  One is much higher than the other.

yes just   Delta*Spot/CallPrice *BetaStock. Not strike /call. That is Elasticity * BetaStock, first mention by Black-Scholes 73 paper as Alan refer to. The DDeltaDVol in my paper could be interesting if you are interested in the sensitivity of Beta in relation to vol you input to calculate option beta.

Do I have to build in any sort of secondary, such as gamma?  How would I do this?  The approximation might be good enough.