Why Martingales Simplify Risk-Neutral Pricing

Summary

The document explains why risk-neutral pricing commonly expresses an asset’s discounted value as a martingale and takes its expectation. Pricing from a process with drift is also possible, but the response says it tends to require more complicated derivations, illustrated by the contrast between martingale and partial differential equation approaches to Black–Scholes.

Martingale methods also make certain analytical tools easier to apply, including calculations involving barrier hitting times and expectations. The discussion is conceptual and brief: it offers no formal derivation, examples beyond the Black–Scholes comparison, or detailed conditions for when the methods apply. Its central point is that the martingale framework is preferred for tractability, rather than because expectations involving drift cannot produce pricing formulas.

Key ideas

• Risk-neutral pricing can be formulated using expectations of martingales.
• Processes with drift can also be used to derive pricing formulas, but the derivations may be more involved.
• The martingale approach can make analytical tools such as barrier hitting-time calculations more accessible.
• The comparison with Black–Scholes is illustrative rather than a full derivation.

Tags

# Risk Neutral Pricing and the Drift


# Risk Neutral Pricing and the Drift












For risk neutral pricing, why do we want to compute expectation of a martingale? why is this so important? Why do we dislike the drift so much?

Avoid math heavy answers please.

## Answer by alexprice (score 4, accepted)

https://quant.stackexchange.com/a/50493

You can compute expectation of drifted processes as well and derive same pricing formulas,but usually its more complicated (compare derivation of Black Scholes using martinglaes and through PDE. PDE proof ,where drift is explicit, is much longer)

With martingale representations you have more analytical mathematical tools/formulas available (e.g. barrier hitting times,easy expectation calculation ,etc).

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