JSFHJSFHJSFHJump-diffusion stochastic processes with MapleIgor Hlivka - Senior Quantitative AnalystMUSI, LondonJump diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognized in several scientific branches including physics, chemistry, but also economics and finance. This application is primarily focused on the last-mentioned field where the theory of jump-diffusion has been particularly actively researched and applied.
We will discuss various jump processes in finance, their applications to option pricing and risk. For this we will rely heavily on Finance package which contains essential ingredients for creation of efficient stochastic routines and together with other Maple's tools provide powerful platform for the study, analysis and application of stochastic processes to economic or scientific problems.JSFH
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>Jump-diffusion in financial literature is associated primarily with Merton who was the first to introduce the jump-diffusion into stock option pricing. From the pure theoretical perspective, jumps are not trivial. Why? Because the martingale theory - the heart of arbitrage-free option pricing - is not fully preserved. When underlying process (for stock price, FX rate or interest rate) jumps, the market turns incomplete due to the presence of two stochastic factors (underlying + jumps), but only one tradeable asset (underlying).
Merton back in 70s argued that since both processes are uncorrelated and independent, jumps can be hedged away through portfolio diversification.Why do we need jumps? Two main arguments: (i) reflection of reality and (ii) extension of traditional log-normal processes in finance where financial returns are not exactly normally distributed. Jump would serve here as "correction" to the limitation of traditional theory.Definitions of jumps:
Usually defined as Poisson process with intensity parameter \316\273. This ensures that jumps are independent of one anotherProbability of jumps occurring in short time internal \316\224t is \316\273\316\224t +o(\316\224t)Jumps are drawn from Poisson probability density => 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The mean waiting time for jumps = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRJyYjOTU1O0YnLyUnaXRhbGljR1EmZmFsc2VGJ0YyL0Y8USV0cnVlRicvRjNRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZGLyUpYmV2ZWxsZWRHRj1GMg==When jump occurs, the underlying value changes to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2JC1GIzYkLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GOUY5LUYsNiVRIkpGJ0YvRjJGOQ== where J is the magnitude of jump. This quantity is usually independent from the jump intensity and can be made stochastic, i.e. drawn from pre-defined probability distribution. We will review several jump types.Jump-diffusion stochastic differential equation:If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= solves the standard SDE 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 then it naturally solves its extension with jumps:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZTRjUtRiw2JVEnJiM5NTY7RicvRjBGPUY5LUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRi9GMkY5RjktRiw2JVEjZHRGJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZbcC1GLDYlUScmIzk2MztGJ0ZYRjlGWS1GLDYlUSNkV0YnRi9GMkY1RmdvRjUtRiw2JVEkSmRORidGL0YyRjk= where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= denotes the number of jumps that occurred \134with magnitude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.
<Text-field style="Heading 1" layout="Heading 1">Stochastic schemes for jump-diffusion process</Text-field>Here we review several stochastic routines applicable to jump-diffusion SDEs. We will call Maple's stochastic routines defined in the Finance package and demonstrate that Maple has almost unlimited potential to define various types of jump-diffusions with constant, probabilistic jumps (and intensities) and also stochastic Poisson processes.JSFH
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Normal mean-reverting process</Font></Text-field>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print(); # input placeholderNormal jump magnitude => jump sizes are drawn from normal distribution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImJGJ0YvRjJGQUZBRkE=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print(); # input placeholderWe can visualize the process paths by assigning numerical values.
