ma599_mini_project_exotic options (1)
TRANSCRIPT
Exotic Options - Products, Applications and Pricing
Emily Chan, Jack Daly, Jeffrey Fung, Jeff Ma, Alan McCabe
March 6, 2015
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Contents
1 Introduction 6
1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Project Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Information Gathering and Resource Methods . . . . . . . . . . . . . . . . 6
1.4 Forward Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3 Features of a Forward Contract . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Uses of Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Options: Call and Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8.2 Key Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9 Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9.1 General Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9.2 Profit from Holding or Writing a Call Option . . . . . . . . . . . . . . . . 11
1.9.3 Holder: Profit for a Call and Put Option . . . . . . . . . . . . . . . . . . 12
1.9.4 Writer: Profit for a Call and Put Option . . . . . . . . . . . . . . . . . . . 12
1.9.5 Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10.1 Profit for Holding or Writing a Put Option . . . . . . . . . . . . . . . . . 13
1.10.2 Holder and Writer Profit for A Put Option . . . . . . . . . . . . . . . . . 13
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1.11 Risks, Strategies and Rewards for Call and Put Options . . . . . . . . . 14
1.11.1 Risk of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.11.2 Hedging, Speculation and Arbitrage . . . . . . . . . . . . . . . . . . . . . 14
1.12 Forwards versus Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 The Binomial Tree Pricing Method 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Factors for Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Pricing using Binomial Tree Method . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 1-time Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 2-time Period Binomial Tree . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 An n-time Period Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Exotic Options 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 Merits of Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.4 Risk Management Using Asian Options . . . . . . . . . . . . . . . . . . . 24
3.2.5 Averaging Method for Asian Options . . . . . . . . . . . . . . . . . . . . . 24
3.2.6 General Outline for the Pay-off of Asian Options . . . . . . . . . . . . . . 25
3.2.7 Is it a Call or a Put? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.8 Arithmetic Averaging or Geometric Averaging? . . . . . . . . . . . . . . . 26
3.2.9 Average Price or Average Strike? . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.10 The Binomial Option Pricing Model for Asian Options . . . . . . . . . . . 26
3.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.3 Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.4 Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Compound Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Call on Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.4 Binomial Tree Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.5 Generic Call on Call Example in Pricing . . . . . . . . . . . . . . . . . . . 31
3.4.6 Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.7 Call on Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.8 Put on Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.9 Binomial Tree Option Pricing - Worked Example . . . . . . . . . . . . . . 33
3.4.10 Put on Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5.2 The Value of Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5.4 Risk of Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.5 Pricing Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.6 Comparing Lookback Prices to Vanilla Options . . . . . . . . . . . . . . . 37
3.6 Chooser Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6.2 Empirical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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3.6.3 Simple Chooser Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6.4 Complex Chooser Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.5 Chooser Option vs Straddle Option . . . . . . . . . . . . . . . . . . . . . 43
3.6.6 Relationship between Strike Price and Option Price . . . . . . . . . . . . 44
4 Software 45
4.1 Software Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Conclusion 46
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1 Introduction
1.1 Abstract
The 5 families of exotics that will be explored will be; ‘Asian, Barrier, Compound, Lookbackand Chooser’, as well as Plain Vanilla Options and Forward Contracts. They will be discussedin detail, as well as priced, not only in the software, but worked examples will be illustrated inthe document as well.
1.2 Project Introduction
All derivative contracts, as well as exotic options (exotics) are a global risk management toolutilised by both corporations and individuals. They are complex, difficult to understand, goodknowledge of the market is needed in order to use them effectively and some of them are expensive.They are modern derivatives, some of which are created using technology that is not widelyavailable to the general public.
This project will include explanations, descriptions and pricing methods of plain vanillaoptions and exotics as well as a brief introduction to a forward contract. The pricing softwarewill be Excel-based and will be included in the final chapter.
The binomial tree model will be used to demonstrate the pricing of a plain vanilla optionand will feature in the pricing of the exotics. A binomial tree graphic will be used illustratedthe price movements of the stock price associated with the option being shown, as well as themethod in which the option price is established.
The risks and rewards associated in the implementation of the exotics will be detailed, andexplanations of why these exotics are required in the market place, will be demonstrated.
1.3 Information Gathering and Resource Methods
All relevant references will be included in the ‘Refernces’ section at the end of the project.
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1.4 Forward Contract
1.4.1 Introduction
A forward contract is an agreement between two parties whereby one contracts to buy a specifiedasset from the other for a specified price on a specified date in the future.[4]
The buyer and seller agree the price and date of trading at the beginning of the contract.At the specified date (delivery date), the buyer pays the seller the agreed price (forward price)and the seller delivers the underlying asset. The contract is able to be tailored to each partiespersonal requirements.
The two parties that are involved in the forward contract are the long party and the shortparty. The long party is the party that has agreed to buy the asset, whereas the short party hasagreed to sell the asset.
1.4.2 Example
Person A wants to sell a pen and person B is looking to buy a pen. A, agrees to sell the pen toB for £1 in one month’s time. If the market price of the pen increases to £1.10 on the deliverydate, B would gain 10p from buying the pen from A, instead of selling it in the market place for£1.10. In this case A loses 10p, as he could have sold the pen in the market place for £1.10.
Instead of physically exchanging the pen with B, A can pay 10p to B in order to settle thecontract, then sell the pen in the market place. The above example assumes B is a speculator.Speculation is a strategy that involves taking advantage of expected price movements. We assumethat B is expecting the price of the pen to increase in the future, thus he would be looking atselling the pen and making a profit in the future.
In the scenario where the market price of the pen decreases to 90p by maturity, A, gains 10pas he agreed to sell the pen to B for £1, thus making a profit of 10p. Instead of buying the penfrom A, B has the option to pay the outstanding 10p to A without the physical exchange of thepen.
1.4.3 Features of a Forward Contract
As forward contracts are traded over the counter they need not necessarily be protected byregulation. This means that the two paries are able to trade directly with each other. This typeof contract is traded over the counter (OTC), that is, two parties trading directly with each othervia an exchange that is not a centralised exchange.
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At the beginning of the contract, the two parties agree the price in a specified currency, theexpiry date and the delivery date of the asset. The two parties are able to set their own terms andconditions of which would suit each party. There are also no margin requirements for forwardcontracts which implies that a deposit does not need to be paid at the beginning of the contract.
