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Black-Scholes-Merton Model

NNath

Active Member
Consider a 145-day put option at 30 on a stock selling at 27 with an annualized standard deviation of 0.30 when the continuously compounded risk-free rate is 4 percent. The value of the put option is closest to: [round d1 and d2 rather than interpolate for N(.)].

PT = [Xe-r (T) × (1 - N(d2))] - [ST × (1 - N(d1))]

where:

d1 = [ln(St / X) + [r + σ2/2](T) ] / σ √(T-t)

d2 = d1 - σ √(T)

Cumulative Standard Normal Probability:

--------------------------------------------------------------------
0.06 | 0.07 | 0.08 | 0.09​
-----|-------------------------------------------------------------
0.3 | 0.6406 | 0.6443 | 0.648 | 0.6517
0.4 | 0.6772 | 0.6808 | 0.6844 | 0.6879
0.5 | 0.7123 | 0.7157 | 0.719 | 0.7224
--------------------------------------------------------------------



A)$3.32.
B)$3.64.
C)$3.97.
D)$4.07.
Your answer: B was correct!

T=145/365 = 0.39726

d1 = [ln(27/30) + [.04 + .32/2](.39726)] / (.3√.39726)

= (-.10536052 + .0337671) / .18908569

= -.07159342 / .18908569

= -0.37863
d1 = -0.37863 ≈ -0.38 N(d1) = 1 -0.6480 = 0.3520



d2 = -0.37863 - .3√.39726

= -0.37863 - .18908569

= -.56771569

= -.56772
d2 = -0.56772 ≈ -0.57 N(d2) = 1 - 0.7157 = 0.2843



PT = 30e-.04(.39726) (1-.2843) – 27(1-.352)

= (29.527056 × .7157) – 17.496

= 21.1325 – 17.496
p = $3.64

----------------------------------------------

Is the above solution correct. If yes I did not understand why N(d1) was calculated as "1-" and again in the Put Price formula it was "1-". Please help - Thanks Nik
 
Last edited:

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @nikks99 Yes, I don't know the source but it's an excellent question in my opinion, the writer did a neat thing: the question gave you CDF lookup values only for positive z; i.e., the right-hand side of the standard normal distribution. Many textbooks only show the right-hand side: due to symmetry of the normal (skew = 0), you don't need the left hand side articulated.

So when you calculate z = d1 = -0.3786, the apparent problem is that negative z values are not displayed in the given lookup table snippet. However, the normal is symmetric, such that the left tail's area under the curve N(-z) = 1 - N(z) which is the corresponding and identically sized area in the right tail, so N(-0.3786) = 1 - N(0.3786), which is here necessary only because we need to use the positive Z values. I get a nearby value for the put; I get $3.62547 but mine does not round the d1, so (B) looks good. I hope that helps,
 

NNath

Active Member
Hi David, In the Practice Question for Black-Scholes-Merton Model specifically Hull problem Problem 14.16 (Page 60), the price of the call is given and we are to calculate the implied volatility. Its was mentioned previously the the implied volatility is the the volatility where the model price is equal to the market price and this can be calculated only by iterative process.

If such a question is asked in the FRM. What is the best way to approach ?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @NNath

Right, a question on implied volatility would not appear on the exam in the same format as problem 14.16. As you suggest, the iteration required (a real-life task!) is simply beyond current testable (exam) capabilities. I nevertheless think it's a good question: to solve it is to understand implied volatility in a concrete way that mere (qualitative) words generally cannot. So, I would say: you won't see this exact question on the exam. Rather, most likely is simply a qualitative question which aims to query an understanding of implied volatility and volatility smile/smirk/skew.

Or, if I were GARP, a not-too-subtle approach would be a question like the following (I am just making this up on the fly, I don't recall seeing this exact format in the FRM):
Question: A one-year at-the-money (ATM) European call option trades at a market price of $3.97 while the underlying non-dividend-paying stock price is $40.00 and the risk-free rate is 4.0%. If d1 is 0.30 and d2 is 0.10 such that N(d1) is 0.6179 and N(d2) is 0.5398, which is nearest to the option's implied volatility?

a. 10.0%
b. 20.%
c. 30.0%
d. 40.0%

I feel like this would be a good query of implied volatility, see how it avoids a real-life iteration by only giving four choices (and I also like my own question because it gives some unnecessary--red herrings--assumptions)?
 
