Pricing European Options on Jumping, Long-Term Decaying Underlyings

Summary

The document considers how to price European calls and puts on an underlying that can jump, fluctuates with short-term supply and demand, and tends to lose value over longer horizons. It also raises practical questions about physical delivery and indivisible contracts, while asking whether a model based on price, volatility, time, and decay would be suitable.

The response recommends the Merton jump-diffusion model for pricing European options when the underlying can experience jumps. It also notes that the negative drift is generally not central to option pricing because valuation is performed under the risk-neutral measure. The document does not supply formulas, calibration steps, model assumptions in detail, or a worked valuation, so it offers model direction rather than a complete implementation guide. Physical delivery and contract indivisibility are characterized as practical considerations, without further analysis of their effect on valuation.

Key ideas

• The Merton jump-diffusion model is suggested for European options when the underlying can jump.
• Risk-neutral valuation makes the physical expected drift distinct from the drift used for pricing.
• Long-term decay and short-term fluctuations are features of the described underlying, but are not modeled in detail.
• The response gives no formulas, calibration procedure, or worked pricing example.

Tags

# How would you price this kind of derivative?


# How would you price this kind of derivative?












I am somewhat familiar with options but am wondering how to price calls/puts on this one:

- European exercise

- "Jumps" in underlying may occur

- Takes physical delivery upon exercise (is this even relevant?)

- Fractional contracts not allowed (technically, though it should be arbitrage-free in reality)

- Underlying's value fluctuates in short-term (i.e. supply/demand) but generally decays over long-term

I am thinking you would input underlying price, volatility, time, and discount rate (decay), accepting some form of GBM. Not really sure though. Can someone please share 'what' this option is and the pricing models available? It seems something like a commodity or futures option but my knowledge in this arena is limited. I would like some formulas (or guidance towards) on how to price this. I have Excel, Matlab, and Maple at my disposal. Thanks in advance!

## Answer by emcor (score 1)

https://quant.stackexchange.com/a/14392

You can use the "Merton Jump Diffusion Model" to price European Options with jumps.

The other points of your question are rather of practical relevance only. The negative drift of the underlying is usually not important, because the pricing goes under the riskneutral measure $Q$.

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