EurexOption Seminar
Introduction to Theoretical Pricing Models What is the theoretical value of a roulette bet?
1, 2, 3, .….. , 34, 35, 36, 0, 00 Choose one of 38 numbers
If your number doesn’t come up you receive nothing
If your number does come up you receive $36
Expected value (expected return)
$36 / 38 ≈ 95¢ Expected value
•depends primarily on the laws of probability
•does not have to correspond to a possible outcome
•is only reliable in the long run The price for the privilege of choosing a number is $1.00
The casino has an edge of $1.00 -95¢= 5¢
The price for the privilege of choosing a number is 88¢
The player has an edge of
95¢-88¢= 7¢ Theoretical value (theoretical price, fair value, fair price):
The price you would be willing to pay now in order to just break even in the long run.
• expected value
• other considerations If your number doesn’t come up you receive receive nothing.
If your number does come up you receive $36, to be paid in two months. interest rates = 12.00% theoretical value = present value of 95¢ = 95¢/ (1 + 2/12*12%) = 95¢/ 1.02 ≈ 93¢ Theoretical edge The difference between the price of a proposition and its theoretical value $1.00 -93¢= 7¢
Positive theoretical edge
•buy at a price lower than theoretical value or •sell at a price higher than theoretical value Intelligent trading of options requires us to …..
•calculate a theoretical value
•choose an appropriate strategy
•control the risk A model is a representation of the real world
A model is unlikely to be an exact representation of the real world
A model is limited by
•the accuracy of the assumptions on which the model is based
•the accuracy of the inputs into the model Theoretical option pricing model
A mathematical model used to determine the theoretical value of an option contract under some set of assumptions about
•market conditions
• the terms of the option contract Theoretical Pricing Models exercise price time to expiration underlying price pricing theoretical model value interest rate volatility
(dividends) 90 95 100 105 110
20% 20% 20% 20% 20% long an underlying contract expected value? 20%*90 + ...... + 20%*110= 100 long a 100 call expected value? 20% * 5 + 20% * 10= 3.00 90 95 100 105 110
10% 20% 40% 20% 10% long an underlying contract expected value? 10%*90 + ...... + 10%*110= 100 long a 100 call expected value? 20% * 5 + 10% * 10= 2.00 Expected Value Exercise
124 126 128 130 132 134 136
4% 12% 19% 22% 21% 16% 6% Using the above prices and probabilities for an underlying contract, what are the expected values for the following contracts:
underlying 126 call 130 call 133 call
126 put 130 put 133 put
What do you notice about the difference between the values of calls and puts at the same exercise price? EurexOption Seminar
Understanding Volatility underlying prices
probabilities
normal distribution 100 +–.25 each day value =.05 90 days to +–1.50 each day expiration value =.75 120 call +–10.00 each day option value value = 8.00 standard deviation (σ) mean (μ)–where –how fast the curve the peak of the spreads out. curve is located
All normal distributions are defined by their mean and their standard deviation. mean
half of the half of the distribution distribution is to the left is to the right of the mean of the mean low standard deviation high standard deviation high peak low peak narrow body wide body ±1 S.D. ≈ +1 S.D. ≈ 34% mean -1 S.D. ≈ 34% 68% (2/3) +2 S.D. ≈ 47.5% ±2 S.D. ≈ -2 S.D. ≈ 47.5% 95% (19/20)
-1 S.D. +1 S.D.
-2 S.D. +2 S.D. We would expect to see an occurrence •within 1 standard deviation of the mean approx. 2 times out of 3
•within 2 standard deviations of the mean approx. 19 times out of 20
•more than 1 standard deviation from the mean approx. 1 time in 3
•more than 2 standard deviations from the mean approx. 1 time in 20 exercise price time to expiration mean? underlying price interest rate standard deviation? volatility
(dividends) Mean –forward price (underlying price, time to expiration, interest rates, dividends)
Standard deviation –volatility
Volatility:one standard deviation, in percent, over a one year period. 1-year forward price = 100.00 volatility = 20% One year from now: •2/3 chance the contract will be between 80 and 120 (100 ± 20%)
•19/20 chance the contract will be between 60 to 140 (100 ± 2*20%)
•1/20 chance the contract will be less than 60 or more than 140 What does an annual volatility tell us about movement over some other time period?
monthly price movement? weekly price movement? daily price movement?
Volatilityt = Volatilityannual * √ t Daily volatility (standard deviation)
Trading days in a year? 250 –260
Assume 256 trading days
t = 1/256 √ t = √ 1/256 = 1/16
Volatilitydaily = Volatilityannual / 16 current price = 100.00 volatilitydaily ≈ 20% / 16 = 1¼%
One trading day from now:
•22//33 chance the contract will be between 98.75 and 101.25 (100 ± 1¼%)
•19/20 chance the contract will be between 97.50 and 102.50 (100 ± 2*1¼%) Weekly volatility: t = 1/52 √ t = √ 1/52 ≈ 1/7.2
Volatilityweekly ≈ Volatilityannual / 7.2
Monthly volatility: t = 1/12 √ t = √ 1/12 ≈ 1/3.5
Volatilitymonthly ≈ Volatilityannual / 3.5 stock = 68.50; volatility = 42.0% daily standard deviation ≈ 68.50 * 42% / 16 =68.50 * 2.625% ≈ 1.80 weekly standard deviation ≈ 68.50 * 42% / 7.2 =68.50 * 5.83% ≈ 4.00 stock = 68.50; volatility = 42.0% daily standard deviation = 1.80
+.70 +1.25 -.95 -1.60 +.35
Is 42% a reasonable volatility estimate?
