Option P&L Attribution and Pricing

Transcription

1 Option P&L Attribution and Pricing Liuren Wu joint with Peter Carr Baruch College March 23, 2018 Stony Brook University Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

2 The classic perspective of derivative pricing The approach: Evaluating payoffs on (risk-neutral) dynamics Specify the full risk-neutral dynamics for the underlying security price. Take expectation of terminal payoffs according to the same dynamics. Example: Time changed Lévy processes The emphasis: cross-sectional consistency The dynamics specification provides a single yardstick (pricing menu) for valuing all options on the same underlying security. The valuations on different contracts are consistent with one another, in the sense that they are all derived from the same yardstick. Even if the yardstick is wrong, the valuations remain consistent with one another. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

3 The classic perspective of derivative pricing The approach: Evaluating payoffs on (risk-neutral) dynamics Specify the full risk-neutral dynamics for the underlying security price. Take expectation of terminal payoffs according to the same dynamics. Example: Time changed Lévy processes The emphasis: cross-sectional consistency The dynamics specification provides a single yardstick (pricing menu) for valuing all options on the same underlying security. The valuations on different contracts are consistent with one another, in the sense that they are all derived from the same yardstick. Even if the yardstick is wrong, the valuations remain consistent with one another. They are just consistently wrong. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

4 Limitations of the classic perspective It is hard to get everything under one roof. Pricing long-dated contracts requires unrealistically long projections. Short-term variations often look incompatible with long-run (stationarity) assumptions (Giglio & Kelly, 2017) It is not always desirable to get everything under one roof Pooling can be useful for information extraction, but can be limiting for individual contract pricing/investment Disturbance on one contract affects everything else. Disengagement between Q-quants and P-quants Q: Pricing guarantees cross-sectional consistency, but P: Time-series variation, daily P&L attribution, risk management Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

5 A new decentralized pricing framework What do investors want? Different types of investors have different domain expertise and focus on different segments of the markets. They need a model that can lever their domain expertise Short-term investors do not care much about the terminal payoffs: All she worries about is the P&L on her particular investment over the short investment horizon. Long-term investors must also worry about their daily P&L fluctuation. We develop a new framework that generates pricing implications based on short-term (daily) investment risk on a particular set of contracts. The short-term focus allows the investor to make near-term risk projections without worrying about long-run dynamics. The particular-set focus allows the investor to make projections and draw inferences only on the contracts she has domain expertise on, without generating implications on other contracts. Bridge (P) statistical risk-return analysis with (Q) risk-neutral option pricing. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

6 Different transformation can highlight different insights BMS is a common/nice choice, but one can explore others... Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28 Value representation of an option contract Imagine we have a position in a single vanilla option contract. We represent the option via the BMS pricing equation, B(t, S t, I t ; K, T ) (K, T ) capture the contract characteristics. The value of the contract can vary with calendar time t, the underlying security price S t, and the option s BMS implied volatility I t. Appropriate representation/transformation is important for highlighting information/risk source, stabilizing quotation... The at-the-money implied volatility term structure reflects the return variance expectation. The implied volatility smile/skew shape across strike reveals the non-normality of the underlying distribution. As long as the option price does not allow arbitrage against the underlying and cash, there existence of a positive I t to match the price. (Hodges, 96)

7 P&L attribution of a short-term option investment With the BMS representation, we start by performing a short-term (e.g., daily) P&L attribution analysis on the option position. We expand the option value change via Ito s lemma: db = [[B t dt + B S ds t + B I di t ] ] B SS (ds t ) B II (di t ) 2 + B IS (ds t di t ) + Jump, The expansion can stop at second order for diffusive moves. The Ito expansion underlies the brilliant idea of dynamic hedging. Diffusive risks can be removed via frequent updating of hedging positions with a few instruments. It is harder to attribute jump risk via expansion. (A lot of) options positioning is needed to hedge jump risk. The effects of large jumps are better analyzed under scenario analysis/stress tests. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

8 Risk-neutral valuation of the investment return Take risk-neutral expectation on the investment P&L, and divide by the investment horizon dt, we can attribute the expected investment return as: [ ] db E t = B t + B I I t µ t + 1 dt 2 B SSSt 2 σt B II It 2 ωt 2 + B IS I t S t γ t (1) µ t, σt 2, ωt 2, γ t denote the time-t conditional mean, variance, and covariance: [ ] [ ( ) ] 2 di µ t E t t I t /dt, σt 2 ds E t t S t /dt, ω 2 t E t [ ( di t I t ) 2 ] [( )] ds /dt, γ t E t t S t, dit I t /dt. With zero financing assumption, the risk-neutral expected return on the security is 0, so is the expected return on the option: 0 = B t + B I I t µ t B SSS 2 t σ 2 t B2 II ω 2 t + B IS I t S t γ t. This can be regarded as a pricing equation: The value of the option must satisfy this equality to exclude dynamic arbitrage. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

