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            Financial Quantitative Algorithms

Is it never optimal to exercise non-dividend American calls ?

 .... and how can we make profits in case it is not?

 

    It is intuitive to think that American style options are more expensive than the European counterpart, since the option holder has the freedom to exercise the option during the option life time.

    However, in the case of non-dividend paying calls, it is widely accepted [1] that is never optimal to exercise the American style. Moreover, it can be demonstrated that the price of a non-dividend paying American call option is the same as the European counterpart.

    This result might seem rather counter intuitive at first sight, and is analyzed in this work by means of a comparison with results obtained with the Monte Carlo (MC) technique.

    When the early exercise (Boolean) function is evaluated in each trajectory backwards in time, the results obtained with MC coincide reasonably with the Binary Trees technique and the Barone Adesi Whaley approximation. When 1) this early exercise evaluation is performed forwards in time or 2) achieving the optimal discounted price for the American call option, substantial differences are found, in a fashion that the classical results underestimate the price by a range that could reach from 25% to 100%.

 

Pricing American call option

 

     In general there are no closed expression to evaluate the price of American style options. An exception to that is the Barone Adesi Whaley approximation valid for American calls and puts. In this case. the price obtained for American calls is in general higher than the European.

    One of the most popular methods to price options is the Binary Tree method, due to the simplicity and speed. Here I will proceed to overview it briefly, in order to analyze the conditions on early exercise.

 

Binary Trees technique

 

    In the Binary Trees technique we have a parent node (i,k) (k indicates node number and i time-layer) with known underlying price S0 and unknown call option price c0.

    The parent node has two children: node (i+1,k) with a underlying price Su = S0 exp(+σ √(δt)), and node (i+1,k-1) with Sd = S0 exp(-σ √(δt)), where σ is the implied volatility and √(δt)=√(T/N) is the square root of the time step, T is the maturity of the option subdivided in N steps.

 

 

Fig. 1: Typical configuration of nodes: the node (i,k) is the parent, with two children nodes (i+1,k) and (i+1,k-1). The price of the underlying is known in the three nodes, the price of the call option is known in the children. We need to determine the price of the option in the parent node.

 

    The call option prices at the children nodes is given by:

            cu,d = max(Su,d-K,0)

    In order to calculate the price of the option c0 , we need to compare two functions:

1) F = max(S0-K,0),

2) G = exp(-r δt) [q cu + (1-q) cd],

and pick the one with the highest price. In the case that F>G, it is said that the option is exercised earlier and the price obtained is higher than the European version, while the case G>F corresponds to the case of an European style option.

    In order to get a more intuitive view on the condition of early exercise, we can approximate the function G in power series of δt, since T/N is typically a small number.

     By definition we know that Su > S0 > Sd , but the relative position of the strike K will determine the condition of early exercise.

    Case a) Su > S0 > Sd > K :

            F ≈ (S0-K)

            G ≈ (S0-K) + S0(r √δt/σ - r δt ) + K r δt + O(δt3/2)

        So unless the unrealistic case σ ≥ 1/√δt, and even when the option is deep in the money, it should be not exercised earlier.

 

    Case b) Su > S0 > K > Sd :

            F ≈ (S0-K)

            G ≈ (1 - r δt) (S0 - K + S0 r δt) (1/2+r √δt/σ)

               ≈ (S0-K)/2

        The option is at the money in the money, and it can happen that optimal to exercise early, since G<F.

 

    Case c) Su > K > S0 > Sd

            F = 0

            G > 0

        The option is at the money, and it is not convenient to exercise early in this case. The price obtained c0 corresponds to the European case.

 

    d) Finally, when K > Su > S0 > Sd we obtain:

            F = 0

            G = 0

        The Option is deeply out of the money, and is not convenient to exercise early.

 

    Although out of the four cases considered, we have found that in one of them is possible to exercise early, this importance is only apparent. In Practice, the majority of the nodes will have prices corresponding to cases a) and d), while the cases b) and c) correspond only to border conditions in one dimension (that is negligible when N is large).

    Under this framework we can say that the condition of early exercise in American call options is almost never fulfilled, and thus the price of an American call is the same as the corresponding European.

