Note on simulation pricing of π -options
NNOTE ON SIMULATION PRICING OF π -OPTIONS ZBIGNIEW PALMOWSKI AND TOMASZ SERAFIN
Abstract.
In this work, we adapt a Monte Carlo algorithm introduced by [Broadie and Glasserman (1997)]to price a π -option. This method is based on the simulated price tree that comes from discretization andreplication of possible trajectories of the underlying asset’s price. As a result this algorithm produces thelower and the upper bounds that converge to the true price with the increasing depth of the tree. Underspecific parametrization, this π -option is related to relative maximum drawdown and can be used in thereal-market environment to protect a portfolio against volatile and unexpected price drops. We also providesome numerical analysis. Keywords. π -option (cid:63) American-type option (cid:63) optimal stopping (cid:63)
Monte Carlo simulation
Date : August 26, 2020.This work is partially supported by National Science Centre Grant No. 2016/23/B/HS4/00566 (2017-2020). a r X i v : . [ q -f i n . C P ] A ug Z. Palmowski — T. Serafin Introduction
In this paper we analyze π -options introduced by [Guo and Zervos (2010)] that depends on so-calledrelative drawdown and can be used in hedging against volatile and unexpected price drops or by speculatorsbetting on falling prices. These options are the contracts with a payoff function:(1) g ( S T ) = ( M aT S bT − K ) + in case of the call option and(2) g ( S T ) = ( K − M aT S bT ) + in the case of put option, where(3) S t = S exp (cid:18)(cid:18) r − σ (cid:19) t + σB t (cid:19) is an asset price in the Black-Scholes model under martingale measure, that is, r is a risk-free interestrate, σ is an asset’s volatility and B t is a Brownian motion. Moreover, M t = sup w ≤ t S w is a running maximum of the asset price and T is its maturity. Finally, a and b are some chosen parameters.Few very well known options are particular cases of a π -option. In particular, taking a = 0 and b = 1 produces an American option and by choosing a = 1 and b = 0 we derive a lookback option. Anotherinteresting case, related to the concept of drawdown (see Figure 1), is when − a = b = 1 and K = 1 .Then the pay-out function ( K − M aT S bT ) + = 1 − S bT M − aT = M T − S t M T = D RT equals the relative drawdown D Rt ,defined as a quotient of the difference between maximum price and the present value of the asset and thepast maximum price. In other words, D Rt corresponds to the percentage drop in price from its maximum.We take a closer look at this specific parametrization of the π -option in the later sections, starting fromSection 3.2.Monte Carlo simulations are widely used in pricing in financial markets they have proved to be valu-able and flexible computational tools to calculate the value of various options as witnessed by the con-tributions of [Barraquand and Martineau (1995), Boyle (1977), Boyle et al. (1997), Broadie et al. (1997),Clément et al. (2002), Caflisch (1998), Dyer and Jacob (1991), Geske and Shastri (1985)],[Glasserman (2004), Jäckel (2002), Resenburg and Torrie (1993), Longstaff and Schwartz (2001)],[Joy et al. (1996), Niederreiter (1992), Raymar and Zwecher (1997), Rogers (2002), Tilley (1993)],[Clément et al. (2002), Tsitsiklis and van Roy (2001), Villani (2010)]. One of the first attempts of MonteCarlo simulation for American options is by [Tsitsiklis and van Roy (1999)] where the backward inductionalgorithm was introduced. However, as appears later, Tilley method suffers from exponentially increasingcomputational cost as the number of dimensions (assets) increases. [Broadie et al. (1997)] to overcomethis problem offered a non recombining binomial simulation approach instead combined with some pruningtechnique to reduce computation burden and other variance reduction techniques to increase precision. Inthe same year [Broadie and Glasserman (1997)] construct computationally cheap lower and upper boundsto the American option price. This method is used in this paper. An alternative way to formulate theAmerican option pricing problem is in terms of optimal stopping times. This is done in [Carriere (1996)],where it was proved that finding the price of American option can be based on a backwards inductionand calculating a number of conditional expectations. This observation gives another breakthrough inpricing early exercise derivatives by Monte Carlo done by [Longstaff and Schwartz (2001)]. They proposeleast square Monte Carlo (LSM) method which has proved to be versatile and easy to implement. Theidea is to estimate the conditional expectation of the payoff from continuing to keep the option alive ateach possible exercise point from a cross-sectional least squares regression using the information in thesimulated paths. To do so we have to then solve some minimization problem. Therefore this method isstill computationally expensive. Some improvements of this method have been also proposed; see also[Stentoft (2004a), Stentoft (2004b)] who gave theoretical foundation of LSM and properties of its estima-tor. ricing of watermark and π options 3 Figure 1.
