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3.3.3 Evaluate an expectancy or eliminate the uncertainty $ \spadesuit $

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The Itô lemma shows how it is possible to superpose infinitesimal increments $ df$ to mimic the evolution of the value of a financial derivative $ f(X(t),t)$ , which is a known function of the stochastic spot price $ X(t)$ . Starting from an initial (alt. terminal) value that is known at a time $ T$ , a finite number of incremental changes $ dV$ can in be accumulated to approximate a single possible outcome at a later (alt. earlier) time: the implementation of the so-called Monte-Carlo method will discussed later with a practical example (sect.4.5). At the end, the fair price for the derivative is calculated as the expectancy from a large number of possible outcomes, i.e. by performing a statistical average where each payoff is properly weighted with the number of times this value has been reached.
The main drawback of a statistical method is the slow convergence ( $ \propto 1/\sqrt{N}$ ) with the number of samples. The problem can be traced back to the difficulty of integrating the stochastic term in the Itô differential (3.3.2#eq.2). By combining anti-correlated assets, it is however possible to reduce the amount of fluctuations in a portfolio. Sometimes, it is even possible to completely eliminate the uncertainty through delta-hedging, in effect transforming the stochastic differential equation (SDE) into a partial differential equation (PDE) that is much simpler to solve. For that

  1. Create a portfolio, combining one derivative (e.g. an option) of value $ f(X(t),t)$ with a yet unspecified, but constant number $ -\Delta$ of the underlying asset. The initial value of this portfolio and its incremental change per time-step are

    $\displaystyle \Pi=f -\Delta X, \qquad\qquad d\Pi=df -\Delta dX$ (3.3.3#eq.1)

    where the Itô differential (3.3.2#eq.2) can be used to substitute $ df$ and the stochastic differential (3.3.1#eq.1) for $ dX$ .
  2. Choose the right amount $ \Delta$ of the underlying so as to exactly cancel the random component, which is proportional to $ dW(t)$ in the Itô differential

    $\displaystyle \Delta=\frac{\partial f}{\partial X}$ (3.3.3#eq.2)

    With this choice, the total value of the portfolio becomes deterministic, i.e. the remaining equation has no term left in $ dW(t)$ .
  3. No-arbitrage arguments show that without taking any risk, the portfolio has to earn the same as the risk-free interest rate $ r(t)$

    $\displaystyle d\Pi=r\Pi dt$ (3.3.3#eq.3)

    Indeed, if this was not the case and the earnings were larger (alt. smaller), arbitrageurs would immediately borrow money from (alt. lend money to) the market until the derivative expires and make a risk less profit from the difference in the returns.
  4. Reassemble the small deterministic incremental values into a partial differential equation, which can be solved more efficiently to obtain the present value of the derivative $ f(X(t),t)$ .
Of course, the amount $ \Delta$ will change after a short time and the portfolio has to be continuously re-hedged to obtain a meaningful value for the derivative-which is not quite possible in the real world. Two examples illustrate the procedure in the coming sections, using delta-hedging to calculate the price of derivatives in the stock and the bond markets.

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