CONTENTS

Solution: Merton Jump-Diffusion: mechanical calculation

Exercise: Merton Jump-Diffusion: mechanical calculation

Part 1

The relevant definition is the lesson's central relation:

dSt/St−=(μ−λk)dt+σdWt+(J−1)dNtdS_t/S_{t^-}=(\mu-\lambda k)dt+\sigma dW_t+(J-1)dN_t

This formula is meaningful only after the measurement convention is fixed.

Part 2

A minimal calculation uses clean inputs, applies the definition, and checks scale. If the result is a rate, compare it with nearby maturities. If it is a risk or performance measure, compare it with an economically similar asset or strategy. If it is an algorithmic estimate, compare it with a simpler baseline.

Part 3

The most important assumption is usually the convention that turns raw observations into the model input: annualisation, compounding, sampling frequency, objective function, or calibration measure. Changing that convention can change the result without any change in market economics.

Takeaways

  • The definition gives the number, but the convention gives the number meaning.
  • Sanity checks are part of the calculation, not a separate clean-up step.
  • The finance use case here is crash risk and short-maturity smiles.
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