Chapter 9 Options

Call options: an agreement in which the holder/buyer of the option has the right to buy, from the writer/seller of the option, the underlying asset at a pre-specified time and price.

• Holder/buyer of option has the right, not the obligation, to buy the underlying asset

• Seller of the call option is obligated to sell the underlying asset to the buyer of the option if the buyer exercises her/his right to buy the underlying asset

• Underlying asset:the asset on which the option is based

• Expiration date:date on which the option expires

• Exercise (strike) price: price at which the transaction will take place if the option is exercised by the holder

• Premium: up-front price for the option, paid by the buyer to the seller of the option

• Style of call option

  • European: holder of the option can exercise her/his right to buy the underlying asset only on the expiration date of the option

  • American: holder of the option can exercise her/his right to buy the underlying asset any time during the life of the option, up to and including the expiration date of the option

• Payoffs to call option positions

  • Purchased call option payoff = \(\max[0, S_T-K]\)

  • Written call option payoff = \(-\max[0,S_T-k]\)

  • Payoffs on a call option reflect a zero-sum game between the parties

• Profits to call option positions

  • Purchased call option profit = \(\max[0, S_T-K]-FV[C(K,T)]\)

  • Written call option profit = \(FV[C(K,T)]-\max[0, S_T-K]\)

Put options: an agreement in which the holder/buyer of the option has the right to sell, from the writer/seller of the option, the underlying asset at a pre-specified time and price.

• Holder/buyer of option has the right, not the obligation, to sell the underlying asset

• Seller of the call option is obligated to buy the underlying asset to the buyer of the option if the buyer exercises her/his right to buy the underlying asset

• Underlying asset:the asset on which the option is based

• Expiration date:date on which the option expires

• Exercise (strike) price: price at which the transaction will take place if the option is exercised by the holder

• Premium: up-front price for the option, paid by the buyer to the seller of the option

• Style of call option

  • European: holder of the option can exercise her/his right to sell the underlying asset only on the expiration date of the option

  • American: holder of the option can exercise her/his right to sell the underlying asset any time during the life of the option, up to and including the expiration date of the option

• Payoffs to put option positions

  • Purchased put option payoff = \(\max[0, K-S_T]\)

  • Written put option payoff = \(-\max[0,K-S_T]\)

  • Payoffs on a put option reflect a zero-sum game between the parties

• Profits to put option positions

  • Purchased put option profit = \(\max[0, K-S_T]-FV[P(K,T)]\)

  • Written put option profit = \(FV[P(K,T)]-\max[0, K-S_T]\)

Put-call parity

\[Call - Put = PV(Forward~~price) - PV(Strike~~price)\]

\[C(K,T) - P(K,T) = PV(F_{0,T})-PV(K)\]

Binomial tree model

\[f=e^{-rT}[p f_u + (1-p) f_d]\] \[u=e^{\sigma\sqrt{\Delta t}}, d=e^{-\sigma\sqrt{\Delta t}},p=\frac{e^{r\Delta t}-d}{u-d}\]