Relating Spot and Forward Prices in the Bachelier Model

Summary

The document explores whether the Bachelier model, which uses arithmetic Brownian motion, can be applied to options on a spot asset by relating the spot price to a risk-neutral forward price. It starts from a diffusion for the forward price and questions whether a simple relationship based on discounting the spot price is sufficient to make the model’s parameters accessible for calibration.

The author notes that some treatments include additional terms such as an asset’s convenience yield, and asks how those terms affect the no-arbitrage relation and calibration of spot options. The text contains no derivation, calibrated example, or answer to these questions, so it does not establish a pricing formula or calibration procedure. Any application would need to specify the asset’s carry or yield assumptions and the relation between spot and forward prices before fitting model parameters.

Key ideas

• The Bachelier model describes price changes with arithmetic rather than geometric Brownian motion.
• The document asks how to relate a risk-neutral forward price to spot for valuing spot options.
• It raises the possibility that convenience yield affects the spot-to-forward relationship.
• No formula is derived and no calibration results are reported.

Tags

# Calibrating Bachelier Spot price model


# Calibrating Bachelier Spot price model












I am interested in a spot price model based on an arithmetic brownian motion (rather than geometric BM). I don't have much experience with the Bachelier model but as I read the risk neutral formulation for a T-forward price $F_t$ in this model follows the dynamics:

$dF_t = \sigma dW_t$

I am specifically interested in the valuation of call options on the spot price $S_t$ where I encounter the following two questions:

- In order to express $F_t$ in terms of $S_t$, I know the relation $F_t=e^{r(T-t)}S_t$. However, looking in the literature (see e.g. https://arxiv.org/pdf/2104.08686), I found different expressions of $F_t$ in terms of $S_t$ (e.g. including the "convenience yield of the asset"). However, to my understanding, the relation above should be valid since it is based on a simple no-arbitrage argument. The importance of this question is that I need a formula with easily assessible parameters to relate $F_t$ to $S_t$, which is the case in the formula above.



Can you confirm Question 1 and the calibration approach in Question 2? Thank you very much in advance!

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