Gamma Risk from Discrete Hedging and Large Underlying Moves
Summary
The document explains gamma risk through the limits of delta hedging. Delta approximates how an option’s value changes for a small move in the underlying, but that approximation becomes less accurate as the underlying moves farther from the point where delta was measured. Gamma describes how quickly delta changes, so it helps characterize the resulting hedging error when prices move by a finite amount before the hedge is adjusted.
In the idealized Black–Scholes–Merton setting with continuous rehedging and geometric Brownian motion, there are no jumps and this form of discrete hedging error does not arise. In practice, hedges are adjusted at intervals, leaving exposure to price moves between adjustments. The answers distinguish this gamma-related P&L risk from vega, which concerns sensitivity to implied volatility. They provide intuition rather than a formula or numerical example, and the discussion does not develop how jump processes or volatility changes affect hedging in richer models.
Key ideas
- Delta is a local estimate of option price sensitivity to the underlying.
- Gamma measures how quickly delta changes as the underlying price moves.
- Finite price moves between hedge adjustments create gamma-related hedging error.
- Continuous rehedging in the idealized Black–Scholes–Merton model removes this discrete adjustment risk.
- Gamma risk from underlying price moves and vega risk from implied volatility changes are distinct sensitivities.
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Full text
# What is the source of gamma risk? # 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 ## Answer by Bikenfly (score 2, accepted) https://quant.stackexchange.com/a/53732 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. ## Answer by nbbo2 (score 1) https://quant.stackexchange.com/a/53734 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.
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