4. Potential Solutions Using a Local Volatility Model
4.2 Forecasting Using the Local Volatility Model
4.2.3 Method 3: Random Path-Wise Selection
In this algorithm a random expiry tR is selected. It is assumed that the change in
local volatility at (t + 1) is equal to that from tR−1to tR. The proceeding change in
local volatility is equivalent to that from tRto tR+1and so forth until T is reached.
Once the latter is reached another random expiry is selected and the process is continued.
Figure4.4 illustrates the average implied volatilities for one hundred simula- tions of Algorithm3with one hundred thousand stock price paths for at-the-money options for observable market expiries as well as expiries extending to thirty years. The average implied volatilities for market observable expiries lies within the 5%
4.2 Forecasting Using the Local Volatility Model 36
Fig. 4.3:Implied volatility estimates using Method 2 for JSE Top40 at-the-money European call.
and 95% percentile interval and does not vary much from the market values. The variation in the forecasts is fairly narrow.
The three methods outlined produce market-consistent estimates as well as rea- sonable long-term implied volatility estimates. Method 2 has a larger variation between the 5% and 95% confidence interval, which is expected, in comparison to method 1 and 3. This is expected as a random expiry is used to estimate the change in local volatility for each iteration, which will result in more variability between consecutive estimates. It is possible to estimate implied volatility for a range of moneyness levels. The advantages of using these simulation-based extrapolation techniques are firstly, that it is possible to estimate implied volatilities for maturities which are not available in the market. This was not possible for both the traditional estimation techniques and non-parametric estimation techniques which were out- lined. Secondly, it is possible to estimate statistical confidence intervals for the resultant implied volatility estimates. This is important because there is variability in the future outcomes for option implied volatility, thus a point estimate is very unreliable. And thirdly, market implied volatility is used to to extrapolate implied volatilities that are not available in the market, rather than using stock price data. The disadvantage of using these extrapolation techniques is that the change in the proceeding local volatility should be assumed to follow a chosen methodology for each time step.
4.2 Forecasting Using the Local Volatility Model 37
Algorithm 3:Path-wise selection
Input :Number of stock price simulations (N) Number of stock price paths (M) Market local volatilities (σL)
Maturities within local volatility surface (T) Strikes within local volatility surface (K)
Output:Stock price paths (S)
1 Create empty matrix of size [N, M ] and set this equal toσfL.
2 Create empty matrix of size [N + 1, M ] and set this equal to S. 3 Set first row of S equal to S0.
4 Set t equal to T 5 for i = 2 to N do
6 if T(i) <= max T; then
7 Interpolate local volatility surface at the minimum of [ Si−1 ormax
K] and T(i). Set the first row ofσfLequal to result.
8 else
9 Set t = t + δT 10 if t > T then
11 Select a random expiry t = TR.
12 end
13 Interpolate local volatility surface at the minimum of [ S(i − 1) or
max K ] and t. Set σL1 equal to result.
14 Interpolate local volatility surface at the minimum of [ S(i − 1) or
max K ] and (t − δT). Set σL2 equal to result.
15 Set row (i) ofσfLequal toσfL(i − 1)σL1/σL2.
16 end
17 Use Milstein to calculate S(i):
18 S(i) = S(i − 1) + rS(i − 1)δT +σfL(i − 1)S(i − 1)
√
δT Z(i) +12σfL(i −
1)2S(i − 1)(Z(i)2− 1)δT .
4.2 Forecasting Using the Local Volatility Model 38
Fig. 4.4:Implied volatility estimates using Method 3 for JSE Top40 at-the-money European call.
Chapter 5
Conclusion
Estimating long-term implied volatility for equity derivatives is difficult in South Africa, where ’long term’ refers to expiries from ten to thirty years. This due to the lack of liquid tradable long-term derivative instruments. Thus, it is important to be able to estimate and extrapolate fair, market-consistent volatility surfaces. This dissertation evaluates different techniques used to estimate volatilities for a range of moneyness levels and expiries. The methods evaluated include statistical and time-series techniques, non-parametric techniques and simulation-based extrapo- lation techniques which forecast implied volatilities making use of a local volatility model.
The statistical and time-series techniques include historical volatility, EWMA and GJR-GARCH(1,1) which only need a time-series of historical stock price data. These three methods cannot estimate volatility skew or statistical confidence inter- vals for the volatility estimates. These methods result in realised volatility estimates which need to be scaled by an estimated IVHV ratio to obtain implied volatility es- timates. Term structure estimation is possible for these methods but maximum pos- sible expiry for historical and EWMA techniques is limited to the amount of stock price data available. GJR-GARCH(1,1) is able to forecast long-term volatilities but the volatility estimates for longer termed maturities are flat.
The non-parametric techniques include Canonical Valuation and Break-Even volatility which only use a time-series of historical stock price data. Both of these methods estimate implied volatility with a maximum expiry limited to the amount of stock price data available. These methods require an immense amount of histor- ical data to produce reasonable results and are computationally expensive. These methods are able to estimate the volatility surface across a range of expiries and moneyness levels with the ability to estimate a reasonable volatility skew. Two sampling techniques, using overlapping returns and statistical bootstrapping, are considered to increase the size of the return data. The use of statistical bootstrap- ping results in implied volatility estimates which improve on the estimates using non-overlapping returns. Both CV and BEV methods require the stock price return data to be independently and identically distributed.
Three potential models are proposed, using Dupire’s local volatility model, to estimate implied volatilities for expiries beyond what is available in the market. Each method contains different assumptions about how future volatility proceeds beyond what is observable in the market. These methods require a snapshot of option implied volatilities which are used for the estimation of local volatilities.