european call option arbitrage opportunity trading strategy

Options Arbitrage

Atomic number 3 derivative securities, options dissent from futures in a very crucial prise. They correspond rights rather than obligations � calls gives you the right to buy and puts gives you the right-hand to sell. Therefore, a keystone feature of options is that the losses on an option position are limited to what you paid for the option, if you are a buyer. Since there is usually an underlying asset that is traded, you can, every bit with futures, construct positions that essentially are riskfree by combining options with the implicit in plus.

Usage Arbitrage

The easiest arbitrage opportunities in the option grocery store survive when options violate obtuse pricing bounds. No option, for instance, should sell for to a lesser degree its exercise value. With a call choice: Value of call option dangt; Value of Underlying Asset � Strike Price

With a put selection: Treasure of set up dangt; Strike Mary Leontyne Pric � Value of Underlying Asset

For example, a call with a strike cost of $ 30 on a commonplace that is currently trading at $ 40 should never sell for to a lesser degree $ 10. Information technology it did, you could lay down an immediate profits by buying the call for to a lesser degree $ 10 and exercising right away to pee-pee $ 10.

In fact, you can tighten up these bounds for call options, if you are willing to create a portfolio of the rudimentary plus and the option and hold it through the option�s expiration. The bounds then become:

With a call option: Value of call dangt; Esteem of Underlying Plus � Present value of Strike Price

With a put option: Value of put dangt; Acquaint value of Strike Mary Leontyne Pric � Rate of Underlying Plus

Too see why, consider the call option in the previous example. Assume that you have one year to expiration and that the safe rate of interest is 10%.

Present value of Strike Price = $ 30/1.10 = $27.27

Lower Bound on call prize = $ 40 - $27.27 = $12.73

The call has to trade for to a greater extent than $12.73. What would happen if it traded for less, say $ 12? You would grease one's palms the call for $ 12, sell short a share of stock for $ 40 and empower the take-home proceeds of $ 28 ($40 � 12) at the safe rate of 10%. Consider what happens a year from now:

If the banal price dangt; 30: You first collect the proceeds from the riskless investment ($28(1.10) =$30.80), exercise the pick (bargain the share at $ 30) and cover your short sales agreement. You will and so annoy continue the difference of opinion of $0.80.

If the stock Price danlt; 30: You collect the return from the riskless investiture ($30.80), simply a share in the open market for the prevailing price then (which is less than $30) and keep the difference.

Put differently, you induct nothing today and are secured a positive payoff in the future. You could construct a correspondent lesson with puts.

The arbitrage bounds work best for not-dividend paying stocks and for options that can be exercised only at expiration (European options). Near options in the real world can be exercised entirely at termination (American options) and are on stocks that pay dividends. Flatbottom with these options, though, you should non see short term options trading violating these bound by vauntingly margins, partly because employment is so rare even with listed American options and dividends tend to be small. As options become long term and dividends become larger and more uncertain, you may real well observe options that breach these pricing bounds, but you whitethorn not be able to profit off them.

Replicating Portfolio

One of the primal insights that Fischer Black-market and Myron Scholes had virtually options in the 1970s that revolutionized option pricing was that a portfolio composed of the underlying plus and the riskless asset could embody constructed to have incisively the same cash flows as a birdsong or put option. This portfolio is known as the replicating portfolio. In fact, Black and Scholes in use the arbitrage argument to infer their option pricing model by noting that since the replicating portfolio and the traded option had the same cash flows, they would have to sell at the same monetary value.

To understand how reproduction works, let us consider a very simple theoretical account for threadbare prices where prices can jump to one of two points in each period of time. This model, which is called a quantity model, allows us to model the replicating portfolio fairly easily. In the figure down the stairs, we have the binomial distribution of a stock, currently trading at $ 50 for the next two time periods. Note that in two time periods, this stock can be trading for as much as $ 100 surgery as little as $ 25. Put on that the objective is to appreciate a telephone call with a strike price of 50, which is expected to give-up the ghost in ii fourth dimension periods:

� = Number of shares in the replicating portfolio

B = Dollars of borrowing in replicating portfolio

The objective is to flux � shares of stock and B dollars of adoption to replicate the cash flows from the call with a mint price of 50. Since we know the cashflows on the option with certainty at loss, information technology is best to start with the last period and work back done the language unit tree.

Step 1: Initiate with the end nodes and work backward. Note that the cry out option expires at t=2, and the receipts payoff on the alternative will follow the difference between the stock price and the exercise price, if the stock price dangt; exercise price, and zero, if the lineage Price danlt; exert price.

The objective is to construct a portfolio of D shares of stock and B in adoption at t=1, when the stock Leontyne Price is $ 70, that will have the same cashflows at t=2 arsenic the call option with a strike price of 50. Deliberate what the portfolio will generate in cash flows under each of the two stock price scenarios, aft you pay back the borrowing with interest (11% per period) and set the hard cash flows equal to the cash flows you would have received on the call.

If stemm price = $ 100: Portfolio Value = 100 D � 1.11 B = 50

If gillyflower price = $ 50: Portfolio Value = 50 D � 1.11 B = 0

Draft on skills that about of United States have non old since high school, we can solve for some the number of shares of stock you will need to buy in (1) and the amount you leave pauperization to borrow ($ 45) at t=1. Gum olibanum, if the stock price is $70 at t=1, borrowing $45 and buying one portion of the stock will give the same cash flows A buying the call. To prevent arbitrage, the value of the call at t=1, if the stock price is $70, has to be equal to the cost (to you as an investor) of setting up the replicating position:

Value of Call = Be of Replicating Position =

Considering the opposite leg of the binomial tree at t=1,

If the stock price is 35 at t=1, then the outcry is worth nothing.

