Discrete space time option pricing forum fsr

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  • In this paper we presented valuation of some classes of exotic options. Our valuation approach as well as interpretation of the derivatives price is completely different from Black-Scholes benchmark.

    Our derivatives valuation approach as well as interpretation of the notion 'derivatives price' is completely different from Black-Scholes benchmark. Black Scholes defined option price as one that can be drawn from their ‘dynamic’ hedging strategy. From financial point of view this strategy can be interpreted as a settlement price between option buyer and a counterparty which finance option buyer at risk-free interest. Note that the notion price of an instrument in Finance always is interpreted as a settlement price between buyer and seller. The second drawback that immediately stems from the buyer – borrower settlement pricing is incorrect randomization. Black and Scholes focusing on spot option price definition ignored the fact of market risk factor of their theoretical option price. They lost the fact that in perfect following their ‘no-free-lunch’ strategy either buyer or seller of the option are subject of the market risk. The third drawback of their construction is the mathematical error which was discussed in details in paper ‘Derivatives pricing’ here on slideshare.net.

    Our pricing concept is based on seller-buyer settlement pricing. For each market scenario which uniquely specifies derivative underling price at any moment we define market price of the derivative. For example the market price of a European call option is equal to zero if for the scenario the price of the underlying asset is bellow of the specified strike price. If for the chosen scenario the price of the underlying exceeds the strike price then market price of the option is defined based on the rule which states that interest rate on risky underlying and derivatives must be equal. Thus, when investment in option is meaningful this construction eliminates an arbitrage opportunity for each market scenario. Spot price of the option is a number which can be interpreted as weighted average of the primary market factors such as surplus and demand or risk and reward. Then the value of the market risk of the buyer of the call option is associated with the probability that market price is bellow than spot price while the value of the market risk of the option seller is the adjacent probability. This construction of the market risk can be identify as when spot price is overpriced for option buyer or underpriced for option seller correspondingly. One important conclusion from our definition is the fact that there is no ‘perfect’ price of the option. Any spot price including heuristic Black-Scholes’ no arbitrage pricing always implies market risk.
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Discrete space time option pricing forum fsr

  1. 1. fsrforum • jaargang 12 • editie #5Discrete Space-Time Options Pricing As far as PV suggests equal price for two investments one can sell a lower rate instrument over the correspondent subinterval and buy the instrument with higher rate of return instrument.Ilya Gikhman We say that two investments are equal at a moment of time if their instantaneous rates of Let S ( t ) denote an asset spot price at date t, t 0. For-This paper presents a formal ap- of the two scenarios: P u = P ( u ), P d = P ( d ). This return at this moment at are equal. If equality of two investments holds any moment of time mally an European option contract is defined by its payoff distribution is assigned then to the option premium. Thusproach to the derivatives pricing. In over [ t , T ] then these investments are equal on [ t , T ]. This definition represents investment at expiration. The call and put values at expiration T are equality, IE law or principle which will be applied throughout of the paper for definition of a defined by formulasthis paper we will study derivatives derivative price. Next we use cash flow notion that implies a series of transactions as a specifi-pricing in a discrete space-time cation of somewhat broad notion of investment. C ( T , S ( T ) ) = max { S ( T ) – K , 0 } (1.1) It is not difficult to see that this concept of the investment equality is a perfect and more accurate P ( T , S ( T ) ) = max { K – S ( T ) , 0 } Let us consider two stocks that have probabilities P 1 ( u )approximation. The primary princi- than the present value, PV concept. Indeed, if two investments are equal in IE sense then they are = P 2 ( d ) = 0.99 and equal for any possible scenario or simply to say always. On the other hand, it is clear that if two Thus, a buyer of the call option would agree to exercise theple of the pricing theory we intro- P 1 ( d ) = P 2 ( u ) = 0.01 correspondingly. The