1.5 Uses of Forward Contracts
The primary sector uses it to reduce the uncertainty risk of profit. They can be the seller of theforward contract to ensure the buyer will buy their product at the maturity date. The primarysector of an economy is related to manual activities such as agriculture, fishing and mining.
Forward contracts are very useful for the buyer to hegde risk, as it locks in the future priceof asset. So the hedger will not be affected by the price fluctuation.
Speculators usually try to buy a forward contract when the underlying asset’s price is lowand expect the price will increase at maturity date. They will sell the underlying asset to thepublic market after they have received the asset to make profit.
1.6 Risks
Since there are no margin payments required, there is a risk of default between the parties. Theremay also be credit risks associated with one of the party’s as well as a risk of the asset not beingdelivered.
The risks are assumed by both parties. In order for the trade to be ‘trustworthy’ in nature,large institutions such as banks and corporations are involved in the trade.
This makes the amount of goods and cash exchanged in the forward contract large in nature,which implies that smaller individuals would not have the ability to trade in such an environment.
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1.7 Payoff
The payoff of a forward contract is the difference between the price at maturity and the agreedforward price. The calculation is:
Long : fT = ST −K, Short : fT = K − ST , (1)
where:
t = current time
T = maturity date
r = risk free interest rate
fT= is the value of the contract at time T
ST = price at time T of the underlying asset
K = forward price at time t= Ster(T−t) = Ste
r∆t
1.8 Options: Call and Put
1.8.1 Introduction
In finance, an option is a contract which gives the holder the right, but not the obligation, tobuy or sell an underlying asset at a specified strike price on or before a specified date. The writerhas the corresponding obligation to fulfil the transaction – that is to sell or buy – if the holderexercises the option.
The holder pays an upfront premium to the writer for this right. An option which conveys tothe owner the right to buy asset at a specific price is referred to as a CALL option; an optionwhich conveys the right of the owner to sell asset at a specific price is referred to as a PUToption.
1.8.2 Key Definitions
• Strike price: This is the pre-agreed price at which the asset is bought or sold at a futuredate.
• Expiry date: The date of maturity of the option contract. This is the date at when theright to exercise the option comes into effect.
• Premium: This is the amount of upfront money that must be paid in order to hold theright to buy or sell the asset.
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1.8.3 Notation
All of the notation detailed below will be used throughout the project. The variations on thepayoff functions for the exotics will be defined in the relevant later chapters.
• S0 = asset price at time 0
• S1(1) = S0u = asset price at time 1 for an ‘up’ movement
• S1(2) =S0d = asset price at time 1 for a ‘down’ movement
• K = strike price
• t = start date
• T = maturity date
• q = er∆t−du−d is the risk neutral probability
• ct = the price of a call option (the upfront premium)
• pt = the price of a put option (the upfront premium)
• VT = max(ST −K,0) for a call option
• VT = max(K − ST ,0) for a put option
• V1(·) = e−r∆t[qV2(·) + (1− q)V2(·)]
• V0 = e−r∆t[qV1(1) + (1− q)V1(2)]
• σ = magnitue of price movements up or down, volatility
1.9 Call Option
1.9.1 General Outline
If a stock is currently trading at a price of £20 and a call option is bought with a strike price of£25, the holder of that option will be able to purchase that stock from the writer at a date inthe future for £25. This future price is known as the strike price and has to be honoured by thewriter regardless of the future value of the stock price.
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1.9.2 Profit from Holding or Writing a Call Option
Figure 1: Profit from Buying a Call.
Figure 2: Profit from Writing a Call.
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1.9.3 Holder: Profit for a Call and Put Option
ST Call PutST ≤ K -ct -pt + (K − St)ST > K -ct+(St −K) -pt
1.9.4 Writer: Profit for a Call and Put Option
ST Call PutST ≤ K ct pt - (K − St)ST > K ct -(St −K) pt
1.9.5 Payoff
If the stock price increases above the strike price, the holder of the option will exercise their rightto buy the asset and will pay a premium to the writer, as well as the strike price. The holdermay then either sell the asset or keep it an write an option. If the stock does not does not reachthe strike price, the holder will not exercise their right and only the premium will have been paidto the writer.
1.10 Put Option
A put option gives the holder of the option, the right, but not the obligation to sell an underlyingasset at the strike price at a predetermined future date with the writer of the option. The writeris obligated to buy the underlying asset from the buyer if the holder exercises their right. Theholder must pay an upfront premium to the writer for this right.
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1.10.1 Profit for Holding or Writing a Put Option
Figure 3 and 4 showing the profit from holding and writing a put option.
Figure 3: Profit from Buying a Put.
1.10.2 Holder and Writer Profit for A Put Option
Please refer back to diagrams 1.6.3 and 1.6.4 for the profit for both parties.
Figure 4: Profit from Writing a Put.
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1.11 Risks, Strategies and Rewards for Call and Put Options
Risks always exist in any investment and options trading is no different. However, these riskscan be effectively managed if the appropriate trading strategy is adopted.
1.11.1 Risk of Options
Volatility Risk
A stock with a high volatility will have an option with a high volatility. This will be reflectedin the ‘delta’ of the option. ‘Delta’ is the ratio of the option price relative to a change in thestock price. The larger the delta, the larger the price of the option will be. Although, this maymake the option more expensive, this may also provide a larger return on investment for theholder of the option.
Complete Loss
There is a risk of losing money if the stock does not do what it is nticipted to do. The mosteffective way to manage this risk would be to hedge the position. That is, to buy an option inthe opposite direction to the initial option just bought. This is the most simple way of hedging,although there are many more hedging strategies.
Complexities of the Trading Environment
High frequency trading (HFT) has made the trading environment more complex throughtechnology and new exotic derivatives. The speed at which trades are made and the volume ofthese trades are not able to be matched by the individual investor.
Combined with the complex financial instruments being traded, the investor would need tobe extremely experienced, knowledgeable and wealthy in order to make increase their investmentby any significant margin.The closer to the exchange a trader’s server is, the faster a trade isable to be made.[7]
1.11.2 Hedging, Speculation and Arbitrage
Hedging in options consists of offsetting a short position with a long position. An example ofthis would be entering into a put option contract where the price of a stock is anticipated todecrease, whilst simultaneously entering into a call option contract where the price of the stockis anticipated to increase.
The strategy is only effective when the minimum loss is less than the maximum gain fromthe payoff of one of the options. Hedging is a way of managing the risk that is involved whenentering into these contracts. If a position is taken up and no hedge is implemented, this isknown as a ‘naked’ position.
