Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew

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Hugh Wilkerson
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1 Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew Mentor: Christopher Prouty Members: Ping An, Dawei Wang, Rui Yan Shiyi Chen, Fanda Yang, Che Wang Team Website:

2 Table of Contents 1. Background Assumptions Our Model Workflow: Implied Volatility Cubic Spline Interpolation Method Extending Skew Curve Market Implied Distribution Denoise MID Test (Monte Carlo) & Result BS Test MID PDF RVs Test Improvement Conclusion... 15

1 Volatility Skew before 1987 X-axis is the strike price (K), and Y-axis is the volatility (sigma). People found the volatility changes with respect to strike price (K) and time to maturity (T).

3 1. Background Before the Black Monday in 1987, people think that Black-Scholes Model fits for the realized market, or at least, not too bad. But when avalanche of out-of-the money options suddenly became in-the-money, people began to realize that they needed to adjust the relationships between volatility and other parameters. Fig. 1.1 Volatility Skew before 1987 X-axis is the strike price (K), and Y-axis is the volatility (sigma). People found the volatility changes with respect to strike price (K) and time to maturity (T). So people need to trade lower-strike-option with a much higher volatility, and the volatility smile looks like this: Fig. 1.2 Volatility Skew after 1987 X-axis is the strike price (K), and Y-axis is the volatility (sigma). But since we have a volatility smile, if the market moves, the seller of the option may lose or make money on the option account even he/she has already delta-hedged. So the seller needs to sell/buy extra underlying to remain delta-neutral. Our model is to calculate how much is the extra delta that we need to take into consideration, and we called it skew delta. 1

4 2. Assumptions First of all, let s see the Black-Scholes model s assumptions: It is possible to borrow and lend cash at a known constant risk-free interest rate. The price follows a Geometric Brownian motion with constant drift and volatility. There are no transaction costs. The stock does not pay a dividend. All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share). There are no restrictions on short selling. There is no arbitrage opportunity. Our model allows each assumption in the Black-Scholes but the second one. In our model, we assume that: The price follows a implied motion that we can know from our Market Implied Distribution (MID) method. The volatility smile doesn t change its shape or increase/decrease, it just moves paralleled to the left or right. 2

5 3. Our Model 3.1 Workflow: Our workflow looks like this: Fig. 3.1 Workflow of our model 3.2 Implied Volatility The basic idea of this process is trying to use different volatilities into Black-Scholes until a predetermined option price is reached. The first method we tried to calculate implied volatility is Newton s method. This method is recognized by its rapid convergence. In order to obtain a quick return in later Monte Carlo testing process that constitutes more than 1 million trials, implementing Newton s method seems appropriate here. However, due to the intrinsic deficiencies, like local convergence, Newton s method had a bad performance when dealing with some extreme underlying prices. In some cases, it forced computer to crash, which did not show any rapidity. 3

