Some of the arguments for using GBM to model stock prices are: The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality. , This is the original derivation of Black and Scholes [1]. This differs from the formula for continuous semi-martingales by the additional term summing over the jumps of X, which ensures that the jump of the right hand side at time t is Δf(Xt). The payoff of our derivative as described above is the discounted risk-neutral expected value of the payoff…, The support for the region is given by the payoff of the call, between the strike price and infinity, and by the definition of expected value…, This can be separated for simplicity into two integrals…, The second integral is rather easy to compute…, Obviously stock prices have to be less than infinity, and we can substitute ф for the risk-neutral cumulative distribution function…, We know the standard normal distribution is symmetrical…, Distributing the negative and re-applying logarithm rules yields…, We have successfully solved the second integral, let's return to this first…, The goal is to arrive at an integral that is similar to the definition of the standard normal distribution. ) 2 Derivation and assumptions. ∇ The Black-Scholes equation, brainchild of economists Fischer Black and Myron Scholes, provided a rational way to price a financial contract when it still had time to run. ; Uma EDP (sigla para equação diferencial parcial), que (neste modelo) mede tal evolução, utilizando como referência uma Opção do Tipo "europeia". Then, if the value of an option at time t is f (t, St), Itô's lemma gives First, the following stock price transformation…, Using Itô’s lemma (2) to solve for the transformed differential…, We can also easily express this in integral form…, Applying logarithm rules makes this looks like a log return which is normally distributed with a mean and variance computed in tandem with rules from stochastic calculus…, Because the log return is distributed normally the return is distributed lognormally…, Since we are under the conditions of risk-neutrality established by our portfolio to satisfy our no-arbitrage assumption, μ=r where r (the risk-free rate) is an appropriate U.S. Treasury bond…, The risk-free bond price today is a function of time with a payoff of $1 can be written as…, This is the same thing as modifying Geometric Brownian motion as…, Revisiting the log return’s mean under risk-neutrality…, Now looking back to the risk-neutral cumulative distribution function…, Forcing this distribution to look like the log return we know the mean and standard deviation so we can standardize and convert to a standard normal distribution…, Logarithm rules clean this up quite nicely…. The jump part of {\displaystyle \mathbf {G} ={\begin{pmatrix}\sigma _{t}^{1}\\\sigma _{t}^{2}\end{pmatrix}}} Recall the required rate of return for risk from the Capital Asset Pricing Model (CAPM)…, Back to our portfolio, to satisfy our no-arbitrage assumption of no-arbitrage, the portfolio’s return must be equivalent to the risk-free rate…. This article is about a result in stochastic calculus. Itô's lemma can be used to derive the Black–Scholes equation for an option. We have {\displaystyle \mu _{t}={\begin{pmatrix}\mu _{t}^{1}\\\mu _{t}^{2}\end{pmatrix}}} From the model, one can deduce the Black–Scholes formula, which gives the price of European-styleoptions. 1 Like Black-Scholes, Li’s formula enabled the creation of new derivative markets, which became worth north of $60 trillion. t g Black, Merton and Scholes: Their work and its consequences. In general, a semimartingale is a càdlàg process, and an additional term needs to be added to the formula to ensure that the jumps of the process are correctly given by Itô's lemma. View Option_Greeks_Derivation.pdf from SOC 482 at IIT Kanpur. The essence of pricing derivatives lies in risk-neutrality. ( t t , a compensated process and martingale, as, Consider a function

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