3. Some computations related to a stock S(t) following the Merton-Black-Scholes Model. (a) Let S(t) =...
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price at time t = 0. At time t = 1,2,-. ., the stock price is S,ett+σ Σ. 2. the drift where a 0 is known as the volatility and the independently and identically distributed standard Normal N(0,1) random 0 is known as Zi variables are (a) Show that S, = S¢_1e#+oZ¢ _ St St-1 (b) What is the distribution of ln (c) What is...
2. Solving the generalized geometric Brownian motion equation. Let S(t) be a positive stochastic process that satisfies the generalized geometric Brownian motion SDE dS(t) = u(t)S(t) dt + o(t)S(t) dW(t), where u(t) and o(t) are processes adapted to the filtration Ft, t > 0. (a) Use Itô’s lemma to compute d log S(t). Simplify so that you have a formula for d log S(t) that does not involve S(t). (b) Integrate the formula you obtained in (a), and then exponentiate...
Question 1 Consider the derivation of the Black-Scholes model of option pricing. Let S=S(t) be the underlying stock price at time t and let f=f(S, t) be the option price at time t. a) Write down the value P of the portfolio defined in the Black-Scholes model. [2 marks] b) Use Itô’s lemma to find an expression for the change Δf in the discrete time Δt. [5 marks] c) Use the expression you have found in point b) to find...
Let S - $78,0 - 40%,r-6,5%, and 8 - 2% (continuously compounded). Compute the Black-Scholes price for a $70-strike European put option with 3 months until expiration a $11.21 b. $1.95 O $0.00 d. 53.21 O. 52.47
Let S - $53, -27%,r-5.5%, and 8-2% (continuously compounded). Compute the Black-Scholes delta (A) of a $55-strike European call option with 6 months until expiration O a.-0.5619 Ob -0.4977 OC 0.4923 Od-0.6132 O 0.0.4411
$125.00, and r = 5%. Find the Black-Scholes formula for the option paying $10.00 in 3 months if S(T) S Ki or if S(T) 2 K2, and zero otherwise, in the 7. Let S(0) = $100.00, K = $92.00, K2 Black-Scholes continuous-time model. $125.00, and r = 5%. Find the Black-Scholes formula for the option paying $10.00 in 3 months if S(T) S Ki or if S(T) 2 K2, and zero otherwise, in the 7. Let S(0) = $100.00, K...
4. (10 marks) (Black-Scholes) Fill in the details of L9.24. Specifically, a. (3 marks) Assume that C(S, t) satisfies the first PDE of L9.24 (Black-Scholes). Show that U(z,T) satisfies the second PDE of L9.24 (the heat/diffusion equation b. (5 marks) Using L9.23, solve for U Write your final answer using the standard normal CDF N Hint: Replace the lower limit of integration (the -o) with a correctly chosen quantity that lets you get rid of the "positive-part function" in the...
Let S - $57,0 -29%,r-7.5%, and 8 - 2.5% (continuously compounded). Compute the Black-Scholes vega of a $55-strike European call option with 3 months until expiration. (That is, compute the approximate change in the call price given a 1 percentage point increase in O.) O a. 0.1276 O b.0.1092 OC 0.1175 O d. 0.1862 0.0.1041
Assume an asset price St follows the geometric Brownian motion, dSt = u Stdt+oS+dZt, So =s > 0 where u and o are constants, r is the risk-free rate, and Zt is the Brownian motion. 1. Using the Ito's Lemma find to the stochastic differential equation satisfied by the process X+ = St. 2. Compute E[Xt] and Var[Xt]. 3. Using the Ito's Lemma find the stochastic differential equation satisfied by the process Y1 = Sert'. 4. Compute E[Y] and Var[Y].
3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B Find functions g and h such that X, has the same covariance as a Brownian bridge. 3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B...