How often do you expect to see an occurrence greater than one standard deviation? Volatility Exercise I For each contract and volatility below, what would be an approximate daily and weekly standard deviation:
Futures trading at 121.00 4% 5% 6% 7% daily
weekly
Index trading at 2950.00 15% 20% 25% 30% daily
weekly Volatility Exercise II
For each contract, volatility, and time interval below, what would be an an approximate one standard deviation price change:
Futures trading at 128.00
volatility = 5.75%, time = 22 days
volatility = 6.30%, time = 86 days
Index trading at 2725.00
volatility = 25.60%, time = 9 weeks
volatility = 17.95%, time = 27 weeks –∞ +∞
normal lognormal distribution distribution 0 forward price = 100
normal lognormal distribution distribution price
110 call 3.00 3.20 2.90
90 put 3.00 2.80 3.10
Are the options mispriced? Maybe the marketplace thinks the model is wrong. Maybe the marketplace is right. realized volatility: The volatility of the underlying contract over some period of time (historical, future) implied volatility:
derived from the prices of options in the marketplace
the marketplace’s consensus forecast of future volatility implied volatility exercise price time to expiration 31%??? 3.25 underlying price pricing theoretical model value interest rate 2.50 volatility 27% today realized implied volatility volatility
backward forward looking looking
(what has occurred) (what the marketplace thinks will occur) implied volatility = price realized volatility = value (historical, future)
8 April 2011 DAX = 7217.02 June Futures = 7223.00 Time to expiration = 10 weeks Interest rate = 1.00%
June price 15% 20% 25% implied
6700 call 600.00 562.87 599.20 643.72 20.10%
7200 call 233.10 207.74 270.48 333.26 17.02%
7700 call 41.10 45.27 91.89 145.10 14.49%
6700 put 78.00 27.03 63.36 107.88 21.72%
7200 put 210.10 170.94 233.69 296.46 18.12%
7700 put 517.10 507.52 554.14 607.34 16.12% 8 April 2011 DAX = 7217.02 June Futures = 7223.00 Time to expiration = 10 weeks Interest rate = 1.00%
June 15% 20% increase %
6700 call ITM 562.87 599.20 36.33 6%
7200 call ATM 207.74 270.48 62.74 30%
7700 call OTM 45.27 91.89 46.62 103%
6700 put OTM 27.03 63.36 36.33 134%
7200 put ATM 170.94 233.69 62.75 37%
7700 put ITM 507.52 554.14 46.62 9% 8 April 2011 June Futures = 7223 September Futures = 7249 June = 10 weeks September = 23 weeks
June price 15% 20% increase implied
6700 call 600.00 562.87 599.20 8.54 20.10% 7200 call 233.10 207.74 270.48 18.52 17.02%
7700 call 41.10 45.27 91.89 9.54 14.49%
September 6700 call 723.90 656.71 727.60 70.89 19.75% 7200 call 379.90 328.53 423.41 94.88 17.71%
7700 call 144.20 133.19 220.80 87.61 15.65% 1. In total points an at-the-money option is always more sensitive to a change in volatility than an equivalent in-or out-of-the-money option.
2. In percent terms an out-of-the-money option is always more sensitive to a change in volatility than an equivalent in-or at-the-money option.
3. A long-term option is always more sensitive to a change in volatility than an equivalent short-term option. Volatility Characteristics serial correlation – in the absence of other information, the best estimate of volatility over the next time period is the volatility which occurred over the previous time period. mean reversion – over long periods of time volatility tends to revert to its average.
