Week 4 Flashcards
(8 cards)
What is the Black Scholes Merton Model?
The Black-Scholes-Merton Option Pricing Model is an explicit pricing formula. For European options only
C=SN(d1)−Xe−rTN(d2). Normally distributed. Continuous model. Calculates call options
What are the assumptions?
- No transactions costs or taxes
- No constraint on short selling
- Assets are infinitely divisible
- Unlimited borrowing and lending is possible at a single risk-free rate
- Continuous trading
- No dividend is paid on the stock
- Stock prices follow continuous time random walk process called
- Prices are lognormally distributed
- Prices have constant mean and variance
- The no-arbitrage condition applies
What is d1 and d2?
d1= ln(S/X)+(r+σ^ 2/2)T / σ Square root T d2=d1−σ Square root T
Difference between BSM and Binomial?
BSM - multiple periods, only for Europeans, continuous, Normal distribution
Binomial- Two periods, for Europeans + Americans, Binomially distributed,
what are the greeks?
are sensitivities — they tell us how the option price moves when something changes.
What is delta?
Basicslly how much the call option goes up per £1 increase of the stock. The rate of change of the option price with respect to the change of the underlying
asset is given by the delta of the option, ∆. From Black-Scholes-Merton, ( ) 1 =∆ dN
∆ must lie between 0 and 1 (a cumulative probability). Delta ∆ can also be viewed as the fraction of shares required to form a riskless
hedge that replicates the value of a bond, B.
Delta is fixed for a given value of S, ceteris paribus. Delta-neutral hedging maintains the value of the hedged portfolio against small
changes in S.
Delta changes with both the underlying asset price and time.
The portfolio will require re-balancing to retain delta-neutrality.
Delta is the change in the price of a call option for a very small change in the price of
underlying asset.
Delta is the first derivative of the option premium with respect to the underlying asset price
but this is not constant since the time value of the option is a non-linear component of the
price (estimating price changes from delta alone may lead to errors) Delta is the slope
What is gamma?
Γ = ∂²C/∂S² = ∂Δ/∂S. Γ = N’(d₁)/(Sσ√T). Γ is the change in Delta as the stock price changes (the change in the slope of the tangent line. as S increases the call behaves increasingly like the stock itself. Γ is distributed fairly symmetrically around the exercise price (tends to be skewed right for a long call option). When Γ is large it is important to re-balance a portfolio to retain delta-neutrality. Delta-neutrality does not ensure that the portfolio is risk-free for large changes in S. Gamma is the first derivative of delta with respect to the stock price: Γ is the change in the slope of the option premium graph.
Estimating a change in the value of the option using delta alone will result in an error that
can be corrected by simultaneously adjusting for gamma.
Non-linear price effect
What is theta?
Θ = -∂C/∂T or ∂C/∂t. and Θ = -S₀N’(d₁)σ/2√T - rXe⁻ʳᵀN(d₂) < 0 This gives the time decay of the option
* Ceteris paribus the option loses value as time passes
* Time value declines with the square root of T
Time is measured in years in this formula. To obtain theta per calendar day, divide by 365