 

Limitations of the Binary Trees Framework

 

    In the Binary trees technique the underling price at the most parent node is typically the Spot price, and the rest of the nodes are filled by applying the operators u = exp(+σ √δt) and d = exp(-σ √δt), in a sequence like: S(i,k)  = uud ... dududu. It is very interesting to note that the price at an intermediate node will never overcome the highest price at the last layer of nodes: the highest price considered at maturity.

    Since one picture is worth more than thousands words, let's illustrate this case by considering that the spot price is 50$, and we generate 400 nodes filled with prices between the highest price 100$ and a exponential small value. If we will try to plot a sequence of underling prices S as a function of time, the prices will lay inside some cone horizon, like the one depicted in Fig. 2. That means that by construction, trajectories like the red line in Fig. 2, which shows peak in the Underlying price at an intermediate time, higher than the highest price considered at maturity, are paths not considered in the Binary Tree framework. 

    Though this is not a very common situation, is a very plausible case that cannot be considered with this technique.

    Now we can consider the MC technique, where the last problem is not present and we can compare results obtained with both methods. Particularly interesting will be to consider the situation of early exercise.

 

      Fig. 2: A possible time dependence of the Underlying price that can

not be considered with the Binary/Ternary trees scheme.

 

The Monte Carlo method

 

    The solution of the stochastic differential equation:

        dS = S(r dt + r σ dWt) ,

where dWt are independent identically distributed Gaussian random variables with E(dWt)=0 and E(dWt2)=dt, is given by the expression:

        S = S0 exp((r-σ2/2)t + σ Wt).

    Trajectories for S are particularly straight forward to generate with MC, where Wt+dt=Wt+dWt+dt , and dWt+dt is generated with the Polar Marsaglia algorithm, for example.

    In order to compare the two techniques, we are going to consider an example, where the time scale is one year and the underling spot price is S0 =100, the strike K=100, the short-term risk-free interest rate r=0.05, the dividends rate q=0, an implied volatility σ=0.2 and a time to maturity T=1.

    Since the Binary technique involves an initial proposition of prices at maturity and then a "backwards" propagation of underling and option prices to earlier time nodes, we will consider the MC simulation in the following fashion:

     1) Complete trajectories of S will be generated from present time (t=0) to maturity (T=t), where ST is the value of S at maturity.

    2) The exercise valuation will be performed backwards in time as:

 

********* Early Exercise Evaluation 1 ***********

    From t = T until t = 0, decrement t

        If  G = exp(-r (T-t)) max(ST - K,0) > F = max(St -K,0)

            exercise early

        else

            let the option live

    End

 

    Using this scheme to evaluate the call American option price, we performed MC simulations, averaging over 40.000 trajectories and subdividing the time interval in 2.000 steps.

    In Fig. 3 it is shown the price of the American call option as a function of the Spot price S0. The yellow triangles were obtained using the Barone Adesi Whaley Approximation (Series 3), the pink squares (Series 2) corresponds to the Binary Trees method (considering 1000 nodes) and the black diamonds (Series 1) were obtained with the MC technique as explained above.

    Since all the results are in agreement with a reasonable precision, we conclude that the MC algorithm is able to reproduce satisfactorily the results obtained with the two other techniques.

Fig. 3: Pricing American calls: Series 1: MC simulation evaluating the early exercise backwards in time: from maturity to present date (#simulations= 40.000, #Points/curve=2000, Strong convergence order 2).

Series 2: Binary trees scheme with 10.000 nodes.

Series 3: Barone Adesi Whaley approximation.

 

In Fig. 4 we observe around 50 different trajectories of the Underlying price vs. time generated by the MC technique. Notice that there are many curves under S=100$: the price is to low and is not convenient to exercise (case d) ). For S > 100 we can see few shorter trajectories instead: it is convenient in general to exercise only when the time is very close to maturity, when the exercise function is evaluated backwards in time. The cyan curve over S=100 has have a very sharp rise soon the option started to exist, in a way that it was exercised with a very short stopping time.  

