This chapter gives a brief survey of continuous time finance. We categorize diffusion models according to the nature of their volatility coefficient. Models whose volatility coefficient does not exhibit randomness are treated in Sect. 3.1. Models whose volatility coefficient follows a stochastic process are discussed in Sect. 3.2.

Continuous Time Finance

We investigate the effect of discrete sampling and asset price jumps on fair variance and volatility swap strikes. Fair discrete volatility strikes and fair discrete variance strikes are derived in different models of the underlying evolution of the asset price: the Black-Scholes model, the Heston stochastic volatility model, the Merton jump-diffusion model and the Bates and Scott stochastic volatility and jump model. We determine fair discrete and continuous variance strikes analytically and fair discrete and continuous volatility strikes using simulation and variance reduction techniques and numerical integration techniques in all models. Numerical results show that the well-known convexity correction formula may not provide a good approximation of fair volatility strikes in models with jumps in the underlying asset. For realistic contract specifications and model parameters, we find that the effect of discrete sampling is typically small while the effect of jumps can be significant.

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The effect of jumps and discrete sampling on volatility and variance swaps

We consider the problem of dynamic portfolio optimization in a discrete-time, finite-horizon setting. Our general model considers risk aversion, portfolio constraints (e.g., no short positions), return predictability, and transaction costs. This problem is naturally formulated as a stochastic dynamic program. Unfortunately, with nonzero transaction costs, the dimension of the state space is at least as large as the number of assets, and the problem is very difficult to solve with more than one or two assets. In this paper, we consider several easy-to-compute heuristic trading strategies that are based on optimizing simpler models. We complement these heuristics with upper bounds on the performance with an optimal trading strategy. These bounds are based on the dual approach developed in Brown et al. (Brown, D. B., J. E. Smith, P. Sun. 2009. Information relaxations and duality in stochastic dynamic programs. Oper. Res. 58(4) 785--801). In this context, these bounds are given by considering an investor who has access to perfect information about future returns but is penalized for using this advance information. These heuristic strategies and bounds can be evaluated using Monte Carlo simulation. We evaluate these heuristics and bounds in numerical experiments with a risk-free asset and 3 or 10 risky assets. In many cases, the performance of the heuristic strategy is very close to the upper bound, indicating that the heuristic strategies are very nearly optimal.

This paper was accepted by Dimitris Bertsimas, optimization.

/pdf/dynamic-portfolio-optimization-with-transaction-costs-4fsradtckw.pdf

Dynamic Portfolio Optimization with Transaction Costs: Heuristics and Dual Bounds

We develop analytical methodology for pricing and hedging options on the realized variance under the Heston stochastic variance model (1993) augmented with jumps in asset returns and variance. By employing generalized Fourier transform we obtain analytical solutions (up to numerical inversion of Fourier integral) for swaps on the realized volatility and variance and for options on these swaps. We also extend our framework for pricing forward-start options on the realized variance and volatility, including options on the VIX. Our methodology allows us to consistently unify pricing and risk managing of different volatility options. We provide an example of model parameters estimation using both time series of the VIX and the VIX options data and find that the proposed model is in agreement with both historical and implied market data. Finally, we derive a log-normal approximation to the density of the realized variance in the Heston model and obtain accurate approximate solution for volatility options with longer maturities.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1686822_code1229200.pdf?abstractid=1408005&mirid=2

Pricing Options on Realized Variance in the Heston Model with Jumps in Returns and Volatility

We present a valuation framework that captures the main characteristics of employee stock options (ESOs), which financial regulations now require to be expensed in firms' accounting statements. The value of these options is much less than Black–Scholes prices for corresponding market-traded options due to the suboptimal exercising strategies of the holders, which arise from risk aversion, trading and hedging constraints, and job termination risk. We analyze the combined effect of all of these factors along with the standard ESO features of multiple exercising rights, and vesting periods. This leads to the study of a chain of nonlinear free-boundary problems of reaction-diffusion type. We find that job termination risk, vesting, finite maturity and non-zero interest rates are significant contributors to the ESO cost. However, we find that in the presence of vesting, the impact of allowing multiple exercise rights on ESO cost is negligible.

Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options

Volatility is the key variable in option pricing models and for risk management in general. Not surprisingly, this has led to the recognition that volatility uncertainty is an important risk factor. This realization, in turn, has given rise to derivative instruments tied to volatility, such as variance swaps, volatility swaps, and options on both variance and volatility, which are specifically designed to help manage this risk. To price these contracts, a model is needed for the volatility process. This article develops full pricing and risk management models for these instruments in the context of a Heston square root stochastic volatility model, including expressions for all of the standard Greek letters and a couple of new ones for the parameters of the volatility process. In addition, the authors provide a procedure for setting up optimal hedges of variance and volatility contracts using a finite set of options as an operational approximation to the full solution requiring a continuum of options

/pdf/pricing-and-hedging-volatility-derivatives-4guxkholn0.pdf

Pricing and Hedging Volatility Derivatives

We propose a multi-period model to value employee options allowing for the possibility that a risk-averse employee strategically exercises her options over time rather than at a single date. Our results describing the representative employee's option exercise behavior are broadly consistent with existing empirical evidence. The value of options to the employee and their effective cost to the firm are significantly different from the predictions of a constrained model that assumes single date strategic option exercise. The constrained model substantially underestimates the cost of options to the firm when, ceteris paribus, the employee's relative risk aversion and/or the time to maturity and/or the stock volatility exceed respective thresholds. Hence, the incorporation of multiple-date exercise has important economic and accounting consequences.

The Inter-Temporal Exercise and Valuation of Employee Options

We propose a multiperiod model to value employee options allowing for the possibility that a risk‐averse employee strategically exercises her options over time rather than at a single date. Our results describing the representative employee's option exercise behavior are broadly consistent with existing empirical evidence. The value of options to the employee and their effective cost to the firm are significantly different from the predictions of a constrained model that assumes “single date” strategic option exercise. The constrained model substantially underestimates the cost of options to the firm when, ceteris paribus, the employee's relative risk aversion and/or the time to maturity and/or the stock volatility exceed respective thresholds. Hence, the incorporation of “multiple‐date” exercise has important economic and accounting consequences.

Ashish Jain

Papers

The effect of jumps and discrete sampling on volatility and variance swaps

Pricing and Hedging Volatility Derivatives

The Inter-Temporal Exercise and Valuation of Employee Options

The Intertemporal Exercise and Valuation of Employee Options

The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps