- complyorexplain
**Posts:**93**Joined:**

It is often argued that the Black Scholes formula is inappropriate in the case where assets are illiquid, or cannot be short sold etc. Is this correct? Consider the following thought experiment.Asset X is illiquid and cannot be short sold, but there are market observable bid offers from which we can agree upon a mid-price reference rate. Suppose a firm sets up two desks, an option desk and a trading desk. In the outside market, the option desk sells a Euro put option on X, struck at K, to pay the difference between K and the reference rate on expiry, if the rate < K.The option desk delta hedges with the trading desk on a contract for difference basis. Trading desk pays a funding charge to option desk, and receives any a 'dividend' payment based on observable dividends for X. I.e. the effect is to replicate the P/L that the trading desk would receive had it actually transacted in the market (except without bid offer or commission slippage). The trading desk is committed to taking any trade from option desk, either bought or sold. All positions unwound on expiry.How do we value the positions held by each desk? I think (1) the option desk position should be valued on a Black Scholes basis, where the drift terms are the expected funding and dividend rates. P/L at maturity for the option desk will reflect its predictions of volatility (and of dividend stream and funding rates). And (2) the expected value of the trading desk (excluding any accumulated or mark to market P/L) will be zero, on the assumption that the value of any risky trading position is zero. Therefore the sum of the two desk valuations will be Black Scholes consistent. Thus, even though the firm as a whole has an unhedged option position, we should value it as if hedged. Is this logic sound?

Last edited by complyorexplain on May 17th, 2016, 10:00 pm, edited 1 time in total.

One should formally understand what did really prove by B&S. From their derivation it follows that there exists a portfolio long call and short delta shares of stocks, which guarantees risk free instantaneous rate of return. In either case delta depends on call price. How then we can draw a conclusion about option price. We have to assume at date t that we can sell a short position of stocks at t + dt moment then the price of the call option is solution of the BSE at t. In practice it is quite common to assume that t is open moment of the market and t + dt is a close moment of the market at the same day. If investor do not buy B&S hedged portfolio at t for C ( t , S ( t )) - [$]C^{'}[$]( t , S ( t )) S ( t ) and buy naked option then price dynamics of the option on [t , t + dt] is equivalent to dynamics of the [$]C^{'}[$]( t , S ( t )) S ( t ) . If any of BS assumptions is only approximately true then everything about hedging and pricing can be more or less violated.

- Martinghoul
**Posts:**3256**Joined:**

NOOOOOOOOOOOOOOOOOO!!!!!!!!!!!!

Martinghoul, can you any formal thoughts than too emotional NOOOOOOOOOOOOOOOOOO!!!!!!!!!!!!

@OP: how will the delta 1 desk (you call it trading desk) then hedge Its synthetic long exposure to the underlying? I am not following the thought experiment.

Last edited by frolloos on May 17th, 2016, 10:00 pm, edited 1 time in total.

I talked about derivation of the BS price and they as well as other handbooks do not talk about trading desk. Thus if we look on derivation of the BS price then we can see that there is hedged portfolio [$]\Pi ( t ) dt[$] and if C ( t , S ) is BSE solution then d[$]\Pi ( t ) = \Pi ( t ) dt[$]. It means that if investor borrows the value [$]\Pi ( t ) = C ( t , S ( t )) - C ^{'}_{S} ( t , S ( t )) S ( t )[$] at risk free rate r at t then at t + dt investor would be able to return the value ( 1 + r dt ) [$]\Pi ( t )[$]. Nothing more and nothing less than that. There is no evidence why market will follow such instruction though it is clear that it is reasonable strategy. Why can we think that BSE solution is a price of the option? Because the price of the BS portfolio or other complex financial issues are assumed constructed synthetically. Underlying of the synthetic pricing is its perfect liquidity. If there is no perfect liquidity of the stock assume for example that in order to sell short stocks in BS portfolio investor should pay for example $10 at t + dt. You can consider that if investor sells short position of the portfolio at t + dt he will get money in 3 days after. Then the value of the short position at t will be adjusted. In these cases BS price should be adjusted correspondingly. Therefore degree of easiness we can get rid from the short stocks of the BS portfolio at t + dt effects on BS option price.

Last edited by list1 on May 17th, 2016, 10:00 pm, edited 1 time in total.

QuoteOriginally posted by: list1I talked about derivation of the BS price and they as well as other handbooks do not talk about trading desk. Thus if we look on derivation of the BS price then we can see that there is hedged portfolio [$]\Pi ( t ) dt[$] and if C ( t , S ) is BSE solution then d[$]\Pi ( t ) = \Pi ( t ) dt[$]. It means that if investor borrows the value [$]\Pi ( t ) = C ( t , S ( t )) - C ^{'}_{S} ( t , S ( t )) S ( t )[$] at risk free rate r at t then at t + dt investor would be able to return the value ( 1 + r dt ) [$]\Pi ( t )[$]. Nothing more and nothing less than that. There is no evidence why market will follow such instruction though it is clear that it is reasonable strategy. Why can we think that BSE solution is a price of the option? Because the price of the BS portfolio or other complex financial issues are assumed constructed synthetically. Underlying of the synthetic pricing is its perfect liquidity. If there is no perfect liquidity of the stock assume for example that in order to sell short stocks in BS portfolio investor should pay for example $10 at t + dt. You can consider that if investor sells short position of the portfolio at t + dt he will get money in 3 days after. Then the value of the short position at t will be adjusted. In these cases BS price should be adjusted correspondingly. Therefore degree of easiness we can get rid from the short stocks of the BS portfolio at t + dt effects on BS option price.The question is not restricted to Black-Scholes. Why do you almost always state the BS pde in every post?

