Corpfin Mod 23: Homework


Corpfin Mod 23: Homework

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NEAS
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Corporate Finance, Module 23: “Advanced Option Valuation”

Homework Assignment

(The attached PDF file has better formatting.)

Black-Scholes Pricing

A stock’s price volatility is 35%. The annual effective risk-free rate is 8%. A non-dividend paying stock now trades at $80. Six-month European calls and puts are trading with a strike price of $85.

A.    What are the values of the five input parameters to the Black-Scholes model for pricing these call and put options? (The sixth input parameter, the dividends, is not considered in this exercise.)
B.    What is the value of PV(X): the present value of the exercise price? The interest rate uses annual compounding.
C.    What is the value of S ÷ PV(X): the ratio of the stock price to the present value of the exercise price?
D.    What is the value of ln(S/PV(X)): the logarithm of the ratio of the stock price to the present value of the exercise price?
E.    What are the values of d1 and d2? (Work out d1 by formula and d2 from d1.)
F.    What are the values of N(d1), N(–d1), N(d2), and N(–d2)? Use either a cumulative normal distribution table or a built-in spread-sheet function. The Brealey and Myers textbook has a cumulative normal distribution table; Excel has a built-in function.
G.    What is the value of the European call option? (Use the formula for a call option.)
H.    What is the value of the European put option? (Use the formula for a put option; you will use put call parity in the next question.)
I.    Verify that the put call parity relation holds.


The values of d1 and d2 are shown below. You will be given these formulas on the final exam, but you must know how to use them to derive put and call option prices.


Question: For Part A, should we know the five inputs for the final exam?

Answer: Know the five inputs and know also which are stated in the options contract (strike price, time to maturity), which are known financial items (risk-free interest rate, stock price), and which are estimates (volatility).

Question: For Part B, do we use continuous compounding or annual compounding for the Black-Scholes formula?

Answer: We should use continuous compounding. The formula is exact if prices change continuously. To keep the mathematics simple enough for the average college student, Brealey and Myers use annual compounding.

Question: Several candidates ask on the discussion forum whether the stock price volatility is the σ in the Black-Scholes formula or the σ2. Is the volatility the standard deviation or the variance?

Answer: The volatility is the σ. This is the standard deviation per square root of the unit of time. σ2 is the variance rate, or the variance per unit of time.

Question: What does standard deviation per square root of the unit of time mean? This is the standard deviation of the stock price; what does it have to do with time?

Answer: The stock price is a scalar, not a random variable.

●    A random variable has a distribution, which has a standard deviation.
●    A stock price is a single value (a scalar); it does not have a standard deviation.

Question: The homework assignment says the stock price volatility is 35%. The volatility refers to the stock price; it makes no mention of time.

Answer: The stock price now is known. The stock price one year from now is not known with certainty.

●    If we know the current stock price and the expected return, we know the expected stock price one year from now, which is the mean of the stock price distribution in one year.
●    If we know the volatility of the stock price, we know the standard deviation of the stock price distribution in one year.

Question: One year from now, the stock price will be a scalar, just like it is a scalar now. If the stock price now does not have a standard deviation, why does it have a standard deviation one year from now?

Answer: To be rigorous, we should say: “The distribution of the possible stock prices one year from now.” The stock price now is $80. In one year, the stock price may be $50, $85, $150, or some other figure. Each possible stock price has a likelihood. The likelihoods form a probability density function (pdf). The pdf is a lognormal distribution, whose parameters depend on the stock price now, the expected return, and the volatility.

Question: If the volatility is 35%, is the standard deviation of the stock price distribution in one year 35%?

Answer: The stock price in one year has a lognormal distribution, which is e raised to the power of a normal distribution. The normal distribution in the exponent of e has a standard deviation of 35%. The standard deviation of the lognormal distribution itself is a complex expression of μ and σ. (The standard deviation of a lognormal distribution is covered in the loss distributions section of Exam 4 / Course C.)

