SURFACE CALIBRATION IN ILLIQUID MARKET ENVIRONMENT

László Nagy Mihály Ormos Department of Budapest University of Technology and Economics Magyar tudósok körútja 2, Budapest H-1117, Hungary E-mail: [email protected], [email protected]

KEYWORDS Besides, our risk neutral risk assessment should be SVI, SSVI, gSVI, , free consequent. Otherwise, calendar and arbitrage pricing opportunities appear;

ABSTRACT C(K, τ1) < C(K, τ2) if and only if τ1 < τ2 (1)

In this paper, we show the fragility of widely-used 퐾3−퐾1 K2−K1 C(K1, τ) − C(K2, τ) + C(K3, τ) > 0 (2) Stochastic Volatility Inspired (SVI) methodology. K3−K2 K3−K2 Especially, we highlight the sensitivity of SVI to the fitting penalty function. We compare different weight where C(K, τ) represents the price of a European call functions and propose to use a novel methodology, the with strike 퐾 and maturity 휏. implied vega weights. Moreover, we unveil the relationship between vega weights and the minimization Considering the behavior of compound interest the task of observed and fitted price differences. Besides, Black-Scholes log-normal model is applicable. we show that implied vega weights can stabilize SVI surfaces in illiquid market conditions. dSt = rStdt + σStdWt (3)

INTRODUCTION Also we have seen the volatility surface is not flat hence this model needs some adjustments. The most Vanilla options are traded with finite number of strikes straightforward correction leads to the and maturities. Thus, we can observe only some points model (Dupire 1994). of the surface. It is known that vanilla prices are arbitrage free hence traders dSt = rStdt + σ(St, t)StdWt (4) would like to calibrate their prices to vanillas (Dupire 1994). The main difficulty is that calibration methods However, calculating implied and realized volatilities need the implied volatility surface. To overcome this show that Local Volatility is only an idealized perfect problem we have to construct an arbitrage free surface fit, because volatility is stochastic. from the observed points (Schönbucher 1998, Gatheral 2013). In this paper we provide a robust arbitrage free dS = rS dt + σ푆푝표푡푆 dW (5) surface fitting methodology. t t 푡 푡 t dσ퐵푆 = u(k, t)dt + γ(k, t)dW + ∑푛 v (k, t) dWi Chapters are structured as follows: Section 2. is a brief t t i=1 i t overview of SVI. In Section 3. we compare the different 1 where W,W1, … Wn are independent Brownian motions, weight functions and present our implied vega weight 퐿 퐵푆 methodology. In Section 4. we summarize the findings. 푘 is the log- and 𝜎 denotes the Black- Scholes implied volatility.

SVI Schönbucher showed that the spot volatility can not be After the Black Monday in 1987, traders behavior an arbitrary function of implied volatility, because of changed. Implied volatility skew became more the static arbitrage constraints. pronounced. Risk aversion incorporated in the volatility. Hence, risk transfers between tenors and strikes get Spot −훾푘 σ = BS ± more sophisticated. σ (k,T) 2 The changes were in line with human nature, because 퐵푆 푘 푛 2 2 √𝜎 (k, T) + 2 (∑푖=1 vi − γ ) (6) people have different risk appetite in different tenors. (휎퐵푆(푘,푇)) Moreover, extreme high out of money implied volatilities are consequences of risk aversion and fear of Besides the arbitrage constraints, Heston's model sheds the unpredictable. more light on implied volatility modeling. Gatheral et

Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD) al. proposed the so called SVI (Stochastic Volatility 3. In addition, the set of traded strikes is not stable in Inspired) function to estimate all the implied volatility time. Therefore, the estimated surface will be surface; unstable in time.

