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Decomposing Gap Call Payoffs into Digital Options

Article Quant Q&A · Author: stoimparando
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Summary

The response explains how to handle a gap call, whose strike determining exercise differs from the strike used to calculate the payoff. It rewrites the payoff as the underlying asset paid only when the exercise threshold is crossed, less a fixed amount paid under that same condition. This expresses the position as a long asset-or-nothing digital and a short cash-or-nothing digital, allowing the trader to use the standard pricing and Greek formulas for those instruments.

The response points to a related answer for further detail, but does not provide the actual Greek formulas or derive them. It therefore gives a useful decomposition rather than a complete calculation method. Its applicability is limited to the stated long gap call payoff and assumes the digital-option framework is appropriate for the contract.

Key ideas

  • A gap call has separate exercise and payoff strikes.
  • Its payoff can be decomposed into an asset-or-nothing digital less a cash-or-nothing digital.
  • The decomposition lets standard digital-option formulas be used to obtain the gap call’s value and Greeks.
  • The response provides no explicit Greek formulas or derivation.

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OptionsDerivatives pricing
Full text
# What are the formulas to compute the greeks of a gap option?


# What are the formulas to compute the greeks of a gap option?












I'm having a problem to calculate the gap option greeks since there are 2 different exercise prices K1 and K2. Do you know the answer or where can I read about these particular greeks?

## Answer by ir7 (score 3, accepted)

https://quant.stackexchange.com/a/55322

Long gap call option position is the same as long an asset-or-nothing digital option and short a cash-or-nothing digital option, both "classical".

$$ (S_T-K_1)\cdot 1_{S_T > K_2} = S_T\cdot 1_{S_T > K_2} -K_1\cdot 1_{S_T > K_2}$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.