Equivalent Martingale Measure in Asian Geometric Average Option Pricing

The general situation of the Black-Scholes Option Pricing Model was discussed under the assumption of the arbitrage-free market, and the pricing of Asian geometric average options with fixed strike price was analyzed at any valid time. Consequently, the price formula of the Asian geometric average options was drawn using the equivalent martingale measure and the significance of the study was also indicated.


Introduction
Asian option, also known as the average price of options, was one of the derivatives of the stock options, and was firstly introduced by the American Bankers Trust Company (Bankers Trust) in Tokyo, Japan, on the basis of the lessons learned from the option implementations, such as real options, virtual options and stock options.It was a kind of exotic options, which was the most active one in financial derivative market, with the difference of the limitation of the exercise price from the usual stock option, that is, its exercise price was the average secondary market price of the stock price implemented during the current six months.
In this paper, after the Black-Scholes [1] Option Pricing Model was fully understood, the pricing of Asian options was discussed: It was assumed that the underlying asset price was driven by the geometric Brownian motion, that is, lognormal distribution.By using the random variables with the same Second moment driven by the lognormal distribution to approximate the arithmetic average of the underlying asset price, the approximate solution of the arithmetic average price of Asian put and call option with fixed exercise price was obtained, and the application of the equivalent martingale measure in the pricing of financial derivatives was further expanded [2].

Model and Formulas
Generally, the stock market could be described as a probability space with a σ -stream, that is, ( ) , ,P Ω  ; It was supposed that the market could meet the following conditions: (1) The market was an efficient frictionless market including two assets: one was the risk-free assets, known as the bonds, whose price process was denoted by where, µ denotes the expectation of the yield rate, σ denotes firm-value process volatility, T denotes time to expiration of option, r denotes the risk-free interest rate and they all are constants.d P t W denotes the instantaneous increment of the Brownian motion under the probability measurer P at time t; (2) Security trading is continuous and there are no transactions costs or taxes; (3) There are no dividends to be payoff during options being held.

Definition 1 Let ( )
, ,P Ω  be a probability space and 0 be a standard Brownian motion.t  be a σalgebra of spanning . Then measure Q is defined a martingale measure equivalent to measure P, where ( ) P E  denotes the expectation of random varia- ble under probability measure P [3].
Lemma 1 The dynamics of the share price under probability measure Q: , where S denotes the share price now and 0 . Prove Because the share price process satisfy the formula (1), using Ito's theorem, we have: Then we get the dynamics of the share price under probability measure P: where 0 ( ) , by definition 1 and Girsanov's theorem we get that the random process ( ) is a Brownian motion on ( ) , ,Q Ω  and: where ( ) E  denotes the expectation of random variable in probability measure P and ( ) Q P  denotes the probability of random variable in measure P, A I is an indicator function of set A. Substituting (5) into (3), we get: Thus we have that under probability measure Q: The proof is completed.
, then the distribution function of ( ) Definition 2 Using a bond as the denominated unit, was the value process of discounted assets.

Asian Geometric Average Options Pricing
In general, for the contingent claim, the risk-neutral pricing principle [4] was obtained as followed.Theorem 1 It was supposed that the market was arbitrage-free, so that the value of the process of any asset ( ) V t at time t was: Considering one bearish Asian option, its return at expiration time was: , where, Under the conditions of arbitrage-free market and from Theorem 1, the price at time t was: To get the specific expression of ( ) p V t , the key was to obtain the distribution of T J under the condition , then e Y T J = , and: Written as: ( ) e t t J t was driven by the normal distribution where, ( ) The proof was completed.Similarly, the price of Asian call options where, ( ) .
Deduction 2 When the interest rate r and the volatility of stock returns 2 σ were constant, there was:

3
could be deduced from the Lemma 1. Theorem It was supposed that the market was arbitrage-free, then the price of Asian put options

Deduction 1 For
Asian geometric average options, the parity relationship between call option and put option was: