How is an option priced and what does it mean? Patrick Ceresna, CMT Big Picture Trading Inc.

How is an option priced and what does it mean? Patrick Ceresna, CMT Big Picture Trading Inc.

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02 Outline Black-Scholes Pricing Model The Greeks Delta Gamma Theta Vega Rho Capturing Vega Capturing Theta

03 Pricing Models Option pricing models Mathematical formulas Remove subjective decisions from pricing process Assume stock prices in marketplace are random Input Quantifiable factors that create an option s price Output Theoretical values for call and put options Greeks - price sensitivities to changing factors Delta Δ Gamma Γ Theta Θ Vega Κ Rho Ρ

04 Option Pricing Factors Quantifiable Underlying stock price Strike price Volatility Time until expiration Risk free interest rate (e.g., T-bill) Dividends Input for any option pricing model

05 The Black-Scholes Pricing Model Black-Scholes (1973) Earliest and most widely known European options No dividends Various other pricing models exist Extensions of Black-Scholes we will discuss Cox-Ross-Rubenstein (1979) Binomial model American-style options (regular equity contracts) Accounts for early exercise and dividends

06 Assumptions of Black-Scholes European-style contracts Volatility constant Short-term risk free interest rate constant Lognormal distribution of returns No dividends paid No commissions or transaction costs Efficient markets Direction of market or individual stocks not consistently predictable

07 Options in the Marketplace Who makes options prices? All market participants (buyers and sellers) Individual and institutional investors Professional market-makers Best bid/ask is consensus of all bids and offers What is an option ultimately worth? What the market is willing to pay Pricing models used as guideline Supply/demand and market dynamics override theoretical values

08 The Five Greeks Delta Γ Gamma Θ Theta Κ Vega Ρ Rho Expected change in option value with changing underlying stock price Expected change in option delta with changing underlying stock price Expected change in option value with passage of time (time decay) Expected change in option value with changing implied volatility Expected change in option value with changing risk-free interest rate

09 Nature of the Greeks Meaningful only during option s lifetime At expiration they are moot Impact of any Greek is on option s time value An expiring option is worth only intrinsic value (if any) Greeks may affect each other e.g., change in an options theta (time decay) may affect its delta Impact of Greeks differ for each option contract In-the-money vs. at-the-money vs. out-of-the-money Near-term vs. far-term

10 Individual Investors With respect to each of the Greeks More important to understand nature of sensitivities being measured Closer look at each will help Value of Greeks Understanding where your risk can come from Balancing risk vs. reward before positions established Setting your expectations Reducing surprises in option price behavior

12 Delta Delta: Value s sensitivity to stock price Expected percentage change in option value With a short-term $1.00 change up or down in underlying All other pricing factors constant In either decimal form (.50) or whole number (50) Both mean 50% Deltas always range from 0 to 100 or 0 to 100% Each underlying share has a delta of 1.00

13 Delta Characteristics Calls have positive (long) deltas Positive correlation to underlying stock price change Stock price Call price Stock price Call price Call deltas range from 0 to +1.00 Puts have negative (short) deltas Negative correlation to underlying stock price change Stock price Put price Stock price Put price Put deltas range from 0 to 1.00

14 Call Delta Examples On a given day an XYZ call has value $3.50 Delta of.50 (50%) XYZ stock is up $1.00 Call will theoretically increase by 50% of stock move $1.00 x.50 = $0.50 Expected call value = $3.50 + $0.50 = $4.00 XYZ stock is up $0.60 Call will theoretically increase by $0.60 x.50 = $0.30 Expected call value = $3.50 + $0.30 = $3.80

15 Call Delta Examples On a given day an XYZ call has value $2.75 Delta of.40 (40%) XYZ stock is down $1.00 Call will theoretically decrease by 40% of stock move $1.00 x.40 = $0.40 Expected call value = $2.75 $0.40 = $2.35 XYZ stock is down $0.50 Call will theoretically decrease by $0.50 x.40 = $0.20 Expected call value = $2.75 $0.20 = $2.55

16 Put Delta Examples On a given day an XYZ put has value $4.50 Delta of.30 (30%) XYZ stock is up $1.00 Put will theoretically decrease by 30% of stock move $1.00 x.30 = $0.30 Expected put value = $4.50 $0.30 = $4.20 XYZ stock is up $0.30 put will theoretically decrease by $0.30 x.30 = $0.09 expected put value = $4.50 $0.09 = $4.41