6 Therefore, we turned to bisection method for its robustness. The actual method coded into our program was modified to some extent. The first modification was placed to decide the starting points the lower bound and upper bound. Given an arbitrary distance between final lower and upper bound, our program would set lower bound from 0.5 to some value that finally squeezes the desired implied volatility between these two bounds. This process sacrifices time but gives advantage of avoiding improper starting points set by human. Another modification links to the thought of removing infeasible inputs. Given spot, strike, interest rate and tenor constant, the option price function of volatility should have a theoretical minimum output. But the market seems not always follow this rule. When such out-of-range spot associated with strike was used to calculate the implied volatility, neither bisection nor Newton s method would give an ideal outcome. (Try this data: S = 30.66, K = 40.00, P = 9.40, r = 0.25%, T = 0.25) Therefore, it is necessary to remove these extremes from dataset so as to avoid endless loops in our program. The way we adapted is to listen an iteration counter and make sure it never exceed an empirical number, say 100 in our program. 3.3 Cubic Spline Interpolation Method This is a method to interpolate unknown values between given points using piecewise thirdorder polynomials. The essential idea is to fit a piecewise function of the form: For each ff ii is a third-order polynomial defined by: ff 1 (xx) xx 1 xx < xx 2 ff FF(xx) = 2 (xx) xx 2 xx < xx 3 (3.3.1) ff 1 (xx) xx 1 xx < xx ff ii (xx) = αα ii xx 3 + ββ ii xx 2 + γγ ii xx + δδ ii (3.3.2) ff ii (xx ii ) = αα ii xx ii 3 + ββ ii xx ii 2 + γγ ii xx ii + δδ ii = ff ii 1 (xx ii ) = h ii (3.3.3) Also, we want to make the function as soon as possible, therefore we make the first and second derivatives agree at each adjacent segment. ff ii (xx) = 3αα ii (xx xx ii ) 2 + ββ ii (xx xx ii ) + γγ ii = ff ii 1 (xx) (3.3.4) ff ii (xx) = 6aa ii (xx xx ii ) + 2ββ ii = ff ii 1 (xx) (3.3.5) Cubic splines have good properties, first the spline functions will interpolate all data points; secondly, interpolation function is continuous on each interval, so are the first and second derivatives. 4

7 Until now, we have 4mm coefficients and 4mm equations, therefore, from basic algebra, we could get the coefficients, and the value between known points. If we have mm + 1 observations(h 0, h 1,, h mm ), and mm + 1 inputs, we have BB = AA 1 HH (3.3.6) xx 1 xx 1 xx 1 1 xx 1 xx 1 xx xx AA = mm 2 xx mm 2 xx mm 2 1 xx mm 2 xx mm 2 xx mm xx 1 2xx xx 1 2xx xx mm 2 2xx mm xx mm 2 2xx mm xx xx xx mm xx mm xx 0 3 xx 0 2 xx xx 1 xx 1 xx xx mm 2 xx mm 2 xx mm xx mm 1 xx mm 1 xx mm xx xx mm BB = (αα 0, ββ 0, γγ 0, δδ 0, αα 1, ββ 1, γγ 1, δδ 1,, αα mm 1, ββ mm 1, γγ mm 1, δδ mm 1 ) 1 HH = (h 1, h 2,, h mm, 0,0,,0) 1 (3.3.7a) (3.3.7b) (3.3.7c) 3.4 Extending Skew Curve In order to simulate the implied volatility, we have to extend the skew curve that we have obtained from Cubic Spline Interpolation. Now, we choose the Least Squares method to extend the curve. The Least squares assumes that the best-fit curve of a given type is the curve that has the minimal sum of the deviations squared (least square error) from a given set of data. Suppose the curve is second curve to approximate the given set of data (xx 1, yy 1 ), (xx 2, yy 2 ) and the equation is : yy = aa + bbbb + ccxx 2. The best fitting curve ff(xx) = yy has the min least square error: ΠΠ = [yy ii ff(xx ii )] 2 = [yy ii (aa + bbxx ii + ccxx 2 ii )] 2 = aa mmmmmm (3.4.1) The unknown coefficients a, b and c must yield zero first derivatives. 5

8 = 2 [yy ii aa + bbxx ii + ccxx 2 ii ] = 0 = 2 xx ii[yy ii aa + bbxx ii + ccxx 2 ii ] = 0 = 2 xx ii 2 [yy ii aa + bbxx ii + ccxx 2 ii ] = 0 (3.4.2) Expanding the above equations, we have 2 yyii = aa 1 + bb xx ii + cc xx ii 2 3 xx ii yy ii = aa xx ii + bb xx ii + cc xx ii xx ii yy ii = aa xx ii + bb xx ii + cc xx ii (3.4.3) The unknown coefficients a, b and c can hence obtained by solving the above linear equations. AA = xx ii 2 xx ii xx ii xx ii 2 xx ii 3 2 xx ii yyii aa 3 xx ii, XX = bb, bb = xx ii yy ii cc 4 xx ii xx 2 ii yy ii (3.4.4) So, aa bb = AA 1 bb (3.4.5) cc For example, if we have some data for strike price and implied volatilities (Table 3.4): Table 3.4 Extending data example K Vol We can use cubic splines and least squares to generate volatility skew curve for all strike prices 6

Fig. 3.4 Extending Skew Curve X-axis is the strike price (K), and Y-axis is the volatility (sigma). 3.5.