Volatility Forecasting Methods ARIMA –auto-regressive integrated moving average (G)ARCH –(generalized) auto-regressive conditional heteroscedasticity Changes in implied volatility
Average volatility = 35%
Jun implied 35% 45% 25% Sep implied 35% 41% 29% Dec implied 35% 37% 33%
Mean Reversion Term Structure of Volatility i m p li e d v ol a t l ty
time to expiration EurexOption Seminar
Risk Measurement –The “Greeks” Trade an underlying contract:
risk
Trade an option:
directional volatility
risk
time interest rate Delta (Δ) – The number of underlying contracts required to establish a neutral hedge
The directional risk of a position in terms of an equivalent position in the underlying contract
The sensitivity of an option’s theoretical value to a change in the price of the underlying contract
calls have positive deltas / puts have negative deltas Gamma or curvature (Γ) –
The rate of change in an option’s delta with respect to movement in the price of the underlying contract
Usually expressed as the change in the delta per one point change in the price of the underlying contract
All options have positive gamma values Underlying price = 100
110 call: delta = 30 gamma = 2
100 30 99 28 101 32 98 26 102 34 97 24
Underlying price falls to 75 New delta of 110 call? 30 -25*2 = -20 The gamma must be changing All options have positive gamma values underlying price = 100 101 99 100 call 100 put delta = +50 delta = -50 gamma = +5 gamma = +5 +50 +50 -50 -50 +5 -5 +5 -5 +55 +45 -45 -55 Add the gamma as the market rises. Subtract the gamma as the market falls. Delta = speed rate of change in the option value (first derivative)
Gamma = acceleration rate of change in the delta value (second derivative) Theta (Θ) –
The sensitivity of an option’s theoretical value to the passage of time
Usually expressed as the change in value per one day’s passage of time
Often written with a negative sign to represent a loss in value as time passes. Using this notation all options have negative theta values. Vega or Kappa (K) –
The sensitivity of an option’s theoretical value to a change in volatility
Usually expressed as the change in value per one percentage point change in volatility
Often interpreted as the change in price with respect to a change in implied volatility
All options have positive vegavalues Rho (P) –
The sensitivity of an option’s theoretical value to a change in interest rates
Usually expressed as the change in value per one percentage point change in interest rates
The rho of an option may be either positive or negative, depending on the type of option, the underlying contract, and the settlement procedure. measure calls puts underlying
delta positive negative positive gamma positive positive zero
theta negative negative zero
vega positive positive zero
rho positive negative zero (stocks) Risk Measurement Exercise For each option on the following page: 1. If we assume that the delta is constant what will be the new theoretical value if the underlying contract moves by the given amount?
2. What will be the new delta if the underlying contract moves by the given amount?
3. If the underlying contract moves by the given amount what will be the approximate theoretical value if you also include the gamma? (Hint: What is the average delta?)
4. What will be the approximate theoretical value if ten days pass with no movement in the underlying contract?
5. What will be the approximate theoretical value if volatility changes by the given amount? Risk Measurement Exercise
theoretical daily underlying change in value delta gamma theta vega movement volatility a) 8.04 65 3.7 -.036 .24 3.00 3% b) 1.88 -28 2.3 -.021 .30 2.50 7% c) 3.76 50 4.9 -.012 .80 1.44 3.5% d) 17.12 -87 2.9 -.060 .75 2.68 9% e) .95 11 1.9 -.002 .06 .66 2.5%
f) 14.56 -44 8.8 -.045 .92 10.00 6% Risk Measurement Exercise
new theoretical new theoretical if ten if original value using a new average value using the days volatility delta constant delta delta delta average delta pass changes a) 65 b) -28 c) 50 d) -87 e) 11 f) -44 Stock Price = 99.50 Time to June Expiration = 60 days Volatility = 25.00% Interest Rate = 6.00%
