Valuation Of Contingent Claims

Question 1

Probability of up move

Answer 1

= (1 + Rf - D) / (U-D)

Question 2

Replication portfolio for a call option (synthetic call)

Answer 2

Leveraged position in h shares where h=hedge ratio or delta of the option
(Stock is purchased using borrowed funds aka short position in bonds)

Question 3

Replication portfolio for put option (synthetic put)

Answer 3

Long position in risk-free bond and short position in h shares of stock (h=hedge ratio or delta of option)

Question 4

Hedge ratio (delta)

Answer 4

Tells you the fractional share of stock needed in the arbitrage model
h = [(C+) - (C-)] / [(S+) - (S-)]
Delta describes the relationship between changes in asset prices and changes in option prices. Call option deltas are positive because value increases as asset price increases. Put option deltas are opposite.

Question 5

Payer swaptipn

Answer 5

Gives holder the right to enter into a specific swap at some date in the future at a predetermined rate as the fixed-rate payer. Becomes more valuable as interest rates increase. Holder would exercise payer swaption and enter into swap if market rate is greater than exercise rate at expiration

Question 6

Gamma

Answer 6

Measures the rate of change in delta as the underlying stock price changes; captures the curvature of the option value vs stock price relationship. Long positions in calls and puts have positive gammas. Gamma is highest for at the money options; changes with stock price and time to expiration. To lower overall gamma of portfolio, one should short options.

Question 7

Vega

Answer 7

Measures the sensitivity of the option price to changes in the volatility of returns on the underlying asset. Both put and call options are more valuable, all else equal, the higher the volatility, so vega is positive for both calls and puts. Vega is important sensitivity measure. Vega gets larger as the option gets closer to being at the money.

Question 8

Rho

Answer 8

Measures the sensitivity of the option price to changes in the risk free rate. Not a very important sensitivity measure bc risk free rate doesn’t affect option price much. Call option value increases as risk free rate increases, so rho is positive. Put option is opposite.

Question 9

Theta

Answer 9

Measures sensitivity of option price to passage of time. As time passes and options reach maturity, speculative values decline, all else equal. Theta is less than zero.

Question 10

Delta-neutral portfolio/hedge

Answer 10

Combine a long position in stock with a short position in call options or long position in puts
Number of short calls needed to delta hedge = number of shares hedged / delta of call option
Number of long puts needed to delta hedge = -(number of shares/delta of put)

Question 11

Assumptions underlying BSM model

Answer 11

The price of the underlying asset changes smoothly and follows a lognormal distribution.
The risk free rate is constant and known.
The volatility of the underlying asset is constant and known.
Markets are “frictionless.”
The underlying asset generates no cash flows.
The options are European.