Speculation is a strategy of betting on whether a stock price will increase or decrease. This
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leads to volatility in the stock price, but plays a significant role in the marketplace. Speculationprevents shortages of commodities, moves risk to those who can afford it and increases prices inthe short-term in order to prevent larger price hikes in the future.[6]
Speculation is a high-risk high-return strategy. Instead of focusing on an asset’s fundamentalvalue, speculator attempts to profit from the fluctuation in the market value of assets.
Arbitrage means simultaneous purchase and sale of an asset in order to profit from a differencein the price.[6] An arbitrageur may identify an opportunity to buy and sell an asset on twodifferent exchanges (as an example) in different parts of the world. They would buy an asset onthe NYSE and then sell in on the
1.12 Forwards versus Options
Options and forwards have a few similar properties. For example, instead of trading immediately,the price is agreed at the start of the contract and the asset can be traded in the future at thespecified date. They can also both be used to hedge risks. An example of hedging, would be tofix the price of an asset now for purchase at the agreed price in the future.
This could be helpful in times when prices of assets are particularly volatile. It may howeverhave the opposite effect in that the price of the asset may move in the other direction to whichthe hedge was structured.
One of the main differences between options and forwards is that there is an upfront premiumin an option contract, but not in a forward. This however, is assuming the forward price of theasset is correct.
Provided the option holder is able to exercise their right at a later date, they may still choosenot to, whereas is a forward contract, both parties are obligated to either buy or sell the asset,as well as exchanging the money for said asset.
Lastly, forwards can only be sold over the counter with customised terms but options can beeither sold over the counter with customised terms, or through a clearing house with standardisedterms. Thus, forwards are more flexible because the parties can set the terms that suit their ownsituations.
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2 The Binomial Tree Pricing Method
2.1 Introduction
In the global options market, financial and mathematics professionals derived a few methods inorder to price options. Some of these methods are more popular. For example, Black-ScholesModel, Monte Carlo simulation, Binomial Tree etc. In this project, we will mainly use binomialtree to calculate the prices of the options in this project.
This project will utilise the binomial model in one-step, two-steps and then generalise for thenth-step. The binomial tree is a tree diagram which represents how asset prices changes overtime under a binomial model. Binomial trees can evaluate both European and American options.
This is done by looking at up and down movements in the price of an underlying asset onwhich the option is based. The price of the option is found by creating a tree and workingbackwards from the end of a given time period to the beginning of the period. It is then possibleto find the value of the option at t = 0 for the option price.
2.2 Factors for Option Pricing
There are many factors which affect an option’s price. We will be consider the following:
1. The current stock price, S0.
2. The strike price, K.
3. The time to expiration, T .
4. The volatility of stock price, σ.
5. The risk-free interest rate, r.
In the following project, we will use the same set of data. Assume that we have a currentasset price S0 is £100, strike price K is £110, the time to expiration T − t is 2, volatility of thestock price σ is 0.4 and the risk-free interest rate is 5%.
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2.3 Pricing using Binomial Tree Method
The binomial tree method is calculated by dividing the whole duration of the option,[0, T ] inton small time intervals of length 4t where 4t = T
n . At every step of the binomial tree, the assetprice S can move in 2 ways, either up or down with probabilities Pu and Pd respectively, suchthat:
Pu + Pd = 1
so that:1− Pd = Pu
So assuming that S0 is the asset price at time 0, asset price at time 1 can either be:
S1u = S0u;
orS1d = S0d.
This method can be repeated at every step for the entire tree.
Define a to be the risk-free growth factor in each step:
a = er4t
And the size of up movement, u can be parameterized as:
u = eσ√4t
While the size of down movement, d can be define as:
d =1
u= eσ
√4t.
The risk-neutral probabilities, q and (1− q) are given by:
q =a− du− d
.
To price the option start at the end of the tree, calculating backwards. Calculating a calloption, we start with its payoff function:
max(0, ST −K)
On the other hand, for a put we use payoff function:
max(0,K − ST )
This is done by constructing a portfolio consisting of a proportion of an asset (at the pricegiven at t = 0) added to a value of a proportion of a risk-free asset (such as a government bond),and equating it to the value of the option at the same time.
In the case of a 2 – time-period tree we will observe that the tree can be divided into three1-time period trees. With the first 2 deriving the up and down value of the asset price at t = 1working back from t = 2, and then using those values in a final tree to work back to t = 0.
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2.3.1 1-time Period
S0
S1d = S0d
S1u = S0u
(1− p)
p
In the case of a call option we can now evaluate the value of the option in the below tree:
C
Cd = max(S0d−K, 0)
Cu = max(S0u−K, 0)
(1− p)
p
where
• C= Option price.
• Cu= Value of option in up movement.
• Cd= Value of the option in the down movement.
Assuming a continuously compounded risk-free rate of r per unit of time, we now need tocreate a portfolio of asset and cash where the value will be equal to that of the payoff in theup and down movement and the portfolio will give us the price of the option by the law of noarbitrage.
To do this at time 0, we set a position αS, formed from an amount α of the asset and anamount of cash β. We then hold this portfolio until time 1 and equate to the payoffs Cu and Cdto the respective portfolio values giving the simultaneous equations:
φSu + ψer = Cu
φSd + ψer = Cd
These can be solved for values of φ and ψ. The value of the portfolio at time 0:
C = φS0 + ψ
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gives the price of the option at t = 0.
In the case of a put option the process is the same with Cu = max(K − Su, 0) and Cd =max(K − Sd, 0).
2.3.2 2-time Period Binomial Tree
If we have:
S0
S0d
S0d2
S0du
S0u
S0du
S0u2
where the second branch represents a second iteration of the price of the asset going up or down.
To find the option price at t = 0, we must first find the option value at t = 1 where the assetprice is Su and Sd by working out the option value of two 1-time period payoff binomial trees aswe did 1 year. Case 1: If S1 = Su
C0
Cd = max(Sud −K, 0)
Cu = max(Suu −K, 0)
(1− p)
p
giving C = φSu + ψ.
Case 2: If S1 = Sd
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C
Cd = max(Sdd −K, 0)
Cu = max(Sdu −K, 0)
(1− p)
p
giving C = φSd + ψSu
as found in the previous example.