C A L L S P U T S exercise theoretical theoretical price price value delta gamma theta vega price value delta gamma theta vega 90 11.30 11.05 87 2.1 -.0300 .084 .90 .67 -13 2.1 -.0154 .084 95 7.40 7.21 73 3.3 -.0387 .134 2.00 1.78 -27 3.3 -.0233 .134 100 4.35 4.26 54 3.9 -.0416 .160 3.85 3.78 -46 3.9 -.0254 .160 105 2.20 2.26 35 3.7 -.0366 .150 6.65 6.73 -65 3.7 -.0195 .150 110 1.00 1.07 20 2.8 -.0266 .113 10.40 10.49 -80 2.8 -.0087 .113 position theoretical edge delta position gamma position theta position vegaposition long 7 stock contracts 0 +7 x +100 0 0 0 short 10 June 95 calls 10 x +.19 -10 x +73 -10 x +3.3 -10 x -.0387 -10 x +.134 +1.90 -30 -33.0 +.3870 -1.340 long 20 June 105 calls 20 x +.06 +20 x +35 +20 x +3.7 +20 x -.0366 +20 x +.150 short 10 June 100 calls 10 x +.09 -10 x +54 -10 x +3.9 -10 x -.0416 -10 x +.160 +2.10 +160 +35.0 -.3160 +1.400 long 10 June 110 calls 10 x +.07 +10 x +20 +10 x +2.8 +10 x -.0266 +10 x +.113 long 10 June 90 puts 10 x -.23 +10 x -13 +10 x +2.1 +10 x -.0154 +10 x +.084 -1.60 +70 +49.0 -.4200 +1.970 short 20 June 90 calls 20 x +.25 -20 x +87 -20 x +2.1 -20 x -.0300 -20 x +.084 long 20 June 95 calls 20 x -.19 +20 x +73 +20 x +3.3 +20 x -.0387 +20 x +.134 +1.20 -280 +24.0 -.1740 +1.000 long 10 June 90 puts 10 x -.23 +10 x -13 +10 x +2.1 +10 x -.0154 +10 x +.084 short 20 June 95 puts 20 x +.22 -20 x -27 -20 x +3.3 -20 x -.0233 -20 x +.134 long 10 June 100 puts 10 x -.07 +10 x -46 +10 x +3.9 +10 x -.0254 +10 x +.160 +1.40 -50 -6.0 +.0580 -.240 Sell 10 June 95 calls –7.40 (theoretical value = 7.21) Buy 7 stock contracts (700 shares)
theoretical edge delta gamma 0 +7 x 100 0 +10 x .19 -10 x 73 -10 x 3.3 +1.90 -30 -33.0
theta vega +7 x 0 +7 x 0 -10 x -.0387 -10 x .134 +.387 -1.34 Positive Delta: You want the underlying price to rise Negative Delta: You want the underlying price to fall
Positive Gamma: You want the underlying contract to make a big move, or move very quickly Negative Gamma: You want the underlying contract to sit still, or move very slowly
Positive Theta: The passage of time will help Negative Theta: The passage of time will hurt
Positive Vega: You want implied volatility to rise Negative Vega: You want implied volatility to fall theoretical delta gamma theta vega option position edge position position position position
90 C -4 +1.00 -348 -8.4 +.1200 -.336 95 C -12 +2.28 -876 -39.6 +.4644 -1.608 100 C 14 -1.26 756 +54.6 -.5824 +2.240 105 C -17 -1.02 -595 -62.9 +.6222 -2.550 110 C 12 +.84 240 +33.6 -.3192 +1.356 call totals -7 +1.84 -823 -22.7 +.3050 -.898
90 P 13 -2.99 -169 +27.3 -.2002 +1.092 95 P -20 +4.40 +540 -66.0 +.4660 -2.680 100 P -8 +.56 +368 -31.2 +.2032 -1.280 105 P 12 +.96 -780 +44.4 -.2340 +1.800 110 P 8 +.72 -640 +22.4 -.0696 +.904 put totals 5 3.65 -681 -3.1 +.1654 -.164
Stock 13 0 1300 0 0 0
Totals 5.49 -204 -25.8 +.4704 -1.062 Risk Interpretation Exercise Match each position with the corresponding market conditions which will most help the position. position market conditions
+delta / +gamma / -vega no price movement; rising implied volatility
-delta / -gamma / -vega upward price movement; falling implied volatility
0 delta / -gamma / +vega price movement in either direction; rising implied volatility
0 delta / +gamma / +vega swift upward price movement; falling implied volatility
+delta / -gamma / +vega downward price movement
0 delta / +gamma / -vega price movement in either direction; falling implied volatility
-delta / 0 gamma / 0 vega slow upward price movement; rising implied volatility
+delta / 0 gamma / -vega slow downward price movement; falling implied volatility delta characteristics as volatility rises 50 (-50) as volatility falls 50(-50) as time passes 50(-50) implied sensitivity the sensitivity (delta, gamma, theta, vega) which is calculated using the implied volatility. in-the- at-the- out-of-the- money money money gamma theta vega rho An at-the-money option always has a greater gamma, theta, and vegathan an equivalent in-the-money or out-of-the-money option. EurexOption Seminar
Dynamic Hedging stock price = 97.50 time to December expiration = 10 weeks interest rates = 5.00% dividend = 0 volatility = 42.56%
December 100 call ?? theoretical value = 6.55 price = 5.50 buy a call option delta neutral or flat
current underlying -ΔC price ΔC
take an opposing delta position in the determine the option’s delta
t h e o r ti ca l v a ue underlying contract (-ΔC) (ΔC)
underlying price Due to the option’s curvature, as market conditions change the position will become unhedged.
current underlying -ΔC price ΔC t h e o r ti ca l v a ue
{ unhedged amount
underlying price theoretical value Determin hnwdlaofption. Rehedgt posiiontorultaut ral underlying current price underlyi ng price new new - Δ Δ C C theoretical value the lifeofthption. Contiue threhdgigprcstouhout underlying current price underlyi ng price } new new - Δ Δ C C theoretical value the lifeofthption. Contiue threhdgigprcstouhout Δ C underlyi ng price Dynamic Hedgng underlying current price - Δ C Each time the position becomes unhedged there is a potential profit opportunity. Wecan capture this profit by rehedgingthe position.
