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tw
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Posts: 592
Joined: May 10th, 2002, 3:30 pm

hedging option-on-future with restricted liquidity

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
 
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Aaron
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Joined: July 23rd, 2001, 3:46 pm

hedging option-on-future with restricted liquidity

July 15th, 2002, 4:32 pm

The situation you posit appears unrealistic to me. You have a simple price process for a non-storable commodity, you can trade at t0 and from t1 to t2, but not from t0 to t1, and the possibility of trading does not affect the process. You know the volatility, spot price and times for certain.However, given this model your conclusions are correct. There is a BS price, but no arbitrage argument to force the market price to equal it. If someone chooses not to hedge, she will have more risk but no more expected return than someone who hedges to a constant delta exposure equal to the expected average delta exposure of the unhedged position.Mean-variance analysis with explicit risk preferences is a reasonable way to price this option from t0 to t1 if there is net positive or negative natural demand for hedging.
 
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tw
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Joined: May 10th, 2002, 3:30 pm

hedging option-on-future with restricted liquidity

July 16th, 2002, 8:35 am

The situation you posit appears unrealistic to me. You have a simple price process for a non-storable commodity, you can trade at t0 and from t1 to t2, but not from t0 to t1, and the possibility of trading does not affect the process. You know the volatility, spot price and times for certain..... >>Aaron, Thanks for the reply. I agree entirely with your scepticism concerning a model where the complete lackof a market is assumed, for at least a certain period of time, but perfect knowledge of volatility,price process is also assumed. However, the motivation comes from considering the optionality implicit in certain long term energycontracts. For instance, a very long term gas supply contract, or a power station with fuel risk hedged out.At their simplest such assets might be considered as simple options.Since the underlying energy markets often tend to have most of the activitity in the spot,with exponentially decreasing interest in forward markets (with time to delivery), the marketsoften appear as having a window of liquidity of an uncertain, but characteristic duration. One question might be, if the owner of the option reduces his/her risk using what forward marketis available at any particular time, does that incur an implicit cost? Before looking at it, I felt that, in a "pure" case (in some vague sense), the average of the option value should somehow be equal to the average of the payoff, but felt that contrasted with a risk-reward of argument.Obviously the accuracy of this type of calculation must necessarily open to question given the uncertainty of inputs, but I don't think it's a redundant question.Thanks again,Tom
 
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Johnny
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Joined: October 18th, 2001, 3:26 pm

hedging option-on-future with restricted liquidity

July 16th, 2002, 10:10 am

However, the motivation comes from considering the optionality implicit in certain long term energycontracts. For instance, a very long term gas supply contract, or a power station with fuel risk hedged out.At their simplest such assets might be considered as simple options.Since the underlying energy markets often tend to have most of the activitity in the spot,with exponentially decreasing interest in forward markets (with time to delivery), the marketsoften appear as having a window of liquidity of an uncertain, but characteristic duration.This is similar to some of the issues that arose with Metallsgesellschaft and stack hedging. Mellon and Parsons (95) and Neuberger (99) may be worth a look.
 
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Aaron
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Joined: July 23rd, 2001, 3:46 pm

hedging option-on-future with restricted liquidity

July 20th, 2002, 4:37 am

In addition to Johnny's excellent answer, I would add that this is a very important practical question. There is a simple theoretical answer, but it requires the assumption that short-term liquidity will always be available, the process is stationary and there are limits to the steepness of the forward price curve. In these conditions, you can ignore the illiquidity of long-term contracts, because they can be replicated by rolling over short term ones.These assumptions are not unreasonable for interest rates (although no one has eliminated the maturity premium entirely). For trading small amounts, both relative to your liquid assets and the market, in reasonably liquid energy markets, it could be considered a first approximation. For large amounts or less liquid markets, it's very dangerous.