9 P&L attribution relative to pricing expectation It is more informative to represent the P&L attribution in terms of surprises against pricing expectations: ( ) ds db = B S S t di t S t + B I I t t I t µ t dt ( ( ) ) B SSSt 2 ds t S t σ 2 t dt ( ( ) ) B II It 2 di t I t ω 2 t dt (( ) ) ds + B IS S t I t di t t S t I t γ t dt (B S, B I, B SS, B II, B IS ) are risk exposures, surprises are risk. Subjective pricing (P Q) expectation difference defines alpha (or rp). Deviation of realization from expectation defines risk. Price move (ds t ) is the main risk source BMS delta hedge can remove over 90% of variation (Figlewski, 89). Vega hedge using ATM contracts of the same maturity can remove most vol risk. Implied vols at same maturity move mostly together. What to hedge/bet is a risk-return trade-off problem. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

10 Investment returns per unit of dollar gamma exposure BMS risk exposures can be all represented in terms of dollar gamma: B t = 1 2 I 2 t B SS S 2 t, B I I t = I 2 t τb SS S 2 t, B IS I t S t = ( k I 2 t τ ) B SS S 2 t, B II I 2 t = ( k I 2 t τ ) ( k 1 2 I 2 t τ ) B SS S 2 t. k ln K/F t the relative strike of the option z ± ( k ± 1 2 I 2 t τ ) convexity-adjusted moneyness Divide the price change by dollar gamma, ( ) ( ( db B SS = l(z St 2 ) dst S t + It 2 di τ t I t µ t dt + 1 ds t 2 S t ( ( ) z di +z t I t ω 2 t dt ) ) 2 σ 2 t dt ) (( ) ds + z t di t + S t I t γ t dt ), (2) where l(z ) B S B SS S t = N( z I t τ )I t τ/n( z I t τ ) With the dollar gamma scaling, the option investment return exposures can be written explicitly as a function of the implied variance level and the two moneyness (z +, z ) measures. Contracts at z + = 0 has no vanna or volga exposure. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

11 Moment-based implied volatility valuation Start with the pricing relation, 0 = B t + B I I t µ t B SSS 2 t σ 2 t B2 II ω 2 t + B IS I t S t γ t, Plug in the partial derivatives and rearrange, It 2 = [ 2τµ t It 2 + σt 2 ] [ + 2γt z + + ωt 2 ] z + z. This can be regarded as a local no-arbitrage implied volatility valuation on a particular option contract. The no-arbitrage condition determines the fair implied volatility level I t of the option contract in terms of the first and second risk-neutral moments of the joint move ( dst S t, dit I t ): (µ t, σt 2, ωt 2, γ t ). Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

12 Moment-based implied volatility valuation It 2 = [ 2τµ t It 2 + σt 2 ] [ + 2γt z + + ωt 2 ] z + z The no-arbitrage condition determines the time-t fair implied volatility level I t of the option contract in terms of the time-t conditional first and second risk-neutral moments of the joint move ( dst S t, dit I t ): (µ t, σt 2, ωt 2, γ t ). It ties the current value of the contract to the current moment conditions, or risk profile of the contract, with no reference to how the risk profile varies over time, how it behaves in the long run how the risk profile is estimated/derived/modeled The valuation is not linked to any other option contracts, unless we specifically want to make a linkage, e.g., via common factor structures on the risk profiles of these contracts. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

13 Common factor structures on implied volatilities It 2 = [ 2τµ t It 2 + σt 2 ] [ + 2γt z + + ωt 2 ] z + z The above pricing equation links the implied volatility level of one option contract to its own risk profile. Joint valuation of the implied volatility surface requires additional commonality assumptions on their risk profiles. Joint valuation across names requires common factor loading structures across names, e.g., BARRA. One can naturally bring in BARRA-type risk structures to capture the delta risk in this framework. One can develop similar risk/alpha structures for volatility (e.g., Bollerslev, Hood, Huss, Pedersen (2016), Israelov & Kelly (2017)). These option return risk structures have direct implications on the pricing of implied volatility levels. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

14 Common factor structures on an implied volatility surface It 2 = [ 2τµ t It 2 + σt 2 ] [ + 2γt z + + ωt 2 ] z + z Carr&Wu (JFE, 2016): One common-factor governs the short-term movements of the whole surface di (K, T )/I (K, T ) = e ηt(t t) (m t dt + w t dz t ) for all (K, T ) That allows them to characterizes the whole implied volatility surface at any given time with five states (m t, w t, η t, ρ t, σ t ) with no additional model parameters. PCA often identifies 3 major sources of variation on the surface: The overall volatility level Term structure variation (short v. long-dated contracts) Implied volatility smile/skew variation along moneyness (OTM put v. straddle v. OTM call) We highlight how our framework can be applied to any particular segment of the surface. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

