Question

Consider the options written on a tradable underlying asset S that follows the stochastic process frac{dS_t}{S_t} = mu dt + sigma dz_t where mu and sigma are constant factors. Assume also that risk-free interest rate r is constant and flat. (a) Derive the current pricing of a down-and-out European call option with current asset price S_0, strike price K, barrier level L below the strike price and current asset price, and under in-the-money (K < S_0) conditions. (b) Consider a look-back option with payoff at maturity T taken to be f_T = egin{cases} 0, & ext{if } S_{overall} > K \ P, & ext{if } S_{overall} le K end{cases} where K is a predefined strike price and S_{overall} is the overall minimum price of the underlying asset during the life of the option. Suppose it has been issued at current time, derive the current pricing of this look-back option.

          Consider the options written on a tradable underlying asset S that follows the stochastic process
frac{dS_t}{S_t} = mu dt + sigma dz_t
where mu and sigma are constant factors. Assume also that risk-free interest rate r is constant and flat.
(a) Derive the current pricing of a down-and-out European call option with current asset price S_0, strike
price K, barrier level L below the strike price and current asset price, and under in-the-money (K < S_0)
conditions.
(b) Consider a look-back option with payoff at maturity T taken to be
f_T = egin{cases} 0, & 	ext{if } S_{overall} > K \ P, & 	ext{if } S_{overall} le K end{cases}
where K is a predefined strike price and S_{overall} is the overall minimum price of the underlying asset
during the life of the option. Suppose it has been issued at current time, derive the current pricing of
this look-back option.

$Consider the options written on a tradable underlying asset S that follows the stochastic process fracdStSt = mu dt + sigma dzt where mu and sigma are constant factors. Assume also that risk-free interest rate r is constant and flat. (a) Derive the current pricing of a down-and-out European call option with current asset price S0, strike price K, barrier level L below the strike price and current asset price, and under in-the-money (K < S0) conditions. (b) Consider a look-back option with payoff at maturity T taken to be fT = egincases 0, extif Soverall > K P, extif Soverall le K endcases where K is a predefined strike price and Soverall is the overall minimum price of the underlying asset during the life of the option. Suppose it has been issued at current time, derive the current pricing of this look-back option.$

Added by Crystal G.

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