Forwards, Futures, and More Option Pricing

2017 ◽  
Author(s):  
Kerry E. Back
Keyword(s):  
Option Pricing ◽  
Interest Rates ◽  
Stochastic Volatility Model ◽  
Futures Contracts ◽  
Expectations Hypothesis ◽  
Forward Measure ◽  
Volatility Models ◽  
Volatility Model ◽  
Forward Prices ◽  
Exchange Options

Forward measures are defined. Forward and futures contracts are explained. The spot‐forward parity formula is derived. A forward price is a martingale under the forward measure. A futures price is a martingale under a risk neutral probability. Forward prices equal futures prices when interest rates are nonrandom. The expectations hypothesis is explained. The option pricing formulas of Margabe (exchange options), Black (options on forwards), and Merton (random interest rates) are derived. Implied volatilities and local volatility models are explained. Heston’s stochastic volatility model is derived.

scholarly journals Variance and Interest Rate Risk in Unit-Linked Insurance Policies

Risks ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 84
Author(s):  
David Baños ◽  
Marc Lagunas-Merino ◽  
Salvador Ortiz-Latorre
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Stochastic Volatility ◽  
Interest Rate Risk ◽  
Stochastic Volatility Model ◽  
Insurance Company ◽  
Insurance Contracts ◽  
Volatility Models ◽  
Volatility Model ◽  
Vasicek Interest Rate Model

One of the risks derived from selling long-term policies that any insurance company has arises from interest rates. In this paper, we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to obtain the risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedging strategy by completing the market. We conclude with a simulation experiment, where we price unit-linked policies using Norwegian mortality rates. In addition, we compare prices for the classical Black-Scholes model against the Heston stochastic volatility model with a Vasicek interest rate model.

scholarly journals Option Pricing under Double Stochastic Volatility Model with Stochastic Interest Rates and Double Exponential Jumps with Stochastic Intensity

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Ying Chang ◽  
Yiming Wang
Keyword(s):  
Option Pricing ◽  
Interest Rates ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Stochastic Interest Rates ◽  
Stochastic Intensity ◽  
Volatility Model ◽  
Double Exponential ◽  
Stochastic Interest ◽  
The Impact

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

scholarly journals FUNCTIONAL ANALYTIC (IR-)REGULARITY PROPERTIES OF SABR-TYPE PROCESSES

2017 ◽  
Vol 20 (03) ◽  
pp. 1750013 ◽  
Author(s):  
LEIF DÖRING ◽  
BLANKA HORVATH ◽  
JOSEF TEICHMANN
Keyword(s):  
Option Pricing ◽  
Interest Rates ◽  
Heat Kernel ◽  
Kernel Methods ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Continuity Properties ◽  
Regularity Properties ◽  
Volatility Model ◽  
Sabr Model

The stochastic alpha, beta, rho (SABR) model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in the markets, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR, these are rarely satisfied. We examine here regularity properties of the model from this perspective with a view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups and identify suitable Banach and Hilbert spaces for these. Finally, we comment on the short-time and large time asymptotic behavior of SABR-type processes beyond the heat-kernel framework.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

scholarly journals South Africa’s economic response to monetary policy uncertainty

2017 ◽  
Vol 44 (2) ◽  
pp. 282-293 ◽  
Author(s):  
Mehmet Balcilar ◽  
Rangan Gupta ◽  
Charl Jooste
Keyword(s):  
Monetary Policy ◽  
Interest Rates ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Economic Consequences ◽  
Policy Uncertainty ◽  
Dsge Model ◽  
Content Type ◽  
Volatility Model ◽  
Central Bankers

Purpose The purpose of this paper is to study the evolution of monetary policy uncertainty and its impact on the South African economy. Design/methodology/approach The authors use a sign restricted SVAR with an endogenous feedback of stochastic volatility to evaluate the sign and size of uncertainty shocks. The authors use a nonlinear DSGE model to gain deeper insights about the transmission mechanism of monetary policy uncertainty. Findings The authors show that monetary policy volatility is high and constant. Both inflation and interest rates decline in response to uncertainty. Output rebounds quickly after a contemporaneous decrease. The DSGE model shows that the size of the uncertainty shock matters – high uncertainty can lead to a severe contraction in output, inflation and interest rates. Research limitations/implications The authors model only a few variables in the SVAR – thus missing perhaps other possible channels of shock transmission. Practical implications There is a lesson for monetary policy: monetary policy uncertainty, in isolation from general macroeconomic uncertainty, often creates unintended adverse consequences and can perpetuate a weak economic environment. The tasks of central bankers are incredibly difficult. Their models project output and inflation with relatively large uncertainty based on many shocks emanating from various sources. It matters how central bankers react to these expectations and how they communicate the underlying risks associated with setting interest rates. Originality/value This is the first study that looks into monetary policy uncertainty into South Africa using a stochastic volatility model and a nonlinear DSGE model. The results should be very useful for the Central Bank as it highlights how uncertainty, that they create, can have adverse economic consequences.

scholarly journals Removing the Correlation Term in Option Pricing Heston Model: Numerical Analysis and Computing

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
R. Company ◽  
L. Jódar ◽  
M. Fakharany ◽  
M.-C. Casabán
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Diffusion Reaction ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Volatility Model ◽  
Classical Technique ◽  
Domain Consistency ◽  
Positive Coefficients

This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, two-dimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed by means of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.

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