The binomial option pricing model – part 5

This is post #5 on the binomial option pricing model. The purpose of post #5:

    Post #5: Tweak the binomial European option pricing methodology to work for American options.

The work in this post is heavily relying on the work in the binomial option pricing model for European options (multiperiod, one-period and more on one-period).

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Valuing American options

The binomial tree approach of pricing options can also be used to price American options. Recall that a European option can be exercised only at expiration. An American option is one that can be exercised at any time during the life of the option. This means that in a binomial tree, an European option can be exercised only at the final nodes while an American option can be exercised at any node if it is profitable to do so. For an American option, the option value at a given node is obtained by comparing the exercise value (i.e. the value of the option if it is exercised at that node) and the intrinsic value (the value of the option resulting from the binomial model calculation). Thus for an American option, the option value at each node is simply the greater of the exercise value and the intrinsic value. The following 3-step process summarizes the approach in pricing an American option.

    Pricing an American option using a multi-period binomial tree

  1. Build a binomial tree.
  2. Calculate the option values at the last nodes in the tree. For a call, the option value at the end of the tree is either the stock price less the strike price or $0, whichever is greater. For a put, the option value at the end of the tree is either the strike price less the stock price or $0, whichever is greater.
  3. Starting from the option values at the final nodes, work backward to calculate the option value at earlier nodes. The option value at the first node is the price of the option. Keep in mind at each node, the option value is either the intrinsic value (the value calculated using the binomial pricing method) or the exercise value, whichever is the greater.

The three-step process is almost identical to the process of European option valuation discussed in binomial model post #4. The tweak is in Step 3, allowing for early exercise at any node whenever it is advantageous to do so (for the option holder).

In Step 3, we use risk-neutral pricing. The idea is that the option value at each node is the weighted average of the option values in the later two nodes and then discounted at the risk-free interest. The two option values (at the up node and at the down node) are weigted by the risk-neutral probabilities as follows:

    \displaystyle C^*=e^{-r h} \ [p^* \times (C^*)_u + (1-p^*) \times (C^*)_d] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

where C^* is the option value at a given node, and (C^*)_u is the option value at the up move and (C^*)_d is the option value at the down move that follow the node at C^*. The risk-neutral probability p^* for the up move is:

    \displaystyle p^*=\frac{e^{(r-\delta) h}-d}{u-d}

    \displaystyle d=e^{(r-\delta) h - \sigma \sqrt{h}}

    \displaystyle u=e^{(r-\delta) h + \sigma \sqrt{h}}

where r is the annual risk-free interest rate, h is the length (in years) of a period in the binomial tree, u and d are the stock price movement factors and \sigma is the stock price volatility factor. The risk neutral pricing is discussed in binomial model post # 2.

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Examples

The binomial tree pricing process produces more accurate results when the option period is broken up into many binomial periods. Thus the binomial pricing model is best implemented in computer. In order to make a binomial tree a more realistic model for early exercise, it is critical for a binomial tree to have many periods when pricing American options. Thus the examples given here are only for illustration purpose.

Example 1
A 6-month American put option has the following characteristics:

  • Initial stock price is $40.
  • Strike price of the put option is $45.
  • The stock is non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 5%.

Price this put option with a 3-period binomial tree. Compare the American option with the European but otherwise identical put option.

Compare the following two binomial trees. The first one is for the American put option. The second one is for the otherwise identical European put option.

    \text{ }

    Example 1 – the binomial tree and option values – American put
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 59.22258163 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 0 \\        \text{ } & \text{ } & S_{uu}=\$ 51.96108614   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 0   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 46.3561487 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 0 \\     \text{ } & S_u=\$ 45.58994896  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 2.41285153  & \text{ }    & \text{ } \\     S=\$ 40 &  \text{ } & S_{ud}=S_{du}=\$ 40.67225322    & \text{ } \\   C=\$ 6.024433917 &  \text{ } & C_{ud}=\$ 4.585624746    & \text{ } \\    \text{ } & S_d=\$ 35.68528077 \text{ }   &  \text{ } \\   \text{ } & \mathbf{C_d=\$ 9.314719233} \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 36.28501939 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 8.714980615 \\      \text{ } & \text{ } & S_{dd}=\$ 31.83598158   & \text{ } \\     \text{ } & \text{ } & \mathbf{C_{dd}=\$ 13.16401842}   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 28.40189853 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 16.59810147 \\      \end{array}

