Share Example 1 Comparisons
Method 200 000 simulations Bias (abs) Bias (%) CPU Time
Standard Monte Carlo ∆t= 0.01 0.0555575 -0.00896 1202.251 Standard Monte Carlo ∆t= 0.001 0.0445763 -0.00719 11277.391 Brownian Bridge Monte Carlo 0.0063537 -0.00102 12.313
True value: 6.19724
Share Example 2 Comparisons
Method 300 000 simulations Bias (abs) Bias (%) CPU Time
Standard Monte Carlo ∆t= 0.01 0.1138508 0.0171 2156.546 Standard Monte Carlo ∆t= 0.001 0.0861704 0.01287 46151.515 Brownian Bridge Monte Carlo 0.0136171 0.00203 23.204
True value: 6.69756
Table 5.4: Numerical comparison of the standard Monte Carlo method versus the Monte Carlo Brownian bridge method applied to pricing share prices, for different parameter sets.
From the tabled results it can be seen that the standard Monte Carlo method becomes more accurate as our time discretisations ∆t become smaller. However this significantly increases com- putation time. The proposed Brownian bridge algorithm is not only considerably faster but also more accurate as measured by the bias of the estimate. In both equity and CDS examples, both methods take more time to complete and are less accurate for the high volatility parameter set. The reason why the estimates are less accurate is because the variance increases with higher values of the parameter set, thus our estimates have a higher standard error. In order to increase accuracy one needs to increase the number of simulations. The Monte Carlo Brownian bridge method is superior in both accuracy and computation time for all scenarios tested.
5.5
Valuing Equity Options by a Monte Carlo Linear Re-
gression Approach
It is difficult to price an equity option in our structural model framework because equity options are in fact compound options. In order to price this compound option we have to evaluate a conditional expectation, which is a non-trivial problem within a Monte Carlo framework.
Under our structural model framework the price of an equity call option at time 0, with strike Kand maturity T∗< T, is given by30
ϕ0=EQ h e−rT∗ max(ST∗−K,0) i . (5.34)
The stock priceST∗ itself is an option under our structural model framework. From (5.30) and
(5.31)ST∗ can be written as ST∗ =η−1EQ h e−r(T−T∗)max(VT−bT∗,T,0)1{τ∗>T} ¯ ¯ ¯FT∗ i . (5.35)
Thus in order to evaluate (5.34) we need to evaluate the conditional expectation (5.35). The naive approach to obtain a Monte Carlo estimate for equation (5.34) would be to simulate a number of paths for the firm’s asset value from 0 until the option maturityT∗, and then for each of these paths simulate again a number of firm’s asset value paths from T∗ to T (called resimulation). This resimulation procedure requires simulating a large number of sample paths which causes high computation time. For example if we use 100 000 simulated paths until T∗ and then for each of these paths simulate another 100 000 paths from T∗ to T, we end up simulating 10 000 000 000 paths. Figure 5.2 illustrates how the simulated paths31 have to be simulated to evaluate the conditional expectation (5.35).
30In practice, following a bankruptcy announcement by a firm, trading in it’s underlying stock is suspended by
the exchange that lists the firm. When trading in the underlying stock has been halted, trading on the options is also halted. Equity option positions are usually then immediately closed out by the clearinghouse. For simplicity,
5.5. Valuing Equity Options by a Monte Carlo Linear Regression Approach 64 0 0.5 1 1.5 2 2.5 3 3.5 4 8.6 8.8 9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 Time (years)
LN(V) Log of the Firm Asset Value
Figure 5.2: Example of simulated sample paths of a resimulation procedure.
Longstaff & Schwartz (2001) provide an efficient alternative to obtain a Monte Carlo estimate for options that are functions of conditional expectations such as (5.34). They name it the Least Squares Monte Carlo (LSM) approach. For our problem, the LSM approach involves simulating one set a sample paths from 0 untilT. The conditional expectation (5.35) is estimated from the cross-sectional information of the simulated paths, at time T∗ by using least squares regression. This is done by regressing the discounted payoffs32 on a set of basis functions33 of the various values of the firm’s asset value at timeT∗.
5.5.1
The LSM Algorithm
For example, say we need to find a Monte Carlo estimate for the following expression
EQ£g¡EQ[h(V
T)|FT∗]
¢¤
. (5.36)
The LSM approach uses least squares regression to approximate the inner conditional expectation within expression (5.36). Let
f =EQ[h(VT)|FT∗].
