### The Paradox of Arbitrage

_{1}

_{2}

_{1}, dW

_{2}) < 1

My ideas I want to share.

The no arbitrage principle makes sense but when the time dimension is considered on how two identical assets trading at different venues can reach no arbitrage pricing there seam to be a paradox given that there is no party at either venue who can fix price fixing, i.e., the arbitrage argument for two assets some times seams paradoxical when there are two trading venues and trading is competitive (no body can set the price) and the time to converge is considered.

Lets consider two trading venues with supplies of identical products. If there is a miss pricing, arbitragers would buy form one venue cheaper and trader on the other. The supplier with the higher price would see less demand and would lower his price decreasing until the prices converge. But still the supplier can maintain his price or have a different price at this trading venue depending on whether he want to fix the price or his output. Any new information on the asset would create fluctuation in demand and the output would fluctuate if price is kept constant. Alternatively the price can be varied to meet a target demand but it is practically not possible.

The equation would be:

dX = [α (X – Y) + μ] dt

dY = - [β (X – Y) + μ] dt

Now let’s assume that there are many market plays now hold stocks of the assets. If there is counterparty, a trade would happen in the market. On new information on the asset, the supply and demand forces would vary randomly.

The equations

dX = [α (X – Y) + μ] dt + σXdW_{1}

dY = - [β (X – Y) + μ] dt + σYdW_{2}

If information dissipation and analysis is imperfect correl(dW_{1}, dW_{2}) < 1

In this case at least intuitively, it seam very difficult to imagine that there be no arbitrage opportunities. If the distance widens the are pulled back, but the random movements would make them drift in different directions. In a time interval they will not be the same. In the case that there is transport cost, with a certain probability there would be a arbitrage opportunity.

I am trying to work it out or trying to figure out if I overlooked something.

Best regards, Suminda Sirinath Salpitikorala Dharmasena

A market maker might buy stock at the bid and sells at the ask and by doing so creates a market. At the time he gives his bid and ask quotes he does not know what the trader is going to do. The market maker increases liquidity since if a trader wants to trade he does not need to look for the counterparty who want to enter the opposite transaction. The markets maker buys low and sells high and make a profit, i.e., his profit or loss is depended on how low he bought the asset and how high he sold it.

The buying and selling may not happen in the same time therefore he has to maintain an inventory of assets to meet the trading requirements. Also we should be mindful that the market may not be trading on actual valuation. Also when a market maker buys a stock and liquidates his position the market price might have changed.

Therefore the factors affecting his profits are:

- Bid and ask prices at which items were bought
- How fast he can turn over his inventory
- Exposure on his inventory

Out of these factors, what he can explicitly control in the spread again subjected to a maximum.

In order to achieve a high turnover and low inventories (thus reducing market exposures), a market making strategy should concentrate on balancing the liquidity supply and demand through setting the spreads.

If he is to manage his inventory he has to look at the number of sells and buys and put his spreads adjusting for buying and selling pressure in the market and against this inventory at the currently prevailing prices and the liquidity absorbed by other market makers and participants. When supply pressure in the market increases / decreases he should decrease / increase the bid and when demand pressure market increase / decrease he should increase / decrease the ask. Also he would have buy and selling pressure against this inventory based on large order he has to fill and the risk exposure on it.

The market pressures can be determined through the order book interaction with other MMs. The distribution of trades in the order book would help a MM to determine on how to setup his bid and ask. If a MM does not want to participate in either a buy or a sell, he can adjust the bid or ask accordingly so that there is a higher chance that another MM would be chosen undesirable transaction.

A market maker will have to fill various order. This would increase his buying and selling pressure. Also he has to asses the risk on the inventory in terms of price changes during his holding period and liquidity (determines the holding period and how fast he can turn around his inventory.)

To rap up, market makers are not in a position to trade on valuation. The profit he makes is based on how effectively a MM handles the supply and demand pressures through his spreads, not necessarily through setting it through a valuation basis like straddling the top and bottom of the market.

Suminda Sirinath Salpitikorala Dharmasena

Labels: Market Making

No. Simply because if the market expectation of volatility is becomes lower than the option implied volatility, there will be a arbitrage opportunity by shorting the option and delta hedging the position. This would bring down the IM and the VIX also.

