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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.

enter image description here 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

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  • $\begingroup$ this doesn't look necessarily wrong. magnitude of theta will decrease as time to expiration decreases. I think your confusion is in representing the portfolio value of theta, which will always be negative, rather than the magnitude of theta itself. $\endgroup$ – Chris Oct 22 '19 at 21:19
  • $\begingroup$ @Chris, thanks for that. if possible, would you be able to elaborate what you mean by the "portfolio value of theta" and the magnitude of theta.. but thank you for the answer! $\endgroup$ – yungpadewon Oct 22 '19 at 21:21

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