Black-Scholes model

[southsouth]

Hi David,

You show us very intuitive way to understand Black-Scholes model
in your article "Option Pricing Models (OPM). Part 4: Black-Scholes".
but I still have difficulties to understand
1) "Add Volatility" to the formula and
2) the meaning of N(d1), N(d2).

Could you please give me more intuitive explanation?

Thanks!
Steve

 

[David Harper CFA FRM]

Hi southsouth,

Thanks for reading the article. First a caveat that you may already understand: my "intuitive" explanation evolved after several years that we (my firm) tried to explain Black-S to clients. It would not satisfy a mathematician. We were often asked, "how does it work" and the proper technical explanation is a real conversation killer. So, we figured out we could show a straight line to represent the (riskfree) minimal value of an option, then add the wavy line to symbolize volatility as a "plus up" on the minimum value. So we could say, B-S = minimum value + volatility - dividend.

To recap what i wrote elsewhere:

1. Start with the minimum value (the lower bound) of a call option: c = S - K*exp[(-rate)(time)]. In words, stock price minus discounted strike price. If the stock grows at a riskless rate, the present value of the future intrinsic value will be worth exactly this minimum value (before FAS123R, this was the value private companies could use for options, because they had no traded volatility)

2. Wrap the minimum value with two cumulative normal distribution functions: c = S*N(d1) - K*exp[(-rate)(time)]*N(d2). That's the Black-Scholes.

So, my "add volatility" is just a way I teach FRM candidates to memorize the Black-Scholes. Take the minimum value and "wrap in" the N(d1) and N(d2) into it, together they increase the minimum value to higher value: the more volatility, the more value. It is just a memory technique.

But it has justification: if the volatility is zero, you are back to the minimum value - which again, is nothing more that a present value calculation.

The N() are cumulative normal distribution functions. N(1) = 84% because one standard deviation (+1) has 84% of the area under a standard normal curve to the left. Together, the N(d1) and N(d2) probability-adjust the formula. N(d2) is the probability that the call will be exercised (i.e., will finish in the money). Such that KN(d2) is the strike price multiplied by the probability that strike with be paid; and SN(d1)exp(rt) is the expected value of the stock price *if* the call finishes in the money.

You can sort of see these in action at the bottom of:
http://www.editgrid.com/bt/frm2007/LO_34.x_BlackScholes

9/11 UPDATE: I just received latest issue of GARP Risk Review and Gunter Meissner has an article on demystifying (Nd2) [probability of of exercising a call] and N(-d2) [probability of not exercising a call]. This is first good explanation I have read of this. He points out, as illustrated in my model above, that N(d1) >= N(d2), which helps us see why "adding volatility" to minimum value (lower bound) is always a plus up on a Euro call. Much of this is beyond FRM. But what is relevant to FRM is the centrality of the Black-Scholes for both option pricing (derivatives) and the Merton model in credit risk. As Meissner says at the end of the article,"The 1974 Merton corporate model [i.e., that treats equity as a call option on firm assets (analog to stock price) with firm's debt as analog to strike price] is mathematically identical with the famous Black-Scholes-Merton 1973 option model"

 

[woki]

Hi David

Perhaps, this has been mentioned already somewhere else.

Is it a good idea to memorize the formulas for D1/D2? Or would the D-values be typically provided in the exam?

thx

W

 

[David Harper CFA FRM]

Hi W,

I think it is a good idea and the "party line" is to expect they will not be provided; unfortunately, for 2009, the odds are good you will memorize much un-needed materal, so you may memorize and then not need it ...

but I really do recommend memorizing d2 because it effectively is the Merton model, so you are hitting two birds with one stone:
i.e., Merton PD = N(-DD) where DD = d2 and this is the same d2 in BSM except replace firm asset growth with riskless rate

...so personally I only try to recall d2 in BSM; i.e., the prob the option will expire ITM:
[ln(S/K) + (riskless - variance/2)*T]/(volatility*SQRT[T])

then
(1) d1 is just a sign change in front of the variance, and
(2) replace riskless with asset growth (and S with firm value, K with default threshold, etc) and you've got DD

Hope that helps, David