Pricing of a Best Of - Whale Option: a case study

Transcription

1 Pricing of a Best Of - Whale Option: a case study - DRAFT - Emiliano Laruccia Gestioni Bilanciate Sanpaolo AM Luxembourg Introduction The aim of this paper is to provide a pricing model applicable to a particular exotic option, that we will call Best Of Whale (BOW). This option returns a payoff that is linked to the highest performance of two baskets, named Growth Portfolio (GP) and Conservative Portfolio (AP). Since the payoff at maturity is not standard (it is concave, hence the whale naming) and Asiatic, there s no closed form to price this derivatives (even it s really difficult to find out an approximation). For this reason we will approach the pricing via Monte Carlo simulation. The paper is arranged as follow: the first section describes the option, the second the pricing model and parameters. Section three tries to compute option greeks and section four approaches the problem of risk assessment for a portfolio that includes Bestof-Whale options. 1

2 1. The Option The general terms of the options 1 are described in annex A. The following example will clarify how the payoff is determined for a 3 - year maturity option. Let s assume the first strike date to be the 2/01/00. Table 1 summarizes the different closing levels on quarterly valuation dates and their arithmetic average. Table1: payoff determination SX5E SPX EMTXGRT First strike Date (02/01/00) Valuation Dates 02/04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /12/ Average (SAverage) SAverage/S Once determined the arithmetic average, we can compute the values of the two baskets: PA = 35% * %* %* = PC = 15% * %* %* = Hence, the best strategy is PC: the option payoff is determined as ( )/ = 1.97%. The denominator of the formula is equal to the final value of the index, instead of the initial value as in a plain vanilla option. This gives a concave profile ( whale ) to the payoff, as shown in figure 1. Compared to a standard payoff, the whale option is cheaper and performs better returns in cases is not deep in the money. 1 Even the contract is one-off, we will distinguish between Option 1 (the 0 to 4 years maturity) and Option 2 (the 4 to 7 yrs maturity). 2

3 Payoff 45% Figure1: payoff of whale option 40% 35% 30% 25% 20% 15% plain vanilla whale 10% 5% 0% St 2. Pricing Model In order to price this derivative, we will approach a Monte Carlo Simulation. In particular, we have to generate a multivariate random variable that mimics the behaviour of the three assets. Let then assume the returns of the three assets follow a multivariate normal distribution. This implies that the random vector X=(X 1,...,X n ) that satisfies the following equivalent conditions: a) every linear combination Y=a 1 X a n X n is normally distributed; b) there is a random vector Z=(Z 1,...,Z m ), whose components are independent standard normal random variables, a vector µ=(µ 1,...,µ n ) and an n-by-m matrix A such that X=A Z + µ.; c) there is a vector µ=(µ 1,...,µ n ) and a symmetric, positive semidefinite matrix Γ such that X has density fx(x 1,...,x n )dx 1...dx n = (det(2πγ))-1/2 exp ½((X-µ)TΓ-1(X-µ)) dx 1...dx n d) there is a vector µ and a symmetric, positive definite matrix Γ such that the characteristic function of X is: φx(u)=exp(iµtu-½utγu) The vector µ in these conditions is the expected value of X and the matrix Γ=ATA is the covariance matrix of the components Xi. It is important to point out that the covariance matrix must be allowed to be singular. 3