A sample drawdown for the Microsoft Corporation stock is marked with blackarrows and dashed lines. Data is taken from
There are other, various pricing methods in the case of American-type options; we refer [Zhao (2018)]for review. We have to note though that not all of them are good for simulation of prices of general π -options as it is path-dependent product. In particular, in pricing π -options one cannot use finite differencemethod introduced by [Brennan and Schwartz (1978), Schwartz (1977)] which uses a linear combinationof the values of a function at three points to approximate a linear combination of the values of derivativesof the same function at another point. Similarly, the analytic method of lines of [Carr and Faguet (1994)]is not available for pricing general π -options. One can use though a binomial tree algorithm (or trinomialmodel) though which goes backwards in time by first discounting the price along each path and computingthe continuation value. Then this algorithm compares the former with the latter values and decide foreach path whether or not to exercise; see [Broadie and Detemple (1996)] for details and references therein.It is a common belief that Monte-Carlo method is more efficient than binomial tree algorithm in case ofpath-dependent financial instruments. It has another known advantages as handling time-varying variants,asymmetry, abnormal distribution and extreme conditions.In this paper we adapt a Monte Carlo algorithm proposed in 1997 by [Broadie and Glasserman (1997)]to price π -options. This numerical method replicates possible trajectories of the underlying asset’s priceby a simulated price-tree. Then, the values of two estimators, based on the price-tree, are obtained.They create an upper and a lower bound for the true price of the option and, under some additionalconditions, converge to that price. The first estimator compares the early exercise payoff of the contract Z. Palmowski — T. Serafin to its expected continuation value (based on the successor nodes) and decides if it is optimal to hold or toexercise the option. This estimation technique is one of the most popular ones used for pricing American-type derivatives. However, as shown by [Broadie and Glasserman (1997)], it overestimates the true priceof the option. The second estimator also compares the expected continuation value and early exercisepayoff, but in a slightly different way, which results in underestimation of the true price. Both Broadie-Glasserman Algorithms (BGAs) are explained and described precisely in Section 2. The price-tree thatwe need to generate is parameterized by the number of nodes and also by the number of branches in eachnode. Naturally, the bigger the numbers of nodes and branches, the more accurate price estimates we get.The obvious drawback of taking a bigger price-tree is that the computation time increases significantlywith the size of the tree. However, in this paper we show that one can take a relatively small price-treeand still the results are satisfactory.The Monte Carlo simulation presented in this paper can be used in corporate finance and especially inportfolio management and personal finance planning. Having American-type options in the portfolio, theanalyst might use the Monte Carlo simulation to determine its expected value even though the allocatedassets and options have varying degrees of risk, various correlations and many parameters. In fact deter-mining a return profile is a key ingredient of building efficient portfolio. As we show in this paper portfoliowith π -options out-performs typical portfolio with American put options in hedging investment portfoliolosses since it allows investors to lock in profits whenever stock prices reaches its new maximum.In this paper we use BGA to price the π -option on relative drawdown for the Microsoft Corporation’s(MSFT) stock and for the West Texas Intermediate (WTI) crude oil futures. Input parameters for thealgorithm are based on real market data. Moreover, we provide an exemplary situation in which we ex-plain the possible application of the π -option on relative drawdown to the protection against volatile pricemovements. We also compare this type of option to an American put and outline the difference betweenthese two contracts.This paper is organized as follows. In the next section we present the Broadie-Glasserman Algorithm. InSection 3 we utilize this algorithm to numerically study π -options for the Microsoft Corporation’s stockand WTI futures. Finally, in the last section, we state our conclusions and recommendations for furtherresearch in this new and interesting topic.2. Monte Carlo algorithm
Formulas identifying the general price of π -option are known in some special cases and they are given interms of so-called scale functions and hence in terms of the solution of some second order ordinary differen-tial equations; see for example [Christensen (2013), Chap. 5] and [Egami and Oryu (2017)] for details andfurther references. Still, the formulas are complex and a Monte Carlo method of pricing presented in thispaper is very efficient and accurate alternative method. In this section we present a detailed descriptionof the used algorithm. In particular, we give formulas for two estimators, one biased low and one biasedhigh, that under certain conditions converge to the theoretical price of the option.2.1. Preliminary notations.