Suppose we add up all the profit opportunities over the life of the option which result from the rehedgingprocess. What should this equal?
the option’s theoretical value
The rehedgingprocess is a type of statistical arbitrage. EurexOption Seminar
Volatility Strategies Volatility Spread
A spread, usually delta neutral, which is sensitive to either the volatility of the underlying contract (gamma), or to changes in implied volatility (vega) Long Straddle
+1 June 100 call +1 June 100 put
Short Straddle
-1 June 100 call -1 June 100 put Long Straddle
underlying price = 100.00 delta +1 June 100 call +50 +1 June 100 put -50 0 Long Straddle
underlying price = 105.00 delta +1 June 100 call +75 +3 June 100 put -25 0 Ratio Spread Any spread where the number long market contracts and short market contracts are unequal Long Straddle
exercise price Short Straddle
exercise price Long Straddle
delta gamma theta vega 0 + – +
Short Straddle
delta gamma theta vega
0 – + – Long Strangle
+1 June 95 put +1 June 105 call
Short Strangle
-1 June 95 put -1 June 105 call Long Strangle
share price = 100 delta +1 June 105 put -75 +1 June 95 call +75 0 Guts
A strangle where both options are in-the-money Long Strangle
exercise exercise price price Short Strangle
exercise exercise price price Long Strangle
delta gamma theta vega 0 + – +
Short Strangle
delta gamma theta vega
0 – + – Long Butterfly
+1 July 95 call -2 July 100 calls +1 July 105 call
+1 August 90 put -2 August 100 puts +1 August 110 put Short Butterfly
-1 July 95 call wing +2 July 100 calls body -1 July 105 call wing
-1 August 90 put wing +2 August 100 puts body -1 August 110 put wing Long Butterfly 90 110 100
+1 July 95 call 0 +15 +5 -2 July 100 calls 0 -20 0 +1 July 105 call 0 +5 0 0 0 +5 minimum value = 0 maximum value = amount between exercise prices Long Butterfly
exercise exercise exercise price price price Short Butterfly
exercise exercise exercise price price price Long Butterfly
delta gamma theta vega
0 – + –
Short Butterfly
delta gamma theta vega 0 + – + Ratio Spread (buy more than sell)
share price = 100 delta +3 August 105 call 25 -1 August 95 call 75 0
+2 September 95 put -25 -1 September 100 put -50 0 Ratio Spread (sell more than buy)
underlying price = 100 delta -3 August 105 call 25 +1 August 95 call 75 0
-2 June 95 put -25 +1 June 100 put -50 0 share price = 100 price +3 August 105 call 1.00 -1 August 95 call 6.00
120 +3 * 14.00 -1 * 19.00 = +23.00
80 -3 * 1.00 +1 * 6.00 = +3.00
100 -3 * 1.00 +1 * 1.00 = -2.00 Call Ratio Spread (buy more than sell)
exercise exercise price price Call Ratio Spread (sell more than buy)
exercise exercise price price Put Ratio Spread (buy more than sell)
exercise exercise price price Put Ratio Spread (sell more than buy)
exercise exercise price price Ratio Spread – buy more than sell delta gamma theta vega 0 + – +
Ratio Spread – sell more than buy delta gamma theta vega
0 – + – downside upside risk / reward risk / reward long straddle / strangle unlimited unlimited reward reward short straddle / strangle unlimited unlimited risk risk long butterfly limited limited risk risk short butterfly limited limited reward reward downside upside risk / reward risk / reward call ratio spread limited unlimited (buy more than sell) reward reward put ratio spread unlimited limited (buy more than sell) reward reward call ratio spread limited unlimited (sell more than buy) risk risk put ratio spread unlimited limited (sell more than buy) risk risk Long Calendar Spread (Time Spread, Horizontal Spread)
+1 September 100 call -1 July 100 call
+1 November 95 put -1 October 95 put Short Calendar Spread (Time Spread, Horizontal Spread)
-1 September 100 call +1 July 100 call
-1 November 95 put +1 October 95 put +1 September 100 call underlying price = 100 -1 July 100 call
September 4 months 3 months 2 months July 2 months 1 month 0 months
September 3.00 2.60 2.10 July 2.10 1.30 0
.90 1.30 2.10 +1 September 100 call -1 July 100 call
100 150 50 September 3.00 50.05 .05 July 2.10 50.00 0
.90 .05 .05 +1 September 100 call -1 July 100 call
25% 30% 20% September 3.00 3.90 2.10 July 2.10 2.50 1.70
.90 1.40 .40
negative gamma / positive vega Long Calendar Spread
exercise price Short Calendar Spread
exercise price Long Calendar Spread
delta gamma theta vega
0 – + +
Short Calendar Spread
delta gamma theta vega 0 + – – interest rates stock options
Long Call Calendar Spread rho
+1 September 100 call + -1 July 100 call
Long Put Calendar Spread
+1 September 100 put – -1 July 100 put dividend (payable between July and September)
Long Call Calendar Spread dividend risk +1 September 100 call – -1 July 100 call
Long Put Calendar Spread
+1 September 100 put + -1 July 100 put Volatility Spreads
delta gamma theta vega 0 + + – –
– +
+ – Volatility Spreads gamma / vega + + long straddle, long strangle, short butterfly, ratio spread (buy more than sell) – – short straddle, short strangle, long butterfly,ratio spread (sell more than buy) – + long calendar spread + – short calendar spread Volatility Strategy Exercise On this and the following page are several different volatility strategies with some possible changes in market conditions. If the underlying contract is currently trading at 80, for each change in market conditions is the strategy making money (+) or losing money (–). Assume all positions are initially delta neutral.