Now that we have Cu and Cd we can use a final binomial 1-period binomial tree that will, usingthe method we have used twice already, work back from t = 1 to obtain the option price at t = 0:
C0
Cd
Cu
(1− p)
p
which we know we can work out to be,
C = φS0 + ψ
.
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2.4 An n-time Period Generalisation
The above method can be generalised for n-period binomial trees as all that has to be done isto work back through the time periods using the component in 1-period tree until achieving theprice at time t = 0.
Calculations
Current Asset Price £100Strike Price £110
Time to maturity 2Volatility 0.4
Risk-Free interest rate 5%
Table 1: Pricing Factors for Binomial Model
When we have the informations for an underlying asset from table 1, we can calculate thecomponents for calculating the binomial tree. Assume that we are calculating a 2-step Europeanput option binomial tree using the informations from table 1.
We get time interval 4T is 1, risk-free growth factor a is 1.0512, discount factor per step is0.9512, Size of up movement is 1.4918, size of down movement is 0.6703, probability of movingup pu is 0.4637 and probability of moving down pd is 0.5363.
Then, we have to evaluate the binomial tree for the price of the underlying asset, as follow:
£100
£67.03
£44.93
S0d 2
£100
S0duS
0d
£149.18
£100
S0ud
£222.55
S0u2
S0u
Figure 5: 2-steps Binomial Tree of underlying asset prices
Figure 5 has shown us a binomial tree of the underlying asset price. At time 1, we calculatethe asset price when it move up:
S0u = 100× 1.4918 = 149.18;
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And when it move down:S0d = 100× 0.6703 = 67.03.
And it carries on using the same method. When we get all the asset prices, we can calculatingthe option prices now. We start pricing the option at time 2 and calculate backwards using thepayoff function.
£21.43
p1d =£37.60
P2dd = £65.07
p2ud = £10
p1u =£5.10
p2ud = £10
p2uu = £0
Figure 6: 2-steps Binomial Option Pricing
In figure 6, we calculate the European put option price at time 2 p2uu as follow:
max(K − ST , 0) = max(110− 222.55, 0) = £0;
and p2dd as follow:max(110− 44.93, 0) = £65.07
After calculating the option price at time 2, we price the option price at time 1, as follow:
p1u = e−r(q × p2uu + (1− q)× p2ud = 0.9512(0.4637× 0 + 0.5362× 10) = £5.1012;
Andp2ud = 0.9512(0.4637× 10 + 0.5362× 65.0671) = £37.6032.
Then, we apply the same calculation method continuously when we are pricing a n-stepsbinomial tree until we get the option price at time 0.
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3 Exotic Options
3.1 Introduction
Exotic Options are options that are different from the typical call and put options previouslymentioned in this piece. They are more complex in the nature of their payoffs and/or the natureof their underlying asset. These options generally trade, due to their complex nature, over thecounter opposed to on an exchange.
In the following pages, five types of exotic option are discussed regarding their existence, howthey work and how they compare to their vanilla options counterparts.
3.2 Asian Options
3.2.1 Introduction
Asian options (also as known as average options) are one of the most active financial tools amongall exotic derivatives. This is a special type of option contract for which the payoff is determinedby the average underlying price over some pre-set period of time. This is different to a plainvanilla American or European style options, where the payoff is based on the asset price atmaturity.
3.2.2 Etymology
In the 1980s Mark Standish was with the London-based Bankers Trust working on fixed incomederivatives and proprietary arbitrage trading. David Spaughton, worked as systems analyst inthe financial markets with Bankers Trust since 1984 when the Bank of England first gave licencesfor banks to do foreign exchange options in the London market. In 1987 Standish and Spaughtonwere in Tokyo on business when “they developed the first commercially used pricing formula foroptions linked to the average price of crude oil.” They called this exotic option, the Asian option,because they were in Asia.[8]
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3.2.3 Merits of Asian Options
• Asian option is attractive because it tends to cost less than regular American and Europeanoption.
• Also it has low market liquidity risk because it can attract investor quickly.
• Asian option can protect an investor from volatility risk as like averaging reduces.
• Low volatility and market liquidity risk have definitely encouraged more investors addingsome Asian options into their portfolio to manage their risk, finding more beneficial thanjust having a well diversified portfolio.
3.2.4 Risk Management Using Asian Options
Suppose asset A is a relatively new technology stock in the investment market and it is growingrapidly over the first month. However in the third month the company encounter some seriousissues adversely affecting its share price.
Month Price1 £1002 £2003 £50
Table 2: Price of Asset A During a 3 Month Period
Investor Beta brought a normal vanilla call option for 3-months period. At the start of thefirst month with strike price at £100, he was feeling excited because the stock double its pricein month 2 and hoping the stock price will be keep going up in month 3. At the end he couldnot benefit from this call option due to the company issues.
Investor Gamma brought an arithmetic averaging Asian option for a 3-months period, for astrike price £100. The average price of asset A during this 3 months is £116.67 which is higherthan the strike price. He still has a positive payoff even the company encounters issues. This isa simple example of an investor using Asian option to avoid the volatility risk in the market.
3.2.5 Averaging Method for Asian Options
There are 2 types of averaging methods for Asian options
1. Arithmetic average
2. Geometric average
Figure 1 shows the option price by using different type of averaging method are nearly identicalto each other, and their option price is generally lower than vanilla option.
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Figure 7: Price Difference Between the Arithmetic and Geometric Asian Option
3.2.6 General Outline for the Pay-off of Asian Options
There are 8 different types of Asian options.
Type Variation1 Call, Arithmetic, Average Price2 Put, Arithmetic, Average Price3 Call, Arithmetic, Average Strike4 Put, Arithmetic, Average Strike5 Call, Geometric, Average Price6 Put, Geometric, Average Price7 Call, Geometric, Average Strike8 Put, Geometric, Average Strike
Table 3: Asian Options Table
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3.2.7 Is it a Call or a Put?
The payoff will be different depending whether it’s a call or put option.
3.2.8 Arithmetic Averaging or Geometric Averaging?
We use A(S)= 1n
∑Nt=1 St denote arithmetic averaging and
G(S)= n
√∏Nt=1 St denote geometric averaging.
3.2.9 Average Price or Average Strike?
Average price is where the payoff is neither zero or the amount of average price exceed the strike,and average strike is where payoff is based on average of the spot rate over a period of time.
3.2.10 The Binomial Option Pricing Model for Asian Options
The basic binomial tree model is defined in chapter 2.