Last edited by frolloos on May 17th, 2016, 10:00 pm, edited 1 time in total.

- Martinghoul
**Posts:**3256**Joined:**

QuoteOriginally posted by: list1Martinghoul, can you any formal thoughts than too emotional NOOOOOOOOOOOOOOOOOO!!!!!!!!!!!!The answer to your question also happens to be NOOOOOOOOOOO!

QuoteOriginally posted by: list1Martinghoul, can you any formal thoughts than too emotional NOOOOOOOOOOOOOOOOOO!!!!!!!!!!!!I dont want to speak for Martinghoul, but a formal thought could be:If List Then NOOOOOOOOOOOOOOOO!!

- Martinghoul
**Posts:**3256**Joined:**

QuoteOriginally posted by: frolloosQuoteOriginally posted by: list1Martinghoul, can you any formal thoughts than too emotional NOOOOOOOOOOOOOOOOOO!!!!!!!!!!!!I dont want to speak for Martinghoul, but a formal thought could be:If List Then NOOOOOOOOOOOOOOOO!!Ooh, that's good... We can probably come up with some nice flow diagrams even.

QuoteOriginally posted by: frolloos@OP: how will the delta 1 desk (you call it trading desk) then hedge Its synthetic long exposure to the underlying? I am not following the thought experiment.I think you are following it just fine. The problem is indeed that effectively shifting the option position from the option desk to the "trading desk" through dynamic hedging may work great for the option desk but does nothing to change the position of the firm as a whole. So the hedging argument does not support valuation with the Black Scholes model.

- Traden4Alpha
**Posts:**23951**Joined:**

Three issue seem to trouble this hypothetical scenario:1. With an illiquid and unshortable stock, put-call parity would likely fail, the no-arbitrage condition could not be guaranteed, and Black-Scholes becomes unmoored from the math that bounds the options' prices to their "correct" values.2. With a true illiquid stock, the bids and offers are asynchronous which makes the mid-point ill-defined. Moreover, the wide bid-ask spread implies that the realized volatility grows severely in the last seconds in which the price at expiration might snap to the bid or offer depending on any transaction(s) at the last instant.3. As a practical matter, the liquidity of an options market is usually worse than that of the underlying so if the underlying is illiquid , the option market is likely derelict.

If we recall original question "It is often argued that the Black Scholes formula is inappropriate in the case where assets are illiquid, or cannot be short sold " then only T4A answered directly to the question and as it seems to me from rother practical point point of view. I attempted highlight theoretical aspect which also deals with practice. General my point is that not perfect liquidity or illiquidity changes price of the option on theoretical level too. Speaking broadly if liquid asset follows GBM with known coefficients and its illiquid counterpart can be governed other equation and the BS price and its illiquid counterpart can be quite far from each other and pricing argument such as perfect hedging could not be applicable or even wrong.

Last edited by list1 on May 17th, 2016, 10:00 pm, edited 1 time in total.

In case of illiquid asset which primarily realized in increasing ask - bid gap it might make sense to develop a BS pricing taking into account bid ask differential.

- complyorexplain
**Posts:**93**Joined:**

QuoteOriginally posted by: bearishQuoteOriginally posted by: frolloos@OP: how will the delta 1 desk (you call it trading desk) then hedge Its synthetic long exposure to the underlying? I am not following the thought experiment.I think you are following it just fine. The problem is indeed that effectively shifting the option position from the option desk to the "trading desk" through dynamic hedging may work great for the option desk but does nothing to change the position of the firm as a whole. So the hedging argument does not support valuation with the Black Scholes model.The question is about how we value the respective positions of the option and the trading (=delta1) desks. I think we agree about the valuation of the option desk position. This will be based on the best estimate of the future volatility of the reference rate. Yes?So, moving to the valuation of the trading desk position. I claim this will be the sum of realised and unrealised P/L, and therefore at the very outset will be zero. This is based on the assumption that the trading desk is simply following an algorithm given to it by the option desk. Since the algorithm is based on previous information, and is not forward looking, and assuming the weak form of the efficient market hypothesis, I believe we cannot make any assumptions about future profits or losses. Therefore inception value = 0. In summary, if we agree that the valuation of the option desk at inception should be the BS value, and that the valuation of the trading desk position (at inception) is zero, and if we agree that the sum of the two values = the value of the unhedged option position, i.e. the exposure of the two desks combined, then it follows that the valuation as a whole is given by Black Scholes.

Last edited by complyorexplain on May 18th, 2016, 10:00 pm, edited 1 time in total.

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