What to predict in delta-gamma hedging?
What to predict in delta-gamma hedging?
Loading saved threads...
Kilkik · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: Kilkik
Original post: https://quant.stackexchange.com/questions/79069
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
I am working in delta-gamma hedging with machine learning. I guess I have to predict gamma (since predicting gamma tells you how delta will behave) but I don't know why is it needed. I think that a mathematical framework can help me have good understanding of the context.
Suppose we currently have a call option with a current delta of
$0.6$
at time
$t$
. At
$t=t+1$
, if the stock price goes up by
$1$
, then the option's value goes up by
$0.6$
and we need to short
$60$
shares and that's it I have a delta neutral portfolio at that time, and I keep doing this at each time step with the new deltas so that I always have a zero delta portfolio and there is nothing to predict and no need for gamma. By the way, when delta hedging the delta of the portfolio is constant equal to 0 and since gamma is the derivative of delta, isn't gamma forced to be 0 too as the derivative of a constant process ?
Of course I know I am wrong and my reasoning above has many issues so I tried to set up a mathematical framework for this. Let
$S_t, D_t, G_t$
be respectively the stock price, Delta and Gamma of a call option at time
$t$
. These quantities are known at
$t$
.
Let
$S'_t, D'_t, G'_t$
be similar quantities for another option on a same underlying, to be able to do neutral gamma hedging. What we want is at time
$t+1$
,
$P_{d,t+1}=D_{t+1}+yD'_{t+1}-x=0$
and
$P_{g,t+1}=G_{t+1}+yG'_{t+1}=0$
where
$P_{d,t+1}$
is the portfolio's delta,
$P_{d,t+1}$
its gamma, and
$N_{t+1}$
the number of shares we short sell (that have a delta of 1). If the rebalancing is done before time
$t+1$
I would understand why we are interested in predicting. Is it possible to buy a portion
$y$
of an option ?
Quote
Report
KaiSqDist · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: KaiSqDist
Original post: https://quant.stackexchange.com/a/79070
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
It does not make sense to buy the "prime" option, you would only obtain a gamma-neutral portfolio by shorting it, because you would need the option gammas to offset each other.
I don't think you can buy a portion of a option, it makes much more sense to size up the position. For example, if you need to hedge 1
$C$
with 0.4
$C'$
, it is more feasible to long (short) 100
$C$
(40
$C'$
). Then again, it really depends on what is offered on the platform or your broker.
Quote
Report
Post Reply
Quoted from Forex.com.bd-Editorial External question — Quantitative Finance Stack Exchange Author: Kilkik Source score (net votes, not local likes): 0 Original post: https://quant.stackexchange.com/questions/79069 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am working in delta-gamma hedging with machine learning. I guess I have to predict gamma (since predicting gamma tells you how delta will behave) but I don't know why is it needed. I think that a mathematical framework can help me have good understanding of the context. Suppose we currently have a call option with a current delta of $0.6$ at time $t$ . At $t=t+1$ , if the stock price goes up by $1$ , then the option's value goes up by $0.6$ and we need to short $60$ shares and that's it I have a delta neutral portfolio at that time, and I keep doing this at each time step with the new deltas so that I always have a zero delta portfolio and there is nothing to predict and no need for gamma. By the way, when delta hedging the delta of the portfolio is constant equal to 0 and since gamma is the derivative of delta, isn't gamma forced to be 0 too as the derivative of a constant process ? Of course I know I am wrong and my reasoning above has many issues so I tried to set up a mathematical framework for this. Let $S_t, D_t, G_t$ be respectively the stock price, Delta and Gamma of a call option at time $t$ . These quantities are known at $t$ . Let $S'_t, D'_t, G'_t$ be similar quantities for another option on a same underlying, to be able to do neutral gamma hedging. What we want is at time $t+1$ , $P_{d,t+1}=D_{t+1}+yD'_{t+1}-x=0$ and $P_{g,t+1}=G_{t+1}+yG'_{t+1}=0$ where $P_{d,t+1}$ is the portfolio's delta, $P_{d,t+1}$ its gamma, and $N_{t+1}$ the number of shares we short sell (that have a delta of 1). If the rebalancing is done before time $t+1$ I would understand why we are interested in predicting. Is it possible to buy a portion $y$ of an option ?
Checking account access…