BSM Assumptions-Clarification

[PDEM200]

Hi David,

Can you please clarify a point relating to BSM model assumptions. Pg. 66 in the notes (section with green arrow) indicates that BSM does not require an assumption about the probability distribution of returns. However, in an earlier section, it was noted that PRICE CHANGES/RETURNS are assumed to follow a NORMAL distribution;while prices themselves follow a LOGNORMAL distribution. Would appreciate your guidance. Thank you.

 

[David Harper CFA FRM]

Good question. My language in the green arrow could be stronger/better as this is subtle but significant. In short, Black-Scholes does make an assumption about the (stochastic) process of the asset, but it does not need to know everything about that process (specifically it does not need to know the asset's expected return).

So, among the Black-Scholes assumptions (FRM AIM: "List...assumptions underlying the model") is the assumption-requirement that stock prices follow Brownian motion: period returns are normally distributed (i.e., LN(S1/S0) is normal) which is equivalent to saying price levels (S1 or S1/S0) are lognormal . This is "merely" the requisite assumption about *which* stochastic process is employed. At this point, it is merely a distribution but we've haven't characterized (described) it with parameters. Most, IMO, write this as "B-S assumes a lognormal process"

But, and this is the hardest thing for most when learning B-S/OPM, the expected return of the stock does not enter (is not required) to get the option price. At first, intuitively (maybe with a growth model in mind), we expect the stock's expected return to figure in somewhere. But only the volatility/variance enters, not the expected return.

concretely, if we use CAPM on a stock and, say, the stock's expected return is 10%

Because 10% = 4% riskless + (4% equity premium)(stock's beta =1.5)

The Black-scholes model uses the 4% riskless rate and the stock's volatility/variance, but the expected return is *not* needed. This is the counter intuitive risk-neutral valuation idea. This is really the risk-neutral idea in a nutshell: Black-Scholes need the volatility but not the stock's expected return.

David

 

[varadarajan]

Hi David,
In page 61 of the market risk study notes (Hull Chapter 13), the first equation [ln(St / S0) = ...] seems to suggest that the returns themselves are log normally distributed.. am I missing something here?

Further if this is an assumption are we assuming that stock prices will never have negative returns? Please clarify.

 

[David Harper CFA FRM]

Hi varadarajan

Almost. It reflects Brownian motion: log returns (i.e., ln[St/S0]) are normally distributed. Which means: Price levels (St) or wealth relative (St/s0) are lognormally distributed. In short, period returns are normal, but price levels are lognormal.

Per lognormal distribution (which is is never < 0), neither the stock price (St) nor the wealth relative (St/S0) can be negative, but the continuously compounded (period) return can be negative; e.g., stock drops from $10 to $6. Period return is negative but 6/10 or x/10 is limited down at zero

David