CONSTRUCTING NO-ARBITRAGE VOLATILITY CURVES IN LIQUID AND ILLIQUID COMMODITY MARKETS Financial Mathematics Modeling for Graduate Students-Workshop January 6 January 15, 2011 MENTOR: CHRIS PROUTY (Cargill) Authors: Arun Abraham Fanda Yang Feng Wan Leigh Lommen Shu-Ting Chiu Sundeep Bhimireddy Yuchuan Li
Abstract In options markets where there is an implied volatility smile, Black-Scholes presents serious limitations due to the implicit assumption of constant volatility. To address this limitation, one could consider backing out the implied volatility using market data and the Black-Scholes model but this may still be inadequate if the data is stale and/or the options are illiquid. An examination of alternative models has been the subject of many studies and manuscripts among quantitative finance academics and practitioners. In this paper, we will discuss one such methodology, the Stochastic Volatility Inspired (SVI) model (Gatheral, 2006) in commodity markets and the SPX. Further, we will illustrate the pricing of a digital option using SVI and compare it to the analytical Black-Scholes price, as well as the fluctuation of this difference with respect to the moneyness of the option Finally a three dimensional volatility surface is constructed via the SVI methodology.. 1. Introduction The SVI (Gatheral, 2006) model utilizes a parametric formula to fit the observed data to a volatility skew: { } 2 2 2 σ SVI = var( k; a, b, σ, ρ, m) = a + b ρ( k m) + ( k m) + σ, where: a gives the overall level of variance b gives the angle between the left and right asymptotes, where the asymptotes are var ( k; a, b, σ, ρ, m) = a b(1 ρ)( k m) L var R ( k; a, b, σ, ρ, m) = a + b(1 ρ)( k m) k is the log (K/F) or log (Strike Price/Futures Price) σ determines the smoothness of the vertex ρ determines the orientation of the graph and m translates the graph from left to right. Note, that as options become progressive out-of-the-money (either for calls or puts), the variance increases linearly with k. Examples of changes of the parameters are shown below:
To obtain the exact values of the parameter to graph the curve, we minimized the following objective function: min ( σ ) 2 SVI σ Mkt BS Some possible constraints used to implement the model s fit would be as followed: 4 a > 0, b > 0, ρ < 1 & b(1 + ρ ), (Friz, Gatheral, 2004) T where the last is necessary to prevent vertical arbitrage. Data from the agricultural commodity markets for futures options on corn, live cattle, and milk were first obtained. Using the Black s model for Futures Options, we backed out the implied volatility and plotted the available points. Special care was taken to only use the out-of-the-money options since they are more representative of the implied skew. The fact that out-of-the-money options are more representative of the volatility skew is due to the economic reason that it is frequently cheaper for traders to construct an in-the-money option via replication. Why pay the expensive premium for an in-the-money option when it can be replicated by an out-of-the-money option and the underlying contract? Unfortunately, since the commodity data was on several different dated underlying maturities, only numerous skews could be constructed and not a single volatility surface.
Implied volatility was calculated from the Black-Scholes model using both the Newton-Raphson and bisection methods. Under the Newton-Raphson, the method is extremely sensitive to the initial guess of x 0 in the forumula: x = x 1 0 The process becomes an iterative process and generalizes to: x = n+ 1 x n f ( x0) f '( x ) 0 f ( xn) f '( x ) Using this method across several different strikes and maturities required a constant adjustment of the initial guess to bring the convergence to a non-negative value (as implied volatilities are always positive). Ultimately, the Newton- Raphson method was abandoned for the bisection method of obtaining implied volatility in an effort to reduce time via automation. The following diagrams illustrate the volatility skews of the different futures maturities and different commodities markets. The green circles denote implied vols obtained via the bisection iterative procedure with the Black-Scholes model, while the blue lines indicate the volatility skew obtained by the SVI methodology via minimizing the aforementioned objective function. Plots of implied volatility skews vs. SVI volatility skews for Corn: n
Plots of implied volatility skews vs. SVI volatility skews for Live Cattle:
Plots of implied volatility skews vs. SVI volatility skews for Milk: 2. Binary Options A binary (or digital) option is an option where the payoff is either a fixed amount of cash (or fixed amount of an asset) or nothing. Analytically, a binary option can be priced using the Black-Scholes model. The following formulae are for Call and Put, respectively: r* T C = e Φ d2 ( ) rt P = e Φ( d ) As an exercise in comparing the Black-Scholes volatility skew with the SVI volatility skew, we contrasted the price discrepancy of the analytical solution of a digital Call and Put with the SVI priced solutions at different areas along the curve. Note, to price a digital call option using the SVI methodology, it must be done via the replication of a Call bull spread ( ) ( ) lim C K ε = C K ε 0 ε 2
= dcv Cv σ dcv Vega σ1 σ 2 = dk σ K dk K K 1 2 (Taleb, 1997) Note that the last equation condenses to a multiplicative function of Vega and the slope of the volatility skew. Thus, pricing the option using SVI and replication is highly dependent on the moneyness of the option and the slope of the skew. Please see examples of these issues in the following graphs. At-The-Money (ATM), OTM, and deep OTM comparison of digital calls via the 2 methods and their accompanying skew: 3 CZ0 0.4 At the money (Corn) 2.5 0.35 Vol (%) 2 1.5 1 Digital option price 0.3 0.25 0.5 0.2 Replication Black Scholes 0-2 -1.5-1 -0.5 0 0.5 1 log(k/s) 10-3 10-2 10-1 10 0 10 1 10 2 TickSize 0.035 Moderately out of the money (Corn) x 10-3 Deep out of the money (Corn) 0.03 4 3.5 Digital option price 0.025 0.02 Digital option price 3 2.5 2 0.015 Replication Black Scholes 1.5 Replication Black Scholes 0.01 10-3 10-2 10-1 10 0 10 1 10 2 TickSize 1 10-3 10-2 10-1 10 0 10 1 10 2 TickSize Note the far left side of the graphs where the tick size is small. When the tick size is small, there is a good fit ATM, an even better fit at a moderately OTM digital, and a poor fit when the option is deep OTM. This reinforces the graph of the implied volatility versus the SVI fit where the vertices are not exact fits, but just a little OTM volatilities of both methods give extremely similar results. 3. Volatility surface: As a final exercise, we fit SPX option data from January 12, 2011 to an SVI parameterized volatility surface. Data was obtained from the CBOE website and fit using the same minimization of the objective function: min ( σ ) 2 SVI σ Mkt BS
What follows is graphical representation of the results: 4. Conclusion The SVI parameterization can offer good alternative to Black-Scholes when options are illiquid and implied data may be stale. It is especially useful in situations where the implied volatilities are out-of-the-money and the options are certain kinds of non-path dependent exotics, like a digital option.
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