σ퐵푆 = a + b (ρ(k − m) + √(k − m)2 + σ2) (7) Data truncation The simplest approach to solve the problem could be where 푎 controls the level, 푏 the slopes of the wings, 𝜌 just using close ATM prices to fit SVI and then the counter-clockwise rotation, 푚 the location and 𝜎 the extrapolate along wings. The main drawback of this at the money curvature of the smile. The SVI model has method is that OTM dated options contain the compelling fitting results, in addition it implies a static market anticipated tail risk information. Truncating the arbitrage free volatility surface. The only arguable step data stabilizes the surface and implies accurate long in the methodology is the model calibration. Kos et al. term fit, but underestimates tail risk hence underprices (2013) proposed to minimize the square differences exotic products. between observed and fitted volatility, while Homescu (2011) advised a square difference method. Square of price differences Nevertheless West (2005) applied vega weighted square The most popular optimization technique is minimizing volatility differences. Zelida system (2009) used total 퐿2 distances. The main drawback of this approach is implied variances. An other noticeable approach comes fitting to the mean, instead of the median which implies from Gatheral et al. (2013) who minimized squared outlier sensitivity. price differences, but there are further regression based models as well (Romo 2011). Vega weights In this article we propose a new absolute price difference based approach to stabilize SVI in illiquid In order to deal with the skew and price ambiguity we market conditions. propose to use a natural Gaussian based weight function. It turned out that truncating the data do not SENSITICITY ANALAYSIS OF SVI give the appropriate results. Therefore, we have to find a weight function which minimizes 퐿1 distance, At the money options are more liquid than far out of penalizes price ambiguity, but still able to use tail risk money options, hence the bid-ask spread widens along information. the wings. This implies larger price ambiguity of OTM option prices. In order to stabilize the implied volatility Fit BS min ∑휏∈퓣 ∑퐾∈퓚 w(K, τ) | σK,τ − σK,τ | (9) Fit 0 surface we have to penalize price ambiguity. σK,τ∈ C

Uniform weights Note that the above optimization problem is still not Highlighting the problem we can apply uniform general enough, because weight is a function of strike weights. This approach assumes that all of the and maturity. This incorporates a sticky strike information is equally relevant. Thus, we get the usual assumption. However, if we add the spot price 푆0 as square distance optimization task (Zelida 2009, Kos another independent variable to the weight, then we can 2013); get more general penalty functions.

Fit BS 2 min ∑ ∑ w(K,S , τ) | σ − σ | (10) Fit BS Fit 휏∈퓣 퐾∈퓚 0 K,S0,τ K,S0,τ min ∑휏∈퓣 ∑퐾∈퓚( σK,τ − σK,τ ) (8) σ ∈ C0 Fit 0 K,S0,τ σK,τ∈ C Our initial problem is to find an implied volatility where 퓚 and 퓣 represent the sets of the traded strikes BS Fit surface. This means that we would like to penalize and maturities, σK,τ is the implied and σK,τ is the fitted observed and fitted volatility differences. volatility. Practitioners need the surface for trading. Hence, they are interested in the dollar amount of the discrepancies BS 1. Note that for fixed maturity σK,T increases in |K − between fitted and observed volatilities. FT|. The volatility bid-ask spread also widens BS,Ask ∑ ∑ 퐹푖푡 퐵푆 along the wings. This implies that σK,T min 휏∈퓣 퐾∈퓚 |퐶퐾,푆0,휏 − 퐶퐾,푆0,휏| (11) 퐶퐹푖푡 ∈ C0 BS,Bid 퐾,푆0,휏 increases faster than σK,T hence defining the fair value of a deep OTM option from bid and ask After some calculations in Appendix, we can see that price is not straightforward. optimizing the price differences is approximately the same as optimizing vega weighted implied volatility 2. Furthermore, deep out of money implied differences. volatilities are usually higher than ATM volatilities. Therefore, uniform square penalty ∑ ∑ 퐵푆 Fit BS min 휏∈퓣 퐾∈퓚 풱K,S0,τ | σK,S0,τ − σK,S0,τ | (12) overfits the wings and underfits the ATM range. σFit ∈ C0 K,S0,τ

Dividing with 𝜎(0, 푆0, 휏) we get: 퐵푆 Using the definition of 풱K,S0,τ we get the price 2 2 difference implied weight function. −2푘(1 − 𝜎(0, 푆0, 휏)Ψ(푆0, 휏)) + 𝜎(0, 푆0, 휏) + 풪(k ) 2 2(𝜎(0, 푆0, 휏) + Ψ(푆0, 휏)푘 + 풪(k )) −qτ w(K, S0, τ) = S0e φ(d1)√τ (13) This function is rather constant if 푘 is small. Equation where φ(x) represents the standard normal distribution 14. also shows that for bigger |푘| values the weight 2 function, 푑1 is the standard notation from the Black- should decrease with approximately exp(−푘 ). Scholes formula and 푞 is the continuous dividends rate.

