Theta in the Bachelier Normal Model Includes Discounting and Time Decay
Summary
The document explains why a put’s theta in the Bachelier, or normal, model cannot generally be represented by the discounting term alone. For a discounted put price, differentiating with respect to the passage of time gives a contribution from interest rate discounting and a separate contribution from the option’s time value. The answer defines theta using the negative of time to expiry so that it reflects the effect of time passing.
For an at-the-money option, the expression simplifies and illustrates that as expiry approaches, the time-value contribution becomes increasingly negative, while the discounting contribution is positive. This resolves the reported behavior: plotting only the rate times option price omits the main time-decay term. The discussion assumes the stated Bachelier pricing setup and parameter definitions; it does not address alternative theta sign conventions or broader model calibration.
Key ideas
- Bachelier put theta includes both discounting and the change in option time value.
- Differentiating with respect to time passing reverses the sign of differentiation with respect to time to expiry.
- The interest-rate contribution to theta is positive under the convention used in the answer.
- Near expiry, the time-value contribution becomes increasingly negative for the at-the-money example.
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Full text
# computing theta of black normal model?
# computing theta of black normal model?
I've been trying to create a black normal model and have used http://janroman.dhis.org/finance/Swaptions/normal%20swaptions.pdf as a guide.
I am trying to validate the theta formula in this paper - which is effectively (for puts):
θ = -r * option_price
where option_price : $$ e^{−r(T−t)}[(K−F)N(−d1)+\frac{σ\sqrt{T−t}}{\sqrt{2π}}e^{−d^2_1/2}]$$
however - I computed theta for an option where I fixed all parameters but varied the time to expiry (T) by a day. As you can see from the graph below: the theta from this model doesn't conform to typical time decay.
As you can see - theta seems to increase as T -> 0, which is incorrect.
Would anyone be able to provide some insight into where I might have interpreted the paper incorrectly - or provide some papers on how to compute theta for the black scholes normal model?
For reference - these are my parameters: S = 2, K = 2, r = 0.02, T = range from 365 -> 0
## Answer by user35980 (score 2)
https://quant.stackexchange.com/a/77627
Using your definition of a put in the bachelier (normal) model: $$p=[(K-F)N(-d)+\sigma \sqrt{T} n(d)]e^{-rT},$$ theta is given by $$\Theta=\frac{\partial p}{\partial (-T)}=rp-\frac{\sigma n(d)}{2\sqrt{T}}\tag{1}$$ where we take the derivative w.r.t. $-T$ to reflect the decreasing effect of time. The second term in (1) is usually what's referred to as the option's "theta" in the literature, while the $rp$ term is the (usually positive) contribution to theta due to the impact of time on discounting (which is all you seem to be plotting).
In your example with $K=F$, (1) simplifies to $$\Theta=r\sigma\sqrt{T} n(0)e^{-rT}- \frac{\sigma n(0)}{2\sqrt{T}}$$ where $n(0),r,\sigma$ are constants. Plotting this will give you the behavior you expected i.e. as $T\rightarrow 0$, $\Theta$ becomes more negative.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.