By N.H. Chan, H.Y. Wong(auth.)
This authoritative guide illustrates sensible implementation of simulation thoughts within the banking and monetary industries via use of real-world, time-sensitive purposes. extraordinary a stability among idea and perform, it demonstrates how simulation algorithms can be utilized to resolve sensible difficulties and showcases how accuracy and potency in enforcing numerous simulation tools can be utilized as necessary instruments in probability administration. It additionally covers subject matters reminiscent of volatility, fixed-income derivatives, LIBOR marketplace versions, danger measures, and contains over two-dozen famous simulation models.Content:
Chapter 1 An creation to Excel VBA (pages 1–32):
Chapter 2 historical past (pages 33–70):
Chapter three dependent items (pages 71–119):
Chapter four Volatility Modeling (pages 121–175):
Chapter five Fixed?Income Derivatives I: Short?Rate versions (pages 177–216):
Chapter 6 Fixed?Income Derivatives II: LIBOR industry types (pages 217–253):
Chapter 7 credits Derivatives and Counterparty credits probability (pages 255–301):
Chapter eight Value?at?Risk and similar probability Measures (pages 303–341):
Chapter nine The Greeks (pages 343–380):
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Additional resources for Handbook of Financial Risk Management: Simulations and Case Studies
Sample text
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 ) .