17 Put Delta Examples On a given day an XYZ put has value $2.90 Delta of.45 (45%) XYZ stock is down $1.00 Put will theoretically increase by 45% of stock move $1.00 x.45 = $0.45 Expected put value = $2.90 + $0.45 = $3.35 XYZ stock is down $0.60 Put will theoretically increase by $0.60 x.45 = $0.27 Expected put value = $2.90 + $0.27 = $3.17

18 Deltas Are Not Fixed At-the-money options deltas are generally around.50 In-the-money calls Deltas 1.00 as calls become deep in-the-money When delta = 1.00 become substitute for long stock In-the-money puts Deltas 1.00 as puts become deep in-the-money When delta = 1.00 become substitute for short stock Out-of-the-money options Deltas approach 0 when far out-of-the-money Rate at which delta changes is not constant

23 Delta vs. Time Over time In-the-money options become more sensitive to underlying stock price changes Out-of-the-money options become less sensitive to stock price changes Calls Puts Strike 18 Month 6 Month 3 Month 1 Month 75.00.38.19.08.01 70.00.45.29.18.04 65.00.53.42.33.19 60.00.61.56.54.52 55.00.69.71.76.86 50.00.78.84.91.99 45.00.85.93.98 1.00 75.00.66.82.92.99 70.00.58.72.82.96 65.00.49.59.67.81 60.00.40.44.46.48 55.00.32.29.24.14 50.00.23.15.09.01 45.00.15.07.02 0 XYZ $60.00 30% vol. 2% int.

25 Delta as Share Equivalence Another way of using delta is in determining an option position s theoretical underlying share equivalence A single option position or entire option portfolio The calculation: # of options x delta amount x 100 shares (unit of trade) The option position can be expected to perform financially like this equivalent number of underlying shares

26 Call Share Equivalence Example Call position: 1 long contract with delta of.40 Share equivalence = 1 x.40 x 100 = 40 shares Call position should perform like 40 long shares Stock up $1.00 Call up $1.00 x.40 x 100 = $40.00 40 shares up $40.00 Stock down $1.50 Call down $1.50 x.40 x 100 = $60.00 40 shares down $60.00

27 Call Delta as Hedge Ratio Practical application of share equivalence Hedge ratio or Delta hedge To neutralize market risk of a position in that option Call position: 1 short contract that has delta of.45 Share equivalence = 1 x.45 x 100 shares = 45 shares Hedged position: 1 short call hedged by 45 long shares Stock up $1.00 Short call up = $1.00 x.45 x 100 = $45.00 loss 45 long shares = $45.00 profit Net profit/loss = 0

28 Netting Position Deltas Deltas from a portfolio of mixed options on same underlying stock Share equivalences are calculated and netted Measures market risk of portfolio in terms of stock Position deltas used for netting depend whether investor is long or short each contract Long call = Positive delta Short call = Negative delta Long put = Negative delta Short put = Positive delta Calculation for each option series # options x delta x 100 shares

29 Netting Position Delta Example Position Delta Position Delta Total Position Delta Long 40 Nov 55.00 calls.69 +.69 +2,760 Short 45 Nov 60.00 calls.57.57 2,565 Long 60 Jan 55.00 puts.29.29 1,740 Long 60 Jan 60.00 calls.55 +.55 +3,300 Short 2,100 XYZ shares 1.00 1.00 2,100 Net position deltas 345 This portfolio should perform like a position of short 345 XYZ shares Position Delta If Net Positive Value Long shares of stock If Net Negative Value Short shares of stock

30 Gamma Γ Gamma: Delta s sensitivity to stock price Expected percentage change in delta s value With a short-term $1.00 change In underlying stock price up or down All other pricing factors constant In decimal form (e.g.,.02) Adjustment to delta Only options have gamma Stock does not

31 Gamma Characteristics Gamma amount is same for calls and puts Gamma for calls Stock price Delta by gamma amount Stock price Delta by gamma amount Gamma for puts Stock price Delta by gamma amount Stock price Delta by gamma amount Gamma is what option buyers are paying for Acceleration of delta Delta of the delta