call investments are equal in IE sense then they are equal in the PV. The inverse statement is incor- right to buy the underlying asset in case if the value of the option price is the random variable taking values : C i ( 0 ,duce in the paper is the notion of rect. More accurately, if two investments are equal in the PV sense then it is easy to present exam- call option at maturity T is positive, i.e. if C ( T , S ( T ) ) > 2 ; u ) = 1, C i ( 0 , 2 ; d ) = 0 , i = 1, 2. Then ple of a scenario that demonstrates arbitrage opportunity. In this example, the present value of 0. It is clear that there is no sense in realization of the rightequality of investment which based two investments are equal while one investment has a higher rate of return over a time subinter- when C ( T , S ( T ) ) 0. On the other hand a buyer of the *) The average rate of return on 1st stock is equal to 1.48on the investors goal : ‘investing in val and lower over another subinterval than other investment. As far as PV suggests equal price put option would exercise the right to sell the underlying and – 0.43 on 2nd stock. for two investments one can sell a lower rate instrument over the correspondent subinterval and asset in case if S ( T ) < K, i.e. a put holder is interested to **) The average rate of return on call option written on 1sta greater return’. buy the instrument with higher rate of return instrument. Then at the end of this period investor sell asset with price S ( T ) for K when S ( T ) < K. That is stock a holder of the put option can exercise the right to sell the option when P ( T , S ( T ) ) > 0. Otherwise, if P ( T , S ( T ) ) 0 the right will not be exercised. The option pricing problem is to determine the call ( put ) We say that two investments are option price at any moment of time t before the expiration date T. To illustrate pricing methodology we begin with a The difference between possible equal at a moment of time if their simple example. Next we use the terms asset, stock, or secu- rity as synonyms. spot prices is the value of risk instantaneous rates of return at this Example 1. Let a stock price at t = 0 be S ( 0 ) = 2 and at T = 1 stock take the values S ( 1 ) = { 5 , 1 }. Introduce the taken by investors. moment are equal. probability space of scenarios of the problem. Denote ω = { u , d }, where u denotes the scenario { S ( 0 ) = 2, S ( 1 ) = 5 }, and d = { S ( 0 ) = 2, S ( 1 ) = 1 }. Putting strike price K = 2 we enable to define the call option price would sell short higher priced instrument which promises lower rate of return over the next for each scenario. Denote C ( t , x ; T , K , ) the value of period and buy for lower price other instrument which promises higher rate of return. At the end the call option for fixed scenario at the moment t , given of the period the investor has pure profit for the scenario though two investments have equal PV. S ( t ) = x. Here t , x are variables of the function C ( ), while This type of the example illustrates the fact that PV reduction of cash flows insufficient to be used T, K, are interpreted as parameters. The value of param- as a definition of the equality of two cash flows. Nevertheless, PV reduction might be helpful for eters T and K are assumed to be fixed and for the writing construction of the market estimates of the spot or future prices. Bearing in mind that price def- simplicity we will omit them next. Let us specify the value while the average rate of return on call option written on inition depends on a scenario we should be aware that any spot price calculated with the help of of the option along the scenario u . Applying IE principle the 2nd stock is equal to 1.5 %. Assume that market price at PV or other rule implies risk. This risk for buyer is measured by the probability of the events for we arrive at the equation with respect to unknown C ( t = 0 t = 0 is equal to the mean of the option at this moment. which scenario’s price is bellow than spot price. ,S(0)=2; u) The expectations of the options price on the 1st and the 2nd stocks at t = 0 are 1. Plain Vanilla options valuation. Let us introduce the definition of the plain vanilla option contracts which is a class option cov- c1(0,2) = EC1(0,2; ) = 1.2 × 0.99 = 1.188 ered European and American types. An option is a right to buy or sell an asset at a known price, The solution of the equation is C ( 0 , 2 ; u ) = 1.2. Then within a given period of time. The known price, K is called exercise or strike price. The last date, as far as the option payoff for the scenario u is max { 1 - c2(0,2) = EC2(0,2; ) = 1.2 × 0.01 = 0.012 T of the lifetime of the option is called maturity. The right to buy is known as the call option, 2 , 0 } = 0 we put by definition C ( 0 , 2 ; d ) = 0. There while the right to sell is the put option. The price of the option also referred to as premium. is no sense to pay for the option a sum if option value in the Other possible estimate of the spot option prices can be European options can only be exercised at maturity, whereas American type of the options can future moment is 0. Therefore, by definition we put C ( 0 , based on the PV concept. Assuming that the risk free inter- be exercised at any moment up to maturity. 2 ; d ) = 0. The security distribution is the probabilities est is equal to 0 we see that »18 • Discrete Space-Time Options Pricing Discrete Space-Time Options Pricing • 19
