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27 Aug 2008
Mar
03
FRM 2008 Early Bird Episode #8 (option pricing models)
by David Harper, CFA, FRM, CIPM
Contents
- This week's learning outcomes: Option pricing models
- Video tutorial (Episode #8)
- Three practice questions
- Previous Early Bird emails
Learning Objectives for Episode #8 (Intro to option pricing models)
This week's Early Bird screencast episode is a 30-minute introduction to option pricing models.
A note for new members about the Early Bird episodes: these are bonus episodes intended to give a boost to those seeking a quantitative introduction/refresher. They are not "mandatory;" e.g., option pricing models will be treated specifically according to the 2008 learning outcomes.
Put-call parity
For put-call parity, we only need four instruments. We use a simple example in the screencast: a $10 share, a call and a put both striking at $10, and cash ($10 discounted to present value). We then construct two portfolios with equivalent payoffs:
- "Discounted" cash + buy (long) a call
- Own a share + buy (long) a put
Because these two portfolios have equivalent payoffs, we can link the price of the call and the put (e.g., a popular test question asks you to solve for one instrument's price given the other three).
Binomial (dynamic hedge risk-neutral pricing)
I quickly review two approaches under a one-step binomial. The example is simple, but the financial ideas illustrated are profound:
- Hedging a partial share by writing a call option. This (dynamic) hedge underlies the subsequent Black-Scholes
- Risk-neutral pricing
Our one-step binomial values a call option where stock = $10, strike = $10, u = 1.2 (if stock jumps up, then it jumps to $12), and d = 0.8 (if stock jumps down, then it jumps to $8).
We then price a call option on this stock with two different approaches. Under the first approach (no-arbitrage), we construct the no-arbitrage portfolio. The portfolio is long a fractional share (Delta shares) and short one call option (we "write" a call). This is like a covered call, except instead of a full share we own a fractional share (<1.0). The result is really quite amazing. we can price the option with this riskless portfolio!
Under the second approach (risk-neutral pricing), we price the option by discounting the weighted average future option values. The hard part to follow here is that we use a riskless rate to find the probability of an up jump (p) and to discount the weighted average future option value. This is risk-neutral pricing in action and it leads us to an assertion that is hard to believe: our option price, derived from a riskless rate, can be valid in the risk-full world.
Black-Scholes
The hedge in the binomial example (i.e., writing a call while owing a fractional share) actually is a building block for the Black-Scholes. If you are interested in a critique, Nassim Taleb has recently written of the "mathematical impossibility" of this hedge. He also says the model isn't really a valuation because it makes no distributional assumption about the future stock price. His premise is of course accurate and a good thing to keep in mind. For each of these approaches, including both binomial approaches above, the stock's expected future stock return does not enter into the option's valuation; the expected future stock price either gets nullified by the discount rate (risk-neutral pricing) or "sidestepped" by way of a riskless portfolio construction.
If you don't need to understand the mathematical derivation of the BSM (an FRM candidate does not), I favor a pragmatic approach to memorizing the formula: take the minimum value of the call option and wrap in the two N() functions. I like to describe Black-Scholes as "minimum value adjusted up for volatility." That is, the Black-Scholes equals the option's minimum value under the special case where the volatility equals 0. Under that scenario, both N(d1) and N(d2) = 1. Then as we "add volatility," the spread N(d1) - N(d2) increases, so that additional volatility is effectively tweaking up the minimum value.
Video Tutorial
The 30-minute video tutorial is located here.
If you are a paid member, you can also access this in the member section (where you will also find the downloadable slides, if you would like to view those. As well as an ipod format file.)
Practice Questions
This week I wrote three questions that I hope give you good practice for all three concepts (put-call parity, binomial and Black-Scholes). These are meant to give new learners an introductory stretch; we will surely revisit them in the formal program!
Question #1 (put-call parity)
Assume Dell's stock price is currently $20 and you are valuing a European option with a strike price of $20 expiring in one year (T = 1.0). The riskless rate is 4%. In our simplified one-step binomial world, the stock will either jump up to $24 (u= 24/20 = 1.2) or down to 16 (u = 16/20 = 0.8). In summary, the assumptions are:
- Stock (S0) = $20
- Strike (K) = $20
- Term (T) = 1.0
- u = 1.2
- d = 0.8
- r = 4%
(i) Construct a riskless portfolio with a guaranteed payoff. The portfolio includes only two assets: a short position in a SINGLE OPTION on Dell's stock and a long position in delta (D) number of shares. In summary, portfolio = -c + (D)(S). Solve for the number of (D) shares gives a riskless portfolio in a simplified one-step binomial world. Hint: the payoff under the jump up scenario (stock = $24) must equal the payoff under the jump down scenario (stock = $16)
(ii) Use the answer in (i) to solve for the present value of the portfolio and the stock option. Hint: discount the portfolio to PV, then recall Portfolio PV = -c + (D)(S)
(iii) In a risk-neutral world, what is the probability of an up jump (p)?
(iv) Solve for the option value as a function of the probability (p) of an up jump; i.e., as a discounted, weighted average value.
(v) In the real world, the expected return on the stock is greater than the riskless rate. Therefore, the probability of an up jump in the real world (p*) is greater than the probability of an up jump in the risk-neutral world. This creates a higher expected payoff from the option in the real world! How can this be reconciled with our value of the option above, which is twice confirmed by two different approaches?
Question #2 (binomial)
Assume Nortel is a non-dividend-paying stock trading at $9. The riskless rate is 4%. Assume both a 3-month European CALL and a 3-month European PUT. Both the call and the put have a strike price of $8.
(i) Characterize the two portfolios (one containing a call and another containing a put) that create equivalent payoffs, and thereby demonstrate put-call parity. Show both portfolios have the same payoff under two scenarios: the stock price increases to $12 and the stock price decreases to $6.
(ii) Assume the price of the call is $2. According to put-call parity, what is the price of the put?
(iii) What is the minimum value of the call option?
Question #3 (Black-Scholes)
Assume Hewlett-Packard’s (HPQ) stock price is $50 with an annualized volatility of 40%. The riskless rate is 4%. For simplicity’s sake, assume HPQ does not pay a dividend (actually, they do). Consider a one year European call option with a strike of $50. To summarize:
- Stock (S) = $50
- Strike (K) = $50
- Volatility (sigma) = 40%
- Term (T) = 1.0
- Dividend = 0
- Riskless rate (r) = 4%
To perform this calculation, you will need the standard normal cumulative distribution.
Click here to view a lookup table.
You can also get N(z) in Excel with the function =NORMSDIST(z)
(i) What is the option’s delta?
(ii) Use this delta in a sentence (i.e., interpret its meaning)
(iii) If we assume (as we often do) that the stock price follows geometric Brownian motion (GBM), how are the stock’s PERIODIC RETURNS assumed to be distributed?
(iv) Under GBM again, how are the future stock PRICE LEVELS assumed to be distributed?
(v) What does the Black-Scholes option pricing model give for the price of the call option?
Answers to practice questions
Previous Early Bird Episodes
In case you missed them, the previous early bird emails (with the practice questions) are found here:
- Episode #1 (notation, compounding, matrix)
- Episode #2 (distributions: binomial, Poisson, normal)
- Episode #3 (linear regression)
- Episode #4 (functions and factor models)
- Episode #5 (derivatives)
- Episode #6 (volatility)
- Episode #7 (duration)
- And the answers to the practice questions are in this forum area
That's all for this week. Because GARP should soon publish the 2008 Study Guide, I will skip next week's newsletter. So I will see you in two weeks!
David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com
P.S. I don't want to spam you!
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