108156.pdf

290 Mathematics for Finance exercise the American option until expiry, when either both options will turn out worthless, or the ...

Solutions 291 4) Finally, ifX′<X<X′′ <S(T), then (7.9) becomes an equality, S(T)−X=α(S(T)−X′)+(1−α)(S(T)−X′′) becauseX= ...

292 Mathematics for Finance 8.2Ifr=0andS(0) =X= 1, thenCE(0) =−u−udd.Foru=0.05 andd=− 0. 05 we haveu−d=0.1andCE(0) = 0.025 dolla ...

Solutions 293 8.9The delta of a European call becomes equal to 1 at the first stepnsuch that the option will be in the money ind ...

294 Mathematics for Finance The corresponding European and American put prices will be n 01 2 1. 68 1. 68 2. 53 2. 80 < /^33 ...

Solutions 295 8.14Consider the distribution function F(x)=P∗{W(t)<x}=P∗ { V(t)<x+ ( m−r+^12 σ^2 )t σ } = ∫x+(m−r+ (^12) σ ...

296 Mathematics for Finance One day later the shorted shares will be worth 17, 765 × 1 .81 = 32, 154. 64 dollars, whereas the ca ...

Solutions 297 9.8With 95% probability the logarithmic return on the exchange rate satisfies k>m+xσ∼=− 23 .68%,wherex∼=− 1 .64 ...

298 Mathematics for Finance 10.3To achieve a return of 14%, we would have to sell the bond for 0.8700e14%∼= 1 .0007 dollars, whi ...

Solutions 299 timet(note that the bond price grows by a factor of eyt), Dt=eytP^1 (y) ( (τn 1 −t)C 1 e−y(τn^1 −t)+···+(τnN−t)(CN ...

300 Mathematics for Finance andf(0,N)<0 requires that (N+1)y(0,N+1)<Ny(0,N).The border case is wheny(0,N+1)=NN+1y(0,N),whi ...

Solutions 301 0 .64% which allows us to findk(2,3; ud) = 0.20%.The other missing returns can be computed in a similar manner, fi ...

302 Mathematics for Finance 11.9Using formula (11.5) and the short rates given, we find the following structure of bond prices: ...

Bibliography.................................................... Background Reading: Probability and Stochastic Processes Ash, R ...

304 Mathematics for Finance Benninga, S. and Czaczkes, B. (1997),Financial Modeling, MIT Press, Cam- bridge, Mass. Bingham, N. H ...

Glossary of Symbols............................................ A fixed income (risk free) security price; money market account ...

306 Mathematics for Finance M market portfolio m expected returns as a row matrix μ expected return (N cumulative normal distrib ...

Index........................................................... admissible portfolio 5 strategy 79, 88 American call option 14 ...

308 Mathematics for Finance covariance matrix 107 Cox–Ingersoll–Ross model 260 Cox–Ross–Rubinstein formula 181 cum-dividend pric ...

Index 309 martingale probability 63, 250 maturity date 39 minimum variance line 109 portfolio 108 money market 43, 235 no-arbi ...