Figures 1: Vega weights as function of strike, and Figures 2: Implied weights as function of strike, volatility σ ∈ [5,15, … ,45], 푆 = 100, K ∈ [0, … ,200] 0 parameters: S0 = 100, σ0,100,1 = 0.2, K ∈ [0, … ,200], and T = 1 year slopes = (2% ,0.2%)

Supposing that 푞, 푟, 𝜎, 푆0, 휏 are fixed and using the Figure 2. unveils that vega weights take into account definition of 푑1 we get that the weight function has a wings, but the bigger the |푘| the larger the impact of the 푘 Gaussian shape in log moneyness 퐾 = 퐹푒 . exp(−푘2) term which balances the increasing OTM volatility. Hene, vega weights provide a balanced SVI 2 퐹 휎2 1 ln + 휏 fit. − ( 퐾 2 ) 1 2 휎√휏 2 푤(퐾) = S e−qτ e = 풪(푒−푘 ) (14) 0 2휋 Empirical test

This implies that we highly penalize fitting In order to lend more color to the fragility of fitting discrepancies in the ATM range, while we are lenient methodology we simulated illiquid market environment with deep OTM fits. by picking 5 data points in each slice from SPX The next step is to fix 푞, 푒, 푆 , 휏 and use the first order 15/09/2015 option data set (Gatheral 2013). To 0 highlight the outlier-sensitivity we stressed the volatility Taylor approximation of 𝜎(퐾, 푆0, 휏) around ATM log moneyness. of the last tenor (T=1.75), moneyness k=0.2 point by 10%. 푤(𝜎(푘)) = 2 2 2 1 −2푘+(휎(0,푆0,휏)+Ψ(푆0,휏)푘+풪(푘 )) 휏 − ( ) 푆 푒−푞휏 2 2(휎(0,푆 ,휏)+Ψ(푆 ,휏)푘+풪(푘2))√휏 0 푒 0 0 √휏 (15) 2휋

Hence we get;

4 w(σ(k)) ≈ 풪(푒−푘 ) (16)

ATM skew is represented by Ψ(푆0, 휏). The asymptotic behavior of 푤(𝜎(푘)) shows that the vega weighted implied volatility surface would be stable against extreme OTM implied volatilities. Moreover, it also can be seen that if 푘 is close to zero then for implied volatility skew and smile we get rather flat vega weights.

2 2 −2k + (σ(0, S0, τ) + Ψ(S0, τ)k + 풪(k )) = 2 2 −2k(1 − σ(0, S0, τ)Ψ(S0, τ)) + σ(0, S0, τ) + 풪(k ) (17)

C. Homescu. 2011. “Implied volatility surface: construction methodologies and characteristics”. arXiv:1107.1834v1 G. West. 2005. “Calibration of the SABR model in illiquid markets”. Applied Mathematical Finance 12(4):371-385 J. M. Romo. 2011. “Fitting the Skew with an Analytical Local Volatility Function”. International Review of Applied Financial Issues and Economics, Vol. 3, pp. 721-736 J. Gatheral; and A. Jacquier. 2013. “Arbitrage-free SVI volatility surfaces”. arXiv:1204.0646v4 L. R. Goldberg; M. Y. Hayes; O. Mahmoud; and Tamas Matrai. 2011. “Vega Risk in RiskManager Model Overview”. Research insight P. J. Schönbucher.1998. “A market model for stochastic implied volatility” Zeliade Systems. 2009. “Quasi-Explicit Calibration of Gatheral's SVI model”. Zeliade White Paper