We adapt the Monte Carlo method introduced by[Broadie and Glasserman (1997)] for pricing American options. In this algorithm, values of two estimatorsare calculated on the so-called price-tree that represents the underlying’s behavior over time. This treeis parametrized by the number of nodes n and the number of branches in each node - denoted by l . Forexample the tree with parameters n = 2 , l = 3 is depicted in Figure 2.In order to apply the numerical algorithm we have to discretize the price process given in (3), by consideringthe time sequence t = 0 < t < . . . < t n = T with t i = i Tn for i = 0 , . . . , n . By S t l ,...,lii we denote the asset’s price at the time t i = iTn . The upper index l , . . . , l i , associated with t i , describesthe branch selection (see Figure 3) in each of the tree nodes and allows us to uniquely determine the path ricing of watermark and π options 5
100 11510090115110110110 1201201051151251251051151101159711510011595115105115 t t t Figure 2.
An example of the price-tree . Underlying’s price is marked with circles andcorresponding maximums are marked with rectangles.of the underlying’s price process up to time t i . Similarly, we define M t l ,...,lii = max k ≤ i S t l ,...,lkk . We introduce the state variable (cid:101) S t l ,...,lii = ( S t l ,...,lii , M t l ,...,lii ) as well.We relate with it the payoff of an immediate exercise (for π -put) at time t i in the state (cid:101) S t l ,...,lii given by h t i ( (cid:101) S t l ,...,lii ) = ( K − S at l ,...,lii M bt l ,...,lii ) + and the expected value of holding the option from t i to t i +1 , given asset’s value (cid:101) S t l ,...,lii at time t i definedvia g t i ( (cid:101) S t l ,...,lii ) = E (cid:20) e − rn f t i +1 ( (cid:101) S t l ,...,li +1 i +1 ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) S t l ,...,lii (cid:21) , Z. Palmowski — T. Serafin S S t S t S t S t , S t , S t , S t , S t , S t , S t , S t , S t , t t t Figure 3.
Branch selecting.where f t i ( (cid:101) S t l ,...,lii ) = max { h t i ( (cid:101) S t l ,...,lii ) , g t i ( (cid:101) S t l ,...,lii ) } is the option value at time t i in state (cid:101) S t l ,...,lii . Note that f t n ( (cid:101) S t l ,...,lnn ) = f T ( (cid:101) S T l ,...,ln ) = h T ( (cid:101) S T l ,...,ln ) = ( K − S aT l ,...,ln M bT l ,...,ln ) + . Estimators.
We will now give the formulas for the estimators Θ and Φ which overestimate andunderestimate the true price of the option, respectively. Then, we will state the main theorem showingthat both estimators are asymptotically unbiased and that they converge to the theoretical price of the π -option. We also provide a detailed explanation of the estimation procedure based on the exemplaryprice tree. In all calculations we consider a π -put option with parameters a = − , b = 1 and K = 1 .Additionally we assume that the risk free rate used for discounting the payoffs equals . ricing of watermark and π options 7 The Θ estimator. The formula for the estimator is recursive and given by: Θ t i = max h t i ( (cid:101) S t l ,...,lii ) , e − rn l l (cid:88) j =1 Θ t l ,...,li,ji +1 , i = 0 , . . . , n − . At the option’s maturity, T , the value of the estimator is given by Θ T = f T ( (cid:101) S T ) . The Θ estimator, at each node of the price tree, chooses the maximum of the payoff of the option’s earlyexercise at time t i , h t i ( (cid:101) S t l ,...,lii ) , and the expected continuation value, i.e. the discounted average payoffof successor nodes. Figure 4 shows how the value of Θ estimator is obtained given the certain realizationof a price-tree. All calculations are also shown below:
100 11510090115110110110 1201201051151251251051101101109711010011095110105110 t t t (a) Price-tree d a b c t t t (b) Evaluation of Θ estimator Figure 4.
