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BLACK SCHOLES MODEL

scorpiomanoj

New Member
Hi David,

While I once again thank u and your team for enabling many aspirants including me to qualify the prestigious FRM examination, I request you to clarify the following which I encountered when I was brushing up Hull's OFD.

I would like to refer page 294 example 13.6 of Hulls OFD 7th edition, which calculates the price of call and put using BSM for a stock option expiring 6 months with volatility 20% p.a. The solution in the book shows that the volatility of 20% p.a. is directly plugged into the BSM formula..
1. Is the volatility given in the exercise a daily volatiltiy or a 6m volatility or a annual volatility?
2. In any case, should volatility be appropriately scaled to time ( in this case 6 m) before plugging into BSM formula? My understanding is that in BSM, Dr of d1, ie. sigma X sqrt (T) does the job of scaling and similarly the second part of the numerator (r + 0.5sigma^2)T does that for the variance (ie, sigma^2 X T). Is my understading correct?
3. If the question statement explicitly mentions volatiltiy as 6m volatility, should one discard the time scaling factor in the Nr and Dr. of d1, as plugging in would lead to scaling the volatility and variance twice? (provided my understanding in s.no.2 is correct).

Thanks in anticipation.

Manoj Kumar Halan.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi Manoj,

It's my pleasure, truly..good to hear from you!

1. It is annual. IMO, "volatilty" connotes an annualized standard deviation. But Hull is specific b/c he writes "20% per annum" so I think many authors try to stick with the convention of specifying an annualized volatility for the input.
2. Either.
3. Yes, exactly!

Re (2) and (3) because the BSM, in d1 and d2, is effectively employing the square root rule (which assumes i.i.d.; note the constant volatility assumption is the "identical" in i.i.d., so this scaling is presumptive) to produce, in the case of d2, a "per period" distance to default. For example, as 20% is the annualized sigma input:

the numerator scales variance directly with time, so you effectively generate: 10% drift * 0.5 six months - 1/2 of the variance (20%^2) * 0.5 six months.
and the denominator is similarly time-scaling (i.e., variance directly with time, so std deviation directly with SQRT(time)): 20% * 0.5 six months

...so, per your point (3), the question could instead say:
the six-month variance is 0.02 (or the six month standard deviation is SQRT(0.02) = 0.1414)
and then you would be incorrect to additional re-scale in the formula.
You'd want to use 0.02 six month variance * 0.5 only in the numerator and SQRT(0.02) only in the denominator
... so (2) is either as you can do the scaling outside or inside

hope that helps, David
 
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