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complyorexplain
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High school calculus question

September 27th, 2020, 2:12 pm

My calculus is a bit rusty, but you know that.
 
I have a summation in mind that I want to express as an integration. The summation: we divide the time between now and expiry into equal parts delta t. We want to sum the sensitivity to change of time (theta) at each point in time and multiply by delta t to give a time decay. The corresponding integral is trivial. We also want, assuming that the price F is a function of t, to sum the sensitivity to price at each point in time, and multiply by the change in price delta F, to get the total P/L corresponding to change in price. The summation is trivial, but how do we express the integral? I would multiply by dF, but dF is not the variable of integration here.
 
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bearish
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Joined: February 3rd, 2011, 2:19 pm

Re: High school calculus question

September 27th, 2020, 4:49 pm

That, my friend, is not a high school calculus problem. Even in the old Soviet Union.... At least in our standard models, the process for F is not of finite variation, so the usual limit taking arguments in going from a sum to an integral will fail. You also seem to want to make a pathwise statement, which doesn’t work in the Ito calculus which is the basis for the classic continuous time finance models from the previous century. If you back off that particular ambition, pretty much any math finance textbook will provide the math, but it’s hardly done in a Sunday afternoon. The two volumes by Shreve will give you a solid foundation. But it’s two volumes, and your problem isn’t really solved until volume II...
 
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complyorexplain
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Re: High school calculus question

September 27th, 2020, 5:38 pm

Thanks. Is the problem due to the (assumed) stochastic nature of the price process? What if I re-express the problem as of integrating some function f(x, t) by dt, where x is a continuously differentiable function of t? Intuitively I would look at the partial differential of f wrt t, then wrt x, and add up the two. Or does it not work even in standard (definitely High School) calculus?
 
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bearish
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Re: High school calculus question

September 27th, 2020, 6:38 pm

That should work, but has the inherent problem that a continuous and differentiable price path guarantees the presence of arbitrage opportunities unless it is trivial (i.e. equal to the value of the money market account).
 
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complyorexplain
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Re: High school calculus question

September 27th, 2020, 6:51 pm

That should work, but has the inherent problem that a continuous and differentiable price path guarantees the presence of arbitrage opportunities unless it is trivial (i.e. equal to the value of the money market account).
Yes correct.