Assume, the object of our interest is an interest rate, so the corresponding jumps are small. We simulate the 1 year evolution of an interest rate with initial value 2.5%
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We generate the sample values to obtain the statistics for the process.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRLVNhbXBsZVZhbHVlc0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYuLUYsNiVRIlhGJ0YvRjItRlM2JC1GIzYlLUkjbW5HRiQ2JFEiMUYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRKnRpbWVzdGVwc0YnRi9GMi1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GaW42JFEkMjUwRidGOUZeby1GLDYlUS1yZXBsaWNhdGlvbnNGJ0YvRjJGaW8tRmluNiRRJjEwMDAwRidGOUZcb0Y5RjktRjY2LVEiO0YnRjlGO0Zhb0ZARkJGREZGRkhGYm9GTUZcb0Y5print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEsRGF0YVN1bW1hcnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJBRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRj1GQA==print(); # input placeholderLet's compare this result with the non-jump process. In our case, this is just the process LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==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(); # input placeholderThe two samples are now clearly different:Sample A (jump-diffusion) has different (i) mean,(ii) volatility, (iii) skewness, (iv) kurtosis and (v) max value. The non-zero skewness and kurtosis > 3 indicate that the distribution is not any longer normalSample B (O-U process) behaves as normal => see the skewness and kurtosisJSFHLet's compare the two distributions: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%;JSFHJSFH
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Log-normal process with Logistic jumps</Font></Text-field>In this example we define the geometric Brownian motion with Logistic 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print(); # input 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%;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print(); # input 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%;Similarly to the mean-reverting normal process, GBM with jumps is not long-normal as it has different moments and longer tail.
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Square-root process with Laplace jumps</Font></Text-field>JSFHThe square-root process us characterized by the following SDE:
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print(); # input 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%;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print(); # input 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%;JSFH
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Stochastic routines with double Poisson process</Font></Text-field>We can further relax the definition of Poisson process and allow the intensity itself being unknown and stochastic. This will allow us to create the jump-diffusion where both (i) intensity and (ii) jumps magnitude are functions of independent processes.We will demonstrate this withGeometric Brownian Motion with stochastic volatility (following square-root diffusion)Gamma jumpsGumbel intensityLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZALUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEnJiM5NTY7RicvRjBGPUY5LUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGOUY5LUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZpbi9GTkZqbi1GLDYlUSNkdEYnRi9GMi1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORltwLUYsNiVRJyYjOTY1O0YnRlJGOUZTRmBvLUYsNiVRI2RXRidGL0YyLUZUNiYtRiM2JC1JI21uR0YkNiRRIjFGJ0Y5RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GLDYjUSFGJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zqbi8lJmRlcHRoR0ZpcS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRmluRmNvLUYsNiVRKGQmIzk2NTtGJ0YvRjJGNS1GLDYlUScmIzk1NDtGJ0ZSRjlGYG8tRlQ2JC1GIzYmLUYsNiVRJyYjOTUyO0YnRlJGOS1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRmpvRlxwRl1wRjlGOUZgb0Zkb0Znby1GLDYlUScmIzk1ODtGJ0ZSRjlGYG8tSSZtc3FydEdGJDYjLUYjNiQtRiw2JVEidkYnRi9GMkY5RmBvRmBwLUZUNiYtRiM2JC1GaHA2JFEiMkYnRjlGOUY5RltxRl5xRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZicC1JI21pR0YkNiVRKHJlc3RhcnRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI6RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGVC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GUDYmRlJGVUZYL0ZlblElYXV0b0YnLUYsNiVRJXdpdGhGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUShGaW5hbmNlRidGL0YyRjlGOUY1Rk9GW28tRl9vNiQtRiM2JC1GLDYlUStTdGF0aXN0aWNzRidGL0YyRjlGOUY1Rk8tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0ZXL0ZORlctRiw2JVEjdjBGJ0YvRjItRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JI21uR0YkNiRRJjAuMDA1RidGOUY1Rk8tRiw2JVEiVkYnRi9GMkZlcC1GLDYlUTRTcXVhcmVSb290RGlmZnVzaW9uRidGL0YyLUZfbzYkLUYjNixGYnAtRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZgcC9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUScmIzk1NDtGJy9GMEY9RjlGZnEtRiw2JVEnJiM5NTI7RidGX3JGOUZmcS1GLDYlUScmIzk1ODtGJ0ZfckY5RmZxLUYsNiVRInRGJ0YvRjJGOUY5RjVGT0ZdcC1GLDYlUSN5MEYnRi9GMkZlcC1GaXA2JFEmMC4wMjVGJ0Y5RjVGT0ZdcC1GLDYlUSJZRidGL0YyRmVwLUYsNiVROEdlb21ldHJpY0Jyb3duaWFuTW90aW9uRidGL0YyLUZfbzYkLUYjNitGaXJGZnEtRiw2JVEnJiM5NTY7RidGX3JGOUZmcS1JJm1zcXJ0R0YkNiMtRiM2Jy1GLDYjUSFGJy1GIzYmRmF0LUZfbzYoLUYjNixGXHEtRjY2LVEiKEYnRjkvRjxGMUY+L0ZBRjFGQkZERkZGSC9GS1EsMC4xNjY2NjY3ZW1GJy9GTkZgdUZmci1GNjYtUSIpRidGOUZddUY+Rl51RkJGREZGRkhGX3VGYXVGLy8lK2ZvcmVncm91bmRHUStbMCwxNjAsODBdRicvJStleGVjdXRhYmxlR0Y9LyUscGxhY2Vob2xkZXJHRjEvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjFGMkY5L0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZjdkZedkZodUY5RmF0Rmh1RjlGZnFGZnJGaHVGOUY5RjVGTy1GLDYlUSJNRidGL0YyRmVwLUYsNiVRL1JhbmRvbVZhcmlhYmxlRidGL0YyLUZfbzYkLUYjNiUtRiw2JVEmR2FtbWFGJ0ZfckY5LUZfbzYkLUYjNiYtRiw2JVEiYUYnRi9GMkZmcS1GLDYlUSJiRidGL0YyRjlGOUY5RjlGNUZPRl1wLUYsNiVRImNGJ0YvRjJGZXAtRmlwNiRRJTAuMDVGJ0Y5RjVGXXAtRiw2JVEiZEYnRi9GMkZlcEZgeEY1Rl1wRk8tRiw2JVEnJiM5NTU7RidGX3JGOUZlcEZpdi1GX282JC1GIzYmLUYsNiVRJ0d1bWJlbEYnRi9GMi1GX282JC1GIzYnRl14RmZxRmN4Rmh1RjlGOUZodUY5RjlGNUZPRl1wRmd3RmVwLUZpcDYkUSUwLjA0RidGOUY1Rl1wRmp3RmVwLUZpcDYkUSUwLjAzRidGOUY1Rk8tRiw2JVEiSkYnRi9GMkZlcC1GLDYlUS9Qb2lzc29uUHJvY2Vzc0YnRi9GMi1GX282JC1GIzYnRmZ4RmZxRmZ2Rmh1RjlGOUY1Rk8tRiw2JVEiWEYnRi9GMkZlcEZfcy1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORltbbEZqeS1GNjYtUSI7RidGOUY7RmlxRkBGQkZERkZGSEZgcEZNRmh1Rjk=print(); 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# input 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%;%;Again, as the RHS tail shows, the stochastic jump makes the distribution non-log-normal with fatter tail.
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Stochastic routines with stochastic jump process</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This is the special case foe the Poisson process where the jump itself is a defined stochastic process, independent from the underlying.We select the underlying process equal to the CEV processAssume the Poisson process itself is driven by another stochastic process which we assume is O-U processThe underlying process SDE: 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 jump generator: 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print(); # input placeholder
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<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Stochastic jump process with stochastic volatility with jumps</Font></Text-field>JSFHHere we design a more complicated example:Define the stochastic volatility process with jumpsUse that process as subordinate part of the underlying process with added jumpsThis will result in stochastic jumped vol process with jumpsVolatility process - square-root diffusion
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This is also confirmed in practice - users typically resort to either (i) stochastic volatility process or (ii) jump diffusion. Handling both or even in the "double fashion" as the last example demonstrates requires a lot of parameters estimation which itself may be quite cumbersome.
<Text-field style="Heading 1" layout="Heading 1">Probability density of the jump-diffusion process</Text-field>Jump-diffusion processes can lead to quite complex distributional patterns with multiple levels of randomness. Whilst explicit form for such densities may not always exist, we may obtain the estimates of such distribution using pattern matching. Moment matching can be particularly useful in this context.Here we use the lastly generated sample and try to fit it to the known statistical distributions.