Step 2: Now that we bon how much the call will be worth at t=1 ($25 if the stock price goes to $ 70 and $0 if it goes refine to $ 35), we can move backwards to the earlier prison term period and create a replicating portfolio that will provide the values that the option will provide.

In other words, borrowing $22.5 and buying 5/7 of a portion out wtoday ill provide the same cash in on flows every bit a call with a excise price of $50. The value of the call thus has to be the same as the cost of creating this position.

Value of Call = Cost of replicating position =

Consider for the moment the possibilities for arbitrage if the call listed at less than $13.21, sound out $ 13.00. You would buy in the collect $13.00 and sell the replicating portfolio for $13.21 and claim the difference of $0.21. Since the cashflows on the two positions are identical, you would be exposed to no risk and make a certain profit. If the call trade for to a greater extent than $13.21, order $13.50, you would buy the replicating portfolio, sell the cry and claim the $0.29 difference. Again, you would not give birth been unprotected to any risk.

You could construct a similar example using puts. The replicating portfolio in that case would be created by merchandising short on the underlying stock and lending the money at the riskless rate. Again, if puts are priced at a value antithetical from the replicating portfolio, you could capture the difference and be unclothed to zero risk.

What are the assumptions that underlie this arbitrage? The first is that both the traded asset and the alternative are traded and that you stern trade simultaneously in both markets, thus locking in your profits. The second is that there are zero (or leastways really low minutes costs). If transactions costs are large, prices bequeath have to go out outside the set created by these costs for arbitrage to be feasible. The third is that you can borrow at the riskless charge per unit and sell short, if needed. If you cannot, arbitrage Crataegus oxycantha no thirster represent feasible.

Arbitrage across options

When you bear multiple options listed on the same asset, you may represent fit to take advantage of relative mispricing � how one option is priced relative to some other - and shut up riskless profits. We will take care first at the pricing of calls relative to puts and then look at how options with polar exercise prices and maturities should glucinium priced, relative to each opposite.

Put-Call Parity

When you have a put and a call option with the same exercise price and the assonant maturity, you can create a riskless set by selling the call, buying the frame and purchasing the inexplicit plus simultaneously. To see wherefore, regard merchandising a call and buying a put with exercise price K and breathing out day of the month t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and e'er yields K at expiration t. To see this, assume that the stock price at termination is S*. The reward along each of the positions in the portfolio can be inscribed as follows:

Position	Payoffs at t if S*dangt;K	Payoffs at t if S*danlt;K
Sell call	-(S*-K)	0
Buy up put	0	K-S*
Buy up stock	S*	S*
Total	K	K

Since this position yields K with sure thing, the cost of creating this position must be adequate to the present value of K at the riskless rate (K e-rt).

S+P-C = K e-rt

C - P = S - K e-rt

This relationship between put and call prices is called put call parity. If it is profaned, you take arbitrage.

If C-P dangt; S � Ke^-rt, you would sell the call, buy the frame and buy the stock. You would earn to a higher degree the riskless pace on a riskless investment.

If C-P danlt; S � Ke^-rt, you would buy the vociferation, sell the put and deal breakable the stock. You would then invest the proceeds at the riskless order and land up with a riskless lucre at maturity.

Note that put call check bit creates arbitrage alone for options that behind exist exercised only at maturity (Continent options) and Crataegus laevigata not hold if options can be exercise embryonic (American options).

Does order-call parity hold astir in practice Oregon are there arbitrage opportunities? Unrivaled study examined option pricing data from the Chicago Board of Options from 1977 to 1978 and found potential arbitrage opportunities in a some cases. Even so, the arbitrage opportunities were small and persisted only for suddenly periods. What is more, the options examined were American options, where arbitrage may not be viable even if put-call parity is violated. A many recent study away Kamara and Miller of options on the Sdanamp;P 500 (which are European options) 'tween 1986 and 1989 finds fewer violations of put-call off parity.

Mispricing across Strike Prices and Maturities

A spread is a combination of ii surgery more options of the same type (call or put) on the unvaried underlying asset. You stern combine 2 options with the same matureness but diametrical exercise prices (bull and stick out spreads), ii options with the same strike price only unusual maturities (calendar spreads), two options with different exercise prices and maturities (diagonal spreads) and Sir Thomas More than two options (flirt spreads). You may be able to use spreads to take advantage of relative mispricing of options on the one underlying stock.

Walk out Prices : A call with a depress strike price should never sell for less than a call with a higher strike price, assuming that they both have the same maturity. If it did, you could bargain the lower run into price outcry and sell the higher attain Mary Leontyne Pric call, and lock in a riskless profit. Similarly, a assign with a glower strike price should ne'er sell for more than a put with a higher collide with price and the same maturity. If IT did, you could steal the higher strike price put down, sell the lower strike monetary value put and make an arbitrage profit.

Maturity : A birdsong (put option) with a shorter time to expiration should never sell for much a call (put) with the same coin price with a long time to expiration. If it did, you would buy the call (lay out) with the shorter maturity and sell (put) the call with the thirster matureness (i.e, create a calendar spread) and curl in a lucre today. When the outset cry out expires, you will either exercise the second call (and have no cashflows) or sell it (and make a further profit).

Even a casual perusal of the selection prices recorded in the newspaper each Clarence Day should make IT clear that information technology is selfsame unconvincing that pricing violations that are this egregious will exist in a market atomic number 3 liquid as the Chicago Dining table of Options.