# 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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