<Text-field style="Heading 2" layout="Heading 2">Sample fit to Gumbel distribution</Text-field>We will try to fit the jump-diffusion data to Gumbel density: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(); # input placeholderWe compute the fist two moments of the density, do the same for the sample data and solve for density 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print(); # input 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print(); # input 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print(); # input 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVEjcDNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUS5GcmVxdWVuY3lQbG90RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiwtRiw2JVEiQUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJmNvbG9yRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRJWJsdWVGJ0YvRjJGWi1GLDYlUSdsZWdlbmRGJ0YvRjJGX28tRlM2Ji1GIzYmLUYsNiNRIUYnLUkjbXNHRiQ2I1E8SnVtcC1kaWZmdXNpb25+ZGlzdHJpYnV0aW9uRidGXHBGOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRjlGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zpbi8lJmRlcHRoR0ZhcS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRmhuL0ZORmluLUYsNiVRI3A0RidGL0YyRjVGTy1GUzYkLUYjNiwtRiw2JVEjUXNGJ0YvRjJGWkZcb0Zfby1GLDYlUSRyZWRGJ0YvRjJGWkZlb0Zfby1GUzYmLUYjNiZGXHAtRmBwNiNRMkZpdHRlZH5HdW1iZWx+UERGRidGXHBGOUY5RmNwRmZwRjlGOUZpcEZccUZpcS1GLDYlUSZwbG90c0YnRi9GMi1GUzYmLUYjNiQtRiw2JVEoZGlzcGxheUYnRi9GMkY5RjlGY3BGZnAtRlM2JC1GIzYmRitGWkZdckY5RjktRjY2LVEiO0YnRjlGO0ZnbkZARkJGREZGRkhGaG5GTUZpcS8lK2V4ZWN1dGFibGVHRj1GOQ==%;We can see that the fit is close, but not exact, especially on the LH tail and on the skewness. We can see that the two moments match is not enough to fit the double jump-diffusion process.
<Text-field style="Heading 2" layout="Heading 2">Sample fit to higher-moments density</Text-field>We will build a new distribution (Normal mixture) with higher number of parameters: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print(); # input 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(); # input placeholderprint(); # input placeholderJSFHWe again calculate the density moments and solve for four 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print(); # input 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print(); # input placeholderWe select the solution that meets the parameters 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print(); # input placeholderSubstitute parameters into the 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print(); # input placeholderJSFHGenerate plots and compare the densitiesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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%;JSFHAlthough not totally exact, the fit to the Normal mixture is now better.
<Text-field style="Heading 2" layout="Heading 2">Solution via Fourier transform</Text-field>When the jump-diffusion process contains constant parameters, we can obtain the probability density by trying to solve the forward Kolmogorov equation - the PDE that returns the evolution of transitional density over time. We use the integral transform (Fourier in particular) to find the solution.The probability density for jump-diffusion is similar to the "standard" PDE. The only unknown quantity is the identification of jumps.
Jump definition in the PDE
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 where the expectation is taken over probability of jumps LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5ODI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg==this means: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We define the forward Kolmogorov equation for jump-diffusion as follows: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(); # input placeholderApply alias for 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print(); # input placeholderTake the Fourier transform of the PDE with parameter 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print(); # input placeholderChange the function to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==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print(); # input placeholderSolve the ODE with initial condition 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print(); # input placeholderTake the inverse of the Fourier and get the desired densityLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjcTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUStpbnZmb3VyaWVyRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEkcmhzRidGL0YyLUZZNiQtRiM2JC1GLDYlUSNxM0YnRi9GMkY5RjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZKL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJmFscGhhRicvRjBGPUY5RmFvLUYsNiVRInhGJ0YvRjJGOUY5LUY2Ni1RIjtGJ0Y5RjtGZG9GQEZCRkRGRkZIRkpGVC8lK2V4ZWN1dGFibGVHRj1GOQ==print(); # input 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEkcmVzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEidUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInhGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ0RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEnZXhwYW5kRidGL0YyLUZTNiQtRiM2Jy1GLDYlUSNxNUYnRi9GMkZaLUYsNiVRJGV4cEYnL0YwRj1GOUZfb0Y5RjktRjY2LVEiO0YnRjlGO0ZnbkZARkJGREZGRkhGaG5GTUZfb0Y5print(); # input placeholderJSFH
<Text-field style="Heading 1" layout="Heading 1">Application in Finance</Text-field>Jump-diffusion processes with respective densities are highly relevant to finance where users look for alternatives to "traditional" approaches to explore "fat tails" - heavier tail distributions than Normal or LogNormal processes. Jump processes serve these needs well.
