detailed notes for the financial derivatives module. Notes on the different types of financial derivatives, such as call and put options, factors which affect their value and much more.
- Assume a sophisticated model describing how the stock price moves over time
o We will call this the “lognormal model”
- Construct a riskless portfolio using the stock and the derivative
o Risk-premiums are eliminated, we can use no-arbitrage arguments and risk-neutral valuation to
find the fair price the derivative
BSM vs Binomial Model
- The Binomial Model → Discrete states and discrete time
o Finite number of possible values of S
o Finite number of time steps
o Increasing the number of steps = increases the number of possible values for the price
- The BS Model → Continuous states and continuous time
o Infinite number of possible values of stock price
o Infinite number of time steps, i.e. prices change continuously
Assumptions of BSM
- The stock price follows the lognormal model
o The expected return (µ) of the stock and the volatility (σ) of the stock are constant over time
- No arbitrage opportunities
- The risk-free rate is constant and the same for all maturities
- Investors can borrow and lend at the risk-free rate
- No transaction costs or taxes
- Security trading is continuous → investors can buy and sell whenever
- All securities are perfectly divisible → can buy half a share, quarter of share etc.
- No dividends during the life of the option
Lognormal model implies:
- The instantaneous return is normally distributed
o The instantaneous return is the return from holding a stock for the shortest time possible
o Imagine the return one would get by holding the asset just for one second
- For longer horizons, T, the stock price is log-normally distributed
o This means that the logarithm of the stock price follows a normal distribution N
- Black-Scholes showed that we can value the option as if we are living in a risk-neutral world
o Since in a risk-neutral world the return of every asset is the risk-free rate, we can set the
(unknown) expected return, µ, equal to the observed risk-free rate, r, ➔ in risk-neutral world, µ = r
o The European option price can be calculated by the BS formula
Black-Scholes Formula
- The prices of a European call c and European put p written on a non-dividend paying stock S are given by:
- The N(x) Function:
o Is the cumulative distribution function of the standard normal distribution
, o In other words, N(x) gives the probability that a standard normally distributed variable will have a
value less than some level x
▪ The standard normal distribution has a mean of 0 and a variance of 1
▪ The higher the x, the higher the N(x)
o In the BS model, a high N(x) reflects that the call option is more likely to end in-the-money
▪ A low N(x) reflects that the call option is more likely to end out-of-the money
Historical vs implied volatility
- Stock Volatility (σ):
o σ is the annualized standard deviation of the stock price
o σ is the only parameter of the BS formula that is not directly observable
- Historical Volatility:
o Is the estimate of stock volatility calculated using observed (past) return data
o If we trust the BS model, we can calculate σ and plug it in the formula to get the required option
price
- Implied Volatility:
o Given an observed option price, we can find the σ traders use
o Unlike historical volatility, implied volatility is forward looking
- BS implied volatility
o Expected volatility according to the BS model
- VIX (volatility index)
o Expected volatility independent of specific option pricing models
o Not based on specific option pricing model
o Proven to be a popular way of understanding if markets are fearing a crisis in the future
Using the BSM to price options
- European options on assets that pay known cash dividends
- American call options on assets that pay no dividends
- American call options on assets that pay known cash dividends
o This will be an “approximate price”
- European options on assets paying a known dividend yield
o This applies to stock index and currency options
European options for stocks paying a known dividend
- Strategy:
o We calculate the Present Value of all Dividends (PVD) to be paid during the life of the contract
▪ See seminar questions for calculating the PV of dividends
o We compute a “modified” spot price S0* = S0 − PVD
o We apply the BS formula using S0* instead of S0
- Example Problem:
o A stock stands at $75 and the risk-free rate is flat 5%
o The stock is expected to pay a 2 dollar dividend in one, three and six months.
o We want to price a 4-month call option
- Example Solution:
1 3
o 𝑆0∗ = 𝑆0 − 𝑃𝑉𝐷 = 75 − ($2ⅇ −0.05 ⋅ 12 ) − ($2ⅇ −0.05 ⋅ 12) = 71.03
o Where 75 = S0 and 2 is the dividend payment
o Ignore dividends paid in month 6 as contract expires in 4 months
o We apply the BS formula using S0* instead of S0
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