  2. 2. fsrforum • jaargang 12 • editie #5 c1(0,2) = EC1(t=1; ) = 3 × 0.99 = 2.97 Here, the last row C ( 0 , S ( 0 ) ) = C ( 0 , 2 ) represents the are payoffs on call and put options at maturity T. The Ameri- option price at time 0. Each entry in the third row has prob- can option can be exercised at any time up to maturity T. c2(0,2) = EC2(t=1; ) = 3 × 0.01 = 0.03 ability of 1 / 6. The situation represented by the Table is the Therefore, its payoff depends on time interval during which Then the $-value of the call option contract at date t is simplest in sense that the option’s return perfectly replicates the option can be exercised. Assuming for simplicity that risk We can see that there no unique rule to define a ‘fair’ price the stock return. To illustrate more a general case in which free rate equal to 0 it looks reasonable to exercise option at (1.3) to the option as far as any number used as a spot price the possibility perfectly replicates the stock return by the call the date when payoff reaches its maximum. Hence, the exer- implies the risk. The difference between possible spot prices option is impossible we assume, for instance, that K = $ 2.5. cise price of the American option is { S ( t ) - K , 0 }. This formula holds regardless whether the exchange rate q is the value of risk taken by investors. This risk is the The correspondent option payoff at maturity is the row C ( 1, Applying IE for the American call pricing leads to the equa- ( t ) is supposed to be stochastic or deterministic. For market risk which specified by the future behavior of the S ( 1 )) and the price of the option is defined by the third row. tion for American call option value at t = 0 instance, let N = £ 31,250, K = $/ £ 1.50, q ( T ) = $ / £1.55. underlying asset. The risk management problem is a calcu- Its values can be calculated applying IE concept. Thus Then payoff at maturity T is equal to lation of the market risk. S (1) 1 2 3 4 5 6 C ( 1, S ( 1 )) 0 0 0.5 1.5 2.5 3.5 N max { q ( T ) - K , 0 } = £ 31,250 × $ / £ ( 1.55 – 1.5 ) = Risk management. C ( 0, S ( 0 )) 0 0 1/3 3/4 1 7/6 where τ ( ω ) = { t T : = max }. Given distribution S ( $ 1,562.5 *) Consider for example call option written on stock 1. An t ) an investor can establish the level L such that the Ameri- investor pays premium A for the option at t = 0 takes risk Risk management. Mean of the option price at t = 0 is can option would be exercised before T if = L for t T. Now we apply for more complex option problem that involves associated with the scenarios 0.5417. Hence, if the market price of the option is A = 0.5417 Otherwise the option would be exercised at T if S ( T ) > K. intermediate moment of time with more than 2 states at then if outcome is 1 or 2 investor’s losses are its premium, Denote C A ( t , S ( t ) ; T ) American option price at t. Then expiration. Assume that the value of 100 British pounds over i.e. 0.5417. Therefore the loss of 0.5417 occurs with probabil- three dates 0, 1, 2 are given as follow C A ( 0 , S ( 0 ) ; T = 2 ) = C E ( 0 , S ( 0 ) ; T = 1 ) )χ{ S ( 1 ) 3 ity 1/3. Then the probability 1/6 is assign to each of the next S ( 2 ) } + + C E ( 0 , S ( 0 ) ; T = 2 )χ{ S ( 2 ) > S ( 1 ) } This is the set of scenarios for which rate of return on call profit-loss outcomes: [ 0.5 – 1/3 ] = 0.17, [ 1.5 – 3/4 ] = t=0 t=1 t=2 q(2) = 186 p(185, 186 ) = 1/4 option will be lower than the rate of return on underlying 0.75, [ 2.5 – 1 ] = 1.5, [ 3.5 – 7/6 ] = 2.3 that correspond Note, for example, that when the event { S ( 1 ) 3 S ( 2 ) } is q(1) = 185, p(180, 185) = 2/3 q(2) = 182 p(178, 182 ) = 1/8 stock. Indeed, buying call option for A and receiving S ( T ) - to stock values 3, 4, 5, 6. We introduce some useful risk char- true when annualized rate of return on stock over the period q(0) = 180 q(2) = 181 p(178, 181 ) = 1/4 q(1) = 178, p(180, 178) = 1/3 q(2) = 179 p(185, 179 ) = 3/4 K at T implies rate of return equal to the left hand side of the acteristics. These are average profit of the option defined by [0, 1] is higher than over the period [0, 2]. The latter formula q(2) = 176 p(178, 176 ) = 5/8 latter inequality while the right hand side is the rate of the formula expresses American option price through European option return on stock over the same period. In