派生証券の値計算

最近変動率と利益の関係に関して学んでいます。A problem in the Hull textbook’s problems manual discusses the market price of risk for the futures rate and the forward rate. For the forward rate, when the drift (mu) is 0, the market price of risk is noted to be –sigma(T-t). When the market price of risk is 0, the drift becomes sigma^2(T-t). Given the first, the second can be derived by algebra, as,
(mu-r)/sigma = -sigma(T-t) when mu=0 implies r=sigma^2(T-t). For (mu-r)/sigma to be 0 when r is unchanged, mu must equal r, so mu=sigma^2(T-t) in the world where mkt price of risk is 0.
The point is to prove that the forward price at t=0 = futures price(t=0) - sigma^2*T^2/2. The first step, it seems, is to derive the drift and the standard deviation of the diffusion process for the forward rate, dF. dF= mdt +sigma(dz). The diffusion process for the interest rate r is: dr = theta*dt + sigma*dz, by the Ho-Lee formula, where dz is a draw from a standard normal distribution. Setting F=d/dt(ln(P(t,T))), where P(t,T) is the price of a bond maturing at T seen at t, df/dr can be derived to be 1. When the diffusion process is S = mu*S*dt + sigma*S*dz, Ito’s lemma states that the process for F derived from S can be written as dF = (dF/dt + mu*S*dF/dS + (1/2)sigma^2*S^2*d^2F/dS^2)dt + sigma*S*dz. However, since dr is defined without the r multiple, it may be possible to rewrite dF as follows.
dF = (dF/dt + mu*dF/dr + (1/2)sigma^2*d^2F/dr^2)dt + sigma*dz. This depends on the derivation of Ito’s Lemma.
Another point is the definition of “forward risk neutral with respect to P(t,T)”. If this means that the market price of risk of F is equal to the volatility of P, and if the volatility of P is –sigma(T-t), then the problem is solved. Also, Ito’s lemma is needed to ensure that the divisor in the market price of risk formula is the sigma of the instantaneous short rate r.
Another higher-level point involves the conceptual meaning of the forward and futures rate. It seems that the “futures rate” is the rate used to determine the price of a Eurodollar futures contract. The forward rate is the interest rate which holds between times t and T as seen at time 0.
Since the drift (growth rate) is sigma^2(T-t) between 0 and T, the total gain in forward rate is sigma^2T^2 (integrating the rate) in the world where the market price of risk is 0. As the futures rate has no growth and must equal the forward rate at T (as the futures rate is a martingale in a world where the market price of risk is 0), the futures rate = fwd rate + sigma^2T^2/2 at t=0.
The next step is to derive the theta in the Hull-White formula. This is dr/dt and is the expected growth rate is also this theta. This is equivalent to the futures growth rate (unclear why). Thus, theta = Ft(0,t) + sigma^2t. QED
In summary, I understand that the numeraire is a denominator used to convert one derivative (e.g. and option) to another derivative (e.g. option in a foreign currency). However, I do not particularly understand its relation to the market price of risk or risk neutral worlds (perhaps reviewing quantos would be useful).
Interestingly, the "equivalent martingale measure" quantity is equivalent to the "risk neutral measure", which apparently causes the returns on the new security to be risk neutral. A numeraire is just a stochastic process which can be used as a divisor of another stochastic process, for example a money market account or an exchange rate. When one quantity is divided by this quantity, the resulting security is a martingale, meaning that E(ft/gt) = f0/g0.
It is unclear what exactly is meant by change of measure, but it seems that this is a probability distribution used in computing expected value.
Also, a "risk neutral measure" seems to be a probability distribution such that the expected return is the risk-free rate.