Note that since V is a Markov process, f is function of VT∗ and not of any past realisations of
V. Letωi represent the ith sample path of V. If we simulate W number of paths for V, we will
have W number of realisations for f. We can write f as f(VT∗(ω)) to show its dependence on
the realisations ofVT∗(ω). Longstaff and Schwartz assumed thatf can be represented as a linear
combination of a countable set ofFT∗-measurable orthonormal34basis functions35. There are many
possible choices of basis functions. Here are a few types: Laguerre, Hermite, Legendre, Chebyshev and Jacobi polynomials. Section 5.5.2 has numerical comparisons of two different choices of basis function: Laguerre, and a simple order 3 polynomial function. Let{Ln}represent the general form
of our choice of basis functions. With this specification,f(VT∗) can be represented as
f(VT∗) =
∞
X
j=0
βjLj(VT∗),
where the coefficientsβj are constants.
To implement the LSM approach, we approximatef(VT∗) using the firstM <∞basis functions
and denote this truncated approximation byfM(VT∗). If we simulateW number of paths ofV from
we will assume that the equity options are closed out at the options maturity.
31The parameters used to generate Figure 5.2 are: V0= 10000,r= 0.05,σ= 0.2,λ= 1,µA=−0.05,σA= 0.2, T∗= 2 andT = 4. The number of simulated paths are 5 and 10 for the interval [0,2] and [2,4], respectively.
32For our case, the discounted payoffs aree−r(T−T∗)
max(VT−bT∗, T,0)1{τ∗>T}. 33This will be explained in Section 5.5.2.
34See Fraleigh & Beauregard (1995) for the definition of orthonormal functions.
35Some integrability conditions must hold for this assumption to hold: see Longstaff & Schwartz (2001) for further
5.5. Valuing Equity Options by a Monte Carlo Linear Regression Approach 65
0 untilT, we can then estimatefM(VT∗) by regressing realised values ofh(VT(ωi)) onfM(VT∗(ωi))
for eachi= 1, . . . , W. This involves finding the least squares estimates forβj,j= 1, . . . , M, which
we will denote as ˆβj. By substituting ˆβjforβjinfM(VT∗), we get a least squares approximation for
fM(VT∗), which we will denote by ˆfM(VT∗)36. The conditional expectation functionf is estimated
by ˆfM(VT∗). We then estimate (5.36) by averaging the realisationsg
h ˆ fM(VT∗(ωi)) i , i.e. W−1 W X i=1 g h ˆ fM(VT∗(ωi)) i .
In some cases it may be more efficient to use alternative regression techniques, such as weighted least squares or generalised least squares in estimating the conditional expectation function. For example, if the processV has volatility that is a function ofV, then the residuals from regression may be heteroskedastic. In this case these alternative regression techniques may have advantages37. Longstaff and Schwartz point out that numerical tests indicate that the results from the LSM algorithm are remarkably robust to the choice of basis functions. They also notice that few basis functions are needed to closely approximate the conditional expectation function (they useM = 3 for their numerical analysis). It is important to note the numerical implications of the choice basis functions. It could lead to computation overflows depending on the how large the realised values ofVT∗ are. This can be resolved by normalisingV.
For our case38,
h(VT) =e−r(T−T
∗)
max(VT −bT∗,T,0)1{τ∗>T}
and
g=e−rT∗max(h(VT)−K,0).
We calculate all our equity option pricesϕ0, using the LSM algorithm, with Laguerre polynomials as our choice of basis functions (withM = 3).
5.5.2
Numerical Comparison of Basis Functions
In this section we compare two choices of basis function to illustrate that this choice has a negligible impact on our numerical results. We chose a Laguerre polynomial
β0+β1e−V /2+β2e−V /2(1−V) +β3e−V /2(1−2V +V2/2), (5.37) and a simple order 3 polynomial
β0+β1V +β2V2+β3V3, (5.38) for our two choices of basis function. We then apply both these basis functions to price equity op- tions using the described Longstaff and Schwartz method.39. Figure 5.3 illustrates the convergence and variability of the results achieved from the two basis functions. The strike price isK = 3 for all the plots. The parameters used for Plot (a.1) and (b.1) are equivalent to those used in Section 5.4.9 for equity example 1, the equity option maturity isT∗= 0.25. Similarly Plots (a.2) and (b.2) have equivalent parameters to equity example 2, with option maturityT∗= 0.5.
It can be seen from Figure 5.3 that both basis functions produce very similar results. There is a significant difference when the high volatility parameter set is used. From Figure 5.3 it can be seen that the simple polynomial basis functions produces estimates with a higher variance. The low and high parameter set examples converge reasonably after the 500 000 and 600 000 mark respectively. The dotted lines represent the maximum and minimum values after the respective reasonable convergence mark.
36Note that ˆfM(VT∗) converges in probability tofM(VT∗) asW tends towards to infinity. 37See Longstaff & Schwartz (1995) for more details.
38Note that the functionh(VT) is also a function ofτ∗, which depends on other realisations of V. This indicator
can easily be evaluated within the LSM algorithm by noting if any simulated paths cross the barrier.
5.6. Estimation of parameters 66