Understanding Option Pricing
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We present a simple numerical method to find the optimal exercise boundary in an American put option. We formulate an intermediate function with the fixed free boundary that has Lipschitz character near optimal exercise boundary. Employing it, we can easily determine the optimal exercise boundary by solving a quadratic equation in time-recursive way.
We also present several numerical results which illustrate a comparison to other call option valuation table. The owner of a put call option has the right but no obligation to sell buy an underlying asset at the exercise price.
European options can be exercised only on the expiry date, while American options call option valuation table be exercised at any time until the expiry date. Closed-form solutions for the European options are call option valuation table in papers by Black and Scholes [ call option valuation table ] and Merton [ 2 ]. In the case of American options, because of the call option valuation table exercise possibility, the pricing problem leads to complications for analytic calculation.
McKean [ 3 ] and van Moerbeke [ 4 ] show that the valuation of American options constitutes a free boundary problem looking for a boundary changing in time to maturity, mostly called an optimal exercise boundary.
Hence, finance researchers have studied methods to quickly and accurately find the optimal exercise boundary. These methods are basically of two types, that is, analytical approximations such as those developed by Geske and Johnson [ 5 ], MacMillan call option valuation table 6 ], Barone-Adesi and Whaley [ 7 ], and Ju [ 8 ] and numerical methods such as those of Brennan and Schwartz [ 9 ], Hull and White [ 10 ], and Longstaff and Schwartz [ 11 ].
Zhu [ 12 ] finds an exact and explicit solution of the Black-Scholes equation for the valuation of American put options using Taylor series with infinitely many terms. His work is an excellent result for the valuation of American put options; however, it seems difficult to perform his solution numerically.
The infinite sum is likely to yield many computation errors. The majority of numerical methods for pricing American options, such as the finite difference method of Brennan and Schwartz [ 9 ], the binomial method of Cox et al. Their idea is to discretize the lifetime of an option and find its optimal exercise boundary backward in time. Since time-recursive ways yield repeated calculations for every time step, they require fast computation times and small pricing errors.
Also, front-fixing methods developed by Wu and Kwok [ 18 ] and Nielsen et al. A secant method developed by Call option valuation table et al. The main contribution of this paper is the development of a simple numerical method to find optimal exercise boundary in a time-recursive way.
Our result is motivated by the necessity for better understanding of the solution surface near optimal exercise boundary. We adopt the front-fixing transformation [ 18 ] to change the unknown free boundary to a known and fixed boundary.
We exploit an intermediate function with the fixed free boundary that has Lipschitz character which avoids the degeneracy of the solution surface near optimal exercise boundary as call option valuation table Kim et al.
Indeed, our function from the Black-Scholes equation and the boundary conditions transforms the surface above the exercise region onto call option valuation table new Lipschitz surface which forms a call option valuation table large angle with the hyperplane corresponding to the exercise region, thereby making the borderline more easily distinguishable see Figure 2. We use implicit scheme in the continuation region and apply extrapolation near optimal exercise boundary.
Call option valuation table we can determine the optimal exercise boundary by solving a quadratic equation in a call option valuation table way. Our method also provides fast and accurate results for calculating the optimal exercise boundary and pricing American put options.
The structure of the paper is as follows. Section 2 presents the model formulation. The intermediate function with the fixed free boundary to calculate the optimal exercise call option valuation table is presented in Section 3. Numerical results and comparative studies are presented in Section 4.
Section 5 summarizes the paper. Consider an American put option on an underlying asset stock with exercise price and expiration. In risk-neutral probability, an call option valuation table asset price is governed by the following stochastic differentiable equation: The payoff function of the put option at is defined as The call option valuation table of an American put option is denoted bywhere is the time to expiration for and is the underlying asset price for.
As seen in the previous article by McKean [ 3 ], the valuation of an American put option is considered the solution to a free boundary problem call option valuation table a parabolic PDE. We suppose that the optimal exercise call option valuation table is continuously nonincreasing with. The region where it is optimal to hold, generally called the continuation region, is defined asand the region where it is optimal to exercise, generally called the exercise region, is defined as.
Then, and uniquely solve where andand are defined by the infinitesimal generator and partial derivatives, respectively. Here, we assume for using the Black-Scholes equation at. See, for example, Karatzas and Shreve [ 25 ] for general reference. Figure 1 shows an illustration of an optimal exercise boundary with.
The two regions are separated by the optimal exercise boundary. From Figure 1 it is necessary that must satisfy.