László Nagy László Nagy is a PhD student at the Department of Finance, Institute of Business at the School of Economic and Social Sciences, Budapest University of Technology and Economics. His main area of research is financial risk measures and asset pricing. Laszló earned his BSc and MSc in Mathematics with major of Figure 3: fitted SVI surfaces to filtered SPX 15/09/2015 financial mathematics at the School of Natural Sciences data, solid lines: price difference based fit (implied vega at Budapest University of Technology and Economics. weights), dashed lines: square price difference based fit, He is teaching investments and working on his PhD dotted line: square volatility difference based fit thesis. Before his PhD studies he worked at Morgan Stanley on risk modeling. The results show that if we use 퐿2 optimization techniques then we overpenalize outliers. It also can be Mihály Ormos seen using absolute difference based optimization Mihály Ormos is a Professor of Finance at the makes the SVI fit stable. In illiquid market environment Department of Finance, Institute of Business at the it is crucial because using square difference based fits School of Economic and Social Sciences, Budapest only one outlier could have a huge impact on the University of Technology and Economics. His area of affected slice or even the all surface, thus destabilizing research is financial economics especially asset pricing, option prices. risk measures, risk perception and behavioral finance. He serves as one of the contributing editors at Eastern CONCLUSION European Economics published by Taylor and Francis. His teaching activities concentrate on financial We showed that the absolute price difference based SVI economics, investments and accounting. Prof. Ormos fitting methodology is able to stabilize the implied published his research results in Journal of Banking and volatility surface. Moreover, we shed some lights on the Finance, Quantitative Finance, Finance Research asymptotic behavior of the weights and displayed the Letters, Economic Modelling, Empirica, Eastern connection with vega weights. We also stressed that European Economics, Baltic Journal of Economics, absolute price difference based optimization do not PLoS One, Acta Oeconomica and Economic Systems assume any specific stickiness, hence it can be used in amongst others. every volatility regime. APPENDIX ACKNOWLEDGEMENTS 퐹푖푡 퐵푆 Mihály Ormos acknowledges the support by the János min ∑ ∑ |퐶퐾,푆0,휏 − 퐶퐾,푆0,휏| 퐶퐹푖푡 ∈ C0 Bolyai Research Scholarship of the Hungarian Academy 퐾,푆0,휏 휏∈퓣 퐾∈퓚 of Sciences. László Nagy acknowledges the support by −푞휏 푆푉퐼 퐵푆 = min ∑ ∑ |푆0푒 (Φ(푑 ) − Φ(푑 )) 퐶퐹푖푡 ∈ C0 1 1 the Pallas Athéné Domus Scientiae Foundation. This 퐾,푆0,휏 휏∈퓣 퐾∈퓚 research is partially supported by Pallas Athéné Domus 푟휏 푆푉퐼 퐵푆 − e 퐾(Φ(푑 ) − Φ(푑 ))| Scientiae Foundation. 2 2

REFERENCES A. Kos; and B. M. Damghani. 2013. “De-arbitraging With a Weak Smile: Application to Skew Risk”. WILMOTT magazine B. Dupire. 1994. “Pricing with a Smile”. Risk

퐵푆 푆푉퐼 퐾 𝜎 𝜎 퐵푆 푆푉퐼 퐵푆 ln + = min ∑ ∑ |풱퐾,휏(𝜎 − 𝜎 ) 퐹 2 퐶퐹푖푡 ∈ C0 −푞휏 퐵푆 휏 퐾,푆0,휏 = min ∑ ∑ | (푆0푒 휑(푑1 ) 휏∈퓣 퐾∈퓚 퐶퐹푖푡 ∈ C0 𝜎퐵푆𝜎푆푉퐼 휏 푆푉퐼 퐵푆 2 퐾,푆0,휏 휏∈퓣 퐾∈퓚 √ 𝜎 − 𝜎 + 풪 (( ) ) | 퐾 𝜎퐵푆𝜎푆푉퐼 𝜎퐵푆𝜎푆푉퐼√휏 ln − −푟휏 퐵푆 퐹휏 2 푆푉퐼 퐵푆 퐵푆 푆푉퐼 퐵푆 − 퐾푒 휑(푑2 ) ) (𝜎 − 𝜎 ) ≈ min ∑ ∑ |풱퐾,휏(𝜎 − 𝜎 )| 퐵푆 푆푉퐼 퐶퐹푖푡 ∈ C0 𝜎 𝜎 √휏 퐾,푆0,휏 휏∈퓣 퐾∈퓚 2 𝜎푆푉퐼 − 𝜎퐵푆 + 풪 (( ) ) | Note that 풱 푖푠 표(√휏), thus options with short expiry are 𝜎퐵푆𝜎푆푉퐼√휏 not vega sensitive. 퐾 𝜎퐵푆𝜎푆푉퐼 ln + 퐵푆 퐹휏 2 = min ∑ ∑ | (풱퐾,휏 퐶퐹푖푡 ∈ C0 𝜎퐵푆𝜎푆푉퐼√휏 퐾,푆0,휏 휏∈퓣 퐾∈퓚 퐾 𝜎퐵푆𝜎푆푉퐼 ln − 퐵푆 퐹휏 2 푆푉퐼 − 풱퐾,휏 ) (𝜎 𝜎퐵푆𝜎푆푉퐼√휏

2 𝜎푆푉퐼 − 𝜎퐵푆 − 𝜎퐵푆) + 풪 (( ) ) | 𝜎퐵푆𝜎푆푉퐼√휏