July 12th, 2002, 11:34 am
I was recently thinking the following situation, which valuersof real options in the power markets may find familiar.It's concerned with a finite time period of futures liquidityand the related risk issues (is there any literature on this subject?)Suppose I have a call option, for the physical delivery ofa non-storable commodity at time t2 that I wish to value.However, due to liquidity issues for this commodity thereis no forward market for delivery at t2. This means I can'tuse the Black(76) model to value it as I cannot replicatethe option, and dynamically maintain a riskless portfolio.In fact, the situation forces me to take full market risk.(I know there are trading strategies that supposedly mitigatethis risk, but I want to keep this simple)However, from knowledge of the the market I think that attime t1 a forward market for t2 _will_ become available;t2>t1>t0 where t0 is the current time.Suppose I have a model for prices. Suppose further, forthe simplicity of the argument, that this process is geometricBrownian motion. I now have (at least) two strategies. Wait till t1, thencommence dynamic hedging a la Black-Scholes, or simplytrust to luck and take the spot price at expiry. (There arealso a continuum of strategies in between, involving delayingthe start of hedging). The expected value of the first strategy is \int_{0}^{\infty} p(s_{t1}, t1; s_{0}, t0) V(s_{t1}, t2-t1) ds_{t1}where V(s,t) is the Black76 formula and p is the _actual_ densityappropriate to the GBM process (solution to Kolmogorov). However, a risk neutral formalism exists for options-on-futures (it is equivalent to a standard option that pays a negative dividend equal to the interest rate). Hence, the above can be written as\int_{0}^{\infty}\int_{0}^{\infty} p(s_{t1}, t1; s_{0}, t0) p^{RN}(s_{t2}, t2;s_{t1}, t1) max(s_{t2}-E,0) ds_{t1}ds_{t2}where p^{RN} is the risk-neutral probability. In the event thatthe risk-neutral density conincides with the actual density (from the above arguments, surely when drift=interest rate=0)one can use Chapman-Kolmogorov as\int_{0}^{\infty}p(s_{t1}, t1; s_{0}, t0) p^{RN}(s_{t2}, t2;s_{t1}, t1) ds_{t1}=p(s_{t2}, t2; s_{0}, t0)This seems to indicate that the two strategies both have the same expected payoff in these conditions.However, both still have considerable market risk. The second moments of the payoffs are \int_{0}^{\infty} p(s_{t1}, t1; s_{0}, t0) [\int_{0}^{\infty} p^{RN}(s_{t2}, t2;s_{t1}, t1) max(s_{t2}-E,0) ds_{t2}]^2 ds_{t1}(since, under strict Black-Scholes after the dynamic hedging commencesat t1, the return must be equal to the Black-Scholes value, with no uncertainty)and\int_{0}^{\infty} p(s_{t2}, t2; s_{0}, t0) [max(s_{t2}-E,0)]^2 ds_{t2}These two expressions look like they will give different answers (and preliminary spreadsheet calculations suggest they do), the "takingit into the spot" strategy being the most risky. Some questions..Is this a situation where simply risk can be taken for no reward?In the case where the risk neutral probabilities do not coincidewith the actuals, is this a mean-variance scenario,with a risk-reward or mean value vs. std devation of valuecurve which each trader must place themselves on, subject to risk preferences?Tom