Pricing Asian options in a semimartingale model

Večeř, Jan; Xu, Mingxin

doi:10.1080/14697680400000021

Cited by 57 publications

(22 citation statements)

References 19 publications

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“…Remark The function

u^{S} (t, x, y) = {double-struckE}^{S} [V_{S} (T) | A_{S} (t) = x]

satisfies partial integro‐differential equation with the terminal condition

u^{S} (T, x) = f^{S} (x) .

As the Asian forward F has the same dynamics as the average asset A , equation can also be used for pricing the fixed strike Asian option. A similar partial integro‐differential equation appeared in Vecer and Xu () and more recently the efficient method how to determine the solution numerically appeared in Bayraktar and Xing ().…”

Section: Extensions To Other Price Evolutionsmentioning

confidence: 80%

“…This pricing partial differential equation was also found independently by Hoogland and Neumann () who used symmetry arguments. Fouque and Han () extended this approach to stochastic volatility models, and Vecer and Xu () to models with jumps. Bayraktar and Xing () found an iterative numerical method for solving the corresponding partial integro‐differential equation for Asian options.…”

Section: Introductionmentioning

confidence: 99%

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Black–scholes Representation for Asian Options

Večeř

2012

Mathematical Finance

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Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond BT with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.

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“…Remark The function

u^{S} (t, x, y) = {double-struckE}^{S} [V_{S} (T) | A_{S} (t) = x]

satisfies partial integro‐differential equation with the terminal condition

u^{S} (T, x) = f^{S} (x) .

Section: Extensions To Other Price Evolutionsmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Black–scholes Representation for Asian Options

Večeř

2012

Mathematical Finance

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“…In particular, the choice of θ is independent on T , and a single market price P market (0, T ) of a zero coupon bond for one given maturity time T defines uniquely the prices defined by (25) for similar bonds for all other maturity timesT = T , since this formula has to be applied with the same θ. For models with time variable coefficients of equations for B and S, the same approach gives a time dependent solution θ(t) of (17), and the value r defined by (29) for a maturity timeT depends onT .…”

Section: Implied θ From Observed Bond Prices For the Incomplete Marketmentioning

confidence: 99%

“…Kim and Kunitomo (1999) studied asymptotic properties of this price with respect to a particular equivalent martingale measure. Vecer and Xu (2004) applied random numéraire to reduce computational dimension for Asian options and general semimartingales. Issaka and SenGupta (2017) applied random numéraire to reduce computational dimension for variance swap options and models based on both Brownian motion and jump processes.…”

Section: Introductionmentioning

confidence: 99%

On the implied market price of risk under the stochastic numéraire

Dokuchaev

2017

Ann Finance

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This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a numéraire. An equivalent martingale measure is not unique for this market, and there are non-replicable claims. Some rational choices of the equivalent martingale measures are suggested and discussed, including implied measures calculated from bond prices constructed as a risk-free investment with deterministic payoff at the terminal time. This leads to possibility to infer a implied market price of risk process from observed historical bond prices.

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“…Vecer (, 2002) explained how to price both continuously and discretely monitored Asian options in the geometric Brownian motion model based on a partial differential equation (PDE) approach. Vecer and Xu () derived an integrodifferential equation for an Asian option price when the underlying price process is assumed to follow the exponential Lévy process.…”

Section: Introductionmentioning

confidence: 99%

Recursive Formula for Arithmetic Asian Option Prices

Lee

2012

Journal of Futures Markets

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We derive a recursive formula for arithmetic Asian option prices with finite observation times in semimartingale models. The method is based on the relationship between the riskneutral expectation of the quadratic variation of the return process and European option prices. The computation of arithmetic Asian option prices is straightforward whenever European option prices are available. Applications with numerical results under the BlackScholes framework and the exponential Lévy model are proposed.

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Pricing Asian options in a semimartingale model

Cited by 57 publications

References 19 publications

Black–scholes Representation for Asian Options

Black–scholes Representation for Asian Options

On the implied market price of risk under the stochastic numéraire

Recursive Formula for Arithmetic Asian Option Prices

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