Lecture 9 Flashcards
Three desirable economic properties of the stochastic process for stock prices
- Price process should be consistent with weak form of market efficiency, present stock price impounds all information contained in a record of past prices
- The price process should also be scale independent
- Because of limited liability stock prices can never go below zero
The Black-Merton-Scholes formula – Notation
r
the continuously compounded risk less interest rate
The Black-Merton-Scholes formula – Notation
C
the current value of a European call
The Black-Merton-Scholes formula – Notation
S
the current price of the stock; the underlying asset
The Black-Merton-Scholes formula – Notation
K
the exercise price of the call
The Black-Merton-Scholes formula – Notation
T
the time remaining before the expiration date expressed as a fraction of a year
The Black-Merton-Scholes formula – Notation
σ
the standard deviation of the continuously compounded annual rate of return of the stock
The Black-Merton-Scholes formula – Notation
ln(S/K)
the natural logarithm of S/K e = 2.7183
Assumptions behind the Black-Merton - Scholes Formula
- The stochastic process for the stock price is lognormal with constant parameters µ and σ.
- Short selling of securities with full use of proceeds is permitted.
- There are no transaction costs or taxes. All securities are perfectly divisible.
- No dividends during the life of the derivative security
- Security trading is continuous
- The risk free continuously compounded interest rate is constant and the same for all maturities
- There are No Risk Free Arbitrage Opportunities.
Value of a European Put today
P = K e^(-rt) (1-N(d2)) - S e-yt (1-N(d1))
N(d1)
the hedge ratio
the fraction of one share that you invest in in the underlying stock in the replicating portfolio
N(d2):
fraction of the present value of the exercise price that you borrow at the risk free interest rate
- Probability that the call will end up in-the-money at date of maturity using the Martingale probability p
Delta (∆)
European call: Delta call / Delta Stock price
European put: Delta put/ Delta Stock price
Theta (Θ)
European call: Delta call / delta time
European put: Delta put/ Delta time
Gamma (Γ)
European call: Delta hedge ratio / delta Stock price
European put: Delta hedge ratio / delta Stock price
Lambda (Λ)
Vega
European call: Delta call / delta Sigma
European put: Delta put / delta Sigma
Rho (ρ)
European call: Delta call/ delta interest rate
European put: Delta put/ delta interest rate
Omega (Ω) (Elasticity of an
option/leverage)
European call: (Stock price * delta call)/ (Call price * delta stock price)
European put: (Stock price * delta put)/ (Put price * delta stock price)
Cash-or-nothing call price with payoff $1
C=e^(-rT) N(d2)
Asset-or-nothing call price:
C=S*N(d1)
Cash-or-nothing put price with payoff $1:
P=e-rTN(-d2)
Asset-or-nothing put price:
P=S*N(-d1)
Financial option VS Real Option (Notation)
Stock Price = Current Market Value of Asset
Strike Price = UpFront Investment Required
Expiration Date = Final Decision Date
Risk Free Rate = Risk Free Rate
Volatility of Stock = Volatility of Asset Value
Dividends = FCF Lost from Delay
Key Insights from Real Options
Out-of-the-money real options have value
In-the-money real options need not be exercised immediately
Waiting is valuable
Delay investment expenses as much as possible
Create value by exploiting real options
