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Outline

Title
Abstract
Figures
Introduction
Monte Carlo Methods and the Pricing of Options
Conclusion
References
All Topics
Economics
Mathematical Finance

The Art of Modeling Financial Options: Monte Carlo Simulation

Profile image of Antonie KotzéAntonie Kotzé

Abstract

Modeling is important because scientists investigate the world around us by building models that simulate real-world problems. Modeling is neither science nor mathematics; it is the craft that builds bridges between the two. Progress in modeling dynamics has always been closely associated with advances in computing. Monte Carlo simulation/modeling or probability simulation is a technique frequently used in the financial markets to understand complex financial instruments. It is used to scrutinise the impact of risk and uncertainty in financial and other forecasting models. It is very useful when complex financial instruments need to be priced. Exotic options are listed on the JSE on its Can-Do platform. Most listed exotic options are marked-to-model and the JSE needs accurate values at the end of every day. Monte Carlo methods in a local volatility framework are implemented. This paper discusses how Monte Carlo (MC) simulation is implemented when exotic options like Barriers are valued. We further summarise the historical development in modern computing and the development of the Monte Carlo method.

Figures (11)
Figure 2 shows 5 price paths generated with Equation (4.5), each having 25 time steps. Here we have a fixed volatility, interest rate and dividend yield (in the limit as M — ov, all Sp’s will have a normal distribution). If we have a call option with a strike price of 100, ] in Table 1. Equation (5.10) leads to an option value of R11.02. This is shown Figure 2: Price paths for a security with price R100 at time t = to, risk-free rate r = 0.05, dividend yield d = 0.025 (both continuous), volatility of 0.25 and T = 1.0. Further, N = 25 and then At = 0.04 and M =5
Figure 2 shows 5 price paths generated with Equation (4.5), each having 25 time steps. Here we have a fixed volatility, interest rate and dividend yield (in the limit as M — ov, all Sp’s will have a normal distribution). If we have a call option with a strike price of 100, ] in Table 1. Equation (5.10) leads to an option value of R11.02. This is shown Figure 2: Price paths for a security with price R100 at time t = to, risk-free rate r = 0.05, dividend yield d = 0.025 (both continuous), volatility of 0.25 and T = 1.0. Further, N = 25 and then At = 0.04 and M =5
Table 1: Five price paths and the Monte Carlo option price for a vanilla call. The parameters are given below Figure 2
Table 1: Five price paths and the Monte Carlo option price for a vanilla call. The parameters are given below Figure 2
Figures 3 and 4 show the implied and local volatility surfaces for ALSI and US- DZAR options respectively on 28 May 2014. Figure 3: ALSI implied and local volatility surfaces on 28 May 2014
Figures 3 and 4 show the implied and local volatility surfaces for ALSI and US- DZAR options respectively on 28 May 2014. Figure 3: ALSI implied and local volatility surfaces on 28 May 2014
Figure 4: USDZAR implied and local volatility surfaces on 28 May 2014 From Figure 3 we notice that the implied volatility surface does not have a lot of curvature — it is skewed but flat. However, we also see from the local volatility surface that it has more curvature. This shows that the local volatility skew is twice that of the implied volatility skew. Figure 4, shows the USDZAR implied volatility surface that has the currency market’s all familiar smile. Here we also show the local volatility surface with steeper sides.
Figure 4: USDZAR implied and local volatility surfaces on 28 May 2014 From Figure 3 we notice that the implied volatility surface does not have a lot of curvature — it is skewed but flat. However, we also see from the local volatility surface that it has more curvature. This shows that the local volatility skew is twice that of the implied volatility skew. Figure 4, shows the USDZAR implied volatility surface that has the currency market’s all familiar smile. Here we also show the local volatility surface with steeper sides.
Table 2: Price paths under a local volatility regime
Table 2: Price paths under a local volatility regime
Table 3: The Dupire local volatilities as obtained from the ALSI local volatility surface on 28 May 2014
Table 3: The Dupire local volatilities as obtained from the ALSI local volatility surface on 28 May 2014
Figure 5: Price paths for a security with price R100 at time ¢ = to, risk-free rate r = 0.05, dividend yield d = 0.025 (both continuous) and T = 1.0. Further, N = 25 and then At = 0.04 and M = 5. The volatility used is the local volatility for ALSI options on 28 May 2014.
Figure 5: Price paths for a security with price R100 at time ¢ = to, risk-free rate r = 0.05, dividend yield d = 0.025 (both continuous) and T = 1.0. Further, N = 25 and then At = 0.04 and M = 5. The volatility used is the local volatility for ALSI options on 28 May 2014.
Table 4: Input parameters for a down-and-out-put option on the JSE/FTSE Top 40 index
Table 4: Input parameters for a down-and-out-put option on the JSE/FTSE Top 40 index
Figure 6: Price dynamics for a down-and-out-put. The dynamics of the hedge ratio Delta is shown in Figure 7. Here is where the answers differ substantially. Far from the barrier all three methods give the same
Figure 6: Price dynamics for a down-and-out-put. The dynamics of the hedge ratio Delta is shown in Figure 7. Here is where the answers differ substantially. Far from the barrier all three methods give the same
Figure 7: The A-dynamics for a down-and-out-put option.
Figure 7: The A-dynamics for a down-and-out-put option.

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Related topics

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