Find the value of the following portfolios of options at

Find the value of the following portfolios of options at expiry, as a function of the share price:

(a) Long one share, long one put with exercise price E,

(b) Long one call and one put, both with exercise price E,

(c) Long one call, exercise price E1, short one call, exercise price E2, where E1 < E2,

(d) Long one call, exercise price E1, long one put, exercise price E2. There are three cases to consider,

(e) Long two calls, one with exercise price Eand one with exercise price E2, short two calls, both with exercise price E, where E1 < E < E2.

Formulate the following barrier option pricing problems as partial differential

Formulate the following barrier option pricing problems as partial differential equations with suitable boundary and final conditions:

(a) The option has barriers at levels Su and Sd, above and below the initial asset price, respectively. If the asset touches both barriers before expiry, then the option has payoff max(S − E, 0). Otherwise the option does not pay out.

(b) The option has barriers at levels Su and Sd, above and below the initial asset price, respectively. If the asset price first rises to Su and then falls to Sd before expiry, then the option pays out $1 at expiry.

Set up the following problems mathematically (i.e. what equations do

Set up the following problems mathematically (i.e. what equations do they satisfy and with what boundary and final conditions?) The assets are correlated.

(a) An option that pays the positive difference between two share prices S1 and S2 and which expires at time T.

(b) An option that has a call payoff with underlying S1 and strike price E at time T only if S1 > S2 at time T.

(c) An option that has a call payoff with underlying S1 and strike price E1 at time T if S1 > S2 at time T and a put payoff with underlying S2 and strike price E2 at time T if S2 > S1 at time T.

The solution to the initial value problem for the diffusion

The solution to the initial value problem for the diffusion equation is unique (given certain constraints on the behavior, it must be sufficiently smooth and decay sufficiently fast at infinity). This can be shown as follows:

Suppose that there are two solutions u1(x, τ) and u2(x, τ ) to the problem

with
u(x, 0) = u0(x).
Set v(x, τ ) = u1 − u2. This is a solution of the equation with v(x, 0) = 0. Consider

Show that

E(τ ) ≥ 0, E(0) = 0,

and integrate by parts to find that

dE/dτ ≤ 0.

Hence show that E(τ) ≡ 0 and, consequently, u1(x, τ ) ≡ u2(x, τ ).

We have seen how to apply the CreditMetrics methodology to

We have seen how to apply the CreditMetrics methodology to a single risky bond, to apply the ideas to a portfolio of risky bonds is significantly harder since it requires the knowledge of any relationship between the different bonds. This is most easily measured by some sort of correlation.

Suppose that we have a portfolio of two bonds. One, issued by ABC, is currently rated AA and the other, issued by XYZ, is BBB. We can calculate, using the method above, the value of each of these bonds at our time horizon for each of the possible states of the two bonds. If we assume that each bond can be in one of eight states (AAA, AA, . . . , CCC, Default) there are 82 = 64 possible joint states at the time horizon. To calculate the expected value of our portfolio and standard deviation we need to know the probability of each of these joint states occurring. This is where the correlation comes in. There are two stages to determining the probability of any particular future joint state:

1. Calculate the correlations between bonds.

2. Calculate the probability of any joint state.