General - Interpolation

[DC92]

Hi. I've come across a number of questions where you have to calculate probabilities using the standard normal distribution.

Do we have to interpolate? Example, Q&A Bank Q2.8.ii)

I would have rounded P(Z<0.7071067812) to P(Z<0.71) and just read that value from the table (0.76115). The answer used interpolation but I find it quite time consuming.

What is the level of accuracy needed in the exam for this type of thing?

 

[bystander]

No hard and fast rule but using .7071067812 seems excessive. I'd say interpolate to 3dp is enough. Examiners probable expect some interpolation. I'd suggest you show the formula for your z then give the answer rounded (stating explicitly this is what you have done). Examiners will check what you've done ans do long as it's mathematically correct and carried forward you won't be penalised harshly. Let's face it, the more decimal places you use the greater the chance you write it down wrong or put it in your calculator wrong. The other point is to consider materiality on your overall result.

 

[deepakraomore]

In your case up to 3 decimal places
So...
To find P(z<0.7071067812) will be P(z<0.707)
Formula is very simple
P(z<0.707)=P(z<0.70)-0.7{(Pz<0.71)-P(z<0.70)}
In general multiply the difference by the 0.x, where x is number at 3rd decimal place.
This level of accuracy needed

 

[Steve Hales]

Definitely interpolate between the values of \(\Phi\) in the Tables. It's usually safe enough to round to 4dp.
Remember that this is very important when using the Black-Scholes pricing formula for puts and calls, which is essentially just a modified share price minus a modified strike price - the residual is the value of the option. If you introduce too much rounding error when calculating the two \(\Phi\)s, then that's likely to crowd out the true value of the option.

 

[DC92]

Thanks. Will do so.