American and European Call and Put Contracts#

In this lecture, we focus on options, a type of derivative that uses shares of a stock or an exchange-traded fund, as their underlying asset.

Call contracts#

A call option is a financial contract that gives the holder the right, but not the obligation, to sell a specified asset, such as stocks, commodities, or currencies, at a predetermined price within a specified time period. Let’s consider stock as the underlying asset. A single standard call contract controls 100 shares of stock.

In the case of American style call contracts, the option buyer can exercise their right at any point between when the contract is purchased and the expiration date. On the other hand, buyers of European style contracts can only exercise their right on the expiration date.

The business case for buying (or selling) a call contract:

  • Buyer (long): From the buyer’s perspective, call contracts allow an investor to benefit from the price movement of XYZ to the upside without purchasing XYZ. Further, call options (again from the buyer’s perspective) have limited downside risk, i.e., the maximum amount that the holder of the call option can lose is the premium paid for the option. Finally, call options are a mechanism to purchase shares of XYZ at the strike price of \(K\) instead of the market price of \(S\).

  • Seller (short): From the seller’s perspective, the main objective of selling a call contract is to collect the premium \(\mathcal{P}\). Call contracts also allow the seller to benefit from the price movement of XYZ to the downside without purchasing XYZ. However, for a seller, call options have unlimted upside risk; thus, call options are often only sold by investors who already own the required number of shares of XYZ (known as a covered call position). Finally, call options offer the seller the opportunity to sell shares of XYZ at the strike price of \(K\) instead of the market price of \(S\).

Payoff, Profit, Premium and Breakeven#

The payoff per share of a call contract at expiration T days in the future \(V_{c}(K,S(T))\) is defined as:

\[V_{c}(K,S(T)) = \max\left(S(T) - K,~0\right)\]

where \(K\) denotes the strike price and \(S(T)\) represents the share price of the underlying asset T days in the future (at expiration). The right (but not the obligation) to buy shares at the strike K is not free. The contract seller charges the contract buyer a premium for each call contract \(\mathcal{P}_{c}(\dots)\). From the perspective of the buyer, the profit per share for the call contract \(P_{c}\) is the payoff of the contract minus the cost of the contract:

\[P_{c}(K,S(T)) = V_{c}(K,S(T)) - \mathcal{P}_{c}(K,S(0))\]

Thus, from the buyer’s perspective, the share price must fall above the strike price to make up the amount paid for the contract. This is called the breakeven price \(\mathcal{B}_{c}(K, S(T))\):

\[\mathcal{B}_{c}(K,S(0)) = K + \mathcal{P}_{c}(K,S(0))\]

Finally, the premium (cost) for each call contract \(\mathcal{P}_{c}(\dots)\) is defined by the expression:

\[\mathcal{P}_{c}(K,S(0))\geq\mathbb{E}\Bigl(\mathcal{D}^{-1}_{T,0}(\bar{r})\cdot{V_{c}}(K,S(T)\Bigr)\]

where \(\mathcal{D}_{T,0}(\bar{r})\) denotes the discount rate between the time when the contract was purchased 0 and contract expiration T days in the future. Option contracts use risk-neutral pricing; thus, the discount rate \(\bar{r}\) is typically taken as the interest rate on 10-year treasury notes.

Put contracts#

A put option is a financial contract that gives the holder the right, but not the obligation, to sell a specified asset, such as stocks, commodities, or currencies, at a predetermined price within a specified time period. Let’s consider stock as the underlying asset. A single standard put contract controls 100 shares of stock.

In the case of American style put contracts, the option buyer can exercise their right at any point between when the contract is purchased and the expiration date. On the other hand, buyers of European style contracts can only exercise their right on the expiration date.

The business case for buying (or selling) put contracts:

  • Buyer (long): From the buyer’s perspective, put contracts allow an investor to benefit from the price movement of XYZ to the downside without purchasing XYZ. Further, put options (again from the buyer’s perspective) have limited downside risk, i.e., the maximum amount that the holder of the put option can lose is the premium paid for the option. Finally, put contracts are a mechanism to sell shares of XYZ at the strike price of \(K\) instead of the market price of \(S\).

  • Seller (short): From the seller’s perspective, the motivation for selling a put contract is to collect the premium \(\mathcal{P}\). Put contracts also allow the seller to benefit from the price movement of XYZ to the upside without purchasing XYZ. However, for a seller, put options have unlimted downside risk; thus, put options are often only sold by investors who have set aside the required capital to purchase the required number of shares of XYZ (known as a cash secured put position). Finally, put options offer the seller the opportunity to buy shares of XYZ at the strike price of \(K-\mathcal{P}\) instead of the market price of \(S\).

Payoff, Profit, Premium and Breakeven#

The payoff per share of a put contract at expiration T days in the future \(V_{p}(K,S(T))\) is defined as:

\[V_{p}(K,S(T)) = \max\left(K - S(T),~0\right)\]

where \(K\) denotes the strike price and \(S(T)\) represents the share price of the underlying asset T days in the future (at expiration). The right (but not the obligation) to sell shares at the strike K is not free. The contract seller charges the contract buyer a premium for each put contract \(\mathcal{P}_{p}(\dots)\). From the perspective of the buyer, the profit per share for the put contract \(P_{p}\) is the payoff of the contract minus the cost of the contract:

\[P_{p}(K,S(T)) = {V}_{p}(K,S(T)) - \mathcal{P}_{p}(K,S(0))\]

Thus, from the buyer’s perspective, the share price must fall below the strike price to make up the amount paid for the contract. This is called the breakeven price \(\mathcal{B}_{p}(K, S(T))\):

\[\mathcal{B}_{p}(K,S(0)) = K - \mathcal{P}_{p}(K,S(0))\]

Finally, the premium (cost) for each put contract \(\mathcal{P}_{p}(\dots)\) is defined by the expression:

\[\mathcal{P}_{p}(K,S(0))\geq\mathbb{E}\Bigl(\mathcal{D}^{-1}_{T,0}(\bar{r})\cdot{V_{p}}(K,S(T))\Bigr)\]

where \(\mathcal{D}_{T,0}(\bar{r})\) denotes the discount rate between the time when the contract was purchased 0 and contract expiration T days in the future. Option contracts use risk-neutral pricing; thus, the discount rate \(\bar{r}\) is typically taken as the interest rate on 10-year treasury notes.

Put-Call Parity#

Fill me in.

Summary#

In this lecture, we will focused exclusively on options, a type of derivative that use equity, i.e., shares of a stock or an exchange-traded fund, as their underlying asset.

  • Call contracts give the option buyer the right (but not the obligation) to buy an underlying asset, e.g., shares of stock at a predetermined price (called the strike price) within a specified time frame. The option buyer pays the option seller a premium for the right to buy the underlying asset.

  • Put contracts give the option buyer the right (but not the obligation) to sell an underlying asset, e.g., shares of stock at a predetermined price (called the strike price) within a specified time frame. The option buyer pays the option seller a premium for the right to sell the underlying asset.

  • Put-Call parity is a relationship between the prices of a European call option and a European put option with the same strike price and expiration date. The put-call parity relationship is important because it allows us to price one option in terms of the other.