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• The standard normal curve is a member of the family of normal curves with μ = 0.0 and = 1.0. The X-axis on a standard normal curve is often relabeled and called ‘Z’ scores.
• The area under the curve equals 1.0.
• The cumulative probability measures the area to the left of a value of Z. E.g., N(0) = Prob(Z) < 0.0 = 0.50, because the normal distribution is symmetric.
• The BSOPM assumes that the stock price follows a “Geometric Brownian Motion” (see
http://www.stat.umn.edu/~charlie/Stoch/brown.html for a depiction of Brownian Motion).
• In turn, this implies that the distribution of the returns of the stock, at any future date, will be “lognormally” distributed.
• Lognormal returns are realistic for two reasons: – if returns are lognormally distributed, then the lowest possible return in
any period is -100%.
– lognormal returns distributions are "positively skewed," that is, skewed to the right.
• Thus, a realistic depiction of a stock's returns distribution would have a minimum return of -100% and a maximum return well beyond 100%. This is particularly important if T is long.
• What happens to the Black-Scholes call price when the call gets deep-deep-deep in the money?
• How about the corresponding put price in the case above? [Can you verify this using put-call parity?]
• Suppose gets very, very close to zero. What happens to the call price? What happens to the put price?
Hint: Suppose the stock price will not change from time 0 to time T. How much are you willing to pay for an out of the money option? An in the money option?
• Believing that an option is undervalued is tantamount to believing that the volatility of the rate of return on the stock will be less than what the market believes.
• The volatility is the standard deviation of the continuously compounded rate of return of the stock, per year.
1. Take observations S0, S1, . . . , Sn at intervals of t (fractional years); e.g., = 1/52 if we are dealing with weeks; = 1/12 if we are dealing with months.
2. Define the continuously compounded return as:
3. Calculate the average rate of return
4. Calculate the standard deviation, , of the ri’s
• The implied volatility of an option is the volatility for which the BSOPM value equals the market price.
• The is a one-to-one correspondence between prices and implied volatilities.
• Traders and brokers often quote implied volatilities rather than dollar prices.
• Note that the volatility is assumed to be the same across strikes, but it often is not. In practice, there is a “volatility smile”, where implied volatility is often “u-shaped” when plotted as a function of the strike price.
Using Observed Call (or Put) Option Prices to Estimate Implied Volatility
• Take the observed option price as given.• Plug C, S, K, r, T into the BSOPM (or other model).• Solving this is the tricky part) requires either an iterative
technique, or using one of several approximations.
• The Iterative Way:– Plug S, K, r, T, and into the BSOPM.
– Calculate (theoretical value)
– Compare (theoretical) to C (the actual call price).
– If they are really close, stop. If not, change the value of
In this table, the option premium column is the average bid-ask price for June 2000 S&P 500 Index call and put options at the close of trading on May 16, 2000. The May 16, 2000 closing S&P 500 Index level was 1466.04, a riskless interest rate of 5.75%, an estimated dividend yield of 1.5%, and T = 0.08493 year.
call price put price Strike Call price Call IV with = 0.22 Put Price Put IV with = 0.22
• The BOPM actually becomes the BSOPM as the number of periods approaches ∞, and the length of each period approaches 0.
• In addition, there is a relationship between u and d, and σ, so that the stock will follow a Geometric Brownian Motion. If you carve T years into n periods, then:
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• Then, the return generating process is such that each day, the return consists of:– a non-stochastic component, 0.0005 or 0.05%– a random component consisting of:
• The stock's daily standard deviation times the realization of z,
z is drawn from a normal probability distribution with a mean of zero and a variance of one.