Exotic Options Trading

2 Conventional Options, Forwards and Greeks 3
2.1 Call and Put Options and Forwards 3
2.2 Pricing Calls and Puts 6
2.3 Implied Volatility 8
2.4 Determining the Strike of the Forward 8
2.5 Pricing of Stock Options Including Dividends 9
2.6 Pricing Options in Terms of the Forward 10
2.7 Put-Call Parity 11
2.8 Delta 12
2.9 Dynamic Hedging 14
2.10 Gamma 14
2.11 Vega 16
2.12 Theta 18
2.13 Higher Order Derivatives Like Vanna and Vomma 19
2.14 Options’ Interest Rate Exposure in Terms of
Financing the Delta Hedge 21
3 Profit on Gamma and Relation to Theta 23
4 Delta Cash and Gamma Cash 25
4.1 Example: Delta and Gamma Cash 26

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5 Skew 27
5.1 Reasons for Higher Realised Volatility in
Falling Markets 27
5.2 Skew Through Time: ‘The Term Structure of Skew’ 28
5.3 Skew and its Effect on Delta 29
5.4 Skew in FX versus Skew in Equity: ‘Smile versus
Downward Sloping’ 32
5.5 Pricing Options Using the Skew Curve 34
6 Simple Option Strategies 35
6.1 Call Spread 35
6.2 Put Spread 37
6.3 Collar 39
6.4 Straddle 40
6.5 Strangle 42
7 Monte Carlo Processes 45
7.1 Monte Carlo Process Principle 45
7.2 Binomial Tree versus Monte Carlo Process 46
7.3 Binomial Tree Example 46
7.4 The Workings of the Monte Carlo Process 48
8 Chooser Option 49
8.1 Pricing Example: Simple Chooser Option 49
8.2 Rationale Behind Chooser Option Strategies 51
9 Digital Options 53
9.1 Choosing the Strikes 55
9.2 The Call Spread as Proxy for the Digital 55
9.3 Width of the Call Spread versus Gearing 55
10 Barrier Options 57
10.1 Down-and-In Put Option 58
10.2 Delta Change over the Barrier for a Down-and-In
Put Option 58
10.3 Factors Influencing the Magnitude of the Barrier
Shift 60
10.4 Delta Impact of a Barrier Shift 63
10.5 Situations to Buy Shares in Case of a Barrier
Breach of a Long Down-and-In Put 63

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Contents vii
10.6 Up-and-Out Call 63
10.7 Up-and-Out Call Option with Rebate 64
10.8 Vega Exposure Up-and-Out Call Option 64
10.9 Up-and-Out Put 65
10.10 Barrier Parity 65
10.11 Barrier at Maturity Only 65
10.12 Skew and Barrier Options 66
10.13 Double Barriers 68
11 Forward Starting Options 71
11.1 Forward Starting and Regular Options Compared 71
11.2 Hedging the Skew Delta of the Forward
Start Option 72
11.3 The Forward Start Option and the Skew Term
Structure 73
11.4 Analytically Short Skew but Dynamically No
Skew Exposure 74
11.5 Forward Starting Greeks 75
12 Ladder Options 77
12.1 Example: Ladder Option 77
12.2 Pricing the Ladder Option 78
13 Lookback Options 79
13.1 Pricing and Gamma Profile of Fixed Strike
Lookback Options 79
13.2 Pricing and Risk of a Floating Strike Lookback
Option 80
14 Cliquets 83
14.1 The Ratchet Option 83
14.2 Risks of a Ratchet Option 85
15 Reverse Convertibles 87
15.1 Example: Knock-in Reverse Convertible 87
15.2 Pricing the Knock-in Reverse Convertible 89
15.3 Market Conditions for Most Attractive Coupon 89
15.4 Hedging the Reverse Convertible 90
16 Autocallables 93
16.1 Example: Autocallable Reverse Convertible 93

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16.2 Pricing the Autocallable 95
16.3 Autocallable Pricing without Conditional Coupon 97
16.4 Interest/Equity Correlation within
the Autocallable 98
17 Callable and Puttable Reverse Convertibles 99
17.1 Pricing the Callable Reverse Convertible 99
17.2 Pricing the Puttable Reverse Convertible 102
18 Asian Options 105
18.1 Pricing the Geometric Asian Out Option 105
18.2 Pricing the Arithmetic Asian Out Option 107
18.3 Delta Hedging the Arithmetic Asian Out Option 109
18.4 Vega, Gamma and Theta of the Arithmetic Asian
Out Option 110
18.5 Delta Hedging the Asian in Option 110
18.6 Asian in Forward 112
18.7 Pricing the Asian in Forward 114
18.8 Asian in Forward with Optional Early Termination 116
19 Quanto Options 119
19.1 Pricing and Correlation Risk of the Option 119
19.2 Hedging FX Exposure on the Quanto Option 122
20 Composite Options 125
20.1 An Example of the Composite Option 125
20.2 Hedging FX Exposure on the Composite Option 126
21 Outperformance Options 129
21.1 Example of an Outperformance Option 129
21.2 Outperformance Option Described as a Composite
Option 130
21.3 Correlation Position of the Outperformance
Option 131
21.4 Hedging of Outperformance Options 132
22 Best of and Worst of Options 135
22.1 Correlation Risk for the Best of Option 135
22.2 Correlation Risk for the Worst of Option 137
22.3 Hybrids 138

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Contents ix
23 Variance Swaps 139
23.1 Variance Swap Payoff Example 140
23.2 Replicating the Variance Swap with Options 140
23.3 Greeks of the Variance Swap 142
23.4 Mystery of Gamma Without Delta 144
23.5 Realised Variance Volatility versus Standard
Deviation 145
23.6 Event Risk of a Variance Swap versus a Single
Option 146
23.7 Relation Between Vega Exposure and Variance
Notional 147
23.8 Skew Delta 147
23.9 Vega Convexity 148
24 Dispersion 151
24.1 Pricing Basket Options 151
24.2 Basket Volatility Derived From its Constituents 152
24.3 Trading Dispersion 153
24.4 Quoting Dispersion in Terms of Correlation 153
24.5 Dispersion Means Trading a Combination of
Volatility and Correlation 153
24.6 Ratio’d Vega Dispersion 155
24.7 Skew Delta Position Embedded in Dispersion 156
25 Engineering Financial Structures 157
25.1 Capital Guaranteed Products 157
25.2 Attractive Market Conditions for Capital
Guaranteed Products 158
25.3 Exposure Products for the Cautious Equity
Investor 160
25.4 Leveraged Products for the Risk Seeking Investor 163