Unformatted text preview: FINC 782: Energy Markets Portfolio Analysis AB Freeman School of Business, Tulane University, New Orleans, Fall 2007 Leslie McNew Email at [email protected] 865-5036 Options, Exotic Options, Model Risk, Pricing 1 What is an Option? An OPTION gives the PURCHASER the RIGHT, but not the obligation, to buy or sell the underlying commodity at a certain price (the exercise or STRIKE PRICE) on or before an agreed date. An OPTION is a DERIVATIVE 2 What is an Option? An OPTION is a contract whereby one party has the right to complete a transaction in the future (with a previously agreed upon amount, date and price) if he/she chooses, is not obligated to do so. Holder Seller For the right to decide he pays a PREMIUM
3 What is an Option? An OPTION is a form of INSURANCE for the hedger: it is not called upon is circumstances are satisfactory. For example, cheaper market prices in the future or no hurricane event risk. Holder PREMIUM as Insurance 4 5 Factors Affect the Price of the Option There are 5 factors affecting the price of the option: 1. Current underlying price (U) 2. Strike price of the option (S) 3. Time until expiration of option (t) 4. Volatility of the underlying price (v) 5. Risk-free interest rate (r)
C = e-rt [UetN(d1) SN(d2)] d1 = [ ln (Uet/S) + 1/2v2t] / vt1/2 d2 = d1 - v? t
C P r t U S v N(x) N'(x) call premium put premium short-term interest rate time to expiration in years underlying commodity price strike price implied volatility standard normal distribution function of x (1/[2 ] ^ ) x e (x2/2) 5 Definitions of the 5 Factors Underlying: Strike (Exercise): the actual commodity price the price at which the option holder may buy or sell the underlying commodity Time Until Expiration: the length of time until the option expires Volatility: Short-Rate: the market's price range and movement within the time until expiration period the short-term interest rate 6 Basic Option Model To calculate the premium
Underlying Price Strike Price Time Until Expiration Market Volatility Short-term Interest Rate Option Pricing Model PREMIUM To calculate the implied volatility
Underlying Price Strike Price Time Until Expiration Implied Volatility Short-term Interest Rate Option Pricing Model PREMIUM
7 Option Building Blocks Call: an option which gives the buyer (holder) the right, but not the obligation, to BUY the underlying for a specific price within a specified period of time in exchange for a one-time payment - THE PREMIUM $
Call Option: Holder Premium paid 25 20 15 10 5 0 -5 -10 Upside Potential 8 Option Building Blocks Put: an option which gives the buyer (holder) the right, but not the obligation, to SELL the underlying for a specific price within a specified period of time in exchange for a one-time payment - THE PREMIUM $
Put Option: Holder
25 20 15 10 5 0 -5 -10 Upside Potential Premium paid 9 Option Descriptions In-the-money: Out-of-the-money: At-the-money:
PARITY VALUE (Intrinsic Value) option yields a positive value option has no value exercise for commodity (exchange
At the money Out of the money In the money UNDERLYING PRICE This is a call option
10 Option Descriptions PARITY VALUE (Intrinsic Value) At the money In the money Out of the money UNDERLYING PRICE This is a put option
11 Call Example Buy a call for $100. Three months from now there a 3 possible choices: $105 Buy call for $100 $100 $95 In-the-money, sell the call back and make profit At-the-money, receive the commodity, but loss of initial premium (cost of insurance) Out-of-the-money, option is worthless ($0), but have paid the price of premium 12 Intrinsic vs. Time Value CALL VALUE Time value Profit Intrinsic value in the money amount Loss Intrinsic value = 0 Time value at maximum at the strike price UNDERLYING PRICE Time Value Intrinsic Value Out of the money Time Value At the money Time Value In the money 13 Intrinsic vs. Time Value Methodology: Traditional Now NPV analysis intrinsic and extrinsic option valuation
The net benefit to be derived from time values for volatility values. ATM options have larger extrinsic values than ITM or OTM options. Extrinsic values are larger when volatility is higher. Extrinsic values are larger when the time until expiration is larger. The net benefit to be derived from exercising an option immediately (the amount by which the option is in-themoney) 14 EXTRINSIC INTRINSIC What are Option Greeks? OPTION GREEKS are TOOLS to measure the RISKS of options. They answer the question: if THIS happens, what will happen to OPTION VALUE? Market Change Option Price Change 15 What are Option Greeks? The GREEKS predict how option value will change when the FACTORS change: Underlying Price Exercise Price (Strike Price) Time until expiration Volatility Short-term interest rates Theta Rho Delta Gamma Vega (Kappa, Tau)
16 What are Option Greeks? DEFINITIONS:
Delta: the rate of change of the option price with respect to the price of the underlying Gamma: the rate of change of the option's delta with respect to the price of the underlying Theta: the rate of change of the option price with respect to time Vega (Kappa/Tau): the rate of change of the option price with respect to the change in implied volatility in the underlying Rho: the rate of change of the option price with respect to a change in interest rates FACTOR CHANGE OPTION PRICE CHANGE $$
17 Delta Delta: the rate of change of the option price with respect to the price of the underlying DELTA =
Option Price Change in Option Price Change in Underlying Price Underlying Delta is not linear 18 Behavior of Delta Delta: the rate of change of the option price with respect to the price of the underlying Delta moves toward 0 as option moves out-of-the-money. Delta moves toward 1 as option moves in-the-money.
PARITY VALUE (Intrinsic Value) At the money Out of the money In the money Delta = 0 Delta = 1
UNDERLYING PRICE Long Call 19 Behavior of Delta Delta: the rate of change of the option price with respect to the price of the underlying Delta is POSITIVE for LONG CALLS and SHORT PUTS. Delta is NEGATIVE for SHORT CALLS and LONG PUTS. Delta Long Calls Short Calls Long Puts Short Puts + +
20 Delta Predicts Change in Option Value Delta: the rate of change of the option price with respect to the price of the underlying OPTION PRICE moves with UNDERLYING PRICE. Example: Delta = 0.6 to $2.60.
$101 Underlying $2.60 Option $100 Underlying $2.00 Option Option Price = $2 Underlying Price = $100 If the underlying moves $1 to $101, the option should move $0.60 21 Delta Neutral DELTA HEDGING is making a neutral or zero delta.
An option is delta-hedged when a position has been taken in the underlying which matches its delta Example: Delta = 0.6 Long Call Position Option Price = $2 Underlying Price = $100 100 calls: Delta = 0.6 (100) = 60 Hedge by selling 60 underlying. Delta = 60 + (-60) = 0 Dynamic Hedge -- must be adjusted over time. Delta is not linear
22 Gamma -- How Fast Does the Delta Move? Gamma: the rate of change of the option's delta with respect to the price of the underlying GAMMA = Change in Option Delta Change in Underlying Price Option Price Underlying 23 Gamma -- How Fast Does the Delta Move? Gamma: the rate of change of the option's delta with respect to the price of the underlying Gamma is high when option is at-the-money. Gamma is low when option is far in-the-money or out-of-the-money.
PARITY VALUE (Intrinsic Value) At the High Gamma money Out of the money In the money Low Gamma Low Gamma Long Call UNDERLYING PRICE 24 Gamma Neutral GAMMA HEDGING is making a neutral or zero gamma. Delta Neutral: use futures, forwards and swaps. HOWEVER: GAMMA is created only by OPTIONS. GAMMA can only be hedged with OPTIONS. 25 Gamma Neutral GAMMA HEDGING is making a neutral or zero gamma.
Example: Delta Neutral/Gamma Neutral Trade a. Looking at a call with a delta = 0.63, gamma = 1.5 b. Our portfolio has an existing gamma = -4,000 1. Set portfolio gamma neutral first: (-1 x portfolio gamma / option gamma) = 2,667 buy a long position of 2,667 calls 2. Adjust Delta of portfolio to be neutral including the new acquisition (2.667 x option delta) = (2.667 x .63) = 1.68 x 1,000 = 1,680 sell 1,680 units of underlying to keep delta neutrality The higher the gamma, the more DYNAMIC the hedging. 26 Theta -- Time Decay Theta: the rate of change of the option price with respect to time THETA = Change in Option Price Change in Time to Expiration (1 day) Theta is loss of time value of an option. Time value is highest when an option is first sold, and decreases as the option gets closer to Time Value expiration.
T0 Expiration Time Decay is like paying a premium for gamma. 27 Theta -- Time Decay Theta: the rate of change of the option price with respect to time Theta is low when there is a long time to expiration (decays slowly). Theta is high when there is a short time to expiration (decays quickly).
T0 Lower Gamma, rate of decay slower T90 T0 T30 Higher Gamma, rate of decay faster
28 Vega (a.k.a. Kappa, Tau) Vega: the rate of change of the option price with respect to the change in underlying volatility VEGA = Change in Option Price Change in Underlying Volatility If Vega is high, the option is very sensitive to changes in volatility. If Vega is low, changing volatility has little effect on option price. Making a portfolio delta, gamma, and vega neutral at the same time is hypothetically possible but practically improbable
29 Rho Rho: the rate of change of the option price with respect to a change in interest rates RHO = Change in Option Price Change in Interest Rates Interest rates have only a minor effect on option prices. Rho is only significant for very long-term portfolios A Rho Neutral portfolio is not usually needed or desired.
30 Portfolio Immunization Immunized Portfolio: Hypothetically will not lose money when price, volatility, or time to expiration changes To immunize a portfolio as much as possible, make it delta/gamma/ vega neutral. Gamma, vega, and theta can only be hedged with options. Delta can be hedged with underlying. The portfolio must be constantly and carefully "rebalanced" as the factors change. This is DYNAMIC HEDGING. Dynamic hedging is a delicate art.