<Text-field style="Heading 2" layout="Heading 2">Option pricing</Text-field>JSFHFreedom in defining various jump-diffusions in Maple enables user to price and value a number of options with various 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print(); # input placeholderProvide process parameters and compute Monte-Carlo expectation - call and put option prices with option maturity T and strike price KJSFHQzY+SSN5MEc2IiQiIiQhIiMhIiI+SSNtdUdGJSQiI0QhIiRGKT5JJmthcHBhR0YlJCIjN0YoRik+SSZzaWdtYUdGJSQiI00hIiVGKT5JJ2xhbWJkYUdGJSQiIzpGLkYpPkkiVEdGJSIiIkYpPkkiS0dGJSQiI05GLkYpPkkickdGJSQiJCZHRjdGKS9JJWNhbGxHRiUqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJkZERj5GPUY+RilGPi0mSShGaW5hbmNlR0ZMNiNJLkV4cGVjdGVkVmFsdWVHRiU2Ji1JJG1heEdGTTYkLCYtSSJYR0YlNiNGPUY+RkBGKSIiIS9JKnRpbWVzdGVwc0dGJSIkXSMvSS1yZXBsaWNhdGlvbnNHRiUtSSJeR0ZNNiQiIzUiIiYvSSdvdXRwdXRHRiVJJnZhbHVlR0ZMRj5GPi9JJHB1dEdGJSomRkpGPi1GUzYmLUZZNiQsJkZARj5GZm5GKUZpbkZqbkZdb0Zkb0Y+Rj4=print(); # input placeholderWe can compare these prices to non-jump stochastic process LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==QyYvSSxub25qdW1wY2FsbEc2IiomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComSSJyR0YlIiIiSSJUR0YlRjAhIiJGMC0mSShGaW5hbmNlR0YpNiNJLkV4cGVjdGVkVmFsdWVHRiU2Ji1JJG1heEdGKjYkLCYtSSJZR0YlNiNGMUYwSSJLR0YlRjIiIiEvSSp0aW1lc3RlcHNHRiUiJF0jL0ktcmVwbGljYXRpb25zR0YlLUkiXkdGKjYkIiM1IiImL0knb3V0cHV0R0YlSSZ2YWx1ZUdGKUYwRjAvSStub25qdW1wcHV0R0YlKiZGJ0YwLUY0NiYtRjo2JCwmRkBGMEY9RjJGQUZCRkVGTEYwRjA=print(); # input placeholderCall option price for jump-diffusioon is higher than the non-jump equivalent. Put values behave differently. This reflects the skewness of the distribution.We can also easily compute the Asian payoff from these processes: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print(); # input placeholderAgain, we can see the valuation differences. Call under jump-diffusion generates higher options prices, as expected. The degree of variation is controlled by both jump intensity \316\273 and jump size distribution parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJyYjOTQ1O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NDY7RidGPUY/Rj9GP0Y/
<Text-field style="Heading 2" layout="Heading 2">Risk</Text-field>Risk applications are related to VaR and extended VaR profile - also knows as Expected Shortfall.We use the jump-diffusion process to calculate the various Risk staticVaR @ various confidence levels - using both jump-diffusion and standard process
