A semi-analytic pricing formula for lookback options under a general stochastic volatility model

2013 ◽  
Vol 83 (11) ◽  
pp. 2537-2543 ◽  
Author(s):  
Sang-Hyeon Park ◽  
Jeong-Hoon Kim
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Lookback Options ◽  
Volatility Model ◽  
General Stochastic ◽  
Pricing Formula

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 Pricing European options with stochastic volatility under the minimal entropy martingale measure

2015 ◽  
Vol 27 (2) ◽  
pp. 233-247 ◽  
Author(s):  
XIN-JIANG HE ◽  
SONG-PING ZHU
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Martingale Measure ◽  
Minimal Entropy Martingale Measure ◽  
Volatility Model ◽  
In Series ◽  
Expected Utility Maximization ◽  
Equivalent Martingale ◽  
The One ◽  
Pricing Formula

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.

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