31 Exotic Options: Modified Payoffs Major Modifications to the Classical Payoff Definition: 1. Change in the underlying price definition to include price trajectory during life of option 2. The underlying is two or more commodities 3. The underlying is itself an option 4. The payoff is event driven (event defined in terms of prices, usually one or more underlying) Underlying Price Strike Price Time Until Expiration Market Volatility Short-term Interest Rate Option Pricing Model PREMIUM
32 Exotic Options: Path Dependency Path Dependency. The option payoff depends not only on the price of the underlying at exercise (which is simultaneous with expiration for a European option) but also on the price trajectory during the entire life of the option (or some part of it). Examples include: Asian Options. The payoff is defined as the average price of the underlying instrument calculated over a specified period of time. Average Strike Options. The call payoff is equal to the price at the horizon less the average price over a certain period of time. Look-back Options. These options offer the opportunity to obtain the best price that occurs during the life of the option. Barrier Options. Options which are extinguished or activated contingent on the occurrence of a certain event defined in terms of the price of the underlying or defined in terms of the price of an entirely different asset. Example: an option on natural gas may have a barrier defined in terms of the price of residual fuel oil. Option Type Traditional Asian Look-back Barrier Option Underlying Price Underlying Price of Instrument (U) Average of U over a specified time period (t) Best price of U over life of option U of a different commodity 33 Exotic Options: Multiple-Commodity Multiple-commodity. These options have payoffs that depend on the prices of two or more commodities. Most common are the dual-commodity options. Examples include: Spread Options. The payoff of the option depends on the difference (spread) between two prices. Basket Options. One or more of the underlying assets that determine the payoff of the option comprise(s) a basket of commodities. Exchange Options. The payoff is the better of the two assets (exchange one for the other). Option Type Traditional Spread Basket Exchange Option Underlying Price Underlying Price of Instrument (U) Underlying is spread of two prices Underlying is a basket of assets not just one Payoff dependent on better of two underlying assets 34 Exotic Options: Underlying is an option Compound. Options in which the underlying is itself an option Option Type Traditional Compound Option Underlying Price Underlying Price of Instrument (U) Not an underlying price but an option 35 Exotic Options: Event Driven Digital / Binary. Options with predetermined payoffs that depend on the occurrence of events usually defined in terms of one or more prices. Option Type Traditional Ditigal Option Underlying Price Underlying Price of Instrument (U) Payoff Determined by an event (in terms of price(s)) 36 Model Risk Underlying Price Strike Price Time Until Expiration Market Volatility Short-term Interest Rate Option Pricing Model PREMIUM Model Risk: is the risk of financial loss in earnings, cash flow, or fair values that arise due to inaccurate or incomplete characterizations of a transaction or portfolio value because fundamental deficiencies in the model(s) being used to perform portfolio valuations, forecast load and system and/or market prices, etc. 37 Model Risk Mitigation Use an independent source to verify and/or validate, to review back-testing results and documentation. Provide a complete document on mathematical model and philosophy: include constraints. 1. 2. 3. 4. 5. 6. 7. 8. Independently verify and/or validate the mathematical models and philosophies Access adequacy of development environment Access selection of programming language (higher-order language needed) State clearly the requirements of the model Consider rapid prototyping Baseline if possible the software approach SOFTWARE TRANSLATION RISK Software has its own peculiarities. Software has a tendency to change dramatically during the development cycle when compared to hardware. Use steps 2 8 to reduce the risk of GIGO. Clearly state the testing philosophy (involve many different kind of `experts') Clearly state the development philosophy Final Step: validate whole process to see that it is what you expected 38 Market Pricing Vs. Model Pricing C = e-rt [UetN(d1) SN(d2)] d1 = [ ln (Uet/S) + 1/2v2t] / vt1/2 d2 = d1 - v? t Value of a Call Option C P r t U S v N(x) N'(x) call premium put premium short-term interest rate time to expiration in years underlying commodity price strike price implied volatility standard normal distribution function of x (1/[2 ] ^ ) x e (x2/2) " Are you buying them or selling them ?? Don't Care About Equations --- Make me a BID !! Note the difference between option VALUATION and option PRICING
39 Then we can talk about what a call option is really worth" Market Pricing Vs. Model Pricing VALUATION : Option Models (consistent with price of underlying) PRICING : The Real World The Trader's World of No Assumptions
no dynamic riskless arbitrage strategy (expectations, risk aversion, market imperfections) no finding the value of the option as a component of the arbitrage portfolio arbitrage strategies are neither riskless or costless in real life positions cannot be rebalanced when market is closed transaction costs model depends on underlying volatility (in itself an estimate) PRICING IS DETERMINED BY SUPPLY AND DEMAND
Call Option Demand offer participation in upside movement Call Option Supply generation of income when market is not volatile Use of Models By Traders:
Use model as a "rule of thumb" rather than an exact Review implied volatility that sets option equal to market price Calculate the option delta (hedge ratio) use models for hedging not pricing
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