CFA Level 2 2015 - Derivatives - Reading 49 - Options Markets and Contracts
Terms in this set (41)
Put-call parity for European options:
put + share
call + zero-coupon riskless bon
One-period binomial model:
Risk-neutral probability of up- and down- moves:
Valuing a one-period call with X = $30
Lower bounds for European/American call:
Lower bounds for European put:
Lower bounds for American put:
Arbitrage with a one-period binomial model
The result is
no. of shares per option
A delta-neutral portfolio is a risk-free combination of a long stock position and short calls where the number of calls to sell is equal:
number of options needed to delta hedge = number of shares hedged / delta of call option
To price options on bonds:
An interest rate
is similar to...
a call option on interest rates
An interest rate
is similar to...
a put option on interest rates
To value a cap (floor), you simply value each caplet (floorlet) and then add them up
!!! Caps and floors pay in arrears: the payoff occurs one period after the expiration of the cap or floor.
Eg. a 2-year cap is a...
2-year caplet and a
6 assumptions of the Black Scholes model
(1) The price of the underlying follows a lognormal distribution
(2) The risk-free rate is constant and known
(3) The volatility of the underlying is constant and known
(4) Markets are frictionless
(5) The underlying has no cash flows
(6) The options valued are European options
BSM formula call:
BSM formula put:
Comparison of the 2 BSM formulas:
To value the put, use the put-call parity:
The five inputs to the BSM model:
(1) Asset price
(2) Exercise price
(3) Asset price volatility
(4) Time to expiration
(5) Risk-free rate
measures the sensitivity of the option price to changes in the volatility of returns on the underlying asset
The price of a European call option does not change much if we use different inputs for the risk-free rate, so rho is not a very important sensitivity measure.
Delta of a call:
Estimate delta using BSM:
N(d1) is the delta from the BSM model
European call option price payoff:
Delta is the slope of the prior-to-expiration curve
When the call is in=the
Deep in the money puts have a delta of...
Deep out of the money calls have a delta of...
The call option delta is between 0 and 1. If the call option is:
, the call delta *moves closer to 0 as time passes, assuming the price of the underlying doesn't change
, the call delta
moves closer to 1
as time passes, assuming the price of the underlying doesn't change
rate of change in delta as the underlying stock price changes
Hedges with at-the-money options will have higher gammas, and small changes in stock price will lead large changes in delta and frequent rebalancing.
In contrast, hedges with deep in- or deep out-of-the money options will have small gammas and stock price changes will not affect the delta of the hedge significantly. This lowers rebalancing and transaction costs.
A gamma of 0.04 means...
that a $1.00 increase in the price of the underlying stock will cause a call option's delta to increase by 0.04, making the call option more sensitive to changes in the stock price
it is largest when a call or put option is at-the-money and close to expiration
All else equal, the existence of cash flows on the underlying asset will:
- Decrease the value of a call option
- Increase the value of a put option
Revised put-call parity for options on underlying assets with cash flows:
Put-call parity for options on forwards (or futures)
A call on the forward
A pure-discount bond that pays X-Ft at time T
The payoff for this portfolio at expiration will depend on whether the option is in or out of the money. The bond will always pay X-Ft:
- If the call is out of the money (S<X), the payoff will be from the bond:
- If the call is in the money (S>X), the payoff is: *(St-X)+(X-Ft) = St-Ft
Put-call parity for options on forwards (or futures) (2)
Equivalent portfolio 2:
A put on the forward
A long position in the forward
The cost of this portfolio is P0, since its costless to enter into a forward
The payoff at expiration:
- If the put is in the money (S<X):
(X-St) + (St-Ft) = X - Ft
- If the put is out of the money (S>X):
The payoffs of the two positions are identical!
Why are American options on futures more valuable than comparable European ones?
Because exercising an in the money option will generate cash from the mark to market. This cash can earn interest, while the futures position will gain or lose from movements in the futures price.
The Black Model can be used to price European options on forwards and futures:
It is just the BSM model with e^RfxT x Ft substituted for S0
Comparison between BSM and Black