Though there is a put-call parity that restricts the relationship between these two, but the real market doesn't

9 Fig. 3.4 Extending Skew Curve X-axis is the strike price (K), and Y-axis is the volatility (sigma) Market Implied Distribution First of all, our model use put option data rather than call option data. Though there is a put-call parity that restricts the relationship between these two, but the real market doesn't follow it perfectly. Fig. 3.5 Volatility skew of put (left) and call (right) X-axis is the strike price (K), and Y-axis is the volatility (sigma). 7

10 As we can see from figure 3.5, using call data brings more uncertainty of concavity of the volatility skew. When we use Market Implied Distribution method, we have to take the second derivative of the skew, and the concavity causes a negative of probability which against the definition of probability. The price of a European put option on an asset with strike price K and maturity T is given by PP = ee rrrr (KK SS) + pp(ss)dddd 0 KK = ee rrrr KKKK(SS)dddd 0 KK ee rrrr SSSS(SS)dddd 0 (3.5.1) Which implies = ee rrrr pp(ss)dddd 0 KK (3.5.2) 2 PP KK 2 = ee rrrr pp(kk) (3.5.3) pp(kk) = ee rrrr 2 PP KK 2 (3.5.4) Where the second derivative of put prices with respect to strike PP(KK) is the discounted market implied density. This risk-neutral probability distribution is estimated from implied volatility smiles. In this project, we used finite differences to approximate the second derivatives. 2 PP PP(KK + KK) 2PP(KK) + PP(KK KK) KK2 ( KK) 2 (3.5.5) 3.6. Denoise MID When we use market data to calculate the implied volatility, the skew curve is not absolute convex at every points on the curve. Therefore, we might come up with negative market implied density at certain strike price. For this kind of bad data, we called them noise in our program. Then we made a method called Denoise, this method threw away the bad data that have negative probabilities. As figure 3.6A shows, after eliminating all bad data which cause a negative probability, we can have a shape much more like a PDF. And we can see in figure 3.6B, after Denoise, our skew curve looks much better than before (figure 3.5 left). 8

By now, we have the market implied distribution of the underlying price, we could find out the expected underlying price at the

11 Fig. 3.6A PDF before Denoise (left) and after Denoise (right) X-axis is the underlying price (S), and Y-axis is the Probability (Pr). Fig. 3.6B Volatility Skew after Denoise X-axis is the strike price (K), and Y-axis is the volatility (sigma). By now, we have the market implied distribution of the underlying price, we could find out the expected underlying price at the maturity by calculating the integration below: EEEE = SS pp(ss) dddd 0 (3.6.1) Then we just go back to our skew curve to find the implied volatility (EEEE) at EEEE 9

12 4. Test (Monte Carlo) & Result 4.1 BS Test After we get the underlying price distribution through MID method, we want to use Monte Carlo Method testing the validity of our model. Firstly we generate the underlying price process from the B-S model s assumption: dddd SS = μμμμμμ + σσσσσσ (4.1.1) Because we assume the skew curve moved parallel, so we can got the skew curve of every day. Use the skew curve to calculate implied volatility, then option price, Delta and Vega can be calculated though the B-S model. Because we know the expectation of next period s underlying price, we can get the expectation of next period s volatility. VVVVVVVV (EEEE σσ) is the loss of portfolio from the change of volatility, and we have the expectation of underlying price change (EEEE SS), so we calculate the skew-delta as: DDDDDDDDaa SSSSSSSS = VVVVVVVV (EEEE σσ) EEEE SS (4.12) to cover the loss. The new delta is equal to the old delta plus skew-delta. After we get all information of the path, we calculate the P&L every day and get the statistics, mean and standard deviation. Undelying Price (S) Volatility (σ) new Delta (old Delta + skewdelta) Vega, Delta, Option Price Fig. 4.1 Test workflow 10

