Introduction Pricing Effects Greeks Summary. Vol Target Options. Rob Coles. February 7, 2014


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1 February 7, 2014
2 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
3 The Basic Idea Basket split between risky asset and cash Chose weight of risky asset w to keep volatility of the basket constant. w = σtarget σ hist Systematic differences between current and historical volatility prevent the basket behaving as true constant vol asset.
4 The Basic Idea Basket split between risky asset and cash Chose weight of risky asset w to keep volatility of the basket constant. w = σtarget σ hist Systematic differences between current and historical volatility prevent the basket behaving as true constant vol asset.
5 The Basic Idea Basket split between risky asset and cash Chose weight of risky asset w to keep volatility of the basket constant. w = σtarget σ hist Systematic differences between current and historical volatility prevent the basket behaving as true constant vol asset.
6 Simple product I n+1 = I n + w ni n S n (S n+1 S n ) w n = 252 m m 1 i=0 σ target ( ) log 2 S n i S n i 1
7 Motivation The value of an option on a vol target is roughly the Black Scholes value with σ eff σ target V BS (I, σ eff, K, T ) Why derive closed form approximations to greeks and price? Improve Intuition. Hedge simulations
8 Motivation The value of an option on a vol target is roughly the Black Scholes value with σ eff σ target V BS (I, σ eff, K, T ) Why derive closed form approximations to greeks and price? Improve Intuition. Hedge simulations
9 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
10  Intuition Even in Black Scholes model σ hist is random. E[σ hist 2 ] = σ BS2 Convexity raises the effective volatility [ ] ( ) 1 m 1 E = m 2 σ hist 2 σ BS2
11  Intuition Even in Black Scholes model σ hist is random. E[σ hist 2 ] = σ BS2 Convexity raises the effective volatility [ ] ( ) 1 m 1 E = m 2 σ hist 2 σ BS2
12  Approximate result Business day volatility m σ target σ target m 2 Calendar day vol. Proportion of variance that falls over weekend = α σ target σ target m m 2(5α (1 α)2 )
13  Approximate result Business day volatility m σ target σ target m 2 Calendar day vol. Proportion of variance that falls over weekend = α σ target σ target m m 2(5α (1 α)2 )
14  Demonstration
15  Removal Use a continuous time version of historical vol, over time period, to derive formulae without discretisation effect. σt hist 1 t = σ 2 udu Historical volatility averaging period t m 252
16 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
17  Intuition As underlying S drifts down its volatility tends to increase. Historical volatility takes time to catch up. Risky weight w too high and σ eff of the index is larger than σ target for paths where the index drops.
18  Intuition As underlying S drifts down its volatility tends to increase. Historical volatility takes time to catch up. Risky weight w too high and σ eff of the index is larger than σ target for paths where the index drops.
19  Intuition As underlying S drifts down its volatility tends to increase. Historical volatility takes time to catch up. Risky weight w too high and σ eff of the index is larger than σ target for paths where the index drops.
20  Approximate result Define skew to be the change in implied volatility for a 10% move in underlying. Assume constant skew in implied vol surface of S. T
