Negative Rho on an American call warrant 50 year maturity

[Morgane57]

Hello,

I am new to the forum and preparing for the Level 2. I have a question (somewhat not directly linked to the exam) that remains unanswered in spite of my researches. According to my knowledge, the rho of a long call warrant (at least European) is always positive whereas a long put warrant will be negative. Is it also true for American call warrants? Apparently, it seems that the Rho of an American call warrant on SX5E with maturity 50 years is negative.
According to what I have understood, there are two impacts related to an increase in interest rate:
1. An impact on the discount (which is positive)
2: An impact on the forward (which is negative)
In my case, the impact on the forward would apparently outweigh the impact on the discount. Does it make sense?
I spoke with a front trader that told me this was impossible since if interest rates rise, the money saved from the leverage he gained by buying the call warrant instead of directly buying the underlying, would earn more interest and therefore the rho has to be positive.
I am now quite lost...If somebody can help...

 

[ShaktiRathore]

There is generally the case with stock call options that the value of the call increases with increase in interest rates. The call option gives right to the option purchaser to buy the underlying asset at a specific price called the strike price before time to maturity for american option and at the expiration of the option for the European options. It is like option holder is a stock holder who sells the stock and buys the call option and deposit the remaining proceedings at interest rate r whenever the interest rate rises. This increase the demand for the call options and thus increases their value whenever interest rate rises. The money gain in this way far outweigh rather than just holding the underlying. As interest rate falls the call option is exercised and deposit is withdrawn to exercise the option and buy back the stock. This decreases the value of the option.I think this line of reasoning is correct and the trader is correct to an extent.
Similar line of reasoning can be applied to put options. So rho is positive.
Mathematically (please take this proof as approximation i mean not exact just to give an idea),
c= SN{ln(S/X)+T(r+.5sigma^2)}-Xexp(-rT)N{ln(S/X)+T(r-.5sigma^2)}
d1=ln(S/X)+T(r+.5sigma^2)
d2=ln(S/X)+T(r-.5sigma^2)
now when r increases; d1 increases=> N(d1) increases
also d2 increases=> N(d2) increases
look at exp(-rT) as r increases exp(-rT) decreases
Now seeing the above formulas we see that increase of r has the same affect on d1 and d2 that is both increase by T*r so they have almost same effect on N(d1) and N(d2) now the differentiating factor exp(-rT) decreases that means Xexp(-rT)N{ln(S/X)+T(r-.5sigma^2)} decreases more than SN{ln(S/X)+T(r+.5sigma^2)} as a result the value of c increase as r increases.
In the end the call options certainly have positive rho but there can be some exceptions as you cited with long maturity. However this is what i know that i can explain to you. I hope this is helpful to you.
thanks

 

[David Harper CFA FRM]

I don't know (would require a numerical demonstration-yes?). With respect to European call options, as rho = K*Term*exp(-rT)*N(d2) and given all four factors must be positive, European call rho must be positive (but tending to zero with longer maturity due to discounting). Intuitively, the trader's argument makes sense to me, I don't intuit why a longer term would invert this outcome in the presence of an early exercise ...