4 In order to generate the vector X then we need the following input: a) the vector µ b) the covariance matrix Γ We will now analyze how these parameters are determined, focusing on our case, where the variables are: - Eurostoxx 50 (SX5E) - Standard and Poor s 500 (SPX) - Euro MTS Global Index (EURMTX) In a risk neutral world, the drift of the process should be determined as the difference between the risk free rate and the dividend yield. The first member can be approximated by the Euro/US zero rate at the option maturity, while the second can be easily obtained through dividing the annulalized dividend by the index current price. In addition, as the basket includes an asset measured in USD, even thought the contract is settled in EUR, we have to correct the SPX drift for the factor ρσ spx σ e (see Hull p. 499), where ρ is the correlation between the Euro Dollar exchange rate (e.r) and the SPSX, and σ e is the standard deviation of the e.r. Finally, the drift for the third asset is the risk free rate correspondent to the duration of the Euro MTS Index R EUR (D) d 1 = R EUR (T) - DY SX5E d 2 = R USD (T) - DY SPX + ρσ spx σ e d 3 = R EUR (D) The covariance matrix can be estimated using the historical returns of the three assets. Alternatively, it can be treated as the product of the correlation matrix Ρ and the variance vector σ. The first component is more stable and can be computed by considering the past returns, while the second can be approximated by the implied volatility on at the money options on the different underlying 2. Once the parameters are set, the first step of the simulation is to generate a random vector X=(X1,...,Xn) of returns and from that vector the price of the three securities. In order to complete the simulation, it is necessary to generate N (>1000) random vectors X and for each of them compute the payoff at maturity: The option price will be equal to the present value of the expected payoff: C = Z* E(P) The following examples will clarify the process. 2 For the EUROMTS, we will consider historical volatility. This second approach is more market oriented, since it considers data implied on traded options. 4

5 Example 1: Best of 0 to 4 th year The first example considers the first option (4 years maturity). Let s now populate the parameters of Monte Carlo Simulation, starting from drifts 3. d 1 = R EUR (4) - DY SX5E = 3.27% % = 0.47% d 2 = R USD (4) - DY SPX + ρσ spx σ e = 3.65% % = 1.44% d 3 = R EUR ( ) = 3.27% In particular: R EUR (4) R EUR (5.69) R USD (4) DY SX5E DY Spx ρ σ e σ SPX = Euro zero rate 4 yrs maturity = Euro zero rate 5.69 yrs maturity = USD zero rate 4 yrs maturity = Dividend yield on SXE Index (source Bloomberg) = Dividend yield on SPX Index (source Bloomberg) = Correlation between EURUSD and SPX (historical, last 3 yrs weekly data) = E.r. implied 1 year volatility (EUUSV1Y Index) = Spot implied volatility on SPX traded option Correlations are estimated considering historical returns (last 3 yrs weekly data), while standard deviations are implied volatilities. 1 Ρ = σ = [ ] The first step of the simulation is to generate a random vector X=(X 1,...,Xn) of returns and from that vector the price of the three securities as shown in Table 2. Table2: random path for the three securities SX5E SPX EMTXGRT First strike Date Valuation Dates T T T Average (SAverage) SAverage/S Data 1/10/04 4 That is the Duration of the EURMTX 5

6 We then compute the two baskets final value: BG = % % % = 1.57 BC = % % % = 1.30 The best strategy for this run is BC and the option payoff is equal to (1.57-1/1.57) = 36.44%. In order to complete the simulation, random vectors are generated and for each of them computed the payoff at maturity. The option price will equal to the present value of the expected payoff: C 1 Z* E(P) = = 6.90 Example 2: Best of 4 th to 7 th year The first example considers the second (3 years maturity). The only difference with option 1 is represented by input parameters. In particular, drifts have to consider forward interest rates (4 to 7 years) and volatilities have to be corrected, as forward volatilities are higher than spot vola. Let then assume d 1 = R EUR (4 to 7) - DY SX5E = 4.51% % = 1.71% d 2 = R USD (4 to 7) - DY SPX + ρσ spx σ e = 5.59% % = 3.39% d 3 = R EUR ( ) = 3.27% In particular: R EUR (4) R USD (4) R EUR (5.69) = Euro forward rate 4 to 7 yrs maturity = USD forward rate 4 to 7 yrs maturity = Euro zero rate 5.69 yrs maturity σ = [ ] With these parameters and adjusting the discount factor (0.77), the option price becomes Option sensitivity to input parameters In this chapter we will try to measure the sensitivities (greeks) of the option price to the different parameters. In particular, as there s not a closed form to determine the first derivative, we will analyze how the option price changes in respect to a 1% of changes in the parameters (for simplicity we will consider only option 1). Let s start with the underlying assets (see table 3): note that as the strike price for option 2 is forward fixed, deltas are null. The highest impact on option prices is played by EURMTX. 5 That is the Duration of the EURMTX 6