Explanation of Θ estimator Z. Palmowski — T. Serafin • a (cid:13) (cid:40) Holding value: +03 e − . ≈ . Early exercise: • b (cid:13) (cid:40) Holding value: +0+ e − . ≈ . Early exercise: ≈ . • c (cid:13) (cid:40) Holding value: + + e − . ≈ . Early exercise: ≈ . • d (cid:13) (cid:40) Holding value: . . . e − . ≈ . Early exercise: ≈ . The Φ estimator. The Φ estimator is also defined recursively. Before we give the formula we needto introduce an auxiliary function ξ by(4) ξ jt l ,...,lii = h t i ( (cid:101) S t l ,...,lii ) , if h t i ( (cid:101) S t l ,...,lii ) ≥ e − rn l − l (cid:88) k =1 k (cid:54) = j Φ t l ,...,li,ki +1 e − rn Φ t l ,...,li,ji +1 , if h t i ( (cid:101) S t l ,...,lii ) < e − rn l − l (cid:88) k =1 k (cid:54) = j Φ t l ,...,li,ki +1 for j = 1 , . . . , l . Now we can define the Φ estimator in the following way:(5) Φ t l ,...,lii = 1 l l (cid:88) j =1 ξ jt l ,...,lii Φ T = f T ( (cid:101) S T ) . The formula for this estimator is more complicated. Therefore, we provide a detailed explanation of themechanism behind the algorithm in the following part of this section. In our explanation we refer to Figure . Note that in the following example, underlined numbers correspond to the final values associated withthe specific branches of the tree. • a (cid:13) Early exercise: Holding value for branch j = 1 : . e − . ≈ . > → Holding value for branch j = 2 : e − . = 0 ≤ h i ( (cid:101) S t i ) → Holding value for branch j = 3 : . e − . ≈ . > → For the branch j = 1 we look at the two remaining ones to determine whether early exercising(payoff = 0) or holding the option (payoff = . e − . ) is more profitable. Obviously, earlyexercise is not optimal so we hold the option and thus, as the value of ξ t we take the payoff of thebranch j = 1 which is 0.For the branch j = 2 both early exercise value and holding value from two other branches equals0. Thus from (4) the value of ξ t equals the payoff of early exercise, which is 0.For the third branch, again holding the option is a more profitable decision (based on the payoffs ofthe two remaining branches). Thus ξ t takes the value corresponding to the branch j = 3 and it is 0. ricing of watermark and π options 9
100 11510090115110110110 1201201051151251251051101101109711010011095110105110 t t t (a) Price-tree d a b c t t t (b) Evaluation of Φ estimator Figure 5.
Explanation of Φ estimatorNow the value of the estimator for node a (cid:13) is the sum of ξ jt across all branches: Φ t = 13 (cid:88) j =1 ξ jt = 0 . Similarly, we have the following values of our estimator. • b (cid:13) Early exercise: ≈ . Holding value for branch j = 1 : . e − . ≈ . < . → . Holding value for branch j = 2 : . . e − . ≈ . < . → . Holding value for branch j = 3 : . e − . ≈ . < . → . In this case, the value of the estimator for the b (cid:13) node equals 0.091. Z. Palmowski — T. Serafin • c (cid:13) Early exercise: ≈ . Holding value for branch j = 1 : . . e − . ≈ . < . → . Holding value for branch j = 2 : . . e − . ≈ . < . → . Holding value for branch j = 3 : . . e − . ≈ . < . → . For node c (cid:13) the value of the estimator is 0.182. • d (cid:13) Early exercise: ≈ . Holding value for branch j = 1 : . . e − . ≈ . > . → Holding value for branch j = 2 : . . e − . ≈ . < . → . Holding value for branch j = 3 : . e − . ≈ . < . → . The value of the estimator for this node equals . . = 0 . . This is also the (un-der)estimated value of the option.Following arguments of [Broadie and Glasserman (1997)], one can easily prove the following crucial fact. Theorem 1.