And the general formula of Asian option price will be;
Ct−∆t,i = e−r∆t(pCt,i+1 + (1− p)Ct,i−1)
where,
Ct,i is the option’s payoff for the ith at time t ,
Barrier Options
3.3 Barrier Options
3.3.1 Introduction
“The option can either come into existance or become worthless if the underlying asset reachessome prescribed value before expiry.” [9]
A Barrier option almost like a vanilla option. It can be either a call or a put option and canbe exercised in the American or European style. However, the effectiveness of a barrier optionis determined by whether the current price of the underlying asset reaches the barrier duringa certain period of time. The barrier is the level of price that determines whether the barrieroption is effective or not.
26
There are two main types of barrier options. They are knock-in options or knock-out options.A knock-in option becomes effective when the current price reaches the barrier. A knock-outoption become worthless if the current price reaches the barrier.
There are two types of knock-in options, one is called an up-and-in option, which is effectivewhen the price of the underlying asset rises to reach the barrier. The other, a down-and-inoption, is effective when the current price of the asset fails to reach the barrier.
The same applies to the knock-out option. An up-and-out option is worthless if the assetprice rises reaching the barrier. The down-and-in option is worthless if the asset price fails toreach the barrier. As a barrier option can be either a call or a put option, it means there are 8types of barrier options.
If the underlying asset price of an up and out option reaches the out-barrier, the barrieroption worth zero. The option does not reactivate if the price falls below the barrier level afterreaching the out barrier. The price of a barrier option is cheaper compared to the correspondingvanilla option, as it has a chance of becoming worthless if it reaches the out barrier or if it doesreach the in-barrier before expiry.
A rebate exists in a select few barrier options. A rebate is a fixed amount paid to the holderif the underlying asset price reaches the out-barrier.
3.3.2 Example
The current price of an asset is £100. Person A thinks that the price of an asset is not going tofall to any value below £90. So they purchase a down-and-out put option with a barrier of £90.After one month, the price of the asset falls to £85.
A’s barrier option becomes worthless. The month after, the price of the asset increases to£95. As the price of the asset reached the out barrier before maturity, A’s barrier option is stillworth zero.
Although it is possible to hedge using barrier options, the hedger rarely uses barrier options,because there is a probability that the option become worthless. It is usually used by thespeculator, because it is cheaper than a vanilla option, thus they can gain more profit from abarrier option.
3.3.3 Risks
Risk of a barrier option can be an unexpected event which affects the underlying assets price.The option then might not be able to reach the barrier level as expected or fall below the out-barrier and become worthless. Also, asset with high volatility can make barrier option riskybecuase the price of the asset fluctuates largely, and so does the option price.
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3.3.4 Price
X=barrier level
I=indicator function
Payoff for barrier call option:
max(ST −K, 0)I(ST < X)
Payoff for barrier put option:
max(K − ST , 0)I(ST < X)
Pricing equation:exp−r(Tn−T0)EQ(max(ST −K), 0)
Pricing Factors
Current Stock Price £100Strike Price £110
Interest Rate 5%Volatility 0.4
Time to Maturity 2Number of Steps 2
Up barrier 110Down barrier 95
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Using the above pricing factors, the Barrier option price will be :
Barrier options Option price
up-and-in call £3.67up-and-out call £0
down-and-in call £0down-and-out call £3.67
up-and-in put £1.37up-and-out put £5.84
down-and-in put £5.84down-and-out put £1.37
3.4 Compound Options
3.4.1 Introduction
This part of the project will include a brief explanation on what a compound option is, as wellas the pricing of the option. Compound options are one of the main families of exotics options.There are 4 types of compound options. They are call on call, put on put, call on put and puton call. Only call on call and put on put calculations will be shown in detail, whilst the othertwo will be briefly described.
Compound options are best suited for future hedging possibilities. These may include man-ufacturers who would like to be protected against raw materials price increases, companies whoare bidding for a tender in a foreign currecy wishing to be protected against future exchangerate fluctuations, and research and development (R&D) companies who are developing productsthat have not yet been bought, thus needing to generate future success from products not yetdeveloped.[5]
A brief explanation of what types of risk these options are intending on mitigating will alsobe provided in each subsection.
3.4.2 Notation
• K1 = compound option strike price
• K2 = vanilla option strike price
• M1(1) =value of the compound option contract at time 1 at the up node
• M1(2) =value of the compound option contract at time 1 at the down node
• M0 = present value of the compund option
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3.4.3 Call on Call
This contract gives the holder right to buy an underlying asset, which is a plain vanilla Europeanoption at some future date. The plain vanilla European option is a call option.
These are extremely effective options in markets where stock prices are increasing over time(a bull market). They are cheaper than the corresponding call option over the same time periodrelative to the compound option.
If the stock prices rapidly decrease, the hedger would be set to lose a lower premium than ifthey had only a plain vanilla option.
3.4.4 Binomial Tree Option Pricing
The binomial tree detailed below is the binomial model for the pricing of the compound optionusing the value of the above vanilla option.
M1(1) = max(V1(1)−K1, 0) (2)
M1(2) = max(V1(2)−K2, 0) (3)
M0 = e−r∆t[qM1(1) + (1− q)M1(2)] (4)
M0
M1(2)
M1(1)
(1−q)
q
The methods behind obtaining the values of the different types of compound options aredetailed in the body of the document. Please note that errors may occur in rounding.
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3.4.5 Generic Call on Call Example in Pricing
The compound option has two strike prices and two maturity dates. The first strike price is thestrike price of the compound option, K1 at T1 (the first maturity date) and the second is thestrike price K2 of the vanilla call option at T2 (the second maturity date).
The price of the vanilla call option is calculated as in chapter 2;
The compound option value is found as follows;
(2)
and
(3).
This is done because the vanilla option is merely an asset (the underlying in this case). Asis seen from the pricing of a vanilla option, the option price, is found by subrtacting the strikeprice from the asset price.
The compound option premium, M0, is then found by discounting M1 back by the risk-freerate of return.
M0 is then found using equation (1).
The price of a call on call option option is cheaper than that of a plain vanilla option at time0. This allows the holder to extend the option to buy the right to another call at a later stage.[5]
3.4.6 Worked Example
• Using the vanilla option price V1(1) = 50.49 from Chapter 2
• K1 = 20
• all other notation as in chapter 2
The compound option price at the up and down nodes is now able to be calculated below;
M1(1) = max(50.49− 20, 0) = 30.49
and
M1(2) = max(0− 10, 0) = 0.