15 At-the-money contracts We define at-the-money as contracts with z + = 0, or k = 1 2 I 2 t τ Such contracts have zero volga and vanna. The implied volatility level only depends on its expected move (µ t ), but not its variance/covariance: A 2 t = 2τµ t A 2 t + σ 2 t. Applications: 1 contract: Infer risk-neutral drift from the slope against instantaneous variance: µ t = A2 t σ2 t 2A 2 t τ. 2 contracts: Infer locally common drift from the slope of nearby at-the-money contracts: µ t = A 2 t (τ 2 ) A 2 t (τ 1 ) 2(A 2 t (τ 2 )τ 2 A 2 t (τ 1 )τ 1 ). Locally constant drift leads to locally linear term structure Drift estimates can be tied to local linear (nonparametric) regression fitting of the term structure. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

16 The at-the-money implied variance term structure A 2 t = 2τµ t A 2 t + σ 2 t. Build the at-the-money term structure function over the whole maturity range or a particular maturity segment based on commonality assumptions on their drifts. Example: All ATM implied variance share the same mean-reversion and proportional variance structure at time t: da 2 t (τ) = κ t ( θ t A 2 t (τ))dt + 2ω t A 2 t (τ)dz t Implication on the implied variance term structure function: A 2 t (τ) = 1 e κtτ κ t τ ( σ 2 t θ t ) + θt. Assumptions similar to standard stochastic volatility models lead to similar term structure implications, but still with more freedom: If the common dynamics assumptions/estimates change tomorrow, so will the term structure. Different from standard option pricing models, the current term structure pricing does not depend on its future risk profile variation. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

17 The implied volatility smile at a single maturity To highlight the implied volatility smile at a certain maturity, we can vega hedge the option with the at-the-money contract of the same maturity, assuming they strongly co-move. Take the ATM implied variance A 2 t as given and focus on the implied variance deviation of other contracts from the ATM variance level. Assume proportional drift at the same maturity: µ t I 2 t = µ t A 2 t. Plug in the at-the-money implied variance to highlight the implied volatility smile effect at the single maturity, I 2 t A 2 t = 2γ t z + + ω 2 t z + z, (3) The smile slope is determined by the covariance rate γ t of the contract. The smile curvature is determined by the variance rate ω 2 t Application: Assuming common proportional implied volatility movements within a particular moneyness range, we can identify the common moment conditions (ω 2 t, γ t ) by regressing I 2 t A 2 t against (2z +, z + z ). Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

18 Application: Extrapolate observed smile to long maturity Exchange-traded contracts are short-dated, but many OTC deals are long term. How to extrapolate exchange quotes to price long-dated OTC deals? The classic approach Calibrate a standard stochastic vol model to observed quotes (say up to 2 years), price long-dated options (say 10 years) with the model parameter estimates. The at-the-money vol level will flatten out to the long-run mean (roughly matching the 2-year ATM vol level) The implied volatility smile/skew will flatten due to central limit theorem (and the fact that volatility converges to its long-run mean). Our pricing approach: If the at-the-money vol is flat extrapolated from 2 to 10 years, the 2-year and 10-year at-the-money implied variance will vary by the same amount same (γ t, ω 2 t ). Empirically, we do observe long-dated contracts vary much more than suggested by estimated mean reversion processes. The smile/skew shape must be extrapolated from 2 to 10 years as well! Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

19 Empirical analysis We use exchange-traded SPX options to Distinguish between changes in floating implied vol series and implied volatility changes of a fixed option contract The latter involves sliding along time to maturity (shrinking) and moneyness (as spot moves). Traditional models are more related to the dynamics of the former, our approach is based on the moments of the latter. P&L attribution: What percentage of variation is explained by which type of risk exposure? Verify the linkage between the moment conditions and the implied volatility surface shape Implied volatility drift v. the at-the-money term structure Variance/covariance estimates v. the implied volatility smile What can we do when P and Q are misaligned? Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

20 Constructing floating series of IV levels and changes The exchange-listed options have fixed strike and expiry. The standard approach is to interpolate the implied volatilities of these contracts to obtain floating series at fixed time to maturities and moneyness OptionMetrics provides floating implied vol series at 1,2,3,6,12 months and different deltas. OTC market often provide indicative quotes on floating time-to-maturity and relative strike grids. Analyzing variations of these floating series provides insights for option pricing modeling Our model depends on the implied volatility variation of a fixed contract, making it necessary to construct implied volatility changes of fixed contracts At each date, construct log implied volatility change over the next business date on each option contract. Interpolate (via Gaussian kernel smoothing) the changes to floating time to maturity (1,2,3,6,12 months) and moneyness points (k/ τ = 0, ±10%, ±20%, and z + = 0). Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

