Black-Scholes-Merton Model

Summaries

A. BSM Model is a continuous version of the binomial model used to evaluate the options. Thus, it uses a instantaneously risk-free portfolio (compared to 1 period risk-free).

Assumptions:

1. Price of underlying assets has lognormal distribution
2. risk-free rate is known and constant (so not good for interest and bond options)
3. Volatility of underlying assets is constant and known.
4. Market is frictionless
5. European options only

B. Equations:

C₀ = S₀N(d₁) – X exp(-R­_f^cT)N(d₂)

d₁ = [ln(S­₀/X)+(R_f^c + (0.5sigma²))T]/(sigma * sqrt(T))

d₂ = d₁ – (sigma * sqrt(T)

T time to maturity (365 days basis)

S₀ asset price

X exercise price

R_f^c  continuously compounded risk-free rate

Sigma volatility of continuously compounded return on the stock

N() cumulative normal probability

C. So you can see the BSM model has 5 inputs: Asset price, Exercise Price, Time to maturity, Risk free rate and Volatility of return.

Greeks: (For European option with no cash flow)

Delta – relationship between Option price and Asset price (+ve for call, -ve for put)

Vega – relationship between Option price and Volatility of Return (+ve for call, +ve for put) (Most sensitive)

Rho – relationship between Option price and Risk free rate (+ve for call, -ve for put)

Theta – relationship between Option price and Passage of time (-ve for call, -ve for put)

Exercise price (-ve for call, +ve for put)

D. Delta ~ N(d₁), Change of option ~ N(d₁) x change of stock

Must understand the European Call/Put option payoff curves at/prior to expiration.

E. Dynamic Hedging

Just as what we learnt in binomial model, we can form a portfolio (long stock and short call) to hedge the risk of stock value change. By definition,

Delta = change of option value / change of stock value

Therefore, if we are talking about hedging the stock, the hedge ratio = 1/Delta. It means we have to short “hedge ratio amount” of option for each stock in order to hedge the risk. (If we are hedging options, then the hedge ratio should be Delta because then we are calculating the # of stock for each option to be hedge)

Since delta changes as the stock price changes, the hedging ratio is changing. Therefore dynamic hedging is required to maintain a risk free portfolio.

F. Gamma measures the change of Delta as the stock changes (so it is the 2^nd derivative of option price)

Put and Call have the same Gamma curves.

Gamma is maximum at-the-money (and closed to expiration) (thus needs very active dynamic hedging management)

G. When there is cash flow for the stocks (dividend), the call option value will decrease and put option value increase. This is similar to what we have learnt from binomial model. We can substitute S0 => S0 exp(-d_c*T)