32 Call Gamma Examples For purpose of adjusting delta amounts round gamma to two decimal places A call has delta of.54 and gamma of.042 (.04) Stock is up $1.00 Delta will become more positive by gamma amount New delta value:.58 Another call has delta of.75 and gamma of.034 (.03) Stock is down $1.00 Delta will become less positive by gamma amount New delta value:.72

33 Put Gamma Examples For purpose of adjusting delta amounts round gamma to two decimal places A put has delta of.54 and gamma of.020 (.02) Stock is up $1.00 Delta will become less negative by gamma amount New delta value:.52 Another put has delta of.27 and gamma of.067 (.07) Stock is down $1.00 Delta will become more negative by gamma amount New delta value:.34

34 Gamma Is Not Constant Gamma Delta.050.040.030.020.010 Gamma vs. Stock Price.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 90-day option 1.00.80.60.40.20 Stock Price Call Delta vs. Stock Price.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 Stock Price Option gamma (call or put) is greatest when stock is at the strike price Option is at-the-money As stock moves up or down (in-the-money or out-of-the-money): Gamma will decrease Delta will change at a decreasing rate 90 days 30% vol. 2% int.

35 Gamma Is Not Constant Gamma.050.040.030.020 Gamma vs. Stock Price Gamma of both deep in-the-money and far out-of-the-money options decrease close to zero.010.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 90-day option 1.00.80 Stock Price Call Delta vs. Stock Price Delta.60.40.20.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 Stock Price 90 days 30% vol. 2% int.

36 Gamma s Greatest Impact Gamma.120.100.080.060.040.020 Gamma vs. Stock Price.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 2-week option 1.00 Stock Price XYZ 60 Call Gamma effect greatest on at-the-money options, close to expiration They can move in-themoney to out-of-themoney very quickly Their deltas change the quickest Delta.80.60.40.20.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 Stock Price 14 days 30% vol. 2% int.

37 Gamma vs. Time Gamma vs. Time Gamma.080.060.040 1 month 3 month 6 month 18 month.020.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 Strike Price XYZ $60.00 30% vol. 2% int. As expiration nears Gamma of at-the-money calls and puts increases Gammas of both in-the-money and out-of-the-money calls and puts decreases

38 Gamma vs. Volatility Gamma vs. Volatility Gamma.080.060.040 40% volatility 60% volatility.020.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 Strike Price XYZ $60.00 90 days 2% int. As volatility decreases Gamma of at-the-money calls and puts increases Gammas of both in-the-money and out-of-the-money calls and puts decreases

39 Netting Position Gamma Gammas from a portfolio of mixed options on same underlying stock may be netted May offset or augment an existing position delta Gamma amounts used depends whether investor is long or short each contract All long contracts have positive gamma All short contracts have negative gamma The calculation for each option series # of options x gamma amount x 100 shares

40 Netting Position Gamma Position Gamma Position Gamma Total Position Gamma Long 40 Nov 55.00 calls.089 +.089 +356 Short 45 Nov 60.00 calls.073.073 328.50 Long 60 Jan 55.00 puts.059 +.059 +354 Net XYZ position gamma +381.50 Position Gamma If Net Positive Value Net long option position If Net Negative Value Net short option position A portfolio with net long position gamma Gets longer to the upside and shorter to the downside The added performance that enhances profits Net short position gamma performs in the opposite manner Gets shorter to the upside and longer to downside Very risky scenario

41 Theta Θ Theta: Option value s sensitivity to time Expected time decay in option value With the passage of 1 day Expressed in decimal form (.080) Represents cash amount per share All other pricing factors constant Calls and puts both have negative theta amounts

42 More on Theta Theta may be measured for periods longer than 1 day e.g., 2-day theta or 10-day theta Decay is per calendar day, not per trading day Theta represents rent buyers pay and writers receive Buyers want time for favorable move in stock price Writers sell that time Buyers hope delta gains will offset theta loss Only time value decays At expiration option worth only intrinsic value (if any)

43 Theta Calculation An option is trading today at $3.50 Theta of $.030 ( $.03) Contract is worth $3.50 x 100 shares = $350.00 Option s expected value tomorrow = $3.50 $.03 = $3.47 Contract is worth $3.47 x 100 shares = $347.00 Theta $.03 $3.00 loss per contract Assuming other pricing factors constant

