Derivatives

Basic Derivatives Forwards Futures Options Swaps Underlying Assets Interest rate based Equity based Foreign exchange Commodities A derivative is a financial security with value based on or derived from an underlying financial asset The usage is often risk management

Forwards Derivative contract to receive or deliver an underlying asset at a particular price, quality, quantity, and place at a future date – Long: Obligation to buy and take delivery of an asset for $K at time T – Short: Obligation to sell and deliver an asset for $K at time T – ISDA Definition ISDA Definition Often a risk management contract – The counterparty may be hedging risk also, may be speculating, may be an arbitrageur or may be a dealer that ‘lays off’ the risk in its net position

Forwards

Forward Price Over the counter market e.g., banks Arbitrage pricing – Only risk free returns without taking risk – Forward prices are not based on ‘forecasted’ prices Example: spot price of gold is $1200/oz, interest rate on money is 4%, storage cost of gold is.5%, and gold lease rate is.125%. storage cost of gold gold lease rate

Arbitrage Pricing Say a dealer offers a gold forward (bid and offer) price, F, of $1300, to be settled in gold 1 year from now At time 0 Short a forward contract with forward price $1300 Borrow $1200 at 4% for a year Buy gold at $1200 spot Store the Flat position: You have obligations, but have not used any of your funds At 1 year Pay loan and interest $1200(1+.04) Pay storage fee $1200(1+.005) Deliver the stored gold and receive $1300 Arbitrage profit of $46.00

Arbitrage Pricing Say a dealer offers a gold forward (bid and offer) price, F, of $1150 At time 0 ‘Go long’ (buy) a forward contract with forward price $1150 Borrow (lease) gold Sell the gold in spot market for $1200 Loan the $1200 at 4% Flat position At 1 year Take delivery on gold and pay $1150 Return gold and pay lease fee Receive $1200 4% Arbitrage profit of $46.50 at no risk

Arbitrage Pricing F = S ( 1 + r T + s T) F = $1200 ( ) = $ F = S ( 1 + r T – g T) F = $1200 ( ) = $ The forward formula indicates that the $1300 contract is too expensive forward price = spot price + FV(costs) – FV(benefits) Sell contracts that are expensive and buy contracts that are cheap when characterized by arbitrage pricing. The forward formula indicates that the $1150 contract is too cheap

Futures Standardized, exchange traded ‘forward’ contracts Eliminates counterparty risk CME

The Five Pillars of Finance 10 Nobel Prize winner and former Univ. of Chicago professor, Merton Miller, published a paper called the “The History of Finance” Miller identified five “pillars on which the field of finance rests” These includeThe History of Finance 1.Miller-Modigliani Propositions Merton Miller 1990 Franco Modigliani Capital Asset Pricing Model William Sharpe Efficient Market Hypothesis Eugene Fama, Robert Shiller Modern Portfolio Theory Harry Markowitz Options Myron Scholes and Robert Merton 1997

Options – Value at Expiry 11 Long put P T = max(K – S T, 0) Long call C T = max(S T -K, 0) Short call -C T = min(K-S T, 0) Short put -P T = min(S T –K, 0)

Basic Options – Profit at Expiry 12 Long put P T = max(K – S T, 0)-P 0 Long call C T = max(S T -K, 0)-C 0 Short call C T = min(K-S T, 0)+C 0 Short put P T = min(S T –K, 0)+P 0

Options vs Forwards Forward Long – Obligation to buy and take delivery of an asset for $K at time T Short – Obligation to sell and deliver an asset for $K at time T Option Call – Long Right to buy an asset at price $K at time T – Short Obligation to sell an asset at price $K at time T Put – Long Right to sell an asset at price $K at time T – Short Obligation to buy an asset at price $K at time T 13

Price of European Call Option Price the call to create a portfolio that returns the risk free rate

Option Pricing 1 Period Binomial Lattice Method Cash flows at time T Solve for h and B Cash flows at time 0 Galitz uses the following future value factor instead

Option Pricing 1 Period Binomial Lattice Method ‘Risk neutral’ probability of move upward Present value of future expected cash flow discounted at risk free rate Recommended calculation of a call option on pages 231 to 233 of handout from Financial Engineering by Lawrence Galitz. Return rate and future value factor notation

Option Pricing 1 Period Binomial Lattice Method

Example on pages 231 to 233 of handout from Financial Engineering by Lawrence Galitz. Same as question 4 on quiz

Black Scholes Eqn & Solution European Call Options A fully hedged portfolio returns the risk free rate S: spot price of underlying asset V: value of derivative  : std deviation of underlying return rates t: continuous time r * is the expected risk-free rate of return (continuously compounded) This formula is the solution to the B-S PDE for the European call option with its initial and boundary conditions

Options vs Forwards 21

Put – Call Parity 22 Portfolio of one share of stock, S, one long put, P, one short call, C Same strike, K, and time to expiry T  T = S T + P T – C T S T ≤ K  T = S T + ( K – S T ) – 0 = K S T > K  T = S T ( S T - K ) = K  0 = K e – r T  0 = S 0 + P 0 – C 0  0 = K e – r T =S 0 + P 0 – C 0 K e – r T = S 0 + P 0 – C 0 C 0 – P 0 = S 0 - K e – r T Long Stock Short Call Long Put K

Option Value Components 23 In the moneyOut of the money Intrinsic Value Time Value At expiry Prior to expiry K StSt Value of a forward with contract price K

Call Value as Expiry Approaches

Call & Put Price Example Not on Quiz 25 Current stock price, S 0 = $40.00 Expected (continuously compounded) rate of return,  * = % Annual volatility,  = 20% Strike price, K: $45.00 Risk free (continuously compounded) rate of return, r * : 6% Time to expiry, T = 1.0 years

Option Pricing 26 If this variable increases The call priceThe put price Stock price, SIncreasesDecreases Exercise price, KDecreasesIncreases Volatility of asset,  Increases Time to expiry, T-tIncreasesEither Risk free interest rate, r IncreasesDecreases Dividend payoutDecreasesIncreases

Put – Call Parity and Forwards at Expiry 27 Long call = Long put + long forward Long forward = Long call + short put Long put = Long call + short forward Short forward = Long put + short call

Put – Call Parity and Forwards at Expiry 28 Long call = Long put + long forward Long forward = Long call + short put Long put = Long call + short forward Short forward = Long put + short call

Protective Put 29

Covered Call 30

Put – Call Parity and Forwards before Expiry 31 Long call = Long put + long forward Long forward = Long call + short put Long put = Long call + short forward Short forward = Long put + short call