Buy a Call = Pay a fee of $c to obtain the right to purchase the underlying asset at $X on or before the expiration date. Buy a Put = Pay a fee of $p to obtain the right to sell the underlying asset at $X on or before the expiration date.

Write a Call = Accept a fee of $c in exchange for the obligation to supply the underlying asset on demand at $X on or before the expiration date. Write a Put = Accept a fee of $p in exchange for the obligation to purchase the underlying asset on demand at $X on or before the expiration date.

Closing contract = To close a buy, write an identical contract; to close a write, buy an identical contract; normally not at the same price but at a profit or loss. Closings constitute the majority of trading activity at the options exchange. Such trades are carried out by speculators and arbitrageurs who provide depth and liquidity to those that will ultimately buy or sell the underlying asset.

Option exchanges = CBOE (the Chicago Board of Options Exchange), CME (the Chicago Mercantile Exchange), PSE (Pacific Stock Exchange), Amex (American Stock Exchange). The CME deals only in options on SP500 index futures. The other three exchanges trade in stock options as well as index futures and options.

Expiration dates = Normally the third Friday of each month with a time horizon of three months. 'Long term' options are traded on some exchanges on longer time horizons but these are not traded in large volumes.

Exercise price X = $2.5 increments for stocks selling under $10, $5 increments for stocks selling under $50, $10 increments otherwise.

Standard deviation s = The measure of risk or volatility of the value of the underlying asset. Options derive their value from volatility or uncertainty about future prices. If there were no uncertainty, options would be worthless. This is because an option trade is actually a trade in risk. The writer of the contract assumes some of the risk of price volatility and is compensated for doing so by $p or $c as the case may be; and takes on the obligation of the promised transaction at $X if the buyer decides to to so. The buyer of the contract pays the writer $p or $c as the case may be to be protected against price volatility and be guaranteed the price of $X. The more volatile the value of the underlying asset the greater is the value of the option written against it. This is because the downside risk of the buyer is limited to the price of the option but the upside 'risk' increases without a ceiling as the volatility increases.

Option valuation: $c and $p The question is therefore what is the appropriate amount of compensation for the risk being transacted in a free and efficient capital market? The valuation can be derived algebraically by forming a riskless hedge and then setting the rate of return of the riskless hedge to the default free rate of government bonds.

Call price $c A riskless portfolio can be formed with a long position in the stock and a short position in the corresponding call. Limiting the rate of return on this portfolio to krf results in the so-called Black Scholes option model presented in the spreadsheet model in the accompanying handout. The algebraic derivation assumes normally distributed stock prices, frictionless markets, and the ability to instantaneously (and costlessly) re-adjust the riskless hedge as prices change.

Put price $p If you buy a stock and a put and write a call, you will also obtain a riskless hedge. If you set the returns of this portfolio to krf you can derive Stoll's put-call parity equation which states, c - p = P - Xe-rt where r represents the risk free rate krf. Using Stoll's put-call parity, we can extend the Black Scholes option model to puts since if you know the price of a call, you can always compute the price of the corresponding put using the relationship above.

Hedge Portfolios Options are not normally held in isolation but in combinations that protect the trader and offer profits either in bull markets (bull hedge), bear markets (bear hedge) or in volatile markets (spread hedge). The most common strategies consist of Bull Spreads (buy a put and a call with the X of the put less than X of the call), Bear Spreads (buy a put and a call with the X of the put more than X of the call), Straddles (buy a put and a call at the same X), Straps (two calls and one put), Strips (two puts and one call, and Butterfly Spreads (combine a bull spread and a bear spread). The spreadsheet model provided in class may be used to graph and evaluate all hedging strategies and to check their sensitivity to the valuation parameters.

Imputed Variance and Arbitrage For any quoted trading price on the stock and its options, one can use the models above to compute the stock price variance that corresponds to the option prices being quoted. This variance is called the 'imputed' or the 'implied' price volatility. Arbitrageurs use discrepancies in implied volatility between expiration dates and between exchanges to formulate arbitrage trades.

Some option pricing links

• monte carlo option pricing
• option pricing terminology in 4 languages
• jamal's monte carlo model of options