# What is the source of gamma risk?

I have two quasi definitions or interpretations of gamma risk in the context of the BSM model (please correct me if these don't make sense):

1) it is the option's sensitivity to jumps in the underlying

2) it is the option's sensitivity to realized volatility in the underlying

What I don't quite understand is this idea of "jump risk" in (1). What is jump risk? Or what is the source of jump risk in reality?

In addition, how is this risk any different to vega risk? I would have thought movements in implied vols would also incorporate the risk of jumps, in which case, why are vega and gamma seen as separate risks?

Thanks for the help on this

• The BMS model is a diffusion model, no jumps, hence there is no jump risk whatsoever in the pure BMS model. The BMS formula though, is generally used in the market to quote options prices. Even so, gamma is not really a Greek for jump risk, it is simply how fast your delta changes as the spot moves. Jump risk can only be hedged by trading other options. Gamma is related to realised volatility risk, whereas vega is more implied volatility risk. Apr 30, 2020 at 12:16
• @ilovevolatility, what is the source of gamma/realised volatility risk? In other words, why do some options have more gamma risk than others is what I'm trying to understand? Apr 30, 2020 at 12:27
• Instead of Jump Risk (which, as said, does not exist in GBM) you might think of it as the sensitivity of the hedged P&L to a finite move $\Delta S$ in the stock price. This risk only shows up in a discrete rehedging situation, not in the theoretical BSM situation. Apr 30, 2020 at 12:36
• @noob2 right I see Apr 30, 2020 at 12:45
• "why do some options have more gamma risk than others is what I'm trying to understand? "- options that are close to the strike price, especially close to expiration, have the most gamma.
– dm63
Apr 30, 2020 at 13:30

Bear in mind I am a business guy, not a quant -jump risk is the inaccuracy of the Delta caused by a large discontinuous move in the underlying. From what I recall of calculus 20+ years ago, Delta is the slope of the tangent line on the underlying (UL) price vs. option price curve. The tangent line's slope - Delta, is only completely valid at that one point. The further away from that point, you go, the less accurate Delta will be and you will need to apply a "Gamma" adjustment. I think of Gamma as the "tracking error" of Delta, how quickly the Delta becomes inaccurate as the underlying's price changes. Read up on "pin risk" and the concept of Gamma will become clear. Over small price moves Delta is not a bad estimator of option price changes as the UL price changes, but as the UL price "jumps" noticeably, the estimate is less and less accurate - and this "less accuracy" can be measured by Gamma.

• Bikenfly: this is an incorrect characterization of Gamma according to @ilovevolatility, apologies for leading you astray Apr 30, 2020 at 12:29
• @AShortSqueeze What Bikenfly wrote is not incorrect per se. What I wrote is basically that jump risk does not exist in a pure Black Scholes model. But of course reality does not follow Black-Scholes and prices do jump (if only because of exchanges closing /trading halts and so forth). As prices "jump" your delta changes and the change can be characterised by BS gamma. If you are getting confused, don't worry. We all are at times. Apr 30, 2020 at 13:22
• @ ilovevolatility - it is very confusing, I think we are debating over technicalities here. I would have thought in practice for instance, gamma risk captures the risk that a stock gets taken over, or for instance the company comes out with a downgrade to guidance - but based on the answers here this does not appear to be the case. Apr 30, 2020 at 13:31
• @Bikenfly - Gamma is the "delta hedge error" then if i've understood you correctly? Apr 30, 2020 at 13:53
• A take over which makes the stock price jump is certainly a good example in practice of "hedging error" and "gamma risk". And it is also an example of a violation of the theoretical assumptions of Black Scholes Merton 1973 (which Merton himself immediately understood and wrote about a few years later in his paper on jumps). Hopefully it is all clear now? ;) May 1, 2020 at 10:58

In the theoretical BSM case, where you are hedging continuously, there is no such risk. And in Geometric Brownian Motion there are no jumps.

However once you rehedge at discrete time intervals (no matter how small) Gamma Risk shows up. It can be defined as the (first order estimate) of the P&L if the stock price moves by a finite amount $$\Delta S$$ in the next arbitrarily small time interval, i.e. you fail to rehedge while the stock price moves by this amount.

This risk is of course very important in practice, since no one can hedge continuously.