Last edited:

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @NNath Exactly! To be honest, I wrote the question above on the fly, thinking the solution would found by solving for volatility (sigma) in d1, as in: d1 = [ln(1) + r + σ^2/2]/σ --> d1 = (r + σ^2/2]/σ, since ln(ATM option) = ln(1) = 0, and sqrt(t) = sqrt(1) = 1, but a quadratic formula remains, which is clearly solvable with modest iteration ...
... I wasn't even thinking about the clearly superior application of d2=d1-σ*sqrt(t), such that in this case, σ = d1 - d2. That's awesome, now I *love* this question. And, yes, you can see that 20.0% is the answer as in:
  • Where c = S*exp(-qT)*N(d1) - K*exp(-rT)*N(d2), for a sigma (σ) of 0.20, q = 0, and T = 1.0, we have:
  • $3.970 = 40*0.6179 - 40*exp(-0.04)*0.5398
And the point of implied volatility is not that we computed a historical volatility and used 20% as in input into the model, but rather than we are given $3.970 (i.e., "the market price') such that we are looking to find the sigma that produces a BSM model output ("model price") that is equal to $3.970. In this way, under this model and its assumptions, $3.970 as a market price is "telling us" the implied volatility is 20%. Thanks!
 

tosuhn

Active Member
Hi @David Harper CFA FRM CIPM , at the moment, I am going through the practice questions for BSM. I am having problems knowing which equations to apply. I must say I do not understand this topic very well. It would be great if you can summarise the key points I will need to know for the exam under this topic. Much appreciated.

Examples of question:
*I do not know when I should be using the absolute normal VaR equation*
3.1. A bank's cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 1.0 per month and variance rate of 3.0 per month. If the bank wants to be 99.0% confident that its cash position at the end of one year will be non-negative (i.e., at least zero), what is the bank's required cash position at the beginning of the year?
a) zero
b) 1.98 million
c) 6.32 million
d) 13.96 million

AND
*How do we know that in this case, d2 would be the probability that option will expire ITM? There are no notes on d1 & d2 in the notes*
3.4. A non-dividend-paying stock has a current price of $20, an expected return (drift) of 9% per annum and a volatility of 40%. The riskless rate is 3%. If the stock follows a geometric Brownian motion (GBM), what is the real-world probability that a European call option, on this stock, with an exercise price of $26 and maturity in one year will be exercised? (please use a statistical lookup table for the standard normal cumulative distribution function, CDF).
a) 26.4%
b) 33.3%
c) 41.2%
d) 48.9%

Although I do roughly understand after looking at the answers, but I am afraid that I will have problems tackling such questions during the exam.
Hope to hear from you soon!
Regards,
Sun
 
Last edited:

Nicole Seaman

Director of FRM Operations
Staff member
Subscriber
Hello @Jhamby,

The spreadsheets that you are looking for are located under the topic review in Topic 4. There are different spreadsheet bundles that you can download, and these spreadsheets will be located in the P1.T4.b spreadsheet bundle.

Thank you,

Nicole
 

desh

New Member
hi @David Harper CFA FRM how to calculate value of N(d1) if d1=0.74644 without rounding off, as per my knowledge and idea it should beb N(0.74)+0.644((N(0.75)-N(0.74)) = 0.7704+0.644(0.7734-0.7704) = 0.772332 but in book correct value is given 0.7731. I don't know where I am wrong in my calculation? Can anyone help
 

Deepak Chitnis

Active Member
Subscriber
Hi @desh, I dont know if I understand your question correctly, but if you ask me after calculating d1, I will directly go to the Statndard normal z table and find the value 0.75 it gives me N(d1)=0.7734. Not sure if this solves your problem, but just my thought. I think @David Harper CFA FRM can clarify more.
Thank you:)!
 

desh

New Member
Hi @desh, I dont know if I understand your question correctly, but if you ask me after calculating d1, I will directly go to the Statndard normal z table and find the value 0.75 it gives me N(d1)=0.7734. Not sure if this solves your problem, but just my thought. I think @David Harper CFA FRM can clarify more.
Thank you:)!
@Deepak Chitnis I want to calculate N(d1) for N(0.74644) actual figure not round off to 0.75 it will differ the answer
 

brian.field

Well-Known Member
Subscriber
You can use excel or you can use a financial calculator. This would not be an issue for the exam and I would caution you (@desh) on getting too caught up on something this minor. The test will provide the D1 or the N(d1) values if there is a BSM question that requires it, or, they would provide a lookup table.

With excel, it would be: NORM.DIST(0.74644,0,1,1) = 0.772299
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
@desh Your approach looks correct to me, specifically:
  • If the probabilities are not rounded (raw)--e.g., N(0.740) = 0.77035--then I get an interpolated N(0.74644) = 0.77230. But I guess we if we need to interpolate then we don't have precise lookup values in the first place!
  • Such that given rounded "lookup" probabilities of N(0.740) = 0.77040 and N(0.750) = 0.77340, interpolated N(0.74644) = 0.77040 + .644*(0.7734 - 0.7704) = 0.772332; i.e., same as you! I hope that helps!
 
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