45 Theta vs. Volatility Theta vs. Volatility Theta.040.030.020 Call - 40% vol. Put - 40% vol. Call - 20% vol. Put - 20% vol..010.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 Strike Price Implied volatility impacts theta amounts With rising volatility Theta increases With decreasing volatility Theta decreases XYZ $60.00 90 days 2% int. Higher volatility more time value more to decay by expiration

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47 Netting Position Theta Thetas from a portfolio of mixed options on same underlying stock may be netted Theta amounts used depends whether investor is long or short each contract All long contracts have negative theta All short contracts have positive theta The calculation for each option series # of options x theta amount x 100 shares

48 Netting Position Theta Position Theta Position Theta Total Position Theta Long 40 Nov 55.00 calls.0207.0207 82.80 Short 45 Nov 60.00 calls.0144 +.0144 +64.80 Long 60 Jan 55.00 puts.0125.0125 75.00 Net position theta 93.00 At close of business today, tomorrow this position could expect to see a loss of $93.00 from time decay Position Theta If Net Positive Value Gain from time decay If Net Negative Value Loss from time decay A portfolio with net positive position theta can expect a profit in 1 day of net theta amount A portfolio with net negative position theta can expect a loss in 1 day of net theta amount

49 Vega Κ Vega: Option value s sensitivity to volatility Expected change in option value With a 1%-point change in implied volatility up or down Expressed in decimal form (.080) Represents cash amount per share All other pricing factors constant Calls and puts both have positive vega amounts implied option value by vega amount implied option value by vega amount

50 Vega Characteristics Also known as kappa (Greek letter) thus symbol K Volatility considered most influential price factor Great impact in dollars on option values possible Changes often occur intra-day Changes may be abrupt and significant Increase in implied or vega does not require change in stock price

51 Vega Examples Vega impact greatest In dollar amounts on at-the-money contracts In percentage terms on out-of-the-money contracts Option today valued at $4.50 with vega +.030 (.03) volatility 1 pt. option price $4.50 +.03 = $4.53 volatility 1 pt. option price $4.50.03 = $4.47 Option today valued at $2.75 with vega +.017 (.02) volatility 1 pt. option price $2.75 +.02 = $2.77 volatility 1 pt. option price $2.75.02 = $2.73

52 Vega vs. Time Vega.300.250.200.150.100.050 Vega vs. Time.000 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 18 month 6 month 3 month 1 month Strike Price 80.00 More time until expiration Greater the vega Vega size related to amount of time value More time value greater the vega Long term options vega may be significantly higher XYZ $60.00 30% vol. 2% int.

53 Netting Position Vega Vegas from a portfolio of mixed options on same underlying stock may be netted Vega amounts used depends whether investor is long or short each contract All long contracts have positive vega All short contracts have negative vega The calculation for each option series # of options x vega amount x 100 shares

54 Netting Position Vega Position Vega Position Vega Total Position Vega Long 40 Nov 55.00 calls +.068 +.068 +272.00 Short 45 Nov 60.00 calls.052.052 234.00 Long 60 Jan 55.00 puts +.118 +.118 +708.00 Net position vega +746.00 If today the implied volatility for XYZ options increased 1 point this portfolio could expect to see profit of $746.00 If the implied volatility decreased 1 point expect to see loss of $746.00 Position Vega If Net Positive Value Gain from implied increase If Net Negative Value Gain from implied decrease A portfolio with net positive position vega (long vega) should expect net vega amount profit with 1 pt. implied increase A portfolio with net negative position vega (short vega) expect net vega amount profit with 1 pt. implied decrease

55 Rho Ρ Rho: Option value s sensitivity to interest rate Expected change in option value With a 1% point change in risk-free interest rate up or down Expressed in decimal form (.080) Represents cash amount per share All other pricing factors constant Considered the least significant of all pricing factors Component of cost of carry Small portion of any option s total premium

56 Rho Characteristics Rho amounts generated by pricing model Calls have positive rho Puts have negative rho Rho is largest for in-the-money calls and puts Decreases as options move out-of-the-money Rho increases with higher priced underlying stocks Rho increases with more time until expiration For shorter-term options little impact For longer-term options more significant

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