rks] 4. Consider a two factor model in which the stock price dynamics St; follows Geometric Brownian Motion and the stock variance vt is itself stochastic and follows a square root Document Preview:...

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You may assume throughout this examination that dX is an increment in a standard Brownian motion X and up to Mean Square Convergence   2 E dX =dt Detailed working must be presented to obtain maximum credit. If a question asks for a particular method to be used, any other technique employed will result in the loss of marks. The questions in this exam are mathematical based and computer programs or spreadsheets should not be used. 1. Using the expansion of sin(kx) when x is small, show that  
sin(x)sin(x)  2 2 lim !   2 x!0 x sin(x) 6 Note: You are not permitted to use L?Hospital?s Rule at any stage. You may use any expansions without proof [4 Marks]   x 2. Consider the function z(x;y) = (x+y)ln ; where x and y are independent variables. y (i) Show that @z @z x +y =z: @x @y 2 2 @ z @ z (ii) By di¤erentiating the expression in (i), ?nd a relationship between and : [6 Marks] 2 2 @x @y 3. By considering two column operations, show that the determinant of the matrix 0 1 1 1 1 @ A x y z M = 2 2 2 x y z can be expressed as(xy)(yz)(zx): [4 Marks] 4. Consider a two factor model in which the stock price dynamicsS ; follows Geometric Brownian Motion t and the stock variance v is itself stochastic and follows a square root process t p dS =S dt+ v S dX (t); t t t t 1 p dv =(v v)dt+ v dX (t): t t t 2 The two processes have a correlation coe¢ cient , i.e. dX dX =dt 1 2 The parameters ; ; v and  are all constant. LetF =F(t;S ;v ):UsingItô, considertheSDEfordF andintegrateover [0;t]toobtainanexpression t t for F(t;S ;v ): [5 Marks] t t 15. A perpetual American call option V gives its holder the right to buy the underlying asset S for a given constant strike price E at any time in the future. It has no expiry date. Assuming that the price of this option satis?es the Cauchy-Euler type equation 2 1 d V dV 2 2   S +(rD)S rV = 0; 0
0 is the constant interest rate, D...