VOLATILITY SURFACE CALIBRATION IN ILLIQUID MARKET ENVIRONMENT László Nagy Mihály Ormos Department of Finance 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, stochastic volatility, arbitrage free consequent. Otherwise, calendar and butterfly 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 option 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 Local Volatility and maturities. Thus, we can observe only some points model (Dupire 1994). of the implied volatility surface. It is known that vanilla prices are arbitrage free hence exotic option 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-moneyness and 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 √ (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, just using close ATM prices to fit SVI and then the counter-clockwise rotation, 푚 the location and the extrapolate along wings. The main drawback of this at the money curvature of the smile. The SVI model has method is that OTM short 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 (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 − (0, 푆0, 휏)Ψ(푆0, 휏)) + (0, 푆0, 휏) + 풪(k ) 2 2((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, , 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.