Recent Advances in Financial Mathematics

Link for registration

https://docs.google.com/forms/d/1ra3TSRYtuXxZFPOYhLLyfezzCw6tIWoPHRwnRtjJ-fM

Detailed Program

(Abstracts below)

Day 1

Day 2

Speakers and Abstracts

Valuation and Optimal Surrender of Variable Annuities with Guaranteed Minimum Benefits

Abstract:

This work studies the valuation and optimal surrender of variable (equity-linked) annuities under a L\'evy-driven equity market with mortality risk. We consider a practical periodic fee structure which can vary over time, and is assessed as a proportion of the fund value. At maturity, the fund value is returned to the policyholder according to a guaranteed minimum accumulation benefit (GMAB). Mortality risk is also modeled discretely, and the contract offers a guaranteed minimum death benefit (GMBD) prior to maturity. The benefits accommodate caps on the growth of funds (in addition to the rising floor) to reduce the fee level, and as a disincentive to early surrender. Interest rates are modeled via a deterministic discounting term structure, which can be calibrated (bootstrapped) to the rates market, according to market convention.

An efficient and accurate valuation framework is developed, along with closed form pricing formulas in the case where policy surrender is not permitted. Numerous experiments are conducted to illustrate the interplay between contract parameters and the decision to surrender, and we provide an extensive analysis that investigates how to structure contracts to disincentivize early surrender.

A general approach for solving behavior optimal investment problems with non-concave utility and probability weighting

Abstract:

We consider the optimal investment problem with both probability distortion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has an inverse S-shaped probability weighting and a S-shaped utility function (i.e. one inflection point). We propose a step-wise relaxation method and have applied it to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yarri’s dual model and the Browne's goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.

Neural network based approximation algorithm for nonlinear PDEs with application to pricing in finance

Abstract:

In this talk, we first introduce the mathematical concept of neural networks and its universal approximation property. Then we show how one can combine neural networks together with tools from Stochastic Calculus, particularly the Feynman-Kac representation, to build a neural network based algorithm that can approximately solve high-dimensional nonlinear PDEs in up to 10’000 dimensions with short run times. We apply this algorithm to price high-dimensional financial derivatives under default risk

Puzzeling Option Prices of Pegged Exchange Rates

Abstract:

IForeign exchange markets are by far the largest markets in the world in terms of volume. Option pricing there relies on the principle that the underlying -- the exchange rate -- is a free floating one for which the classical Black and Scholes and derivates thereof are used.

It is an interesting fact that many foreign exchange markets are eventually pegged -- Bulgaria, China, Hong-Kong, Switzerland, Thailand -- at least for a long period of time. Surprisingly, even though the underlying is fixed, there is an active option market taking place on those currencies. We document this puzzling fact and propose a simple model to explain it.

We calibrate this model on two major examples: the Hong-Kong/US dollars and the Euro/Swiss franc and provide some insights on what could be the motivation of agents on this markets to trade such options.

This talks is based on joint works with Tan Wang and Tao Wang from Shanghai Advanced Institute of Finance as well as Yunbo Zhang from SJTU

Behavioral Portfolio Selection with Periodic Evaluations

Abstract:

We consider a continuous-time portfolio optimization problem featuring a behavioral trader who maximizes the utilities derived from the profit-and-loss relative to a performance target in each accounting period over an infinite horizon. The combination of non-concave utility function and infinite horizon makes the problem mathematically non-trivial. The value function at the beginning of each period is first characterized as the unique fixed point to a discrete-time Bellman equation, and then the value function at the intermediate time points within a period are identified via martingale duality. Imposing an excessively aggressive performance target on the trader can be detrimental, where the ruin probability of the portfolio can be strictly positive and the trader may severely underinvest in the good state of the economy.

Option Pricing And CVA Calculations Using The Monte Carlo-Tree (MC-Tree) Method With The Distribution Correction Factor

Abstract:

This presentation introduces the MC-Tree method with the distribution correction factor for pricing options with high accuracy. MC-Tree combines the MC method with the recombining binomial tree based on Pascal’s triangle for pricing single asset options. Similarly, MC-Tree combines the MC method with recombining multinomial trees based on Pascal’s Simplex for pricing the multi-asset options. Recombining multinomial trees based on Pascal’s Simplex in higher dimension generalizes a well-known recombining binomial tree based on Pascal’s triangle. The generalization of this model comes naturally and is set up in a complete market. MC-Tree allows generating probability parameters and other parameters randomly with a chosen probability distribution (mixing distribution).

MC-Tree method holds all benefits of both the MC method and the tree method. We do mixing distribution on the tree parameters and obtain compound distribution on the tree outcomes. Compound density turns out to be non-Gaussian, but we can get close using entropy maximization. We consider a linear convex combination of mixing densities, then apply optimization algorithms to find optimal coefficients of this combination so that the entropy of compound density is maximized. As well known in the literature review, the standard Gaussian density has the maximal entropy at 1.418939. We find the entropy at 1.418777 after applying our optimization algorithm. We also work with a much simpler mixing density which the entropy is reached at 1.418767. The difference between those two values is possibly insignificant of 10−5.

We can use the tree method for option pricing using the well-known logarithmic transformation of the share prices, bringing log-normal down to Gaussian. We introduce the usage of the bias correction and the distribution correction factor on the MC-Tree method for pricing European options and the algorithm for CVA calculations on an American put option using MC-Tree. The bias-correction technique gives the resulting tree model complete and obtains the risk-neutral probability, as shown in the paper by Sierag and Hanzon. The compound density is close to Gaussian density but not equal exactly. To handle this problem, we employ a technique, namely the distribution correction factor. The distribution correction factor is the ratio of the standard Gaussian density to the compound density. It is observed that the compound density is closer to the Standard Gaussian density when increasing the tree depth N and fixing the power of mixing density m. The compound density and the standard Gaussian density are almost the same in the central distribution but dissimilar in the tails.

Based on our numerical results, the MC-Tree method is more accurate than the well-known Least Square Monte Carlo (LSM) proposed by Longstaff Schwartz (2001) at the same numbers of simulations. Also, the MC-Tree method with the distribution correction factor dramatically improved the accuracy. For example, given model parameters such as initial stock price=100, strike price=95, expiration date=1 year, and interest rate=0.03, the price of a European put option is as follows: MC-Tree with the distribution correction factor (4.3720), MC-Tree with bias-correction (4.3828), CRR( 4.3657), JR (4.3600), MC (4.4107), compared with the analytical solution from Black-Scholes model at 4.3720, using the same tree depth at N=50, and the same numbers of simulations at M=100000. Clearly, the prices of the European option from MC-Tree with the distribution correction factor are the same as the analytical solutions of the Black-Scholes model.

Created by ICASM team