Equity-Based Insurance Guarantees Conference Nov. 5-6, 2018 Chicago, IL

Being Honest About Pricing: What the Market Can Tell You and Why it Matters

Jeff Greco

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Sponsored by Being Honest About Option Pricing: What the Market Can Tell You and Why it Matters

JEFF GRECO, FRM Senior Director – Head of Strategy Research Portfolio Management Group Milliman Financial Risk Management LLC

November 5, 2018 Limitations & Disclosures

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2 Outline 1. Market Prices and Option Models 2. Problem Setup 3. Model Calibration 4. Model Application

3 Market Prices and Option Models

4 Fundamental Theorem of Asset Pricing (FTAP) • Arbitrage-free market prices equivalent to existence of a state price density—known as the risk-neutral measure • Central theory backing /option pricing models ℚ • Tells us how a model relates to market prices and why it is a reasonable thing to build

5 Selecting an Option Pricing Model • Which models fit the market? • How do we choose? • What about calibration?

 Black-Scholes (classic, practitioner)  Jump Models (Bates, Variance Gamma, …)  …  Stochastic (Heston, SVI, …) 

6 What Can the Market Tell Us? • FTAP informs us risk-neutral density alone determines prices = 𝑑𝑑ℚ � −𝑟𝑟𝑟𝑟 + 0 = 𝑇𝑇 𝐶𝐶 𝐾𝐾 𝑒𝑒 𝔼𝔼 +∞𝑆𝑆 − 𝐾𝐾 −𝑟𝑟𝑟𝑟 • Therefore (static) market𝑒𝑒 �prices𝑆𝑆 𝑇𝑇yield− 𝐾𝐾 information 𝑑𝑑� 𝑆𝑆𝑇𝑇 about only specific marginal𝐾𝐾 distributions of the risk- neutral measure

7 What the Market Cannot Tell Us • A model can be judged by the fit of its generated probability distributions—market consistent pricing • However typically models provide much more • Underlying stochastic behavior • Joint dynamics, e.g. correlation • Constructive replicating strategies across multiple expirations • Dynamic hedge ratios • prices • Excluding further assumptions, market prices alone offer no assistance in targeting these properties

8 Where to Start? • Liquid quotes typically confined to vanilla options • Recover marginal densities across expirations , consistent with vanilla quotes (a.k.a. the volatility surface) 𝑑𝑑� 𝑆𝑆𝑇𝑇 𝑇𝑇 • Impose assumptions minimally but as needed to improve results • Extract as much information as possible • Become better equipped to select and evaluate a model

9 Problem Setup

10 Fitting the Volatility Surface • Requires inverting non-linear mapping from model parameters to prices • Significantly smaller number of model parameters than market prices • Calibration thus an overdetermined inverse problem

11 Calibration Challenges • Fitting all prices infeasible • Computationally expensive • Enforcing no-arbitrage conditions can be difficult • Outlier identification is hard • Not identifying outliers can generate poor outcome • Accounting for dividends • Estimating financing interest rates

12 Possible Compromises Used • Quotes can be thinned for computational speed • Full bid/ask information can be discarded and replaced by mid prices for tractability • Break calibration into smaller separate problems for each , isolating full reach of each quote’s information across entire volatility surface

13 Model Calibration

14 Financing Rates and Dividends

• Bounds from linear program (linear inequalities in discount factors and forwards) • Interpolate financing spread above risk free rate and dividend yield curves

15 Outlier Identification • Search for existence of static arbitrage violations • If violations detected then how are bad apples separated from the bunch? • Can be identified by another linear program • Minimally perturb bids/asks (linear: ) • So that arbitrage eliminated (linear: 0 ) • So that calendar arbitrage eliminated𝐵𝐵 (linear:𝑇𝑇 ≤𝐶𝐶 ≤ 𝐴𝐴 ) 𝑑𝑑ℚ ≥ 1 2 𝑟𝑟𝑇𝑇1 𝑟𝑟𝑇𝑇2 𝑇𝑇 ≤ 𝑇𝑇 ⇒ 𝑒𝑒 𝐶𝐶𝑇𝑇1 𝐾𝐾 ≤𝑒𝑒 𝐶𝐶𝑇𝑇2 𝐾𝐾 16 Example: Fitting Heston to Market Mids