Compute the mean and standard deviation of terminal valueQygtSS1TYW1wbGVWYWx1ZXNHNiI2JS1JIlhHRiU2I0kiVEdGJS9JKnRpbWVzdGVwc0dGJSIkXSMvSS1yZXBsaWNhdGlvbnNHRiUtSSJeRyUqcHJvdGVjdGVkRzYkIiM1IiIlISIiPkkjbW5HRiUtSSVNZWFuR0YlNiNJIiVHRiUiIiI+SSRzdGRHRiUtSTJTdGFuZGFyZERldmlhdGlvbkdGJTYjSSMlJUdGJUY9print(); # input placeholderQygtSS1TYW1wbGVWYWx1ZXNHNiI2JS1JIllHRiU2I0kiVEdGJS9JKnRpbWVzdGVwc0dGJSIkXSMvSS1yZXBsaWNhdGlvbnNHRiUtSSJeRyUqcHJvdGVjdGVkRzYkIiM1IiIlISIiPkklbW5OSkdGJS1JJU1lYW5HRiU2I0kiJUdGJSIiIj5JJnN0ZE5KR0YlLUkyU3RhbmRhcmREZXZpYXRpb25HRiU2I0kjJSVHRiVGPQ==print(); # input placeholderand calculate the VaR Z-scope for each processQyw3Iy1JJHNlcUclKnByb3RlY3RlZEc2JComLCZJI21uRzYiIiIiJkkkdmFyR0YrNiRJImlHRisiIiMhIiJGLEkkc3RkR0YrRjIvRjA7RiwiIipGMj5JJHAxMUdGKy1JKkxpbmVDaGFydEdGKzYlSSIlR0YrL0knbGVnZW5kR0YrNyNRO0p1bXAtZGlmZnVzaW9uflZhUn5aLXNjb3JlRisvSSdjb2xvdXJHRitJJHJlZEdGK0YyNyMtRiU2JComLCZJJW1uTkpHRitGLCZJK3Zhck5vSnVtcHNHRitGL0YyRixJJnN0ZE5KR0YrRjJGNEYyPkkkcDEyR0YrLUY6NiVGPC9GPjcjUTNOb3JtYWx+VmFSflotc2NvcmVGKy9GQkklYmx1ZUdGK0YyLSZJJnBsb3RzR0YrNiNJKGRpc3BsYXlHRis2JEY4Rk5GLA==%;As shown on the chart, the jump-diffusion process exhibits both (i) lower Z-score values and (ii) flatter slope. This is due to higher volatility and higher skewness that moves the distribution away from Normal.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=%;We also look at Expected Shortfall, the risk behavior in the tail region falling below VaR.
This will be achieved in two steps: (i) define the confidence limits for a given ES level and (ii) loss profile in the tail regionQyY3Iy1JJHNlcUclKnByb3RlY3RlZEc2JTckSSJpRzYiLUkyRXhwZWN0ZWRTaG9ydGZhbGxHRio2Ji1JIlhHRio2I0kiVEdGKi9JKnRpbWVzdGVwc0dGKiIkXSMvSS1yZXBsaWNhdGlvbnNHRiotSSJeR0YmNiQiIzUiIiUvSTBjb25maWRlbmNlbGV2ZWxHRipGKS9GKTskIiQhKiohIiQkIiQqKipGQiQiIiJGQiEiIj5JJ0VTZGF0YUdGKi1JKGNvbnZlcnRHRiY2JEkiJUdGKkkmQXJyYXlHRiZGRg==print(); # input placeholderJSFHQyY+SS52YXJjb25maWRlbmNlRzYiJCIkISoqISIkISIiPkkpdmFybGltaXRHRiUtJkktQ3VydmVGaXR0aW5nR0YlNiNJM0FycmF5SW50ZXJwb2xhdGlvbkdGJTYlSSdFU2RhdGFHRiVGJC9JJ21ldGhvZEdGJUknc3BsaW5lR0YlIiIiprint(); # input placeholderQyQvSS1FeHBTaG9ydGZhbGxHNiItJkkoRmluYW5jZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSS5FeHBlY3RlZFZhbHVlR0YlNiYtSSRtaW5HRio2JCwmKiYtSSJYR0YlNiNJIlRHRiUiIiJJKXZhcmxpbWl0R0YlISIiRjhGOkY4IiIhL0kqdGltZXN0ZXBzR0YlIiRdIy9JLXJlcGxpY2F0aW9uc0dGJS1JIl5HRio2JCIjNSIiJS9JJ291dHB1dEdGJUkmdmFsdWVHRilGOA==print(); # input placeholderThis is the Expected shortfall static - the average loss in the tail region when measured on the entire distribution sample.We can also compute alternative shortfall measure - the average loss when only data smaller than the VaR limit are taken into