7 Table 3: Option sensitivities to securities prices Changes in Option price for 1% Delta 6 change in price Delta SX5E 2.19% 0.15 Delta SPX 2.19% 0.15 Delta EURMTX 5.22% 0.35 The impacts of changes in volatilities are lower, except for the volatility of EURMTX (see table 4). An increase in the EURUSD volatilities has a negative effect, since it reduces the drift fot SPX (because of negative correlation) Table 4: Option sensitivities to volatilities Changes in Option price for 1% change in volatility Option 1 Option 2 σ SX5E 0.80% 0.57% σ SPX 0.66% 0.48% σ EURMTX 2.20% 1.46% σ ΕUR -0.22% -0.21% Price sensitivities to small correlations changes are negligible (see table 5) Table 5: Option sensitivities to correlations Change in Option price for 1% change in correlation Option 1 Option 2 ρ SX5E /SPX 0.06% 0.07% ρ SX5E/ EURMTX 0.01% 0.01% ρ SX5E/ EURMTX % % ρ SPX/ EUR 0.09% 0.09% Finally, we can compute the option sensitivity to interest rates. As expected, an increase in interest rates produces an higher drift (for the euro interest rate the effect is higher than the decrease of the discount factor). Table 6: Option sensitivities to interest rates Change in Option price for 1% change in correlation Option 1 Option 2 R EUR 15.96% 9.91% R USD 4.74% 3.14% 6 measured as C/ S 7

8 4. Volatility of a portfolio that includes Best of-whale Option In order to value the volatility of a portfolio tha includes BOW option, such the Doppia Opportunità Dic 2001 it is possible to proceed as follows: given the amount of money available for the purchase of the options (0 4 years, 4 7 years) calculate the weights of each security: bond portfolio, best of whale options underlying indexes (narrowing the analysis to the Growth portfolio with the aim to asses just the worst case scenario) and cash; estimate the historical volatility for the above mentioned indexes; construct the covariance matrix. Let s assume that the sum available for the two options deal is equal to 7.87% of the NAV (as of 4th October 2004), so the percentage of government bond required for capital protection, certain coupons and management fees coverage is simply equal to 100% % = 92.13%. Assessing the implied weights of the indexes underlying the options is more difficult. The first step is to determine the equity participation for both options, that is the number of options that can be acquired. Assuming the prices we obtained are tradable, this participation is equal to 7.87/( ) = 57.45% In order to determine the underlying indexes implied weights, we need the option deltas (as determined in chapter 3) and the options participation (57.45%, as defined above). Then, the implied weight of i-th index is calculated applying the formula: iw i = p * δ i where w i is the implied weight of the i-th index, p is the option participation and di is the estimated delta of i-th index. Applying the known data we obtain: iw SX5E 8.62% iw SPX 8.62% iw EURMTX 20.11% Finally, reasonably assuming that the forward starting (4 7 years) option has deltas equal to zero and his price is affected just by forward interest rate and forward volatility, it should be compared to cash investment, thus obtaining the following portfolio. Government bonds 92.13% Cash 3.92% Eurostoxx % S&P % EuroMTS Global Index 20.11% Total % 8

9 The estimation of the historical underlying indexes volatility (JPM EMU 5 7 for government bonds, JPM EMU 3 Months Cash for the cash, JPM EMU All Maturities for EuroMTS Global Index, Eurostoxx50 and S&P500) and the resulting covariance matrix eventually allows the evaluation of the fund volatility, equal to 4.33%. 9