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Yor. Making markov martingales meet marginals: With explicit constructions. Bernoulli, 8(4):509–536, 2002. ´ j. The skorokhod embedding problem and its offspring. Probability [9] J. Oblo Surveys, Volume 1:321–392, 2004. ´ j. Financial derivatives ii: Problem sheet 1 & correction. Mathematical [10] J. Oblo Institute, University of Oxford, 2011. 38 BIBLIOGRAPHY 39 [11] K. Oleszkiewicz. On fake brownian motions. Statistics and Probability Letters, Volume 78:1251–1254, 2008. [12] Z. Qian. Lecture notes: Stochastic differential equations.

We will see that for European options, the stock prices via Az´ema-Yor and reversed Az´ema-Yor fake GBM result in the same price as indicated by the famous Black & Scholes formula. However, some differences appear when we price the path-dependent options. 1 European call option We compute European call option price, for which the payoff at maturity T is defined by (ST − K)+ . By risk neutral valuation, the price at time 0 is C0euro = e−rT EQ ((ST − K)+ |F0 ). We find that this call option price depends only on the stock price at maturity, in which case our Az´ema-Yor and reversed Az´ema-Yor fake GBM have the same lognormal distribution as the stock prices following the classical geometric Brownian motion.

6: Histograms of the path average for the three processes. 35 CHAPTER 4. 4 Variance swap A variance swap ([10]) is a contract with payoff at maturity [X]T − κvar , where Xt = ln(St /S0 ) and [X]t is the quadratic variation of (Xt ). The constant κvar is predetermined in order that there is no cash flow when the contract is set. Under the Black & Scholes setting, κvar can be computed and equal to σ 2 T . For our fake GBM processes, we approach κvar by computing for a large N N −1 N −1 2 (Xti+1 − Xti ) = i=0 (ln i=0 Sti+1 2 ) Sti for each path and then get the result via Monte Carlo simulation.

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Fake Geometric Brownian Motion And Its Option Pricing by Xingjian Xu

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