other words this price and the conclusion that follows from this formula does where p ( a, b ) denotes transition probability from the state not coincide with a well-known statement that the current ‘a’ to state ‘b’. Assume that all transitions are mutually inde- prices of American and European options on no dividend pendent. Consider European call option with the strike price asset are identical. The last formula can be easily extended on K = 180. We begin with calculations of the option price by multiple steps economy. moving backward in time. Applying the method that we used The American option can be exercised at any time up In one-step economy, let us briefly outline the construction above over period it is easy to see that of the call option price. Let t and T denote initial moment to maturity T. Therefore, its payoff depends on time and option maturity and let underlying values at T and strike price satisfy inequalities: S 1 < S 2 < …. < S p K Sp interval during which the option can be exercised. + 1 < … < S n . Then call option premium is defined as a and random variable Assuming for simplicity that risk free rate equal to 0 it looks reasonable to exercise option at the date Then when payoff reaches its maximum. j = p + 1 , … , n. If c 0 is a market price of the option then the risk connected to the price is a chance that realized sce- nario belongs to the set Here, p (a, b, c) = P { q (0) = a, q (1) = b, q (2) = c }and {a} {b} is the union of two states ‘a’ and ‘b’. We summarize cal- risk set of scenarios forms buyer risk when investors pay < Profit ( 0 , T ; K ) > = E C ( 0, S ( 0 )) χ { C ( 1, S ( 1 )) > K } where is a solution of the equation culations in the table higher price for stock that implies by the market. Thus the and average losses probability P { ω risk-buyer ( A ) } is a measure of the buyer . This equation specifies one- to- one C( 0, 180 ) C ( 1, ω) C ( 2, ω) p( ) ω risk. Let us consider are more complex case when random 5.807 5.968 6 1/6 < Loss ( 0 , T ; K ) > = E C ( 0, S ( 0 )) χ { C ( 1, S ( 1 )) K} correspondence between S to the option price c. If the value 1.978 1.956 2 1/24 stock admits multiple values. of the underlying at maturity T will be below than S then this 0.994 0.983 1 1/12 Example 2. Let us study the rolling dice example to illustrate 0 0 0 17/24 The profit-loss ratio is < Profit ( 0 , T ; K ) > / < Loss ( 0 , T ; K ) scenario is an element of the risky set risk ( c ) associated the multiple values stochastic security in the option pricing >. These are primary risk characteristics of the option. with the investor’s market risk. The probabilities in the fourth column related to the events problem. Let again assume that time takes two value t = 0, 1 Let us highlight the difference between European and American We consider now options written on exchange rate. This in each cell in the row. Now let us investigate a possible which are the initial and expiration dates of the option. The option prices. Consider American option written on stock. The problem is similar to the problems studied above neverthe- investor’s strategy. The average return on the exchange rate set 1, 2..., 6 represents possible values of the stock and prob- IE rule applying for the European call or put options at t, t < T less some peculiarities are needed to be specified. We will over abilities of the events { S ( 1 ) = j }, j = 1, 2, ... 6 are equal in either discrete or continuous time is a solution of the equation use cross currency exchange as underlying of the option to 1 / 6. The payoff at the maturity is defined C ( 1 , S ( 1 ) ) = contracts. Let K denote strike price measured in $ / £ and q max { S ( 1 ) – K , 0 } and let K = $ 0.8. The value S ( 0 ) = ( t ) denotes $ / £ - exchange rate at time t. That is £1 ( t ) = An investor who might interested in calculation of the value $2 can be interpreted as a price to roll the dice . Applying the $ q ( t ) and therefore a £1 can be interpreted as a portion of (1.2) of the option price which expected return would be not worse IE concept we arrive at the definition of the call option price. asset that can be sold or bought on $-market. All contracts then 1.0148. This price is a solution of the equation The option price is a random variable taking different values are settled by delivery of the underlying currency. By defini- E C ( 1 , ω ) / x = 1.0148. Solving this equation for x yields j - 0. 8, j = 1, 2, ... 6. We express the theoretical price of the tion, the contract payoff at maturity T is N max { Q ( T ) – K game with the help of the table 0 t T. Here , 0 }, where N denotes a contract size. For instance, the size of a British pound call option contract traded on PLHX is N S(1) 1 2 3 4 5 6 Hence, the premium of 1.14 on call option with strike K = C ( 1, S ( 1 )) 0.2 1.2 2.2 3.2 4.2 5.2 = £ 31,250. The call option equation (1.2) can be rewritten 180 approximately in average promises the return of 1.48% . C ( 0, S ( 0 )) 0.4 1.2 1.47 1.6 1.68 1.73 in the form The risk of buying option for $1.14 is the probability »20 • Discrete Space-Time Options Pricing Discrete Space-Time Options Pricing • 21
