Week – 2 . Topics: Lognormal Distribution Black Scholes Formula Estimating Volatility from historic data Greeks Elasticity Sharpe Ratio. Lognormal Distribution. If X ~Normal(m, v) and Y = Then Y ~Lognormal(m, v) For Y: Median: Mean: E[Y] = Mode:
For Y: Median:
Mean: E[Y] =
Variance: Var(Y) =
Note: multiplying lognormal random variables is similar to adding normal random variables.
We say: The continuously compounded returns on stocks are normally distributed and the stock price is log normally distributed.
Let be the stock price at time t. It is a random variable, such that =
A ~normal(α-δ, σ)
/ ~ lognormal(m,v)
Therefore: (α-δ)t= m+.5, v = σ
Or m=(α-δ-.5)t and v = σ
Used for pricing options
S = current stock price
K = strike price
r = annual risk-free interest rate
T = Time to expiration
= annual volatility of stock returns
= annual dividend rate
Calculate the annul volatility.
For European Options:
(T-t) = Time to expiration