Quotes thinned, bid/ask replaced by mid, and independent Heston calibration per expiration (improves fit but can be difficult to enforce calendar spread no-arbitrage)

17 Example: Fitting Heston to Market Mids

Total variance lines cross, implying calendar spread arbitrage

18 Non-Parametric Fit: No Compromises • Adds degrees-of-freedom • Can eliminate misfit to market and arbitrage violations • Must take care to avoid overfitting • Very sensitive to outliers, elimination is a must! • Important to use guiding principle as an offset, such as smoothing/limiting neighboring price differences • Helpful to balance market fit with smoothness

19 Example: Non-Parametric Fit

• Illustrates single expiration date • All bids/asks used • Butterfly and calendar spread no- arbitrage enforced • Outliers identified and eliminated • Single calibration across full volatility surface

20 Example: Non-Parametric Fit, Full Surface

21 Model Application

22 Hedge Ratio (In)consistency • Euler Theorem implies delta = invariant across strike-space homogeneous models 𝜕𝜕𝐶𝐶 =∆ 𝜕𝜕𝑆𝑆+ • However for models including addition𝜕𝜕𝐶𝐶 𝜕𝜕𝐶𝐶 state variables (e.g. ) as well𝐶𝐶 as 𝑆𝑆inhomogeneous𝜕𝜕𝑆𝑆 𝐾𝐾𝜕𝜕𝐾𝐾 models, effective delta varies = + + 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜎𝜎 • Optimal replication/minimum variance� hedging relatively consistent across jump∆-free𝜕𝜕 models𝜕𝜕 𝜕𝜕𝜎𝜎 𝜕𝜕for𝜕𝜕 short⋯ expiration options that are near-the-money

23 Pricing Exotics • Replicating strategies closely linked to exotic option pricing • Exotic price discrepancies can exist between models even when calibrated to the same (vanilla) volatility surface and when capturing comparable hedge ratios • Therefore additional information needed to narrow down exotic pricing

24 Risk-Neutral Volatility Term Structure

• Three days before Brexit market • One day before Brexit market shows anticipates elevated volatility for the possibly bimodal distribution upcoming vote

25 References

26 References [1] Andersen, Torben G., and Oleg Bondarenko. Construction and interpretation of model-free . No. w13449. National Bureau of Economic Research, 2007. [2] Andersen, Torben G., Nicola Fusari, and Viktor Todorov. "Short‐Term Market Risks Implied by Weekly Options." The Journal of 72, no. 3 (2017): 1335-1386. [3] Ayache, Elie, Philippe Henrotte, Sonia Nassar, and Xuewen Wang. "Can anyone solve the smile problem?." The Best of Wilmott (2004): 229. [4] Carr, Peter, and Dilip Madan. "Towards a theory of volatility trading." Volatility: New estimation techniques for pricing derivatives 29 (1998): 417-427. [5] Corlay, Sylvain. "B-spline techniques for volatility modeling." (2016). [6] Dupire, Bruno. "Pricing with a smile." Risk 7, no. 1 (1994): 18-20. [7] Gatheral, Jim. "A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives." Presentation at Global Derivatives & Risk Management, Madrid (2004). [8] Gatheral, Jim, and Antoine Jacquier. "Arbitrage-free SVI volatility surfaces." Quantitative Finance 14, no. 1 (2014): 59-71. [9] Heston, Steven L. "A closed-form solution for options with stochastic volatility with applications to bond and currency options." The review of financial studies 6, no. 2 (1993): 327-343. [10] Klassen, Timothy. "Necessary and Sufficient No-Arbitrage Conditions for the SSVI/S3 Volatility Curve." (2016). [11] Lee, Roger W. "The moment formula for implied volatility at extreme strikes." Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics14, no. 3 (2004): 469-480. [12] Monteiro, Ana Margarida, Reha H. Tütüncü, and Luís N. Vicente. "Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity." European Journal of Operational Research187, no. 2 (2008): 525-542.