  3. 3. fsrforum • jaargang 12 • editie #5 Ccn ( 0, 180 ) Ccn ( 1, ω) Ccn ( 2, ω) p( ) ω 0.9677 0.9946 1 1/6 0.989 0.978 1 1/24 The exotics or non-standard options are those 0.9945 0.9834 1 1/12 0 0 0 17/24 This might be a high risk for an investor. Note that one can which payoff cannot be reduced to American or reach an arbitrary high average return by chosen the option Each raw in this table represents a path of the call option for price sufficiently small but the risk of any price will be not European options. They are divided into two a fixed scenario ω 0 = { q ( 0, ω 0 ), q ( 1, ω 0 ), q ( 2, ω 0 ) } and less than 17/24. We can use data provided by the latter Table therefore for the fixed scenario ω 0 the option’s rates of return to present calculation for the put option. Consider European primary classes referred to as path-dependent coincide with the correspondent rates of return of the under- put option with the strike K = 182. Then lying exchange rate. Similar class of exotics is assets-or-noth- ing call and put options payoff at maturity are defined as and path-independent. Can ( T , q ( T )) = q ( T ) χ { q ( T ) > K } and Pan ( T , q ( T )) = q ( T ) χ { q ( T ) < K } The pricing formulas can be derived from the general pricing Then formulas Similarly, χ{q(T) K 1 } = 1 -χ{ q ( T ) > K 1 } Can ( t , q ( t )) = N q ( t ) χ { q ( T ) > K } Ppl ( t , q ( t )) = ( K - q ( T ) ) χ{ q ( T ) > K } χ{q(T) ( K1 , K2 ] } = χ { q ( T ) > K1 } - χ { q ( T ) > K2 } Pan ( t , q ( t )) = N q ( t ) χ { q ( T ) < K } In the next Table we enclose the valuation of the paylater call one can see that payoff of the collar can be presented as following The mean and standard deviation of the put premium are option when underlying is the value of foreign currency unit 2.8697, 2.0598 correspondingly. Let for example, investor Gap options are contacts for which European call payoff is be which value given by the Table on page 7. I ( T ) = K1 - K1 χ { q ( T ) > K1 } + q ( T ) χ { q ( T ) > K1 } - pays $1 premium for the put option then the risk to receive written in the form at expiration less return than invested is 7/24. This loss is C pl ( 0, 180 ) C pl ( 1, ω) C pl ( 2, ω) p( ) ω - q ( T ) χ { q ( T ) > K2 } + K2 χ { q ( T ) > K2 } = K1 + associated with the scenario 0.2.911 2.9919 3.089 1/6 Cg ( T , q ( T )) = ( q ( T ) – R ) χ { q ( T ) > K } 0.9944 0.9889 1.0056 1/24 { q (2) = 186, 182, or 181 } If the put premium is $4 then the 0.4978 0.4958 0.5022 1/12 + [ q ( T ) - K1 ] χ { q ( T ) > K1 } - [ q ( T ) - K2 ] χ { q ( T ) > K2 } risk is 0 0 0 17/24 where K, R are known constants and K > R. The value of the P [ q ( 2 ) = { 186, 182, 181, 179 }] = 19/24. contracts can be represented by the cash-or-nothing option The right hand side of this equality is equal to a portfolio Indeed, applying formula (2.3) we see that solution where X = q ( T ) – R. The gap-put payoff is holding $K1 cash, long European call with the strike price 2. Exotics options. K1 , and short European call with the strike price K2. This In this section, we introduce option pricing formulas for Pg ( T , q ( T )) = ( R - q ( T )) χ { q ( T ) < K } decomposition of the collar payoff is not unique. Indeed, one some popular exotic classes. The exotics or non-standard can be easily verify other payoff’s representation options are those which payoff cannot be reduced to where K < R. Then the gap-put pricing formula can be per- American or European options. They are divided onto two form by the second formula (2.1) where X = R – q ( T ). I ( T ) = K1 + K2 - q ( T ) + [ q ( T ) - K1 ] χ { q ( T ) > K1 } - primary classes referred to as to path-dependent and path-independent. Exotic options are generic name of Paylater options call and put payoffs are defined by formulas - [ K2 - q ( T ) ] χ { q ( T ) < K2 } these derivatives. Exotic options are referred to as path- independent if their payoff does not depend on the path Cpl ( T , q ( T )) = [ q ( T ) - K - Cpl ( t , q ( t )) ] χ { q ( T ) > K } Thus collar payoff is equivalent now to the value of the portfolio during the lifetime of the option. (2.2) that contains $( K1 + K2 ) cash , short stock , long European call, Cash-or-nothing options also known as digital or binary Ppl ( T , q ( T )) = [ K - q ( T ) - Ppl ( t , q ( t )) ] χ { q ( T ) < K } and short European put. The price of a collar contract at any options. The call and put digital options are defined by time prior expiration coincides with the value of the portfolio. their payoff at maturity as where Cpl ( t , q ( t )) , Ppl ( t , q ( t )) are