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### Odd result

Posted: August 21st, 2020, 12:25 pm
Let phi be the normal density function (i.e. the Gaussian), and let F be the forward price, K the strike, V the volatility and T time to expiry. Define d1 and d2 as follows

d1 = (LN(F / K) + V ^ 2 * T / 2) / (V * SQRT(T))
d2 = (LN(F / K) - V ^ 2 * T / 2) / (V * SQRT(T))

The odd result is that

F phi(d1)  =  K phi(d2)

It's easily proved by substituting the standard exponential formula for the Gaussian. Then there is a load of cancellation and you find the ratio of phi(d1) to phi(d2) = exp(ln(K/F))  =  K/F

Is there any textbook reference for this, or has it been noted by others?

### Re: Odd result

Posted: August 21st, 2020, 2:12 pm
Yes, it gets noticed by everyone who works with the BS or Black formula. Once you start computing Greeks, you naturally want to get rid of one of the d's in terms of the other and the result follows.

### Re: Odd result

Posted: August 21st, 2020, 5:17 pm
Lots of unusual results have been derived. For example, if you put a negative vol into the BS call price formula you get the negative of the put price.

### Re: Odd result

Posted: August 21st, 2020, 5:30 pm
I think that was the basis of several old Star Trek episodes -- Mirror World

### Re: Odd result

Posted: August 21st, 2020, 10:13 pm
Let phi be the normal density function (i.e. the Gaussian), and let F be the forward price, K the strike, V the volatility and T time to expiry. Define d1 and d2 as follows

d1 = (LN(F / K) + V ^ 2 * T / 2) / (V * SQRT(T))
d2 = (LN(F / K) - V ^ 2 * T / 2) / (V * SQRT(T))

The odd result is that

F phi(d1)  =  K phi(d2)

It's easily proved by substituting the standard exponential formula for the Gaussian. Then there is a load of cancellation and you find the ratio of phi(d1) to phi(d2) = exp(ln(K/F))  =  K/F

Is there any textbook reference for this, or has it been noted by others?

Peter Carr is usually a good source for insights of this sort. E.g., take a look at his paper Deriving Derivatives of Derivative Securities.

### Re: Odd result

Posted: August 22nd, 2020, 9:18 am
I think that was the basis of several old Star Trek episodes -- Mirror World
Not with you I'm afraid. Where does Black Scholes feature in the old Star Trek?

### Re: Odd result

Posted: August 22nd, 2020, 1:18 pm
I think that was the basis of several old Star Trek episodes -- Mirror World
Not with you I'm afraid. Where does Black Scholes feature in the old Star Trek?
It doesn't -- small joke -- very small. Neg. vol => mirror world, get it?

### Re: Odd result

Posted: August 22nd, 2020, 3:14 pm
I think that was the basis of several old Star Trek episodes -- Mirror World
Not with you I'm afraid. Where does Black Scholes feature in the old Star Trek?
It doesn't -- small joke -- very small. Neg. vol => mirror world, get it?
Even I can't remember that one .. 4th episode of season 2. It was always on TV on Friday afternoons around 4 pm.

Another mirror image from the same era.

### Re: Odd result

Posted: August 22nd, 2020, 4:14 pm
That's enough Star Trek. The question is whether we would need to prove that result in a paper, or just assume it as a standard result, perhaps with a reference.

### Re: Odd result

Posted: August 22nd, 2020, 10:25 pm
@comply,
If it was my paper, I would just say,
"familiar computations show F phi(d1)  =  K phi(d2)".

@Daniel,
Good Prisoner. Amazon prime has early season Star Trek's, so happened to watch it recently.

### Re: Odd result

Posted: August 23rd, 2020, 9:24 am
Thanks Alan. We worked out how to fit the proof into a footnote.

### Re: Odd result

Posted: August 23rd, 2020, 3:27 pm
@comply,
If it was my paper, I would just say,
"familiar computations show F phi(d1)  =  K phi(d2)".
So long as it's not
1) its is obvious
2) a 6-year old child can do it.
3) using simple but mind-numbing algebra.

### Re: Odd result

Posted: August 23rd, 2020, 5:19 pm
Lots of unusual results have been derived. For example, if you put a negative vol into the BS call price formula you get the negative of the put price.
Take a call spread option PDE, for example. Taking [$]\sigma1 = -0.4, \sigma2 = -0.2[$] gives the same result as  [$]\sigma1 = 0.4, \sigma2 = 0.2[$]. Not surprising I suppose because PDE only 'sees' [$]\sigma^2[$].

Ditto conclusion for put spread. In fact, my FDM gives the same results as Kirk 1995. Even for [$]\sigma1 = +0.4, \sigma2 = -0.2[$] .

Is formula BS 'wrong'?

Getting a negative put price in a PDE would be heresy in the current context. Pardon me if I misread your post.

### Re: Odd result

Posted: August 23rd, 2020, 7:47 pm
I could see a 'correct' model producing garbage output when supplied with garbage input. Good argument validation would take care of that particular one. Anyway, the result with the negative of the put price emerging from a BS call formula supplied with a negative vol is in a paper by Rolf Poulsen from the University of Copenhagen titled Four Things You Might Not Know About the Black Scholes Model. I think there is a formal proof of the result if memory serves me correct. And at least one typo (in the pseudo code).

### Re: Odd result

Posted: August 24th, 2020, 9:59 am
Something else (thinking out loud) is that the SDEs

dS/S  = rdt + sig dW
dS/S = rdt - sig dW

give the same option value (antithetic variates) ? It would be consistent with the PDE approach?

//
Poulson paper,
https://papers.ssrn.com/sol3/papers.cfm ... _id=991344

[$]c(-\sigma) = - p(\sigma)[$]