Both Θ and Φ are consistent and asymptotically unbiased estimators of the option value.They both converge to the true price of the option as the number of price-tree branches, l , increases toinfinity. For a finite l : • The bias of the Θ estimator is always positive, i.e., E [Θ ( l )] ≥ f ( (cid:101) S ) . • The bias of the Φ estimator is always negative, i.e., E [Φ ( l )] ≤ f ( (cid:101) S ) . On every realization of the price-tree, the low estimator Φ is always less than or equal to the high estimator Θ , i.e., P (Φ t l ,...,lii ≤ Θ t l ,...,lii ) = 1 . Numerical analysis
In this section, we will present results of the numerical analysis. First, we use the algorithm describedabove to price the American option with arbitrary parameters. This will allow us to confirm that ourMonte Carlo algorithm produces precise estimates of options’ prices. We focus on options related toMicrosoft Corporation stock. Next, we price π -options for a number of combinations of parameters. Wealso consider π -option on drawdown using the real-market data and we compare it with an American put,which is one of the most popular tool for protecting our portfolio against price drops.3.1. American options.
First of all we decided to check the robustness of the Monte Carlo pricingalgorithm. We estimate prices of the American call options with different strike prices. In the example,the uderlying asset price S equals 100, σ = 20% , risk free rate r = 5% and the maturity is 30 days.In Table 1 we present the results of the estimation. Note that when utilizing the Broadie-Glassermanalgorithm, we obtain the upper and the lower boundaries of the option price. To obtain the Americanoption price estimate we average both values.3.2. π -options. We will analyze put π -option for various combinations of parameters a and b . We assumethat parameter a is varying from − . to − . and b parameter between . and . . The ranges of theseparameters have been chosen arbitrarily for illustrative purposes. All input parameters for options pricing, S , M , volatility and interest rate are taken from the real-market data for the Microsoft Corporation stock(MSFT) and are given in Table 2. The numerical results are presented in Figure 6. ricing of watermark and π options 11 Strike Low Est. High Est. Estimated Price Real Price Abs. Perc. Err.$80 $20.16 $20.55 $20.36 $20.33 0.14%$85 $15.10 $15.54 $15.32 $15.35 0.19%$90 $10.28 $10.62 $10.45 $10.43 0.19%$95 $5.84 $6.02 $5.93 $5.89 0.68%$100 $2.54 $2.60 $2.57 $2.51 2.36%$105 $0.76 $0.77 $0.77 $0.73 5.33%$110 $0.16 $0.16 $0.16 $0.14 13.35%
Table 1.
Comparison of the estimated and ’real’ American option prices with differentstrikes. Absolute percentage errors are also included.Parameter Value S M σ r l K Table 2.
Input parameters for pricing π -put option on the Microsoft Corporation stock. -0.9-0.94 a parameter -0.98-1.02-1.06-1.10.90.940.98 b parameter O p t i o n v a l u e Figure 6. π -option price estimations for varying a and b - Microsoft Corporation stock. Z. Palmowski — T. Serafin π -options on relative drawdown. Recall that for a = − and b = 1 the payoff of the π -optionequals(6) (cid:18) K − S t M t (cid:19) + , where S t /M t is the current value of the relative drawdown of the underlying asset. We believe that suchcontracts could be very efficiently used for hedging and managing portfolio risk against the volatile dropsin underlying’s price (see Section 3.4). One can adjust the payoff function (6) by the appropriate choice ofthe strike K . The choice is arbitrary and solely dependent on the risk management goals of the option’sbuyer. It allows to set the minimal size of drawdown we would like to protect against and let the buyeradjust and fully control the level of our exposure at risk associated with unexpected price drops. Forexample by setting K = , the payoff of our option becomes greater than zero only if the drop in theprice of the underlying from its maximum exceeds . Of course, the bigger the value of K , the moreexpensive the option is.We take a closer look at the impact of M t and K on the price of this special case of π -option. Here, weassume that the maximum price M t is between and and K ranges between . and . This time,the remaining parameters, namely S , r and σ , have been arbitrarily chosen for illustrative purposes andare given in Table 3. The results are shown in Figure 7.Parameter Value S σ
20 % r l Table 3.