Question: Is the standard deviation of the stock price in half a year = ½ × 35% = 17.5%?

Answer: If the variance in a year is 35%2 = 12.25%, the variance in half a year is ½ × 12.25% = 6.125%. The standard deviation in half a year is 6.125%0.5 = 24.749%. The volatility is the standard deviation per square root of the unit of time.

Question: Is this material covered in the corporate finance course?

Answer: This material is covered on the actuarial exams (CAS 2, 3, 4 and SOA FM, M, C). We mention it here because some candidates asked about it on the discussion forum.

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for call potion:

c = S*N(d1) - pv(K)*N(d2)

for put option:

p = -S*N(-d1) + pv(K)*N(-d2)

where d1 and d2 were defined from HW.

[NEAS: Yes]


ming
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um..... anyone can help me?
B) X = $85, PV(X)=85/1.0393=81.7913
C) 80/81.7913
D)=-0.02214
E) d1= 0.15623, d2=-0.2621

F) N(d1)= N(0.15623)=?

How Can I find the N?

ming
sundgaard
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N refers to the normal distribution.
use microsoft office EXCEL spreadsheet and the formula command is =NORMSDIST(d1) and =NORMSDIST(d2)or use the table in the back of the text

[NEAS: Yes.  Excel (or any statistical package) is the easiest way to find values of the cumulative normal distribution. You can also use the tables at the back of Brealey and Myers: Appendix Table 6 on page 973 of the Eighth edition.]

 


el_torero
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When a problem about Black-Scholes says "Volatility = x%", can we automatically assume that Volatility = (sigma)^2? I'm getting this definition from one of the sample problems, however, I cannot find volatility in the book at all. Then again, I'm also using an older edition of the book. Can someone please verify this for me? Thanks!

[NEAS: The volatility is σ.]


nic
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I'm not getting the same answers for part E.  I thought that t (time intervals until maturity) would be 1 since the risk-free rate is now in terms of 6-month compounding-intervals, and the options mature in 6-months.  Moreover, I thought that sigma would equal the square-root of the volitility.  Given that, I got d1=.2584 and d2=-.3332.

[NEAS: Time is in years; sigma is the volatility, which is the standard deviation for a one year horizon. If the options mature in six months, t = 0.5 and the volatility in the Black Scholes equation is multilplied by the square root of t. The rationale for this is that the variance of the stock price is proportional to time, so the standard deviation is proportional to the square root of time.]


NereusRen
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It's easiest to leave everything in terms of years, and use time = 1/2. You are right that you could convert the risk-free rate to a 6-month rate (1.08^(0.5)) and use t=1 instead, but you would also have to convert the volatility to a 6-month rather than annual number.

Using t=1/2 does those exact same conversions implictly within the formula anyway, so if you are calculating everything by hand there will be no difference, except that you are less likely to make a conversion error if you leave everything as annual and apply the formula directly.

Also, I believe that sigma is equal to the volatility (not the square root).
nybcnow
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I looke up online (http://en.wikipedia.org/wiki/Volatility) and it seems that volatility is Standard Deviation).  If that is the case, then d1 and d2 and few other things need to be recalculated.

[NEAS: Precise definition is standard deviation per square root of time.]



Regards,

Greg


daria
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According to pg. 556 of the 8th edition, "variance (i.e., volatility)...per period is sigma squared" (though they use symbols, I don't see how to do that here).  So that looks pretty clear, volatility is variance, and equals sigma squared as the practice problems show.  The table that shows that as sigma increases, so do call and put values--well, sigma increases when sigma squared increases.
JCaelum
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this is very confusing because in this thread they said volatility = sigma. But in practice problems exercise 23.1, they said volatilty = sigma squared. any help would be much appreciated. thanks!

[NEAS: Volatility = sigma per year (or per unit of time).  The one year volatility = the one year standard deviation.]


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