the values of the We introduce direct evaluation of the collar contract applying options at their date of origination date t and paid only on the formula (2.4). It follows that the collar payoff (2.4) is the basket Ccn ( T , q ( T )) = X χ { q ( T ) > K } exercise of the options. These are up-front payments paid at of the three hypothetical financial instruments with payoffs at Note that we omitted for writing simplicity index ‘pl’ that date t. We show that the paylater payoff can be negative. To specifies paylater option. One might note that the risk char- Pcn ( T , q ( T )) = X χ { q ( T ) < K } produce the valuation of the problem one needs to use the I1 ( T ) = K 1 χ { q ( T ) K1 } acteristics of the paylater call option as well as other exotics benchmark formula (1.3). The solution of this equation call option with the same strike price have been introduced where X is a predetermined constant and q ( t ) can be inter- when N = 1 can be presented in the form I2 ( T ) = q ( T ) χ { q ( T ) ( K1, K2 ]} preted as a spot exchange rate in dollars per unit of foreign above coincide with the correspondent risk characteristics of currency at time t, t T. Note, that in contrast to the con- the standard European option with the same strike price. All Cpl ( t , q ( t )) = Cpl ( T , q ( T )) I3 ( T ) = K 2 χ { q ( T ) > K 2 } tinuous payoff of the European or American options the dig- these options offered the same return though their premi- ital options have discontinuous payoff. The constant X is ums and payoffs are different. Bearing in mind formula (1.3) the above equation can be rep- with the same maturity T. Then the collar contract price at t is usually assumed equal to 1. The valuation of the options con- A collar contract payoff at maturity T is defined by a formula resented in the form tracts can be represented by the formula I(t) = I1(t) +I2(t) + I3(t), I ( T ) = min { max { q ( T ) , K1 } , K 2 }}. Cpl ( t , q ( t )) = [ q ( T ) - K - Cpl ( t , q ( t )) ] χ { q ( T ) > K } Ccn ( T , q ( T )) = N X χ{ q ( T ) > K } where Note that this payoff can be rewritten in a more comprehen- (2.1) Solving the equation for Cpl ( t , q ( T )) we arrive at the call sive form paylater option price P cn ( T , q ( T )) = N X χ{ q ( T ) < K } I ( T ) = K1 χ { q ( T ) K1 } + q ( T ) χ { q ( T ) ( K1 , K2 ] } + Cpl ( t , q ( t )) = (q(T) - K) Here N is the contract size expressed in foreign currency, K is the strike price, q ( T ) is the currency exchange rate at + K2 χ { q ( T ) > K2 } (2.4) χ{ q ( T ) > K } = (2.3) date T. Let us the numeric example. Assume that the under- lying security data is given by the Table on page 7 and N = X Below we will introduce standard arguments that perform = ( q ( T ) - K ) χ{ q ( T ) > K } the valuation of the collar contract. Using identities = 1. Then using the same algebra one arrives at the table »22 • Discrete Space-Time Options Pricing Discrete Space-Time Options Pricing • 23
  4. 4. fsrforum • jaargang 12 • editie #5A cliquet or ratchet option is a series of at the money options,with periodic settlement, resetting of the strike price atthe reset date spot price level, at which the option locks inthe difference between the old and new strike prices andpays that difference out as the profit. A chooser or as-you-like option is other exotic option Therefore, the return to the chooser option can be repre- exchange rates q ( s ) , s t. An investor buys the ladder type. A holder of this option can choose whether the sented in the form option with a strike price Q = Q 0. Thus a ladder start with the option is a call or put after specified period of time. An height Q and going upwards in the step interval of ε > 0 until interesting point is that the chooser option payoff does Hence, the maximum rung of Q N , Q j = Q + j ε , j = 0, 1, … , N. At not specify it as call or put options. More accurately, this maturity, T buyer of the ladder call option would receive type of derivatives could be named as a forward-choice C ( tji, q ( t ji ) ; t ji + 1 , q ( t ji ) ) = payoff options contract. Consider a chooser option that matures at moment Tch , the maturity of the underlying call and put denote Tc , Tp respectively min (Tc , Tp ) > Tch . Thus, the values of underlying call and put at the date Tch are Using this formula, we can calculate cliquet call value recur- sively. On the other hand, the present value at t of the sto- C ( Tch , q (Tch ) ; Tc , Kc ) and P ( Tch , q (Tch ) ; Tp, Kp ) chastic cash flows generated by the series of 1) the initial From this formula, follows that the ladder payoff takes into values of the forward start options and 2) payoff of these account the maximum value of the underlying price over correspondingly, q ( t ) is the underlying security of the call options are equal to lifetime of the option. To construct ladder call option price and put