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Finally, the price of the compound option at t = 0 is able to be calculated;
M0 = e−0.05∗1[0.46 ∗ 30.49 + (1− 0.46) ∗ 0] = 13.34.
The binomial tree for the compound option is illustrated below.
M0 = 13.34
M1(2) = max(0− 20, 0) = 0
M1(1) = max(50.49− 20, 0) = 30.49
0.54
0.46
3.4.7 Call on Put
The compound option in this case is a call option, which gives the right but not the obligationto buy a plain vanilla put option at a future date. If a manager of a protfolio is concerned abouta decline in the return of a portfolio, he/she will look to purchase a call on put option.
This allows him/her to exercise the call option at a future date in order to protect against adecline in the value of the portfolio. If the value of the portfolio is declining, then the managerwill exercise the right on the vanilla put option and sell his/her shares in the portfolio to prevent further losses.
The advatages of this are that if the portfolio does experience an upward trend, the managercan choose not to exercise the call option for the plain vanilla put option. This example wastaken from [5].
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3.4.8 Put on Put
This contract gives the holder right to sell an underlying asset, which is a plain vanilla Europeanoption at some future date. The plain vanilla European option is a put option.
A put option is bought when there is an expectation of a decrease in a particular stock ofinterest. A compound put on put is cheaper to enter into than a standard put option, relativeto the same time period as the compound option.
In markets where stock prices are increasing, the compound option will result in a smallerpremium for hedgers, but in a market with price decreases, the hedgers will benefit greatly fromthis option.
This example was taken from[5].
3.4.9 Binomial Tree Option Pricing - Worked Example
• Using the vanilla option price V1(1) = 6.45 from Chapter 2
• Using the vanilla option price V1(2) = 44.64 from Chapter 2
• K1 = 70
• all other notation as in chapter 2
The compound option price at the up and down nodes is now able to be calculated below;
V1(1) = max(70− 6.45, 0) = 63.55
and
V1(2) = max(70− 44.64, 0) = 25.36.
Finally, the price of the compound option at t = 0 is able to be calculated;
M0 = e−0.05∗1[0.46 ∗ 63.55 + (1− 0.46) ∗ 25.36] = 60.45.
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The binomial tree for the compound option is illustrated below.
M0 = 60.45
M1(2) = max(70− 44.64, 0) = 25.36
M1(1) = max(70− 6.45, 0) = 63.55
0.54
0.46
3.4.10 Put on Call
The compound option in this case is a put option, which gives the right but not the obligationto sell a plain vanilla call option at a future date. If a firm is loooking at a tender and wouldlike to protect themselves in adverse price movements, they may enter into a compound optionagreement with the writer of the comound option.
A call or put option on interest rates is known as a ’Caption’[5]. The trader may be antici-pating a drop in interest rates in the near future and is looking to sell the call option when theanticipated drop arises.
The company who has entered in to the tender, may not be expecting their bud for the tenderto be realised in the near future, but know that a call option may be necessary in order to hedgeagainst stock price movements that would be unfavourable in the future.
This example was taken from [5].
The pricing of this compound option is a combination of a compound put and a call option.The mathematics is similar to that already detailed in .
3.5 Lookback Options
3.5.1 Introduction
A lookback option is financial derivative that allows the holder to buy or sell the underlying assetat whatever price is most advantageous to the holder over the time period of the option.
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The two types of lookback options that exist are fixed-strike lookback options and floating-strike lookback options. A fixed-strike lookback option’s strike price is fixed from the outset andtherefore at maturity, the holder exercises his option at the optimal price.
This grants the holder the highest difference between the strike price and the value of theunderlying asset across the time period. In the case of a call this would be when the value of theunderlying asset is at its highest, and in a put option, at its lowest.
As the name suggests the strike price of a floating-rate lookback option is floating and atmaturity is set at the most beneficial price it ever reaches across the option life. In the case of afloating-strike lookback call option the holder would exercise his option at maturity, setting thestrike price to be the lowest value the underlying asset reached across the life of the option. Ina put option the highest price would be used.
3.5.2 The Value of Lookback Options
Lookback options can be seen as possible solutions to the problems that all investors face, suchas when to enter and when to exit the market. All investors want to sell their assets at thehighest possible value and buy assets at the lowest possible cost.
The dilemma that arises is when to do this. Is it best to sell now or wait for the price tocontinue to rise, or the converse, and what happens if the price movement is the converse towhat is expected?. These problems are known as the market entry and market exit problems.
A fixed-strike lookback option allows the holder to solve their market exit problem (at leastacross the lifetime of the option) as the price at which the option is exercised, at maturity, isat the best price it reached across the life of the option. In this way the holder has used thecall/put option to negate any regret of buying/selling an asset too early or too late.
This still leaves investors with the market entry problem which can be solved using floating-strike lookback options. With a floating-strike lookback call option, the investor at maturity willbe exercising a call option where the strike price is set at the lowest value the underlying assetreaches, and a put option where the strike price will be set by the highest possible price.
Therefore in effect, buying the underlying asset at the cheapest possible level whilst sellingat the highest. This however does come at a cost which will be outlined in their risks at a later
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stage.
The payoff of each option therefore can be denoted as follows:
• Fixed-strike call: max[0, Smax −K]
• Fixed-strike put: max[0,K − Smin]
• Floating-strike call: max[0, ST − Smin]
• Floating-strike put: max[0, Smax − ST ]
where K = strike price, Smax = the maximum price the asset reaches over the time period andSmin = the lowest price the asset reaches.
3.5.3 Example
A 1 year fixed-strike lookback call option is struck with a strike price of K = £100 where theprice of the underlying asset fluctuates between a range of £70 and £180 which it reaches at thetime of maturity. Then, the holder will exercise their right in taking the price of the underlyingasset when it was at its highest, which in this case is at maturity at the price of £180.
The holder would make £80 profit with the strike price of £100. If in possession of a fixed-strike lookback put option in this position the holder would exercise at the lowest price the assetreaches and sell the asset at £70 with £30 of profit against the strike price .
If in this same scenario it is a floating-strike lookback option instead, a call would leavethe holder exercising the option at maturity, setting the strike price at the lowest value of theunderlying asset across the life of the option, £70, which against the price of the asset at timeof maturity gives the holder £110.
