# Using option pricing methods to model real asset liquidity

Liquidity risk (in the sense of asset exit risk) is warranted on investments that may not be easily divested at the going market or fair-value price. I am looking at a portfolio of private assets which is a mix of real estate, infrastructure and private equity. These consist of a handful of direct investments in real assets (not a fund of funds) and so are highly illiquid and exit/liquidity risk is a primary concern.

My goal is to get a metric indicating whether the liquidity risk of the private asset portfolio as a whole is deterioriating or improving

However, measuring this exit risk precisely is difficult. One method I've seen discussed is to apply real option pricing theory. Suppose a put option existed that gave us the right to sell the real asset at any time. The premium that we'd pay for this option would provide one measure of the "cost" of illiquidity, and could be aggregated for a portfolio of direct investments to give an overall metric for liquidity risk.

I've seen a few papers that briefly suggest this approach, but I haven't seen it actually applied. If it is a valid approach, how would it be practically applied, and how can the characteristics of real assets be substituted into option valuation methods? If there are some pitfalls that invalidate this approach, are there better ways for gauging this risk?

• Can you list some of the papers. I've never seen this, but thinking about it, the strike price you set on you option will obviously dictate the price of the option, so what is the mechanism for determining the price? Is it discrete and unique or is there a suggestion to use some distribution of strike prices, reflecting an unknown sale price ?
– Attack68
Jun 16 '18 at 5:15

I thought a little bit more about your problem and can suggest an analogy. I work in fixed income which deals with IBOR reference rates. One of the outstanding questions is often about the IBOR basis - how much higher a 6M rate should be compared with a 6M period generated by two rolling 3M rates, or 6 rolling 1M rates. From a lending banks' perspective each is equivalent except the rolling loans contain an inherent value option - the lending bank can choose whether to reissue the loan or not. From the borrowers' perspective this represents an inherent cost option. The borrower might suffer from liquidity scarcity and not be able to resource funds for the whole period if they originally chose the rolling strategy.

Why do I cite this? Your problem is semantically similar. A 6M loan represents an illiquid asset for 6M while the 6, 1M loans represent the same asset with more liquidity disposal points. Also, there is a lot of historical information about IBOR so the chance to explore theories is available. In Darbyshire: Pricing and Trading Interest Rate Derivatives, he suggests the following model (it is effective because he can backtest it against historical data and gets good, intuitive results):

Suppose your asset has some daily return, so that the compounded rate over the period gives the total return: $$d_n R_n = (1+d_1 r_1)(...)(1+d_1r_n) - 1$$ where $d_1$ is the DCF for 1 day and r_n is the overnight rate for day $n$. This formula could be used to evaluate all rate tenors, but the problem is that there would be no basis - every strategy would have the same price.

So, now suppose that an event can occur within the period, following a poison distribution. In the case of IBOR it represents the event that a bank will choose not to reissue the loan since the funding is better diverted to another cause. The event is expressed annually so that the expected value of the annualised event over event over a set number, $n$, of days is: $$\Lambda_n = \Lambda_{n-1} + d_1 \lambda = d_1 \sum_i^n \lambda$$ And your adjusted return is then $$d_n R_n + Adj = (1+d_1 r_1 + d_1 \Lambda_1)(...)(1+d_1r_n + d_1 \Lambda_n) - 1$$

There are some nice properties associated with this model. If the event occurs very quickly then its impact is more pronounced since that that event is propagated over terms, if it occurs towards the end then the impact is less. Intuitively this might make sense with your scenario, since if your assets are liquid for a period of time then your concern is mitigated, on the other hand if they become illiquid very quickly then you lose the ability to reassign your funds.

Your asset value might be more complicated since the price is also volatile (and you probably can't delta hedge future periods). Even though interest rates are volatile the pricing technique here uses the information about the current future values, and if a bank was keen on 'locking in' the lambda parameter they could delta hedge the market using interest rate swaps, which exist for any period.

I myself have tried to come up with another type of model that is options based but I have never been able to get close to the empirical results or intuition that this type offers.