the underlying time passes with no implied price rises sharply change in the underlying volatility falls +1 June 80 call +1 June 80 put
the underlying time passes with no implied price falls sharply change in the underlying volatility falls -1 October 80 put +1 July 80 put the underlying time passes with no implied price falls sharply change in the underlying volatility rises +1 August 75 call -2 August 85 calls Volatility Strategy Exercise the underlying time passes with no implied price rises sharply change in the underlying volatility rises -1 September 75 put +2 September 80 puts -1 September 85 put the underlying time passes with no implied price falls sharply change in the underlying volatility falls +1 August 80 call -1 June 80 call
the underlying time passes with no implied price rises sharply change in the underlying volatility rises -1 June 85 call -1 June 75 put
the underlying time passes with no implied price falls sharply change in the underlying volatility falls -3 October 70 puts +1 October 80 put EurexOption Seminar
Directional Strategies Underlying price = 100 90 Long Straddle
+1 December 100 call delta neutral +1 December 100 put
+1 December 95 call Bueallr StStrraaddddlele +1 December 95 put +1 December 100 call bull spread -1 December 110 call
-1 December 100 call bear spread +1 December 110 call
minimum value = 0
maximum value = Xh -Xl +1 December 100 put bull spread -1 December 110 put
-1 December 100 put bear spread +1 December 110 put
minimum value = 0
maximum value = Xh -Xl Bull (Vertical) Spread Buy an option at a lower exercise price Sell an option at a higher exercise price
Bear (Vertical) Spread Buy an option at a higher exercise price Sell an option at a lower exercise price
Both options must be the same type (both calls or both puts) and expire at the same time. Bull (Vertical) Spread
buy the lower exercise price
sell the higher exercise price Bear (Vertical) Spread
buy the higher exercise price
sell the lower exercise price Underlying price = 100 Time to expiration = 6 weeks
Volatility 20%
95 call 5.86 (78) 95 / 100 spread 3.15 (27) 100 call 2.71 (51) 100 / 105 spread 1.75 (27) 105 call .96 (24) Underlying price = 100 Time to expiration = 6 weeks
Volatility 20% 15% 25%
95 call 5.86 5.41 6.38 95 / 100 spread 3.15 3.38 3.00 100 call 2.71 2.03 3.38 100 / 105 spread 1.75 1.56 1.86 105 call .96 .47 1.52 Underlying price = 100 Time to expiration = 6 weeks
Volatility 20% 110 90 100
95 call 5.86 95 / 100 spread 3.15 + + 100 call 2.71 100 / 105 spread 1.75 + 105 call .96 Underlying price = 100 Time to expiration = 6 weeks
Volatility 20% 15% 25%
95 put .86 .41 1.38 95 / 100 spread 1.85 1.62 2.00 100 put 2.71 2.03 3.38 100 / 105 spread 3.25 3.44 3.14 105 put 5.96 5.47 6.52 Bull and Bear Strategy Exercise From the four strategy choicesin each question below select the strategy which best fits the give market conditions (underlying price, directional outlook, and implied volatility). underlying directional implied strategy price outlook volatility choices 90 bullish high long a 90 call / short a 100 call long a 95 call / short a 90 call long an 85 put / short a 90 put long a 95 put / short a 90 put
60 bearish low long a 55 put / short a 60 put long a 65 call / short a 55 call long an 65 call / short a 70 call long a 60 put / short a 55 put
150 bearish high long a 140 call / short a 160 call long a 150 put / short a 140 put long a 150 call / short a 140 call long a 160 put / short a 150 put
2800 bullish low long a 2800 put / short a 2900 put long a 2900 call / short a 2800 call long a 2700 put / short a 2800 put long a 2700 call / short a 2800 call EurexOption Seminar
Synthetics Buy a call and sell a put with the same exercise price and expiration date…. +1 December 100 call -1 December 100 put
above 100: put is worthless / exercise call
buy underlying at 100
below 100: call is worthless / assigned on put
buy underlying at 100 Buy a call and sell a put with the same exercise price and expiration date…. +1 December 100 call -1 December 100 put
Regardless of whether the underlying is above or below the exercise price at expiration…… buy underlying at 100
Synthetic Long Underlying long call short put exercise price
long underlying long call + short put ≈ synthetic long underlying short call + long put ≈ synthetic short underlying
delta of an underlying contract? 100
+1 December 100 call +30 +80 -1 December 100 put -70 -20 long call + short put ≈ synthetic long underlying short call + long put ≈ synthetic short underlying
gamma of an underlying contract? 0
+1 December 100 call 5 2 -1 December 100 put 5 2 long call + short put ≈ synthetic long underlying short call + long put ≈ synthetic short underlying
vegaof an underlying contract? 0
+1 December 100 call .15 .40 -1 December 100 put .15 .40 long put + long underlying ≈ synthetic long call short put + short underlying ≈ synthetic short call long call + short underyling ≈ synthetic long put short call + long underlying ≈ synthetic short put Synthetic Equivalent Exercise Match each position with its synthetic equivalent.
