Hi I was hoping someone could help me with something that is confusing the life out of me and honestly doesn't seem to be explained anywhere on the entire internet...

I need an option model for pricing options on normally distributed underlying as opposed to log normally distributed i.e. Black Scholes. I have been using Louis Bachelors 1900 model and finding there is no put call parity when using and interest rate that is not 0%. The original Bachelier model had no interest rate factor but from the book The Complete Guide To Option Pricing Formulas by Espen Gaarder Haug I managed to get some VBA code for a modified version of the model which would incorporate interest rates. The formulas are as follows:

S = Underlying asset price
X = Strike
T = Time to expiration in years
v = Volatility of underlying asset
N(x) = The cumulative normal distribution
n(x) = The standard normal density function

Bachelier Call = (S - X)N(d1)  +  v*sqrt(T)*n(d1)
Bachelier Put = (X - S)N(-d1)  +  v*sqrt(T)*n(d1)

where d1 = (S - X) / v*sqrt(T)

Espen Gaarder Haug then explains "By using the arguments of BSM but now with arithmetic Brownian motion (normal distributed stock price), we can easily correct the Bachelor model to take into account the time value of money in a risk-neutral world. This yields:"

Modified Bachelier Call = SN(d1) - Xe^-rtN(d1)  +  v*sqrt(T)*n(d1)
Modified Bachelier Put = Xe^-rtN(-d1) - SN(-d1)  +  v*sqrt(T)*n(d1)

Essentially just multiplying the strike price by e^-rt in both cases.

So at first everything seems to be great:

if risk free rate = 0% and S = 100, T = 1, v = 25

where X = 100, Put Price = 9.9736
where X = 100, Call Price = 9.9736
i.e. we have put call parity

where X = 95, Put Price = 7.6724
where X = 105, Call Price = 7.6724
i.e. equidistant out of the money puts and calls are valued identically, which is one of the main aims of my entire project

However!!!!  Changing just the interest rate causes problems I don't understand:

if risk free rate = 5% and S = 100, T = 1, v = 25

where X = 100, Put Price = 7.5350
where X = 100, Call Price = 12.4121
i.e. we do not have put call parity, WHHHHYYYY!!

where X = 95, Put Price = 5.7230
where X = 105, Call Price = 9.8269
i.e. equidistant out of the money puts and calls are valued differently, which I can actually understand since the time value of the purchasing at the two different strike prices is affected by the interest rate.

Can someone please explain what is going wrong here? Thank you very much

However!!!!  Changing just the interest rate causes problems I don't understand:

if risk free rate = 5% and S = 100, T = 1, v = 25

where X = 100, Put Price = 7.5350
where X = 100, Call Price = 12.4121
i.e. we do not have put call parity, WHHHHYYYY!!

but you do have put-call parity!

call - put = S - [$]Xe^{-rT}[$]

I'm guessing you forgot the discount factor on the right-hand side?

Right, yes. That is indeed the formula for put call parity and it does indeed hold true in this formula. But now I'm more confused, is put call parity in Black Scholes different then? surely it is the same formula?

One of the first things I learned in option pricing was that put call parity in Black Scholes requires that put premium is equal to call premium for ATM options regardless of interest rate???  I always learned that a long futures position is identical to a long call and short put with atm strikes.  Sorry but what am I missing here?

Right, yes. That is indeed the formula for put call parity and it does indeed hold true in this formula. But now I'm more confused, is put call parity in Black Scholes different then? surely it is the same formula?

One of the first things I learned in option pricing was that put call parity in Black Scholes requires that put premium is equal to call premium for ATM options regardless of interest rate???  I always learned that a long futures position is identical to a long call and short put with atm strikes.  Sorry but what am I missing here?

I think you are confusing spot and forward. A long position in a call plus a short position in a put equals a long forward position. So the option prices are equal for a strike price that is ATM forward, i.e. equal to the forward price.

