By Alonso Peña
This ebook will introduce you to the most important mathematical types used to cost monetary derivatives, in addition to the implementation of major numerical versions used to unravel them. specifically, fairness, forex, rates of interest, and credits derivatives are mentioned. within the first a part of the ebook, the most mathematical versions utilized in the area of economic derivatives are mentioned. subsequent, the numerical tools used to unravel the mathematical types are awarded. ultimately, either the mathematical types and the numerical tools are used to unravel a few concrete difficulties in fairness, foreign money, rate of interest, and credits derivatives.
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In fact, we can now use all the results from pricing equity derivatives to price structural models of credit risk. Note that the volatility σ is the firm's assets volatility and not the equity volatility. 5σ 2 )T σ T and d 2 = d1 − σ T In equation 12, N() is the cumulative standard normal distribution. This is a useful expression when we need to calculate problems like Initial Public Offer (IPO) of a firm and determine, based on the characteristics of a firm, what should be the fair price of its equity.
Like in MC simulation, time is discretized in steps dt from t=0 to t=T. From one step tp, the next price of the underlying can either go up or down by a factor u or d as shown in the following formulas: At t = 0 : S0 At t = t1 : uS0 or dS0 At t = t2 : u 2 S0 , udS0 , d 2 S0 Thus, the values of the tree at maturity are as follows: ST1 = u 2 S 0 , ST2 = udS 0 , ST3 = d 2 S 0 In general case, we proceed in a similar manner until we arrive at the maturity T, and we have N+1 values for the variable S.
Collocate the stencil to all the nodes of the domain. The following is the initial condition: X [ PD[ H[S N [ H[S N [ Alternatively, the following is the condition with numerical values: X [ ^ ` In the preceding equation, N . The following is the final boundary condition: X W and X W . 4. Iterate the solution in time with the stencil until we cover the full domain. The following are the internal nodes: XL M D XL M D XL M D XL M M XL M D XL M D XL M D XL M L L M XL M D XL M D XL M D XL M L M XL M D XL M D XL M D XL M [ 49 ] Numerical Methods We can arrange the numerical results from our algorithm as shown in the table in the following screenshot, using the transformed variables (upper table) or the original variables (lower table), where we can find that for S=75 and t=0, the option price is £4,20: Example of Finite Difference pricing.