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Grammatic sketch of the Catawba language by Albert Samuel Gatschet

By Albert Samuel Gatschet

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11) where W ∗ is the standard Brownian motion under a certain sense of the probability measure. Let us call it probability measure Q for the moment. Then, c(0, S) = c(0, X ). Consider the process c(t, X (t)) derived from X (t). By Itˆo’s lemma (Eq. 5), dc(t, X (t)) = ∂2 ∂ 1 ∂ c(t, X (t)) + r X (t) c(t, X (t)) + σ 2 X 2 (t) 2 c(t, X (t)) dt ∂t ∂X 2 ∂X ∂ +σ X (t) c(t, X (t)) dW ∗ (t). ∂X The BS equation stipulates that the coefficient of dt is identical to r c(t, X (t)), thereby resulting in dc(t, X (t)) = r c(t, X (t)) dt + σ X (t) ∂ c(t, X (t)) dW ∗ (t).

2 Further information about the Euler and Milstein schemes can be found in Kloeden and Platen (2010), Milstein (1995), and the seminal treatise of Jacod and Protter (2011). 5 The Black–Scholes Equation and Risk-Neutral Valuation The application of Itˆo’s calculus in finance can be effectively illustrated in the derivation of the celebrated BS equation. Suppose that in a frictionless market, an option writer sells a call option and simultaneously holds a portfolio to perfectly hedge the payoff of the call option, that is, the payoff of the call option is financed completely by the proceeds of portfolio liquidation.

Suppose that the instantaneous interest rate is a constant r , and the stock has constant volatility σ . Then the call option price is given by c(t, S) = S (d1 ) − K e−r (T −t) (d2 ), ˆ CALCULUS A BRIEF REVIEW OF MARTINGALES AND ITO’S where (x) is the CDF of a standard normal random variable and √ log d1 = d2 + σ T − t, d2 = Proof: 49 S K + (r − σ2 )(T − t) . √ σ T −t 2 By the risk-neutral valuation formula (Eq. 12), c B S (t, S) = e−r (T −t) EQ [max(ST − K , 0)] = e−r (T −t) EQ ST 1{ST >K } − K Q(ST > K ) .

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