As Chockalingam and Muthuraman [ 26 ] point out, the continuation and exercise regions are determined by which term in the HJB equation is tight. Their method requires iterations till convergence of the boundaries. However, we emphasize that there is no iteration in our method using a square root transformation. Refer to Chockalingam and Muthuraman [ 26 ], and Pham [ 27 ]. A front-fixing method, proposed in Wu and Call option valuation table [ 18 ], uses a change in variables to transform the free boundary problem into a nonlinear problem on a fixed domain.
The following transformation of state variable serves for such a purpose: They derive the equation and the boundary conditions with respect to as follows: Note that is a monotonically decreasing function of with a nontrivial asymptotic limit as follows: Especially, plugging into 6we have the asymptotically optimal exercise boundary as follows: Note that transformation 4 is valid because holds for all.
Refer to Kim [ 28 ]. In this section, we present an intermediate function with the fixed free boundary and can determine the optimal exercise boundary by solving a quadratic equation in a time-recursive way. Under the assumption of the Black-Scholes model, the time call option valuation table optimal call option valuation table can be shown to be the first hitting time of a boundary, the optimal exercise boundary, in the plane consisting of pairs of the underlying asset price and the time to expiration.
Namely, the price curve of an American put option touches the line representing the intrinsic value tangentially.
With a careful examination of the solution surface near optimal exercise boundary, we find a Lipschitz surface which avoids the degeneracy of the solution surface near optimal exercise boundary. To find the optimal exercise boundary, we present an intermediate function with the fixed free boundary that has Lipschitz character near optimal exercise boundary as follows: The transformed function provides that the solution surface in is a horizontal plane, and it is an inclined plain in.
Namely, this function forms a sufficiently large angle with the hyperplane corresponding to the exercise region, thereby making the borderline more easily distinguishable. Hence, we have Figure 2 a shows that is transformed toand Figure 2 b is a magnified view of the optimal exercise boundary. We find the intermediate function with the fixed free boundary to decide the optimal exercise boundary by the Taylor series.
Fromwe obtain the following relations near optimal exercise boundary: Plugging 11 into 5we obtain and then call option valuation table get. More precisely, we have withwhere. Then, we have for some constant because is Lipschitz and a natural candidate for computation in. We obtain an angle between exercise surface and surface such that for some constants and.
We also calculate the partial derivative with respect to in 5 as follows: Hence, plugging call option valuation table into 14we get where and. From 6 we easily show that and are bounded parameters such that for some negative constants and andrespectively. Using the similar arguments, we can obtain the following equations near optimal exercise boundary as follows: Hence, plugging 18 into 17we get. Furthermore, we recognize that is analytic up call option valuation table the optimal exercise boundary and is locally analytic.
Hence, the approximation for at can be written as We introduce the equilibrium call option valuation table which call option valuation table us to adjust the location of optimal exercise boundary in a mesh size. So, plugging, and into 20we obtain as follows: We rewrite 21 with respect to as follows: From 24 we have where respectively.
Hence, we rewrite 25 call option valuation table respect to as follows: When the initial values are given by transformed price of the American put option and optimal exercise boundary atwe can determine optimal exercise boundary at using More importantly, for updating the optimal exercise boundary our method dose not include iteration until sufficient accuracy is obtained. So, we repeat the previously mentioned process until and obtain the optimal exercise boundary in a time-recursive way.
In this section, we provide numerical examples to illustrate our method. We also make runtimes and computation errors compared with the results obtained by other numerical methods such as the binomial method Binomial developed by Cox et al.
A finite difference method with Crank-Nicolson scheme is proposed for our method. The benchmark results are obtained using the Binomial with time steps, and we consider these results to be the exact values of the American put options. In Figure 3we find a numerical optimization. Table 1 also shows the results of the optimal exercise boundary and the values of American put options with safety parameters. One can see from Table 1 that the optimal exercise boundary monotonically increases as the value of increases, but the value of American put options monotonically decreases when increases.
They are so gradual that they are not very susceptible to change in the value of. Table 2 reports the values of the American put options for the specific parameter set associated with the table. In Figure 4 and Table 3we take the parameter values used in Figure 3 except for the discrete mesh and plot runtimes and computational errors compared with various methods.
As is shown in Table 2Figure 4and Table 3although four different methods have similar values of the Call option valuation table put option, our method is computationally faster and more accurate than other methods.
Especially, Figure 4 and Table 3 show the numerical convergence of our method. So our method is superior to the call option valuation table in accuracy and computational efficiency. The front-fixing method suggested by Wu and Kwok [ 18 ] shows a degeneracy near optimal exercise boundary, while our method adopts a square root function to avoid the quadratic behavior of solution surface that causes degeneracy.