I don’t see this happening soon though

I don’t see this happening soon though

Hi,

In order to while away some time in the last few days, I managed to work out a proof that for any time interval where there are 2 markets trading identical assets, there is an arbitrage opportunity in a given time window, where the price is sampled at discrete intervals within that time window.

Generally, when speaking of arbitrage, people do not really talk about the time horizon or the time dimension in which the arbitrage opportunity would disappear. For any time interval where there will definitely be arbitrage opportunities if there is no transportation cost. But if such costs are introduced, the probability of having an arbitrage opportunity can be worked out.

Arbitrage without a time dimension Zeno’s paradox to me at this point of time.

The proof is more intuitive than mathematically rigorous.

I got my personal diary pagers scanned where I made the entry. Can you be good enough to have a look at it and comment? I am wondering whether any body else also has come up with such an argument?

(c) 2008, Suminda Sirinath Salpitikorala Dharmasena, All rights reserved.

This is a response to question

"Well fiance decision making is in a fuzzy context, since the outcome cannot be predicted. I do not know much about your tool, but what I feel is that most of the linear tools are good enough, since the precision brought about by the more accurate tool might not really increase the outcome." - Suminda

"Well fiance decision making is in a fuzzy context, since the outcome cannot be predicted. I do not know much about your tool, but what I feel is that most of the linear tools are good enough, since the precision brought about by the more accurate tool might not really increase the outcome." - Suminda

There is an interesting language for parallelism in trading trad4 (http://trad4.sourceforge.net/). This is very interesting approach in building trading application. Some time back I also tried doing some thing by taking a deeper drive in complexity by designing a processor (http://www.wipo.int/pctdb/en/wo.jsp?wo=2007039837).

Suminda Sirinath Salpitikorala Dharmasena

Suminda Sirinath Salpitikorala Dharmasena

In the advent of Economic Events and Cooperate Action the following can happen:

• Implied volatility can increase leading up to the event and subsequently fall after directional move

• Implied volatility can increase leading up to the event and subsequently increase with the historic volatility also increasing

Depending on the event an option portfolio need to hedged as follows

If the even would result in the option volatility rising up to the then the following need to be checked:

• Is the volatility surface skewed towards the back months, in which case the option’s implied time to expiration should be compared using the forward options implied volatility and by fudging the time to expiration, if the fudged time to expirations are some what than they actually are, this can be used as a candidate for a diagonal spread

• Whether the event would be directional or would create uncertainty

If the event would create uncertainty the ideal position would be to make the portion of options (equity option if it is a cooperate event, equity and index related option if it is an economic event) which would be effected by such event Vega neutral but Volga positive, while being delta neutral. This can be done by going diagonally.

If the uncertainly reduces after the day with a possible large change in asset value, the ideal would be negative Vega.

• Implied volatility can increase leading up to the event and subsequently fall after directional move

• Implied volatility can increase leading up to the event and subsequently increase with the historic volatility also increasing

Depending on the event an option portfolio need to hedged as follows

If the even would result in the option volatility rising up to the then the following need to be checked:

• Is the volatility surface skewed towards the back months, in which case the option’s implied time to expiration should be compared using the forward options implied volatility and by fudging the time to expiration, if the fudged time to expirations are some what than they actually are, this can be used as a candidate for a diagonal spread

• Whether the event would be directional or would create uncertainty

If the event would create uncertainty the ideal position would be to make the portion of options (equity option if it is a cooperate event, equity and index related option if it is an economic event) which would be effected by such event Vega neutral but Volga positive, while being delta neutral. This can be done by going diagonally.

If the uncertainly reduces after the day with a possible large change in asset value, the ideal would be negative Vega.

Labels: Derivatives, Options, Trading

I was initially intrigued when I heard that Taleb was trading on out of the money option. In a way this makes sense since he has exposure to market making. I shall discuss below why this is a sensible strategy from two perspectives.

Market making perspective

In out of the money option pose lesser risk to a market maker than option this are either in or near the money. Since their delta is small, the market maker need not bring down his ask on a small unfavourable price move, but if the price move is large then the liquidity might dry up and might not be able to sell the out of the money option. Also due to liquidity of the options the market maker can lower his bid on them thus gaining a larger spread. Though a market maker need not lower his ask on a downward move he is in a position to increase the ask on a by a greater margin on an upward move. In extreme events the option inventory may end up in the money. The time decay of the options is also less or negligible when considering the market makers holding time.