Input parameters for pricing π -option on relative drawdown.3.4. π -options on relative drawdown - application. We now focus on the potential application of π -options and compare the prices of American put and π -option on relative drawdown. We compare theseparticular instruments due to the fact that their values increase with the decrease of the underlying asset’sprice. As an exemplary environment for the options comparison we choose two time series containingdaily closing prices of the Microsoft Corporation’s stock (see Figure 8) as well as daily closing pricesof the West Texas Intermediate (WTI) crude oil futures (see Figure 9). Both datasets are taken from and span approximately one year, from 6.11.2017 to 9.11.2018. We use the first months (from 6.11.2017 to 3.08.2018) to calibrate the historical volatility for both assets, which is one ofthe input parameters in our pricing algorithm.Then, using the historical volatility, we compute prices of π and American options (using assets’ pricesfrom 3.08.2018), both expiring months after the end of calibration period. Note that the parameters forthe π -option on a relative drawdown are a = − , b = 1 and K = 1 . Input parameters for calculation andestimated options prices for both assets are given in Table 4 and Table 5.Since the payoff of π -option on relative drawdown with K = 1 is always less than , in order tocompensate against the drop in underlying’s price, we need a certain number of these contracts per eachunit of stock in our portfolio. This number has to be equal to M . Note that in Table 4 and Table 5, thereal price of the single π -option on relative drawdown contract should be . for MSFT and . forWTI. However, in order to be able to compare the results to the American put values, we initially needto make the instruments pay the same amount in case of a price drop, therefore we multiply the price ofsingle π -option on drawdown by M ( and for MSFT and WTI respectively). That is why in Table4 in Table 5 the price of π -option equals . ·
110 = 8 . for the stock and . ·
74 = 6 . for the oilfutures contract.It turns out that π -option is more expensive than vanilla put in case of both assets, which is not asurprise as it initially pays the amount equivalent to the present maximum drawdown. However, since the ricing of watermark and π options 13 Strike
Maximum O p t i o n v a l u e Figure 7. π -option on relative drawdown price estimations for varying K and M parameters. MSFTAmerican put π on drawdown Parameter Value Parameter Value K M S σ r l T $ . $ . Table 4.
Input parameters for computation and estimated options’ prices for the MSFT dataset.difference in price between these instruments is rather significant, a question emerges whether there existsa situation in which purchasing π -option on relative drawdown is more profitable than buying a simplevanilla put. In order to answer this question, let us focus on the dashed part of the Microsoft Corporationand WTI futures data from the beginning of this section. In Figures 10 and 11 we show the amount eachinstrument would pay (on each day) throughout the whole -month period until options’ maturity.In order to display the difference more clearly, we construct two portfolios V American and V π , bothconsisting of an underlying asset (a single Microsoft Corporation stock or a barrel of the WTI crude oil)and an option (American put and π -option on relative drawdown, respectively). We observe them at theend of the volatility calibration period. Assets’ prices and options prices are taken from Table 4 and Table5. In Table 6 and Table 7 we show the initial net values of both V American and V π portfolios.Then we analyze the behaviour of the constructed portfolios, by calculating the net value of eachportfolio for each day until options’ maturity; see Figure 12 and Figure 13. Z. Palmowski — T. Serafin
Figure 8.
Daily closing prices of the Microsoft Corporation’s stock. Data spans from6.11.2017 to 9.11.2018. Vertical dashed line indicates the end of volatility calibration period.Option prices are calculated based on the volatility and the stock’s price on 3.08.2018.
WTIAmerican put π on drawdown Parameter Value Parameter Value K M S σ r l T $ . $ . Table 5.
