Numerical Methods for PDE

The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability v_f associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of v_f—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of v_f consists of additive probabilities below v_f whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below v_f, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.

This paper gives an ordering on option prices under various well‐known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q‐optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer (Mathematical Finance, 11(4), 2001) in the special case of the Heston model.

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

We provide a framework for learning risk-neutral measures (Martingale measures) for pricing options from high frequency financial data. In a simple geometric Brownian motion model, a price volatility, a fixed interest rate and a no-arbitrage condition suffice to determine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these assumptions to obtain a class of allowable risk-neutral measures. We then propose a framework for learning the appropriate risk-neural measure. Since the riskneutral measure prices all options simultaneously, we can use all the option contracts on a particular underlying stock for learning. We demonstrate the performance of these models on historical data. In particular, we show that both learning without a no-arbitrage condition and a no-arbitrage condition without learning are worse than our framework; however the combination of learning with a no-arbitrage condition has the best result. These results indicate the potential to learn Martingale measures with a no-arbitrage condition providing just the right constraint. We also compare our approach to standard Binomial models with volatility estimates (historical volatility and GARCH volatility predictors). Finally, we illustrate the power of such a framework by developing a real time trading system based upon these pricing methods.

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.

A Keynesian Approach to Modeling the Long-Term Interest Rate

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives

The sub-fractional CEV model

In this work, we developed methods to price both fixed and floating strike Volume Weighted Average Price (VWAP) options. VWAP options have a payoff that is dependent on both the stock price and volume of stock traded over the lifetime of the option. We concentrate on the valuation of European style, fixed and floating strike call options. Little additional effort is required to adapt the results of this work to value puts. First, introductory mathematical material is presented, followed by a general overview of options and how they are priced. The VWAP is then formally introduced, and some examples of VWAP options in practice are given. Expectations and a partial differential equation that describes the price of the options are then constructed. Throughout, we assume the stock price evolves as a geometric Brownian motion process, and the volume is a fast mean reverting process. Upper and lower bounds are established for the price of both fixed and floating strike VWAP options. These bounds are given as analytic formulae and are independent of the volume model. The bounds are obtained from a simple put-call type parity. We then interpret these bounds further. The upper bounds represent the minimum cost of hedging all of the risk. A Monte Carlo investigation is then performed; this gives a vivid picture of how the VWAP options behave. It also allows later results to be benchmarked. In addition to vanilla Monte Carlo, some effective controls variates are found. We conclude the Monte Carlo investigation by finding the Greeks by the finite difference, pathwise, and likelihood methods. Next, approximations to the prices of the VWAP options are obtained by matching the first two moments of the VWAP to a lognormal distribution. For the fixed strike, a partial differential equation similar to the Black-Scholes-Merton one is obtained. When the market price of risk is constant, an analytic expression is found. For a non-constant market price of risk, a numerical method must be used. The floating strike case gives a two dimensional partial differential equation that needs to be solved numerically. In addition to pricing call options, we also demonstrate a method for the valuation of a VWAP digital option. The partial differential equation which describes the prices of the VWAP options is then solved using finite differences. We use explicit, Crank-Nicolson, and Alternating Direction Implicit schemes. Solving by finite differences is a challenging problem as there are four state variables as well as time, and an incomplete set of boundary conditions are known. Additionally, we report several similarity reductions which are effective in simplifying the partial differential equation for the floating strike option. In the final approach to pricing, a series solution to the price is assumed and the first two terms are found. The analysis assumes that the volume process is a fast mean reverting process. The first term of the series is described by a partial differential equation, which is very similar to the one that arises when pricing the Asian option. Numerical experiments established that the first term of the expansion approximates the true solution well when the mean reversion rate is high. Chapter 10 concludes this work with an overview of the results presented in this thesis, along with possible directions for further work.

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents "noise". The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black–Scholes–Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.

One of the momentous equations in financial mathematics is the Black-Scholes equation, a partial differential equation that governs the value of financial derivatives, such as options. In this paper, we attempt to show the application of Stochastic Process. We have shown how geometric Brownian motion & Ito’s Lemma overlaps on Option Pricing.

Arbitrage-Free Prediction of the Implied Volatility Smile