Also the market maker can scalp using his inventory of out of the money option to make an additional amount of money. (If a trader has DMA then also he is in the position to make money by “juggling” an out of the money option portfolio.) All these do not require any predictive power in the market, which in deed is very difficult to do consistently.

If out of the money options are bought

Reference: http://www.cs.cmu.edu/~softagents/papers/CR_nevmyvaka_sycara_seppi.pdf

http://www.investorschronicle.co.uk/InvestmentGuides/Shares/article/20080423/eaf42688-f417-11dc-96a0-0015171400aa/Shares-Understanding-Level-2-market-makers--DMA.jsp

http://www.investorschronicle.co.uk/InvestmentGuides/default/article/20061124/443df720-e8ff-11db-9f1b-00144f2af8e8/Which-DMA-platform.jsp

Volatility Trading Perspective

dV / V = 1 / V * (∂V/∂S * dS + ∂V/∂σ * dσ + ∂V/∂T * dT + ∂V/∂r * dr)

E(ΔV / V) = 1 / V * Delta * E(ΔS) + 1 / V * Vega * E(Δσ) + 1 / V * Theta * ΔT + 1 / V * Rho * E(Δr)

E(ΔV / V) = Lambda * E(ΔS) + 1 / V * Vega * E(Δσ) + 1 / V * Theta * ΔT + 1 / V * Rho * E(Δr)

Returns can be maximized by buying the Option Greeks at the cheapest weighted by the expected change in the factor.

When trying to maximize the return by buying the Greeks cheap would mean that you end up buying somewhat out of the money.

Thing to watchout here is trasaction cost. This can eat all your profits.

Market making perspective

In out of the money option pose lesser risk to a market maker than option this are either in or near the money. Since their delta is small, the market maker need not bring down his ask on a small unfavourable price move, but if the price move is large then the liquidity might dry up and might not be able to sell the out of the money option. Also due to liquidity of the options the market maker can lower his bid on them thus gaining a larger spread. Though a market maker need not lower his ask on a downward move he is in a position to increase the ask on a by a greater margin on an upward move. In extreme events the option inventory may end up in the money. The time decay of the options is also less or negligible when considering the market makers holding time.

Also the market maker can scalp using his inventory of out of the money option to make an additional amount of money. (If a trader has DMA then also he is in the position to make money by “juggling” an out of the money option portfolio.) All these do not require any predictive power in the market, which in deed is very difficult to do consistently.

If out of the money options are bought

Reference: http://www.cs.cmu.edu/~softagents/papers/CR_nevmyvaka_sycara_seppi.pdf

http://www.investorschronicle.co.uk/InvestmentGuides/Shares/article/20080423/eaf42688-f417-11dc-96a0-0015171400aa/Shares-Understanding-Level-2-market-makers--DMA.jsp

http://www.investorschronicle.co.uk/InvestmentGuides/default/article/20061124/443df720-e8ff-11db-9f1b-00144f2af8e8/Which-DMA-platform.jsp

Volatility Trading Perspective

dV / V = 1 / V * (∂V/∂S * dS + ∂V/∂σ * dσ + ∂V/∂T * dT + ∂V/∂r * dr)

E(ΔV / V) = 1 / V * Delta * E(ΔS) + 1 / V * Vega * E(Δσ) + 1 / V * Theta * ΔT + 1 / V * Rho * E(Δr)

E(ΔV / V) = Lambda * E(ΔS) + 1 / V * Vega * E(Δσ) + 1 / V * Theta * ΔT + 1 / V * Rho * E(Δr)

Returns can be maximized by buying the Option Greeks at the cheapest weighted by the expected change in the factor.

When trying to maximize the return by buying the Greeks cheap would mean that you end up buying somewhat out of the money.

Thing to watchout here is trasaction cost. This can eat all your profits.

Labels: Derivatives, Market Making, Options

Risk

Main risk to consider is downside volatility

Risk adjusted rewards

Intuitive reward to risk ratio: mean(returns) * SD(Upside Risk) / V(Downside Risk)

NB: without adjustment to skew and kurtosis

Hedging error

Many talk about minimising hedging error. But what makes sense is minimising the downside due to the hedging error and maximise the upside hedging error. The upside hedging error can be maximised by holding a long Gamma position.