Input parameters for computation and estimated options’ prices for the WTI dataset.Based on Figures 12 and 13 we can observe that the maximum value of portfolio V American is greaterthan the one for V π . Thus when focusing purely at the possible maximum profit over some period of time, ricing of watermark and π options 15 Figure 9.
Daily closing prices of the West Texas Intermediate crude oil futures contracts.Data spans from 6.11.2017 to 9.11.2018. Vertical dashed line indicates the end of volatilitycalibration period. Option prices are calculated based on the volatility and the asset’s priceon 3.08.2018. Portfolio V American V π Initial asset value 108.13 108.13Option premium -$5.47 -$8.09Option initial payoff $0 $2.68
Portfolio’s net value $102.66 $102.72
Table 6.
Portfolios and their initial net values for the MSFT dataset.Portfolio V American V π Initial asset value 68.49 68.49Option premium -$3.07 -$6.95Option initial payoff $0 $5.66
Portfolio’s net value $65.42 $67.20
Table 7.
Portfolios and their initial net values for the WTI dataset.then the portfolio containing American option performs better. However, we can notice that V American ’svalue over time is much more volatile compared to V π and it directly follows the behavior of underlyingasset (it increases when asset’s price rises and decreases in the opposite situation). The value V π of π -option portfolio is most of the time non-decreasing. Moreover, V π increases its value every time the asset’sprice reaches a new maximum and essentially does not decrease in case of any price drop. In other words,combining the underlying asset and π -option on drawdown allow us to lock in our profit whenever theprice reaches its new maximum. Z. Palmowski — T. Serafin
Figure 10.
Microsoft Corporation stock closing prices ( top ) and the corresponding payoffsof π -option on relative drawdown and American put ( bottom ) with the parameters fromTable 4.This brings us to the conclusion that the purpose of using π -option on relative drawdown and an Americanput is completely different. Vanilla American option protects us from asset price drops and ensures usthat the current worth of our portfolio will not be less than its initial value. Unfortunately, in this caseour portfolio’s value is more volatile and reflects the volatility of the underlying asset. This may result inbigger gains when comparing to the use of π -option on relative drawdown if the price of the underlyingrises significantly and stays on that level until option’s maturity. However, in case of a drop in assetprice after the upswing, we do not benefit from the fact that the new maximum has been reached andthus the value of our portfolio decreases together with the price of the underlying asset. When looking atthe value of V π over time one can notice that combining stock or a commodity and π -option on relativedrawdown protects us against price drops as well but the volatility of our portfolio is reduced significantly.Additionally, the contract allows us to benefit from the underlying’s price upswings and locks in the profitevery time new maximum is reached.We have analyzed two datasets, MSFT and WTI, and the above analysis shows that the behaviour ofa portfolio based on π -option is similar for various choices of underlying assets. ricing of watermark and π options 17 Figure 11.
WTI crude oil futures contract closing prices ( top ) and the correspondingpayoffs of π -option on relative drawdown and American put ( bottom ) with the parametersfrom Table 5. 4. Conclusions
In this paper we focus on the numerical pricing of the new derivative instrument - a π -option. Weadapted the Monte Carlo algorithm proposed by [Broadie and Glasserman (1997)] to price this new option.We focused on a specific parametrization of this option which we call the π -option on drawdown. Weobserved that this specific financial instrument is related to so-called relative maximum drawdown. Weobtained prices of the π -option on relative drawdown for the Microsoft Corporation stock with differentparameters in order to examine the influence of those parameters on option’s premium. Our next stepinvolved the analysis of two portfolios: first one based on a π -option on relative drawdown and thesecond one based on an American put. We used the Microsoft Corporation data as well as the WestTexas Intermediate crude oil futures dataset. It turned out that the portfolios behave in a completelydifferent manner. The value of the portfolio containing the American put was highly correlated with theunderlying’s price movements and thus had an unpredictable and volatile behavior. On the other hand,combining π -option on relative drawdown with the underlying asset not only ensures that the worth of theportfolio will not drop below the initial level, but it also allows us to take advantage of price upswings andto reduce the portfolio’s volatility at the same time. Similar analysis could be carried out for a geometricLévy process of asset price. One can also consider the regime-switching market. Z. Palmowski — T. Serafin
Figure 12.
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