options, and Kc , Kp are the correspondent strike Recall that exchange rate q ( * ) is the underlying process for assume that ω {ω: Q j < Q j + 1 } for some j . prices. The payoff to the chooser option at maturity Tch is admissible scenarios. Therefore the equation for co ( t , q ) Then ladder call option payoff realized for this scenario will could be presented in the form be equal to co (Tch , q (Tch )) = max { C ( Tch , q (Tch ) ; Tc , Kc ) , P ( Tch , q (Tch ) ; Tp, Kp ) } C lad ( T , q ( T )) = max { q ( T ) - Q , Q j - Q } correspondingly. If c is spot price of the option at t then Note that the payoff can be expressed in the form buyer and seller risk can be estimated by probabilities P { C ( Therefore the IE rule brings us to the valuation equation t , x , ω ) < c } , P { PC ( t , x , ω ) < c } which are the meas- co (Tch , q (Tch )) = C ( Tch , q (Tch ) ; Tc , Kc ) × Solving this equation, we figure out that the value of the ure of scenarios when these counterparties pays more than chooser option is the scenarios provide for. × χ { C ( Tch , q ( Tch ) ; Tc , Kc ) P ( Tch , q (Tch ) ; Tp , Kp ) } + A Couple option is a similar type of the cliquet options. As for cliquet option, payoff to a holder could take place either at + P ( Tch , q ( Tch ) ; Tp , Kp ) χ {C ( Tch , q ( Tch ) ; Tc , Kc ) < P ( specified reset dates or at maturity. The only distinction Tch , q (Tch ) ; Tp , Kp )} between couple and cliquet is that the couple options at reset dates switch its value to the smaller of the current spot level A cliquet or ratchet option is a series of at the money options, Using explicit representation of the call and put prices given and the initial strike price. The cash flow generated by the with periodic settlement, resetting of the strike price at the by (1.2.1) it is easy to verify equalities couple call option is Thus reset date spot price level, at which the option locks in the difference between the old and new strike prices and pays that difference out as the profit. This profit might be paid out at each reset date or could be accumulated until maturity. where t = t 0 < t 1 < … < t N = T are reset dates, and min [ q ( Thus, a cliquet option can be thought, as a series of options t j ) , K ] is reset strike price. The price of the call and put that settles periodically at the reset dates is an example of the Remark. Other modification of the ladder call option can be couple options are path-dependent class of options. introduced by assuming that call option payoff is defined as Let us introduce a n-years cliquet option with k-resets annu- following C cp ( t j , q ( t j ) ; t j + 1 , min [ q ( t j ) , K ] ) = ally. Let t j i be reset moments of time j = 0, 1, … , n ; i = 0, 1, …k – 1 and T denotes maturity. The payoff over the period [ t j i , t j i + 1 ] that is due to paid at the t j i + 1 is In this case in which the payoff is similar to the ladder which P cp ( t j , q ( t j ) ; t j + 1 , min [ q ( t j ) , K ] ) = max { q ( t j i + 1 ) - q ( t j i ) , 0 } admits a finite number of values 0, Q 1 - Q, … , Q N – Q with probabilities P j = P{ Q j Q j + 1 }, This formula corresponds to the case when option writer j = 0, 1, … , N - 1, and P N = P{ > Q N }. The pays out periodically at the reset dates. Denote the underly- valuation formula in this case can be obtained from the ing of the cliquet option q ( s ) = q ( s ; t , x ), s t and A ladder option payoff is also similar to a cliquet payoff with above formula by replacing payoff in the brackets by its C ( t , x ; T , Q ) the value of the European cliquet call option exception that the gains are locked in when the asset price modification. at date t with strike price Q and expiration date T. Applying IE breaks through certain predetermine rung. The strike price The purchaser of the ladder put will receive at maturity valuation we arrive at the pricing equation is then intermittently reset. Consider a ladder option on payoff of »24 • Discrete Space-Time Options Pricing Discrete Space-Time Options Pricing • 25
  5. 5. fsrforum • jaargang 12 • editie #5 extendible expiration date. The additional premium of $d is P ew ( Te , q ( Te )) = [ Q - q ( Te )] χ { Q q ( Te ) } + [ Q - q ( These derivative contracts can be interpreted as derivatives paid by the holder in case when extension feature is chosen T ) ] χ { Q q ( Te )} having variable strikes in contrast to a constant strike used to exercise at Te . There are new factors involved to the prob- in the previous examples. The pricing formulas to the con- lem. Valuation equation of the call extendible can be repre- correspondingly. These payoff types give additional benefit to tracts can be obtained using standard IE pricing rule. Indeed, Here, Q – M < Q – M + 1 < …< Q – 1 < Q is a rung sequence. sented in the form buyers of the call or put options. The first term on the right