If it was a floating-strike lookback put option then the holder would sell the underlying assetsat £180 as that is the maximum value the underlying asset reached. This happens to be thesame value as what the asset is worth at maturity so in this case no profit is gained.
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It is important to note that in no case is the holder not in the money, this is the mostsignificant feature of lookback options, but it comes comes at a high price.
3.5.4 Risk of Lookback Options
Due to the fact that there is no market risk with the options themselves, in the sense that a gainof the maximum profit achievable across the lifetime of the option is made, the only considerationtoward risk is the volatility of the underlying asset itself.
With the knowledge of taking the best profit at all times, a more volatile asset has thepotential to be the most rewarding, creating bigger differences between the strike price, the priceat maturity and the minimum and maximum prices of the asset during the life of the option. Allof this contributes to the amount of profit possible available across all four options of fixed call,fixed put, floating call and floating put.
The downside to all this is that the options themselves come at a steep price which reflectsthere rewarding nature of their payoffs.
3.5.5 Pricing Lookback Options
Lookback options can be priced using the binomial tree method, discussed in section (4.3). Inthe same way as Barrier and Asian options, the binomial tree method is used with the differentpayoffs for lookback options seen previously.
For example with a floating strike lookback option for 1 year with a risk free rate of r = 10%and volatility σ = 25% a floating strike lookback call option using the method shown in section(4.3) would give a price at time = 0 of £22.80 and similarly for the put a price of £16.27.
3.5.6 Comparing Lookback Prices to Vanilla Options
As is clear to see from the above table there is a premium placed on Lookback options whichshould come at no surprise due to the nature of their payoff. As they are in a postion of always
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K LC Floating LC Fixed LP Floating LP Fixed Van Call Van Put100 15.05 15.47 5.95 5.53 13.27 3.75
Table 4: Table showing the different option prices with S0 = 100, volatility = 20% and interest= 10% for a 1 year 2 time period maturity date
being in the money and remove market entry and exit problems the value of Lookback options,both call and put, has to increase.
3.6 Chooser Options
3.6.1 Introduction
‘Have you ever purchased a call option ,but after a while, you want to purchase a put optioninstead?’ [1]
A Standard chooser option allows investors to purchase at time t = 0 the right to decidewhether at a future time, t, an option is a call or a put option, but this right must be decidedbefore maturity, T (see figures 8 & 9). Chooser option is also called an, ‘As-you-like’ and‘You-choose option’[1][4].
Chooser options can be divided into two kinds, Simple (Standard) Chooser options andComplex chooser options. The difference between these two is that the strike prices of the calland put can be the same or different. If the strike prices are the same, then it is called a simplechooser. When strike prices are not same, it is a Complex chooser option[1][2][3][4].
Figure 8: Time lines on Standard Option
Purchase an option
0
Option at Maturity
T
Figure 9: Time lines on Chooser Option
Purchase an option
0
Time to choose
t
Option at Maturity
T
Figure 8, shows that a standard option has a simple time line with the option buying time 0;and maturity time with strike price. At time T , the investor has the right to exercise the option.Figure 9, shows that an investor can choose the option to be a call or put option at specifiedtime t, where t < T .
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This kind of option is two-faced, just as the straddle is. However, the chooser option ischeaper because chooser option can end up as being worthless. The straddle always has a payoff.From the description of chooser option above, we can summarise a payoff function for a standardchooser option[1]:
Choosersimple = max[C(St,K, T − t), P (St,K, T − t)] (5)
where St is the asset price at time t, K is strike price, T − t is time to maturity; C(.) is the priceof a vanilla Call option and P (.) is the price of vanilla Put option.
Equation 5 means that, at time t, investor will choose the higher value option. If the investorchoose the call option, then:
C(St,K, T − t) > P (St,K, T − t).
By using put-call parity, we have(Rubinstein 1992, Nelken 2000):
P (St,K, T − t) = C(St,K, T − t) +Ke−r(T−t) − Ste−q(T−t) (6)
where q is the dividend yield.
Combining the above equations, we can rewrite the payoff function as:
max[C(St,K, T − t), C(St,K, T − t) +Ke−r(T−t) − Ste−q(T−t)] (7)
And it can be simplified into:
C(St,K, T − t) + max[0,Ke−r(T−t) − Ste−q(T−t)] (8)
Complex Chooser option share the same concept with Standard Chooser option, butprevious one can allow difference on call or put Strike prices and maturity times. Therefore, thepayoff function of complex chooser is:
Choosercomplex = max[C(St,Kc, Tc, P (St,Kp, Tp)] (9)
3.6.2 Empirical Result
Using the binomial model, the value of a simple chooser option can be calculatedas follows;
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Consider a European simple chooser option on stock X with a maturity time of 2 years. Thestock price is £100, the risk-free rate is 5%, the volatility of the asset is 0.4. Assume that nodividend is paid on this asset. The chooser option provides the choice at time t = 1 of choosingcall and put at the same strike price is £100.
3.6.3 Simple Chooser Option
£41.08
C1(2):£0P1(2):£37.60
3 C2(3):£0; P2(3):£65.07
B
2 C2(2):£0; P2(2):£10B
C1(1):£49.65P1(1):£5.10
2 C2(2):£0; P2(2):£10
A
1 C2(1):£112.55; P2(1):£0A
Figure 10: 2-step Simple Chooser Option pricing
Figure 10, The pricing starts from right to left. A simple vanilla option payoff function isused to calculate call and put options. Thus, the first step is a call option C2(1):
max[0, S0u2 −K];
and the put option is P2(1):max[0,K − S0u
2].
In the next step C1(1) is calulated, and use the information from the first step and find the valueof call using the following function:
e−r∆T [pu × C2(1) + pd × C2(1)];
Finding the value of put option as follows:
e−r∆T (pu × P2(1) + pd × P2(2)).
In process A, we use the above functions to calculate the value of a call option for the secondstep,
0.951229× (0.463724× 112.55 + 0.536276× 0) = £49.65.
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Using the same method, the result of node 2 in figure (4) is obtained.In process B, the value ofput option is calculated at time 1:
0.951229× (0.463724× 0 + 0.536276× 65.07) = £37.60
To calculate the option value at time 0, according to the basic definition of Chooser option:max[c, p]. Therefore, we take:
0.951229× (0.463724×max[49.65, 5.10] + 0.536276×max[0, 37.60])
so, we have:0.951229× (0.463724× 49.65 + 0.536276× 37.60) = £41.08
The simple chooser option value at time 0 is: £41.08.