21  Demonstration.
22  Demonstration.
23 simulations  Morrison, Tadrowski Historical volatility from exponentially weighted mean. Equivalent to 0.8 year unweighted mean. Calibrate jump model to 1.5% skew on sx5e at 2 years. Effective Vol skew around 0.53% ATM vol bump 1.65%
24 simulations  Morrison, Tadrowski Historical volatility from exponentially weighted mean. Equivalent to 0.8 year unweighted mean. Calibrate jump model to 1.5% skew on sx5e at 2 years. Effective Vol skew around 0.53% ATM vol bump 1.65%
25 simulations  Morrison, Tadrowski Historical volatility from exponentially weighted mean. Equivalent to 0.8 year unweighted mean. Calibrate jump model to 1.5% skew on sx5e at 2 years. Effective Vol skew around 0.53% ATM vol bump 1.65%
26 simulations  Nelte, Roche Calibrate jump model to skew on S&P500 at 3 years. σ eff skew under local vol σ eff skew under local vol plus jumps. ATM vol bump 1.8%
27 simulations  Nelte, Roche Calibrate jump model to skew on S&P500 at 3 years. σ eff skew under local vol σ eff skew under local vol plus jumps. ATM vol bump 1.8%
28 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
29 Stochastic Rates  Approximate result Risky and riskless assets grow at short rate following a Hull White process with normal volatility σ short. Negligible speed of mean reversion. ( ) σ eff σ target 1 + ρ σshort σ σ target T + short 2 T 2 σ target 3
30 Stochastic Rates  Approximate result
31 Outline Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
32 Vega  Intuition Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Volatility of underlying rises from σ to σ + δσ. V BS (I, σ eff (δσ), K, T ɛ) While historical volatility is too low the leverage is to high and the Index evolves with σ eff higher than σ target
33 Vega  Approximate result Vega Theta Delta & Gamma Hedge P& L Jump sensitivity For an option maturing after T years. σ eff σ target 1 + δσ σ T Vega BS BS σtarget Vega σ 2T
34 Vega  Demonstration. Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
35 Outline Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
36 Theta  Intuition Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Underlying remains constant for period ɛ. V BS (I, σ eff (ɛ), K, T ɛ) Time decay of option will decrease its value. Period of zero volatility decreases σ hist so σ eff increases.
37 Theta  Intuition Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Underlying remains constant for period ɛ. V BS (I, σ eff (ɛ), K, T ɛ) Time decay of option will decrease its value. Period of zero volatility decreases σ hist so σ eff increases.
38 Theta  Approximate result Vega Theta Delta & Gamma Hedge P& L Jump sensitivity σ eff (ɛ) = σ target σ eff ɛ 1 (1 T ɛ log ɛ ) = σtarget 2T θ BS θ BS BS σeff + Vega ɛ
39 Outline Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
40 Delta  Intuition Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Underlying moves from level at last fixing S 0 to S V BS (I (S), σ eff (S), K, T ) The value of I changes in proportion to ( current weight. ) σ eff changes through addition of log 2 S S 0 term to σ hist 2
41 Delta  Intuition Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Underlying moves from level at last fixing S 0 to S V BS (I (S), σ eff (S), K, T ) The value of I changes in proportion to ( current weight. ) σ eff changes through addition of log 2 S S 0 term to σ hist 2
42 Delta  Approximate result Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Effective volatility σ eff is even in log(s). V S = V σ target I σ
43 Gamma  Result Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
44 Gamma  Result Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
45 Outline Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
46 Hedge P & L Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
47 Outline Introduction Vega Theta Delta & Gamma Hedge P& L Jump sensitivity 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity
48 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity Jump sensitivity  How big is a jump?
49  σ hist measurement period As increases Fair Value decreases because less discretisation effect. Vega increases because takes time for historical volatility to change. increases because historical vol contains values from today. Jump sensitivity decreases.
50  σ hist measurement period As increases Fair Value decreases because less discretisation effect. Vega increases because takes time for historical volatility to change. increases because historical vol contains values from today. Jump sensitivity decreases.
51  σ hist measurement period As increases Fair Value decreases because less discretisation effect. Vega increases because takes time for historical volatility to change. increases because historical vol contains values from today. Jump sensitivity decreases.
52  σ hist measurement period As increases Fair Value decreases because less discretisation effect. Vega increases because takes time for historical volatility to change. increases because historical vol contains values from today. Jump sensitivity decreases.
53  Largest pricing effects Jumps Discretisation
54  Largest pricing effects Jumps Discretisation
55  Largest pricing effects Jumps Discretisation
56 Appendix Some simulation studies Some simulation studies I Steven Morrison, Laura Tadrowski Guarantees and target volatility funds. Barrie and Hibbert research paper Sept Maximilian Nelte, Peter Roche Controlling volaility to reduce uncertainty. Risk Sept 2010.
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