April 2021 Q2(iii)

[AaronD]

Hi there,

In this question we are asked to calculate the implied volatility using the value calculated in part (ii).

There's a nice trick to do this on page 45 of the tables, but my mind went to the black-scholes formula for a call and slotting in the numbers.

Although I'm unsure what value the dividend rate, q, takes in the black scholes formula .

I used a goalseek to find the dividend rate (after slotting in sigma = 44.6%) to return the dividend of c8%. Unsure if this is correct and if so, is this the dividend rate per year 4*2?

Thanks,
Aaron

 

[Steve Hales]

This would be the force of dividends which are paid continuously in the Garman-Kohlhagen formulae. Therefore the dividends aren't paid on the nodes of the binomial tree but continuously across time. I'm guessing you used (T-t) = 0.5 in your equation which would mean that the units are per annum.
Whilst the Examiner's Report did say that marks would be awarded for a Black-Scholes solution here, part (v) of the question demonstrates why this would be difficult.

 

[AaronD]

Got it, thanks Steve!

Related question.

We reduce the share value by the dividend rate at time 6 months by the dividend as the dividend is paid just before the option expires and the option holder doesnt benefit from the dividend

Question 1
Presume we would do the same for a put option?

Question 2
If the dividend were paid just before time 3, how would the answer change - would we reduce the share price at time 3 by the dividend amount? If we reduce the share at time 3 by the dividend would the up/down values at time 6 months be based off the reduced time 3 amount?

Question 3
How would this answer change if the dividend were just after time 3 months?

 

[Steve Hales]

Question 1

Yes. It's the share price that's being modified not the option.

Question 2

It's more complicated when the dividend isn't paid at expiry! Yes, the share price at time at time 3 would need to be reduced, and this would then drive the price at time 6. Remember that dividends don't generate wealth out of nothing - they are just an immediate transfer of wealth from the share 'price' to cash. The 'value' of the share is the 'price' plus the cash. An easier approach is to discount the known dividends back to time zero, deduct them from the initial share price, and then work on the now-non-dividend-paying share.

Question 3

The dividend would then need to be deducted from the time 6 share price since the time 3 share price is unaffected. The deduction would need to be adjusted for the risk-free rate earned between "just after 3" and "just before 6".

 

[AaronD]

Thanks so much Steve, really appreciate it!