considerationQyg+SSNzdkc2Ii1JLVNhbXBsZVZhbHVlc0dGJTYlLUkkbWluRyUqcHJvdGVjdGVkRzYkLCYqJi1JIlhHRiU2I0kiVEdGJSIiIkkpdmFybGltaXRHRiUhIiJGM0Y1RjMiIiEvSSp0aW1lc3RlcHNHRiUiJF0jL0ktcmVwbGljYXRpb25zR0YlLUkiXkdGKzYkIiM1IiIlRjU+SSl0YWlsc3RhdEdGJS1JJ1NlbGVjdEdGJTYkZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjaXNHNiRGK0koX3N5c2xpYkdGJTYjMjkkRjZGJUYlRiVGJEY1L0kwQ29uc2VydmF0aXZlX0VTR0YlLUklTWVhbkdGJTYjRkJGMw==print(); # input placeholderAs we can see this "conservative" loss is now much higher.Expected shortfall measure depends on the VaR confidence setting above. Higher intervals will naturally reduce the loss size.We can investigate the tail distribution further by looking at the data summary statistics to see other characteristics of the loss.QyQtSSxEYXRhU3VtbWFyeUc2IjYjSSl0YWlsc3RhdEdGJSIiIg==print(); # input placeholderThis confirms that the CAS such as Maple offers flexible methods
<Text-field style="Heading 1" layout="Heading 1">Conclusion</Text-field>The example shown above are just simple illustration of the power of Maple for numerical simulation of jump-diffusion processes. The system provides wide flexibility for creation of stochastic schemes required by the user. Financial application - both in option pricing and risk - are particularly well suited for such modelling.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHTTdSMApJN1JUQUJMRV9TQVZFLzQ1MTcwMjczOTRYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISM1IisiIyQiIiohIiIkIiMiKiEiIyQiIyMqRiskIiMkKkYrJCIjJSpGKyQiIyYqRiskIiMnKkYrJCIjKCpGKyQiIykqRiskIiMqKkYrXTNGOUFENTg4RjQyQTQ0QUNdM0Y5QTcxMDIzMUNBMzE2Ml0zRjlBNDEyN0Q4RkExNkU4XTNGOUExMjYwMDQyRDNFNzZdM0Y5OUVBNjFCRDNEOEJCMF0zRjk5NUZDQ0FCMDlDREE5XTNGOThFREZDNThCRTc1NDZdM0Y5ODY1NDY3N0Y2RTZBNV0zRjk3QkYzNDY1NzQxRjZBXTNGOTZFRDVDRTkxNThFMTlGJQ==TTdSMApJN1JUQUJMRV9TQVZFLzQ1MTcwMjE4MjZYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISM1IisiIyQiIiohIiIkIiMiKiEiIyQiIyMqRiskIiMkKkYrJCIjJSpGKyQiIyYqRiskIiMnKkYrJCIjKCpGKyQiIykqRiskIiMqKkYrXTNGOTlGQUMwNTQ1MzlEOTNdM0Y5OTlEREJCNUMwNDBCM10zRjk5NjAwQTFBNDMzREMwXTNGOTk0NUJBRENGQTFBQzVdM0Y5OTA2NTBENzZCMzQzRV0zRjk4QjFFRDAzNDA0NjExXTNGOTg2QzQ2MEQ4NEY5M0VdM0Y5N0YzQTZFOEEwQTBFNV0zRjk3NzMwMzQyOTg5Q0Y2XTNGOTZBMzgyMjBFQkZBMjNGJQ==TTdSMApJN1JUQUJMRV9TQVZFLzQ1MTcwMDc1NTRYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISM1IisiIyQiJCEqKiEiJCQiJCIqKkYoJCIkIyoqRigkIiQkKipGKCQiJCUqKkYoJCIkJioqRigkIiQnKipGKCQiJCgqKkYoJCIkKSoqRigkIiQqKipGKF0zRjk1NzFFNEFBNDQyNTI3XTNGOTU1MDlERkJDMzZBRDNdM0Y5NTNEMzE3ODYxRDcxQ10zRjk1NEI1MTZBMzAzNTkwXTNGOTRBQTlCRUVBQ0ZDNjVdM0Y5NDZGOTZBNjAwM0M4M10zRjk0QjQ5RTc1NDgwRjJBXTNGOTQxN0QxQzk5NTdCNDddM0Y5M0JFNTBFRUNGRjAwQl0zRjkyQzQ3ODExMDI5NzY1RiU=