    \text{ }

    \text{ }

    Example 1 – the binomial tree and option values – European put
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 59.22258163 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 0 \\        \text{ } & \text{ } & S_{uu}=\$ 51.96108614   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 0   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 46.3561487 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 0 \\     \text{ } & S_u=\$ 45.58994896  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 2.41285153  & \text{ }    & \text{ } \\     S=\$ 40 &  \text{ } & S_{ud}=S_{du}=\$ 40.67225322    & \text{ } \\   C=\$ 5.787711996 &  \text{ } & C_{ud}=\$ 4.585624746    & \text{ } \\    \text{ } & S_d=\$ 35.68528077 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 8.864829182 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 36.28501939 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 8.714980615 \\      \text{ } & \text{ } & S_{dd}=\$ 31.83598158   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 12.79057658   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 28.40189853 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 16.59810147 \\      \end{array}

    \text{ }

At the node where the stock price is S_{dd}= 31.83598158, the option value for the American option is in bold and is greater than the option value in the tree for the European option. This is due to the fact that early exercise is possible in the tree for the American option. When early exercise is possible, the put option value at that node is $45 – $31.83598158 = $13.16401842. As a result of the early exercise in one node, the price of the American put option is $6.0044 whereas the price of the option if early exercise is not permitted is $5.7877.

Example 2
Consider Example 3 in the binomial model post #4. That example is to price a 6-month European call option in a 3-period binomial tree. The following shows the specifics of this call option.

  • Initial stock price is $60.
  • Strike price of the call option is $55.
  • The stock is non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 4%.

What is the price if early exercise is possible? The following is the binomial tree for the European call option from Example 3 in the previous post.

    \text{ }

    Example 2 – the binomial tree and option values – European call
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 88.39081 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 33.39081 \\        \text{ } & \text{ } & S_{uu}=\$ 77.68226   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 23.04770   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 69.18742 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 14.19742 \\     \text{ } & S_u=\$ 68.27104  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 14.23394  & \text{ }    & \text{ } \\     S=\$ 60 &  \text{ } & S_{ud}=S_{du}=\$ 60.80536    & \text{ } \\   C=\$ 8.26318 &  \text{ } & C_{ud}=\$ 6.61560    & \text{ } \\    \text{ } & S_d=\$ 53.43878 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 3.08486 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 54.15607 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 0 \\      \text{ } & \text{ } & S_{dd}=\$ 47.59506   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 42.39037 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 0 \\      \end{array}

    \text{ }

Observe that early exercise is optimal at none of the nodes in this binomial tree. In this example, the American call option and the European call option have the same price (when suing a 3-period binomial tree).

Example 3
A 2-year American call option has the following characteristics:

  • Initial stock price is $75.
  • Strike price of the call option is $72.
  • The stock pays continuous dividends at the annual rate of \delta= 0.06.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 3%.

Price this call option in a 3-period binomial tree. Also compute the price for the European call with the same characteristics.

    \text{ }

    Example 3 – the binomial tree and option values – American call
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 147.2799263 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 75.27992628 \\        \text{ } & \text{ } & S_{uu}=\$ 117.6114109   & \text{ } \\   \text{ } & \text{ } & \mathbf{C_{uu}=\$ 45.61141089}   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 90.2367785 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 18.2367785 \\     \text{ } & S_u=\$ 93.91941129  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 23.94529115  & \text{ }    & \text{ } \\     S=\$ 60 &  \text{ } & S_{ud}=S_{du}=\$ 72.05920794    & \text{ } \\   C=\$ 12.16262618 &  \text{ } & C_{ud}=\$ 7.848617166    & \text{ } \\    \text{ } & S_d=\$ 57.54338237 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 3.377832957 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 55.28707407 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 0 \\      \text{ } & \text{ } & S_{dd}=\$ 44.14987805   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 33.87377752 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 0 \\      \end{array}

    \text{ }

Observe that early exercise is optimal at the node where the stock price is S_{uu}= $117.6114109. If early exercise is not allowed, the following is the binomial tree.

    \text{ }

    Example 3 – the binomial tree and option values – European call
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 147.2799263 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 75.27992628 \\        \text{ } & \text{ } & S_{uu}=\$ 117.6114109   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 42.42549702   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 90.2367785 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 18.2367785 \\     \text{ } & S_u=\$ 93.91941129  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 22.57415983  & \text{ }    & \text{ } \\     S=\$ 60 &  \text{ } & S_{ud}=S_{du}=\$ 72.05920794    & \text{ } \\   C=\$ 11.57252827 &  \text{ } & C_{ud}=\$ 7.848617166    & \text{ } \\    \text{ } & S_d=\$ 57.54338237 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 3.377832957 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 55.28707407 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 0 \\      \text{ } & \text{ } & S_{dd}=\$ 44.14987805   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 33.87377752 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 0 \\      \end{array}

    \text{ }

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Practice problems

Practice problems can be found in the companion problem blog.

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\copyright \ \ 2015 \ \text{Dan Ma}

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