13 We simulate underlying price process 1 million times, we get below result: Table 4.1 Result from B-S Test Mean of P&L Mean (BS Delta) Mean of P&L Variance (BS Delta) Mean of P&L Mean (New Delta) Mean of P&L Variance (New Delta) The result of new delta is worse than the old delta, and the data of variance is so extreme. At first we are confused by this bad result, and the reason why we have this bad simulation results is that we based the simulation on the B-S model. The delta from B-S Model is one of the best hedging strategy. 4.2 MID PDF RVs Test After B-S Test, we use another method to generate the underlying price process. We get the return distribution from MID method. Then we use formula to generate the underlying price process. SS tt+1 = SS tt ee rrrrrrrrrrrr rrrrrrrrrrrr (4.2.1) The first target is to get a random variable follow the given distribution. We calculate the CDF through the PDF. Next step is to generate a random number in [0,1], then use the inverse CDF function to get the corresponding random variable x. Fig. 4.2 From PDF to CDF X-axis is the strike price (K), and Y-axis is the Probability (Pr). 11

14 We simulate 1oo thousand times, and then we get the result below: Table 4.2 Result from MID PDF RVs Test Mean of P&L Mean (BS Delta) Mean of P&L Std (BS Delta) Mean of P&L Mean (New Delta) Mean of P&L Std (New Delta) The new mean of P&L s mean is better than the older one at the cost of the worse standard deviation. The advantage is in a long run we will have a better hedging strategy result. But the disadvantage is the much bigger profit or loss in the short run. We hope our result will be a reference to traders who want to hedge options in a long run. 12

5. Improvement Until now, our model consider that the volatility skew doesn t change its shape as time goes by (which is the second assumption we made).

15 5. Improvement Until now, our model consider that the volatility skew doesn t change its shape as time goes by (which is the second assumption we made). But it is not true in the real world, so we need to adjust our model to take the volatility surface into account. In addition to volatility skew, traders use a volatility term structure when pricing options. Using the Cubic Spline Interpolation method, we can interpolate between T, and then plot the 3- D variation of implied volatilities with strike price and time to maturity. The implied volatility surface simultaneously shows both volatility smile and term structure of volatility, Option traders use an implied volatility plot to quickly determine the shape the shape of the implied volatility surface. Fig. 5.1 Volatility Surface Option with different expiration dates and strikes whose underlying is SPY For options of different maturities, we also see characteristic differences in implied volatility. However, in this case, the dominant effect is related to the market s implied impact of upcoming events. For example, it is well-observed that realized volatility for stock prices rises significantly on the day that a company reports its earnings. In our previous MID result, we assume that the expected underlying movement (ES) tomorrow is exactly the same as the expected movement on the maturity date. If we can calculate the 13

16 options which expires every trading day, however, we can use them to get the expected underlying movement on the next trading day. It will greatly improve the accuracy of our model. 14

17 6. Conclusion From this modeling program, we ve learnt: How to improve the B-S model if there is a volatility smile or even a volatility surface. Why we need to use bisection method rather than Newton's method to get the volatility smile from the market, and why sometimes both these two methods can t work. How to interpolate, extend or eliminate some points from a given data in order to maintain its information in the greatest degree but still qualify our standard. How to know the market movement of the future if we know today s market information. How to generate a bunch of random numbers that follows a given Probability Density Function. And we ve also learnt: How to work as a team to focus on a problem, discuss and solve it. How to break programming task into pieces for everybody, define the standard and make it up at last. Though our result doesn t improve the classical B-S model, we still learnt a lot from these ten days, and hoping to have another ten days for further modeling. Our mentor, Chris Prouty, gave us plenty of new ideas and suggestions which keep us standing closer to the industry. We really appreciate his help and patient. 15

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