position synthetic equivalent
long June 100 call / short underlying long June 100 call
long June 100 call / short June 105 call long underlying
short June 100 put / short underlying long June 100 put / short June 105 put
short 2 June 100 puts / short underlying short June 100 call
long June 100 put / long underlying long June 100 put
long June 100 call / short June 100 put short underlying
short June 100 call / long underlying short June 100 put
short June 100 call / long June 100 put short June 100 straddle Buy December 100 straddle:
+1 December 100 call +1 December 100 put
+2 December 100 calls -1 underlying contract
+2 December 100 puts +1 underlying contract Bull spread: buy a lower exercise price sell a higher exercise price
+1 Dec 90 call -1 Dec 100 call
+1 Dec 90 put / +1 underlying -1 Dec 100 put / -1 underlying Synthetic Pricing Put-Call Parity
call price -put price = present value (forward price -exercise price)
Futures: call price -put price = futures price -exercise price
Stock: call price -put price ≈ stock price -exercise price + interest on the exercise price -expected dividends December 100 call 6.00 December 100 put 2.00 underlying contract ???
C - P = F - X 6.002.00 ?? 100 sell December 100 call 6.00 5.75 buy December 100 put 2.00 2.25 buy underlying contract 104.00 103.75
C - P = F - X 6.002.00 103.75 100 4.00 ≠ 3.75 Option Arbitrage Conversion: synthetic long underlying + short underlying short call long underlying + long put Reverse Conversion: synthetic short underlying + long underlying long call short underlying + short put Synthetic Pricing Exercise -Futures For each set of futures options below, fill in the missing value
futures price exercise price call price put price
121.35 118.00 3.66
115.82 2.05 2.48
108.20 1.82 1.00
127.55 122.00 .18
107.70 1.11 1.11
116.39 118.75 1.79
124.08 2.46 5.88
120.50 117.50 .72 Synthetic Pricing Exercise -Stock For each set of stock options below, fill in the missing value. Assume all options are European.
stock days to interest expected exercise call put price expiration rate dividends price price price
68.25 46 3.87% 0 70 1.35
54.60 115 4.15% .35 50 1.90
109.00 88 5.71% 110 7.55 7.55
23 4.89% .78 120 7.60 1.90
28.72 177 .29 30 1.54 2.06
82.55 2.90% 1.10 75 8.10 1.36 EurexOption Seminar
Hedging with Options Options as “insurance”
Practical considerations
How much protection do I need? How much risk am I willing to accept?
Theoretical considerations
What is the cost of the insurance? Does the premium I receive fairly compensate me for the lost opportunity? Options as insurance
Buy a protective option
long stock–buy a put short stock–buy a call
Advantage –absolute, well-defined protection; unlimited potential profit Disadvantage –cost of option; loss of premium long stock long a protective put
exercise price
}option premium
long call short stock long a protective call
exercise price
}option premium
long put Options as insurance
Sell a covered option
long stock–sell a call short stock –sell a put
Advantage –receive premium
Disadvantage –sold option offers only partial protection; limited profit potential long stock buy / write short a covered call
exercise short put price
}option premium short stock short a covered put
exercise price short call
}option premium Options as insurance Simultaneously buy a protective option and sell a covered option long stock –buy a put / sell a call (long collar)
short stock –buy a call / sell a put (short collar)
Zero-cost collar –the price of the bought and sold options are the same long stock long put bull spread short call exercise exercise price price short stock long call bear spread short put exercise exercise price price Hedging strategies tend to reduce the volatility of a portfolio. Is that desirable? year 1 year 2 year 3 average total returns returns returns return return +25% -20% +25% +10% +25%
+29% -34% +44% +13% +22.6%
+16% -6% +17% +9% +27.6%
+8.5% +8.5% +8.5% +8.5% +27.7% EurexOption Seminar
Answers to Exercises Expected Value Exercise (answers)
124 126 128 130 132 134 136
4% 12% 19% 22% 21% 16% 6% Using the above prices and probabilities for an underlying contract, what are the expected values for the following contracts:
underlying 126 call 130 call 133 call 130.32 4.40 1.42 .34
126 put 130 put 133 put .08 1.10 3.02
What do you notice about the difference between the values of calls and puts at the same exercise price? They differ by intrinsic value. Volatility Exercise I (answers) For each contract and volatility below, what would be an approximate daily and weekly standard deviation:
Futures trading at 121.00 4% 5% 6% 7%
daily .30 .38 .45 .53
weekly .67 .84 1.01 1.18
Index trading at 2950.00 15% 20% 25% 30%
daily 27.66 36.88 46.10 55.31
weekly 61.46 81.94 102.43 122.92 Volatility Exercise II (answers)
For each contract, volatility, and time interval below, what would be an an approximate one standard deviation price change:
Futures trading at 128.00
volatility = 5.75%, time = 22 days 128.00*.0575*√(22/365) = 1.81
volatility = 6.30%, time = 86 days 128.00*.0630*√(86/365) = 3.91
