Lets denote the option payoff by H and the assets by S and X respectively and assume the option expires at time T.I am assuming that you mean the payoff H=h(S) is a function of the entire path t->S(t) since obviously path dependent option payoffs are not deterministic functions of the asset price S(T) at expiration only.Now let F_T denote the sigma field generated by the process t->S(t) up to time T.A random variable H is a deterministic function H=h(path(S)) if and only if H is F_T-measurable.So if the second asset price process X generates the same sigma field F_T as S does thenH=g(path(X))is another deterministic function of the path t->X(t). What you might not like is that whilethe first representation H=h(path(S)) might be how the option is defined, the second representationH=g(path(X)) is typically not known. Moreover if the option is path independent "as an option on S"H=h(S(T))(depends only on S at expiry) the same might not be true as an option on X.For example often you can represent S(T) as a stochastic integralS(T)=integral_0^T v(s)dX(s)and then H=h(S(T))=h( integral_0^T v(s)dX(s) )(path independent as an option on S but path dependent as an option on X).Purely mathematically there really is no unique "underlying".There are only options H (F_T-measurable random variables) and assets S.The option H can be replicated by trading in S if we have a representationH=c+integral_0^T u(t)dS(t)which is a very special explicit way of writing H as a deterministic function of thepath t->S(t). The valuation is by the martingale pricing formulac_t = E_t[ H ]with no explicit connection to the assets S. The assets S only determine the equivalent martingalemeasure (simplistically speaking since there might be more than one).Quite possibly this same option can be replicated by trading in Xalso regardless of how S and X are correlated. The correlation between S and X is not what is important here.Easy example: single asset marketdS(t)=sigma(t)S(t)dW(t).with W a Brownian motion on a the probability space (O,F_T,P). If the volatility process is bounded and bounded away from zero0<e<=|sigma(t)|<=K (*)then every P-integrable random variable H can be represented asH=E(H)+integral_0^T u(t)dS(t)(replicated by trading in S). Note that the condition is P-integrability not any explicit connectionto the process S. Consequently we can replace S with any other process X satisfyingdX(t)=nu(t)X(t)dW(t)with volatility nu(t) satisfying (*) and H will be replicable by trading in X.I understand where the confusion comes from.In the frequent case where the option payoff H=h(S(T)) is path independent we can hope that the price process c_t has the formc_t=C(t,S(t))(itself only dependent on the value S(t) and not the whole path of S up to time t).In this case we can then consider the deterministic function C(t,s) and the DELTASdC(t,s)/ds evaluated at s=S(t)and try delta hedging. In general the deltas are not defined so there is no delta hedging.A reasonable procedure of finding a hedge in general is to minimize the hedge variance between hedge trades.There is an easy solution to finding the optimal hedge weigths for each hedge trade.(download the book from http://martingale.berlios.de
, chapter 5, p117). The equations defining the variance minimizing hedge weights (equation 5.13, p129 for the single asset case, equation 5.40, p150 for the basket case) just take H as a random variable regardlesswether H=H(S(T)) or not and compute the hedge weights with respect to the assets S.You simply replace S with X and get the hedge weights with respect to X.In the martingale probability minimizing the variance over each hedge trade also minimizes the total hedge variance.In the market probability the situation is considerably more complicated.There may not be a variance minimizing hedge at all.
Last edited by trc
on September 27th, 2003, 10:00 pm, edited 1 time in total.