only two possibilities are available underlying exchange and The pricing equation for the ladder put is hand side of the call and put option payoffs, correspond to derivatives. If a scenario ω is such that C e ( T ; Δ ,T 0 ) = 0 then “in-the-money” scenarios at Te while the second term implies there is no sense to invest in extreme call. If C e ( T ; Δ ,T 0 ) “out-of-the-money” scenarios. By using extended feature > 0 then there is the unique price to avoid arbitrage. This does not cost or imply more losses for counterparties. The price is defined as a solution of the equation valuation equation of writer extendible call and put options The indicator on the right hand side of the equality contains can be presented in the form union of two events, which signify that at least one of the = possibilities at Te should be strictly positive. Otherwise, the Hence, if ω { ω : < Q – M }. The value of the put ladder option value of C eh ( Te , q ( Te )) and C eh ( t , q ( t )) for this particu- is then lar scenario is 0. Taking this into account and solving call option price equa- Similarly, tion we arrive at the premium formula Extendible options have become popular over recent time for volatile underlying. There are two types of the extendible These pricing equations bring us to the valuation formulas options: holder and writer extendible. A holder extendible option is an option that can be extended by the holder at Other path-dependent option class is Lookback options. The option maturity Te. This possibility is required an additional The formula for the holder extendible put option can be per- extreme exotic options introduced above sometimes are con- premium. The holder of the extendible option on call or put form in the similar way sidered as subclass of lookback options and called it extrema has a choice to get an ordinary call option payoff or by paying Note that European type of the underlying options can also lookback options. Two primary forms of the lookback options a predetermine premium $d to the writer at time Te to get be replaced by American options. exist based on strike price definition. First form is defined as call option with extended maturity. The Extreme or Reverse Extreme exotic options was intro- lookback options with fixed strike price. The payoffs of the Assume that the holder’s choice is based on the maximum duced in 1996. Call extreme options payoff at maturity T is call and put options are value of the option payoffs at Te . That is determined by the difference between maximum values on compliment subintervals constituted the lifetime of an C eh ( Te , q ( Te )) = max {{ q ( Te ) - Q , 0 } , C ( Te , q ( Te ) ; T , underlying asset. Let t < T 0 < T and denote Δ = [ t , T ]. Then K) -d}= payoffs to the call option at maturity for the extreme and A reciprocal problem given option price to estimate the value inverse extreme options are = max { q ( Te ) - Q , C ( Te , q ( Te ) ; T , K ) - d , 0 } , of the premium d is important too. respectively. Applying the same arguments as for extreme A writer extendible option allows a seller of the option, options pricing we arrive at the formulas C e ( T ; Δ ,T 0 ) = max { q(v) - q(u), 0 } P eh ( Te , q ( Te )) = max { Q - q ( Te ) , P ( Te , q ( Te ) ; T , K option writer to extend the option either call or put with zero ) -d,0} cost at the maturity Te if the option is out-of-money. Recall C i e ( T ; Δ , T 0 ) = max { q(u) - q(v), 0} that option call ( put ) is out-of-money at date t if its value at Though for example option buyer at Te may expect over [ Te , this moment is less ( larger) or equal to the strike price. The payoffs to put extreme and put inverse extreme options T ] to get higher overall return by exercise extendibility than Thus, if option have a negative value its can be exercise later at maturity get the spread value between minimum over to get call option payoff at Te and investing it at risk free rate. at a date T. Therefore, writer extendible payoff of the call and The lookback options with floating strike price can be settled adjacent periods, i.e. We do not analyse such possibility. put are equal to in cash or assets in contrast with the fixed strike options in In above formulas C ( Te , x ; K , T ) , P ( Te , x ; K , T ) denote P e ( T ; Δ , T 0 ) = max { q(v) - q(u), 0 } which cash settlement is only admitted. The payoff of the the price of the European call or put options at date Te with C ew ( Te , q ( Te )) = [ q ( Te ) - Q ] χ { q ( Te ) Q}+ [q(T) lookback call and put options with floating strike price are a strike price K that might be equal to Q, and T , T > Te is the - Q ] χ { q ( Te ) < Q } P i e ( T ; Δ , T 0 ) = max { q(u) - q(v), 0} defined as following »26 • Discrete Space-Time Options Pricing Discrete Space-Time Options Pricing • 27

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