Index trading at 2725.00
volatility = 25.60%, time = 9 weeks 2725.00*.2560*√(9/52) = 290.22
volatility = 17.95%, time = 27 weeks 2725.00*.1795*√(27/52) = 352.46 Risk Measurement Exercise (answers)
new theoretical new theoretical if ten if original value using a new average value using the days volatility delta constant delta delta delta average delta pass changes a) 65 9.99 76.1 70.6 10.16 7.68 7.32 b) -28 1.18 -22.3 -25.1 1.25 1.67 3.98 c) 50 3.04 42.9 46.5 3.09 3.64 6.56 d) -87 19.45 -94.8 -90.9 19.56 16.52 10.37 e) 11 1.02 12.3 11.6 1.03 .93 .80 f) -44 18.96 -100 -72.0 21.76 14.11 20.08 Risk Interpretation Exercise (answers) Match each position with the corresponding market conditions which will most help the position. position market conditions
+delta / +gamma / -vega swift upward price movement; falling implied volatility
-delta / -gamma / -vega slow downward price movement; falling implied volatility
0 delta / -gamma / +vega no price movement; rising implied volatility
0 delta / +gamma / +vega price movement in either direction; rising implied volatility
+delta / -gamma / +vega slow upward price movement; rising implied volatility
0 delta / +gamma / -vega price movement in either direction; falling implied volatility
-delta / 0 gamma / 0 vega downward price movement
+delta / 0 gamma / -vega upward price movement; falling implied volatility Volatility Strategy Exercise (answers) On this and the following page are several different volatility strategies with some possible changes in market conditions. If the underlying contract is currently trading at 80, for each change in market conditions is the strategy making money (+) or losing money (–). Assume all positions are initially delta neutral.
the underlying time passes with no implied price rises sharply change in the underlying volatility falls +1 June 80 call +1 June 80 put + – –
the underlying time passes with no implied price falls sharply change in the underlying volatility falls -1 October 80 put – +1 July 80 put + + the underlying time passes with no implied price falls sharply change in the underlying volatility rises +1 August 75 call – – -2 August 85 calls + Volatility Strategy Exercise (answers) the underlying time passes with no implied price rises sharply change in the underlying volatility rises -1 September 75 put +2 September 80 puts + – + -1 September 85 put the underlying time passes with no implied price falls sharply change in the underlying volatility falls +1 August 80 call -1 June 80 call – + –
the underlying time passes with no implied price rises sharply change in the underlying volatility rises -1 June 85 call – – -1 June 75 put +
the underlying time passes with no implied price falls sharply change in the underlying volatility falls -3 October 70 puts – +1 October 80 put + + Bull and Bear Strategy Exercise (answers) From the four strategy choicesin each question below select the strategy which best fits the give market conditions (underlying price, directional outlook, and implied volatility). underlying directional implied strategy price outlook volatility choices 90 bullish high long a 90 call / short a 100 call long a 95 call / short a 90 call long an 85 put / short a 90 put long a 95 put / short a 90 put
60 bearish low long a 55 put / short a 60 put long a 65 call / short a 55 call long an 65 call / short a 70 call long a 60 put / short a 55 put
150 bearish high long a 140 call / short a 160 call long a 150 put / short a 140 put long a 150 call / short a 140 call long a 160 put / short a 150 put
2800 bullish low long a 2800 put / short a 2900 put long a 2900 call / short a 2800 call long a 2700 put / short a 2800 put long a 2700 call / short a 2800 call Synthetic Equivalent Exercise (answers) Match each position with its synthetic equivalent.
position synthetic equivalent
long June 100 call / short underlying long June 100 put
long June 100 call / short 105 call long June 100 put / short June 105 put
short June 100 put / short underlying short June 100 call
short 2 June 100 puts / short underlying short June 100 straddle
long June 100 put / long underlying long June 100 call
long June 100 call / short June 100 put long underlying
short June 100 call / long underlying short June 100 put
short June 100 call / long June 100 put short underlying Synthetic Pricing Exercise -Futures (answers) For each set of futures options below, fill in the missing value
futures price exercise price call price put price
121.35 118.00 3.66 .31
115.82 116.25 2.05 2.48
109.02 108.20 1.82 1.00
127.55 122.00 5.73 .18
107.70 107.70 1.11 1.11
116.39 118.75 1.79 4.15
124.08 127.50 2.46 5.88
120.50 117.50 3.72 .72 Synthetic Pricing Exercise -Stock (answers) For each set of stock options below, fill in the missing value. Assume all options are European.
stock days to interest expected exercise call put price expiration rate dividends price price price
68.25 46 3.87% 0 70 1.35 2.76
54.60 115 4.15% .35 50 6.80 1.90
109.00 88 5.71% 2.51 110 7.55 7.55
125.35 23 4.89% .78 120 7.60 1.90
28.72 177 7.22% .29 30 1.54 2.06
82.55 49 2.90% 1.10 75 8.10 1.36