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3.6.4 Complex Chooser Option
£47.22
C1(2) :£4.41 P1(2) :£37.60
C2(3) : £0 P2(3) : £65.07
C2(2) : £10 P2(2) : £10
C1(1) :£63.57 P1(1) :
£5.10
C2(2) : £10 P2(2) : £10
C2(1) : £132.55 P2(1) : £0
Figure 11: 2-step Complex Chooser Binomial Option Pricing
Consider, a European Complex chooser option on stock X with a maturity time of 2 years.The stock price is £100, the risk-free rate is 5% and the volatility of the asset is 0.4. Assumethat no dividends are paid on this asset. The chooser option provides the choice at time t = 1of choosing the strike price of a call of £90 and the strike price of a put is £ 110. The binomialmodel for complex Chooser options is shown in figure 11.
Unlike Simple chooser options, option pricing of Complex chooser options start with thepayoff function:
max[C(St,Kc, Tc), P (St,Kp, Tp)]
. In this example, it is assumed that the maturity for the call and the put are the same. Suchthat,
max[C(St,Kc, T ), P (St,Kp, T )]
.
For example, the asset price at time 2 with 2 up-movements S0u2 is £222.55. In that node,
both the call and the put options can be calculated, so that, Call:
max(222.55− 90, 0) = £132.55;
and Put:max(110− 222.55, 0) = £0.
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Then, we calculate C1(1) using the equation as follow:
e−r(T−t)(pu × C2(1) + pd × P2(1))
The same method in simple chooser options can be used to calculate the option prices.
3.6.5 Chooser Option vs Straddle Option
There is a financial investment strategy in the derivative market which shares similar characteris-tics with the chooser option. It is called Straddle. A straddle means that there are 2 transactionson the same security, but with opposite positions to each other.
For example, purchasing a call option and a put option on the same underlying asset. Bothoptions have the same strike price and time to maturity. The holder of the straddle can makeprofit if the underlying price moves above or below from the strike price.
Investors can buy a straddle when they that the market is highly volatile and are uncertainin which direction the asset price will move.
There is a very slight, limited risk that the investor might incur the cost of buying both calland put option if the underlying asset price is equal to the strike price. However, there is apotential of unlimited profit.
In figure 10, the straddle is compared with the Simple chooser option as shown,
Straddle (ct + pt) Simple Chooser Option£49.65+£5.10=£54.75 £49.65
Table 5: Straddle and Simple Chooser Option Prices
Table 5 shows us that chooser option is cheaper for investors because chooser option can endup priceless. However, straddle always has a payoff. So that, investor won’t lose much whenbuying straddle, but it is more expensive.
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3.6.6 Relationship between Strike Price and Option Price
Strike Price K 60 80 100 120 140 150Option Price 49.63 41.36 38.18 43.99 49.80 52.71
Table 6: Table of Strike prices and Option price
0 20 40 60 80 100 120 140 1600
20
40
60
80
Strike Price K
Op
tion
Pri
ce(£
)
Value of Simple Chooser Option
In subsection 3.6.6, the figures indicates that when the strike price is approaching the currentasset price, the option price is becoming less. However, when the strike price is decreasing andis higher than the current asset price, the option price will be relatively larger.
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4 Software
The software is an Excel spreadsheet, which details both the pricing of the plain vanilla optionsand the exotic options.
4.1 Software Manual
• Save File ‘MA599 Exotic Options Options Pricing Software’ to computer.
• Open File.
• Click on the ’Title Page’ tab. This can be found at the bottom hand side of the ExcelSpreadsheet.
• The grey area at the top left of the spreadsheet is the area where the required values are tobe input by the user. This is how the user will be able to price the required exotic options.
• Once the user has set the required values, all of the exotic options will be priced accordingly.
• The user has to manually set the strike price for the compound and chooser options, asthey are of a different sequence to the other exotics.
• Once this is done, the compound option and chooser option prices will adjust accordingly.
• To exit the programme, click on the ’Excel Close Programme Button’.
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5 Conclusion
In order to successfully create this project, three main areas needed to be explored. Those were,‘products, applications and pricing’.
This project explored the complex world of both plain vanilla and exotic options. The pricingmethod that was used in both vanilla and exotics was the binomial option pricing model. Volatil-ity features heavily in the pricing of all options. The greater the fluctuation of the underlyingasset, the less expensive the price of the option.
This relatiionship is due to the inverse nature of prices with respect to returns. It is wellknown, that the high price of an income yielding asset gives a low return on investment and viceversa. If the volatility is high, this implies a possible high return on the asset, which gives riseto a lower asset price. However, as this is true for the return, this is also true when consideringthe potential loss by investing in this volatile asset.
There were 5 main families of exotic options that were investigated. Those were; Asian,Barrier, Compound, Lookback and Chooser options. All plain vanilla and exotic options areused to hedge different kinds of risks in relation to their specific underlying assets.
It was shown that some exotics were more expensive than that of their vanilla counterpart,whereas some were cheaper than their vanilla counterpart. The exotics that were more expensivewere those that offered the greater flexibility with a relatively low volatility.
The more expensive exotics offered greater flexibility and less risk, whereas the cheaper oneshad slightly different conditions attached to them than their vanilla counterparts. Having greaterflexibility may allow the investor to actively control the risks associated with the investment.
Whilst each exotic had a different hedging strategy and reason, exotics are extremely im-portant to both large corporations and individuals in order to create effective risk managementstrategies, as well as, well diversified portfolios.
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References
[1] Mark Rubinstein with Eric Reiner, Exotic Options, First Edition, 1992.
[2] http://www.bme.vgtu.lt/index.php/bme/article/viewFile/bme.2012.20/pdf, Chooser Op-tions, PDF Paper, 2012.
[3] Israel Nelken, Pricing, Hedging, and Trading Exotic Options, First Edition, 2000.
[4] John C. Hull, Options, Futures and Other Derivatives, 10th Edition, 2010.
[5] R. Madhumathi, Derivatives and Risk Management, Pearson Education India.
[6] www.investopedia.com, Investopedia, Webpage.
[7] Allen Lane, Flash Boys, 1st Edition, 2014
[8] www.wikipedia.com, Wikipedia, Webpage.
[9] Paul Wilmott, Sam Howison, Jeff Dewynne, The Mathematics of Financial Derivatives AStudent Introduction, 1995.
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