21 At-the-money implied volatility level and changes Maturity Panel A: At-the-money implied volatility level Mean Std Panel B: Daily percentage change of floating series Mean Std Auto Corr Panel C: Daily percentage changes of fixed contracts Mean Std Auto Corr Floating series has little drift; fixed contracts can have larger drift. Floating series vary much more, and more negatively correlated with spot, and more mean-reverting than fixed contracts Sticky delta, sticky strike, or something in between? Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

22 Extract risk-neutral drift from the ATM term structure Maturity Panel A: Summary statistics of risk-neutral drift estimates Mean Std Min Max Auto Panel B: Average difference between P and Q estimates P Diff Risk-neutral rate of change estimates vary greatly over time, from highly negative to highly positive. The estimates show high persistence. Compared to statistical average estimates, risk-neutral drifts on average are higher at short term, but lower at long term. Short short term, long long term? (Egloff, Leippold, Wu (2010)) Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

23 Forecast future movements with the term structure da t+1 = α µ t + βda t Maturity α β R 2, % ( 2.35 ) ( 4.47 ) ( 3.24 ) ( 2.32 ) ( 3.74 ) ( 1.18 ) ( 3.22 ) ( 0.91 ) ( 1.21 ) ( 2.20 ) 0.32 The term structure can be used to predict future rate of change at most maturities. Direction prediction remains difficult... Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

24 P&L variance attribution for at-the-money options Call options Put options Maturity Delta risk Vega Gamma Delta hedge can remove % of the risk. Vega hedge contributes to another 1-3%. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

25 P&L variance attribution for at-the-money options Call options Put options Maturity Delta risk Vega Gamma Delta hedge can remove % of the risk. Vega hedge contributes to another 1-3%. With delta and vega hedge, we can level our domain expertise on smile construction and exploit misalignment between the observed smile and our implied volatility variance/covariance forecasts. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

26 Extract variance and covariance from the smile γ t Maturity A. Risk-neutral estimates extracted from the implied variance smile Mean Stdev B. Rolling-window statistical estimates on at-the-money options Mean Stdev C. Correlation Correlation ω 2 t Skew/covariance are highly correlated, not so for the variance/curvature estimates Potential for enhanced statistical forecasts with observed smile. Variance/covariance extracted from the smile are on average larger in magnitude than statistical estimates potential trading opportunities. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

27 Predicting realized variance/covariance rates Maturity Intercept HE RN R 2, % A. Predicting covariance rate γ t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) B. Predicting variance rate ω t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Skew is particular useful in enhancing prediction on covariance rate. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

28 Statistical arbitrage trading on the smile Take vega-neutral spread positions (against the ATM contract), based on the difference between the observed smile and the smile constructed from the forecasted variance/covariance rates (γ t, ω t ). Perform daily delta hedge Annualized information ratio from the investments: τ / k τ -20% -10% 0 10% 20% Sum The trades are not that profitable at short maturity (one month) Missing contribution from jumps can play a large role at short maturity. The trades are very profitable at long maturity Variance/covariance drives the main delta-hedged P&L of long-dated smiles. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

29 Concluding remarks Traditional pricing models tend to go bottom-up by specifying the dynamics of some basis reference, from which one derives the value for all contracts. We propose a complimentary, decentralized, top-down pricing approach by starting with the particular contract one is interested investing in and by considering its risk profile/exposure one day at a time. It sounds less ambitious, generates narrower implications, but the implications it generates tend to be more relevant and more accurate. For option contracts on the same underlying, one can build smoothed implied volatility surfaces (or a particular segment of the surface) via commonality assumptions, such as locally parallel movements. Across names, one can build up the risk structure via cross-name commonality assumptions, such as industry and other firm characteristics (e.g., BARRA). The framework tightly links P&L analysis/risk management to pricing. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28

30 Future research Theoretical developments Other securities: Bond yields,... Other representation: Bachelier, SVG How to characterize the short-run contribution of jumps? Empirical analysis Implied volatility surface construction with sparse data With common variance/covariance rates for nearby contracts Risk reversal and butterfly trade construction that singles out the difference between risk-neutral and statistical variance/covariance forecasts Barra-type factor structure on single-name option returns Firm characteristics, linked to volatility level projections and their variance/covariance rates. The risk structure has direct implications on the pricing of the implied vol level and spreads source of statistical arbitrage opportunities. Carr and Wu (NYU & Baruch) P&L Attribution and Option Pricing March 23, / 28