Ah blimey, yes, thanks so much, I can’t believe I didn’t think of this. So the bachelier formula that I have is for options on a spot contract. Would you possibly be able to help me tweak it so it works for a forward contract? Ie so that where X=S puts and calls are the same premium, it would be really really helpful. Thanks again

ok I've done it, sorry I've not actually played around with, or evidently, fully understood, the discounting aspects in option pricing but you have helped me a great deal, the formula for an option on a normally distributed futures contract I believe is:

Call = e^-rt((S - X)N(d1)  +  v*sqrt(T)*n(d1))
Put = e^-rt((X - S)N(-d1)  +  v*sqrt(T)*n(d1))

thanks very much for your help

Maybe it would help if you think of Put-Call Parity as:

(F-K) = (C-P)*exp^(rt)

This way when you add things like dividend yields, or foreign/domestic interest rates, it would still work because it would be embedded in the F

thank you for this - actually I have one final thing I wanted to make sure I am doing right:

in my Black Scholes formula

d1 = (Log(S / X) + (b + v ^ 2 / 2) * T) / (v * Sqr(T))
d2 = d1 - v * Sqr(T)
call = S * Exp((b - r) * T) * CND(d1) - X * Exp(-r * T) * CND(d2)

where b = cost of carry and r = risk free rate

for option on futures I simply set b to 0 since the cost of carry element is already built into the futures price.

in my normal distribution model

d1 = (S - X) / (v * Sqr(T))
call = Exp(-r * T) * ((S - X) * CND(d1) + v * Sqr(T) * ND(d1))

I was hoping to tweak the formula for my normal distribution model to incorporate the cost of carry too and there for make the 2 models consistent with each other with the exception of the normal vs lognormal element - you couldnt help with this could you?

actually I understand that i can redistribute the call formula to incorporate the cost of carry, b

my question is really - do I need to incorporate b into my formula for d1?  I ask becasue b features in d1 for the Black Scholes model

With a constant cost-of-carry you need to solve the corresponding evolution PDE:

[$]c_{\tau} = \frac{1}{2} v \, c_{SS} + b S c_S - r c[$]

with the appropriate domain, boundary and initial conditions. If I recall correctly, with non-zero b, you won't get a solution involving the cumulative normal, but instead other special functions come into play. Vaguely recall Airy functions. So the notion of a "d1" likely is meaningless in this case, as well as the thread title.

The only escape from this might be to assume a state-dependent cost-of-carry [$]b(S) = b_1/S[$], where now [$]b_1[$] is a constant. Then, you're back to a normal distribution model. Whether or not this (or indeed the model without the cost-of-carry) makes any sort of financial sense, esp as S crosses 0, is another question.

Working on the assumption that you are looking for a crude approximation of a complicated reality in terms of a simple normal distribution, the most plausible solution is probably to first estimate a forward price, taking into account the interest rate and other carrying costs and then assume that the terminal price has a (risk neutral) normal distribution around this forward price. Whether this meets Alan's "any sort of financial sense" standard is an open question without more context. 

whats up hrad, need it for negative interest rates or temperature?  I implemented a few such models when looking onto some temperature derivatives, this seemed to be a promising ad-hock model when combined with sophisticated Heuristics.

I’m actually using it for options on forward delivery of daily gas and electricity where the underlying can indeed be less than zero and possibly more importantly, the movement we see in forward prices is not at all lognormal, ie 0.10 movement in the underlying price is just as likely when underlying is 5.00 as it is when underlying is 25.00 ... the thing is in some instances we may trade options on the spot price of some other commodities like coal and iron ore too and so I wanted to be able to introduce the cost of carry factor to my normal distribution model in the same way it exists in Black scholes

Try:
https://www.math.nyu.edu/~iwasawa/normal.pdf

To make things simpler, I would replace  (1/sqrt(2*PI)) * exp(-(d1^2/2)) with N'(d1)
where N' is the PDF of the standard normal distribution.

Read also up on more of the Old Masters, such as Vinzenz Bronzin

https://en.wikipedia.org/wiki/Vinzenz_Bronzin

Vinzenz Bronzin's Option Pricing Models, Springer Verlag