Exit Talk of Patrick J. Laub
University of Queensland & Aarhus University
2015 | Aarhus |
2016 Jan-Jul | Brisbane |
2016 Aug-Dec | Aarhus |
2017 | Brisbane/Melbourne |
2018 Jan-Apr (end) | China |
Supervisors: Søren Asmussen, Phil Pollett, and Jens L. Jensen
10101
Sums of random variables
Asymptotic analysis / rare-events
Monte Carlo simulation
Data
Fitted model
Decision
Statistics
App. Prob.
You have some goal..
Estimated financial cost of the natural disasters in the USA which cost over $1bn USD. Source: National Centers for Environmental Information
E.g. guaranteed benefits
Want to know:
- cdf values
- value at risk
- expected shortfall
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of political economy
Fischer Black Myron Scholes
S&P 500 from Oct 1998 to Apr 2008
Google Finance
In words, Stock Price = (Long-term) Trend + (Short-term) Noise
General diffusion processes...
Stochastic volatility processes...
SV with jumps...
SV with jumps governed by a Hawkes process with etc...
For free, you get a confidence interval
where
Start with a multivariate normal
Then set
Then add them up
Easy to calculate interesting things with the density
Example
No closed-form exists for a single lognormal
Asmussen, S., Jensen, J. L., & Rojas-Nandayapa, L. (2016). On the Laplace transform of the lognormal distribution. Methodology and Computing in Applied Probability
Laub, P. J., Asmussen, S., Jensen, J. L., & Rojas-Nandayapa, L. (2016). Approximating the Laplace transform of the sum of dependent lognormals. Advances in Applied Probability
Solve numerically:
Hashorva, E. (2005), 'Asymptotics and bounds for multivariate Gaussian tails', Journal of Theoretical Probability
Find
Expand with 2nd order Taylor series about the maximiser
2. Find its orthogonal polynomial system
3. Construct the polynomial expansion
Pierre-Olivier Goffard
Asmussen, S., Goffard, P. O., & Laub, P. J. (2017). Orthonormal polynomial expansions and lognormal sum densities. Risk and Stochastics - Festschrift for Ragnar Norberg (to appear).
If the reference is Gamma
then the orthonormal system is
For r=1 and m=1,
Final final step: cross fingers & hope that the q's get small quickly...
1. From the moments
2. Monte Carlo Integration
3. (Dramatic foreshadowing) Taking derivatives of the Laplace transform...
Dufresne, D., & Li, H. (2014).
'Pricing Asian options: Convergence of Gram-Charlier series'.
Goffard, P. O., & --- (2017). 'Two numerical methods to evaluate stop-loss premiums'. Scandinavian Actuarial Journal (submitted).
Say you don't know how many summands you have...
Imagine you are an insurance company;
there's a random amount of accidents to pay out (claim frequency),
and each costs a random amount (claim size)
Approximate S using orthogonal polynomial expansion
where
As
and using
we can write
With we can deduce
and just take derivatives
In progress:
Thomas Taimre
Robert Salomone