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stilyo
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Joined: January 12th, 2009, 6:31 pm

Black-Scholes with stochastic interest rates

August 3rd, 2018, 9:16 pm

Hi all,

I was looking at option prices which incorporate stochastic interest rates, as in the following paper - formulas 4 and 5 on p.9:
http://www3.nccu.edu.tw/~liaosl/Publication/28.pdf

So looking at the adjusted volatility term, the higher the correlation between stock returns and bond returns (i.e. rho is close 1 and the stock is "bond-like"), the lower the variance and correspondingly, the lower the option price. Conversely, if rho is close to 0 and stock returns and bond returns are uncorrelated, keeping everything else equal, option prices will be higher.

Why does that make sense? For instance, a company which can respond to inflation quickly by pushing costs to customers should have lower correlation with interest rates. Why would that make an option issued by this company more expensive - shouldn't it be the opposite? Thanks all!
 
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bearish
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Joined: February 3rd, 2011, 2:19 pm

Re: Black-Scholes with stochastic interest rates

August 3rd, 2018, 10:29 pm

First, you should harbor a bit of natural suspicion towards a paper that talks about "Black-Scholes options", which is a non-sensical term. A reasonable reference for this kind of topic is Amin & Jarrow. Additionally, your reference to "an option issued by this company" makes relatively little sense. Unless you are talking about certain very rare warrants, that's just not how options work. So, rather than looking at exceedingly obscure papers floating around on the internet, I suggest you pick up a basic textbook or two. Hull is not the worst place to start.
 
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stilyo
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Joined: January 12th, 2009, 6:31 pm

Re: Black-Scholes with stochastic interest rates

August 3rd, 2018, 11:13 pm

Thanks for your reply, bearish! Point well taken about not relying on obscure papers on the internet. However I did look at a few papers before posting the question and they all suggested that formula, plus I know Margrabe’s formula of which this is a special case where one of the asset is a stock following GBM and the other asset is a bond under some stochasic interest rate process. So it makes sense for the adjusted vol to have that form, doesn’t it?

I was careless with my phrasing saying ‘option issued by the company’ - I don’t mean a warrant, just mean an equity option where the underlying is a stock of the company I referenced. I hope this clarifies my question?
 
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Alan
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Re: Black-Scholes with stochastic interest rates

August 4th, 2018, 2:53 am

Thanks for your reply, bearish! Point well taken about not relying on obscure papers on the internet. However I did look at a few papers before posting the question and they all suggested that formula, plus I know Margrabe’s formula of which this is a special case where one of the asset is a stock following GBM and the other asset is a bond under some stochasic interest rate process. So it makes sense for the adjusted vol to have that form, doesn’t it?
Exactly, so you've answered your own question.