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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:

Variance: Var(Y) =

Note: multiplying lognormal random variables is similar to adding normal random variables.

- Then Y ~Lognormal(m, v)

Lognormal Distribution Contd.

- Using the knowledge from before now we interpret it in different way:
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

Black Scholes

- Second way to calculate the price for options. To get the formulas for Black Scholes you start with the binomial tree model and make the number of periods reach infinity use some stochastic calculus.
- Assumptions: (important for theory type questions on test)
- Continuously compounded returns on stock are normally distributed and independent over time.
- Continuously compounded returns on strike asset (the risk free rate) are known and constant.
- Volatility is known and constant.
- Dividends are known and constant.
- No transaction cost or taxes.
- You can short-sell or borrow any amount of money at risk free rate.

Black-Scholes Model

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

- Pricing European Calls and Puts:

Exam Note: It is recommended to use 4 or 5 decimal places when entering your numbers in the calculator.

- On the MFE exam to calculate you will be given a Normal Distribution Calculator instead of a Normal Distribution Sheet. Here is screen shot of what it looks like and a link from prometric to try it out:
- Click here for the calculator.

Black Scholes Contd.

- Black-Scholes for Options on Stocks with Continuous Dividends simplifies to (it might be useful to remember this as this is the most common question type for the exam):
- Black-Scholes for Options on Currencies (Garman-Kohlhagen Model):
- Standard Substitution of x0 (the exchange rate) for S and rf (risk free rate for foreign currency) for δ:

Estimating Volatility

- These are the stock prices in the past few weeks:
Calculate the annul volatility.

Using STAT, DATA key on TI-30XS multiview:

- This saves a lot of time and makes these type of questions a breeze to solve:
- [data] [data] 4 (this clears the data table )
- (enter the data)
- DATA: (highlight L1) FRQ: (highlight one) (select CALC) [enter] 3 (to get Sx)
- Note: 52 is number of weeks in a year use this unless they specifically mention to use number of work weeks.

Greeks

- Delta – Δ : Change in option value with respect to stock price (first partial):
For European Options:

- Gamma – Γ : Change in Delta with respect to stock price (second partial of value wrt price), measures convexity, always > 0.
- Notes:
- Delta is the same as the one we discussed in binomial tree model i.e. delta is the number of shares needed to replicate the option.
- Delta is the only one that they can use to ask to compute other Greeks you just need to understand what they are and not needed to calculate them. However we will see how to calculate Gamma using Binomial Tree model.
- According to author of ASM books, the graphs for Greeks although in syllabus have not appeared on the exam ever.

Greeks Contd.

- Vega : Change in option price with respect to volatility, always > 0.
- Theta Θ : Change in option price as time to maturity get closer usually < 0.
(T-t) = Time to expiration

- Rho ρ : Change in option price as risk free rate increases ( positive for calls, negative for puts)
- Psi ψ : Change in option price as dividend yield increases ( negative for calls, positive for puts)

Greeks For Binomials

- There can be a question on the exam to solve for theta for a 2 period binomial model:

Elasticity (important)

- Elasticity Ω : Percent change in option price per percent change in stock price
- Volatility of an Option :
- Risk Premium of an Option
- Note: gamma is the expected yield from option.
- r is the risk free rate.

Sharpe Ratio (also important)

- Sharpe Ratio of an Option : risk premium divided by volatility.
- Elasticity and Risk Premium of a Portfolio:
- Note: We have 2 ways of solving elasticity for a portfolio:
- First: Calculate the delta for the portfolio and total price of the portfolio and use the second formula.
- Second: Take the weighted average of individual options. Weight = (# options)*(Price of option)/(price of portfolio)

- Note: We have 2 ways of solving elasticity for a portfolio:

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