Main risk to consider is downside volatility

Risk adjusted rewards

Intuitive reward to risk ratio: mean(returns) * SD(Upside Risk) / V(Downside Risk)

NB: without adjustment to skew and kurtosis

Hedging error

Many talk about minimising hedging error. But what makes sense is minimising the downside due to the hedging error and maximise the upside hedging error. The upside hedging error can be maximised by holding a long Gamma position.

Labels: Hedging Error, Risk

Model break downs could result in substantial loss. An ideal financial model should have a positive expected value and should be hedged against possibility of model breakdowns. This way the losses in extreme events can be tamed and in the long run the trading strategy would prove profitable.

Through it cannot be fully elaborated in the few lines I am writing here, with many of the currently available derivative products, trading strategies can be developed without the large negative exposure to extreme events. These would consistently perform well in normal market conditions but would also would be graceful ant failsafe in turbulent times, while producing higher than average returns.

Suminda Sirinath Salpitikorala Dharmasena

My Profile: http://www.linkedin.com/in/sirinath

http://www.facebook.com/profile.php?id=842840093

http://www.blogger.com/profile/03835227536866539389

(c) Suminda Sirinath Salpitikorala Dharmasena. All rights reserved.

Through it cannot be fully elaborated in the few lines I am writing here, with many of the currently available derivative products, trading strategies can be developed without the large negative exposure to extreme events. These would consistently perform well in normal market conditions but would also would be graceful ant failsafe in turbulent times, while producing higher than average returns.

Suminda Sirinath Salpitikorala Dharmasena

My Profile: http://www.linkedin.com/in/sirinath

http://www.facebook.com/profile.php?id=842840093

http://www.blogger.com/profile/03835227536866539389

(c) Suminda Sirinath Salpitikorala Dharmasena. All rights reserved.

Labels: Investment

The biggest problem in developing trading strategies is that few returns can wipe a portfolio through the trading strategy on average would make positive gains. There are many lines of thoughts on how to overcome these, which are as follows:

• These unexpected events are outliers and can be ignored – this would result in blown up traders

• Try to profit from the rare (Black Swan) type of events – as advocated by Nassim Nicolas Taleb

• Hedge away the possibility or the exposure to these rare events – may end up becoming un profitable due to hedging cost

• Use only strategies which have positive exposures to rare events while the expected value of the trading strategy is also positive

The latter is like card count at a Black Jack table! The money just keeps flowing.

Suminda Sirinath Salpitikorala Dharmasena

My Profile: http://www.linkedin.com/in/sirinath

http://www.facebook.com/profile.php?id=842840093

http://www.blogger.com/profile/03835227536866539389

(c) Suminda Sirinath Salpitikorala Dharmasena. All rights reserved.

• These unexpected events are outliers and can be ignored – this would result in blown up traders

• Try to profit from the rare (Black Swan) type of events – as advocated by Nassim Nicolas Taleb

• Hedge away the possibility or the exposure to these rare events – may end up becoming un profitable due to hedging cost

• Use only strategies which have positive exposures to rare events while the expected value of the trading strategy is also positive

The latter is like card count at a Black Jack table! The money just keeps flowing.

Suminda Sirinath Salpitikorala Dharmasena

My Profile: http://www.linkedin.com/in/sirinath

http://www.facebook.com/profile.php?id=842840093

http://www.blogger.com/profile/03835227536866539389

(c) Suminda Sirinath Salpitikorala Dharmasena. All rights reserved.

Labels: Investment

A rather interesting article on Quant Hedge funds!

The measure of liquidity should be an indication of the matching of supply and demand. The ideal indication of this is a consistent bid ask spread. If the bid ask spread changers regularly, then this is a indication of imbalance of supply and demand.

A fist judgement equation is:

Liquidity = Mean(spread) / SD(spread) * % volume average trade size / MAD(volume size) * % matched orders w.r.t. all bid and ask volume

MAD: See - http://en.wikipedia.org/wiki/Median_absolute_deviation

Suminda Sirinath Salpitikorala Dharmasena

A fist judgement equation is:

Liquidity = Mean(spread) / SD(spread) * % volume average trade size / MAD(volume size) * % matched orders w.r.t. all bid and ask volume

MAD: See - http://en.wikipedia.org/wiki/Median_absolute_deviation

Suminda Sirinath Salpitikorala Dharmasena

Labels: Liquidity