Fig. 4: Trajectories of the Underlying price vs. time generated by the MC technique, plotted until stopping time, with early exercise evaluated backwards in time. Many curves under S=100$: the price is to low and is not convenient to exercise. Curves in the region S > 100: few shorter trajectories, it is convenient to exercise close to or at maturity. The cyan curve over S=100 has have a very sharp rise and was exercised with very short stopping time.

 

Forward and Optimal evaluations of the American call price

 

    Notice that the backwards evaluation of the exercise condition is an artifact of the Binary Trees method. A normal holder of an American option will typically "feel" that the time is flowing from present to maturity and the decision of exercise will be evaluated facing expectations for the future

    We can very simply adapt the MC algorithm to evaluate the exercise function forwards in time:

 

********* Early Exercise Evaluation 2 ***********

    From t = 0 until t = T, increment t

        If  G = exp(-r (T-t)) max(ST - K,0) > F = max(St -K,0)

            exercise early

        else

            let the option live

    End

 

    We will consider too, the American call payoff definition given in the Rennie and Baxter book [2], that says that the option holder should be charged the maximum (discounted) price of the option:

 

********* Early Exercise Evaluation 3 ***********

    Analyze the discounted price of the option all along each trajectory

    The maximum value is the price of the American call

 

    The Fig. 5 shows the American call option prices calculated using MC simulation, under the same conditions as in Fig. 4. The black diamonds (Series 1) where generated using the backwards early exercise. The pink squares (Series 2) were calculated using the early exercise evaluation forwards in time, and the yellow triangles (Series 3) were evaluated considering the maximum discounted value of the option at each trajectory.

    It is remarkable the difference in the prices, beyond the statistical error. Between Series 3 and Series 1 there is factor of almost 2, and Series 2 is almost a 25% higher than Series 1.

Fig. 5: Series 3: MC simulation exercising the option at the most expensive price (optimal strategy). Series 2: MC simulation exercising the option forwards in time (from present date to maturity). Series 1: MC simulation exercising the option backwards in time (from maturity to present date).

 

The Fig. 6 is equivalent to Fig. 4, but using the forwards evaluation of early exercise. (See caption).

 

Fig. 6 shows 50 different trajectories of the Underlying price vs. time generated by the MC technique, when the early exercise evaluation is performed forwards in time. There are many curves over S=100$, with stopping times distributed more or less uniformly. For S < 100 we can see fewer number of trajectories: the underling was not performing well and soon after being at the money is exercised.

 

And finally, in Fig. 7 we can see some trajectories for the optimal case, where the discounted price of the option is a maximum along a trajectory. The paths are plotted from t=0 until the maximum discounted American call option price is realized.

 

Fig. 7 Trajectories of the Underlying price vs. time generated by the MC technique, plotted from present date t=0 until the maximum discounted option price is realized.

 

 

Conclusions

 

    The statements "is never optimal to exercise a non-dividend paying American call option" and "the price of a non-dividend paying American call option is the same as a the corresponding European" are only valid within the approximations introduced by Binary Tree like schemes.

    The Binary Trees method is forced to evaluate the early exercise function backwards in time, in fact that is counter intuitive since the option holder has to make a decision always facing the future.

    In this work it was showed that the Monte Carlo technique is able to reproduce the results of the Barone Adesi Whaley and Binary Trees approximation with remarkable precision, when the early exercise is evaluated backwards in time.

    Using the forwards and optimal evaluations, the results differ from the classical result in 100% and 25%, respectively.

    Taking into account how widely spread are the preconceptions that non dividend paying American call options should have the same price as the European counterpart, one is tempted to explore the following strategy:

    Explore the market for American calls, with spot and strike prices on the same order and a reasonable volatility. An unaware option writer could charge a price close to the European version, in that case purchase it, wait until it is reasonable in the money  an exercise. 

 

References:

[1] "Options, Futures, and Other Derivatives", John C. Hull.

    1) Bloomberg

    2) Yahoo

    3) www.ivolatility.com/calc/

    4) http://janroman.dhis.org/calc/Binomial2.php

    5) "Financial Recipes", Berndt Arne Oedegard.

    6) "Numerical Solution of Sde Through Computer Experiments", Platen, Kloeden and Schurz.

 

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