 Working with FTS Exotic Options Module

How Do we Value an Asian Option?

You can apply the binomial model to value non-standard options.  The options we will consider in this example are Asian options.  The payoff at maturity for an Asian call is max {Sa  - X,0} where Sa  is the average stock price over the life of the option.  This option may be attractive to firms that wish to hedge a stream of transactions over time.  This is because the average price risk becomes a more relevant factor than the price at one time.

Defining the underlying asset in this way greatly complicates the option valuation problem.  Now, its value depends upon the path followed by the underlying asset, and thus it is a path-dependent option.  Note that the problem is not to determine the option value at the end of its life, but rather at the beginning, when we do not know which path the stock price will take.

The beauty of the binomial approach is that the same principles (i.e., riskless hedge, or synthetic option) still apply and provide a way to calculate the value of the option.  The difference is that the tree diagram becomes more complex because it is path dependent.

We will work through a concrete example applying the exotic option module to the problem described in step 1.

Step 1: Select Asian from the drop down menu of different types of exotic options.  Set asset price to equal 10, uptick equal to 2, downtick equal to 0.5, number of steps equal to 2, maturity equal 2, risk free 0.01 (this implies 1% per step because there are 2-steps and maturity is set to 2) and strike price equal to 15.

Note:  Compounding assumption in the tree.  If number of steps equal the maturity then interest rate is applied continuously compounded per step.  If the number of steps equal n and the maturity is set to 1, then interest rate per period is rate*1/n continuously compounded per period.  The module lets you set number of steps and maturity independently to accommodate these two types of problem.

Note:  The module has a box average from node 1 which is left unchecked.  This implies that the averaging process takes place after the first tick is realized.  That is, the stock price in this examples first increases to either 20 or 5 in nodes 2 and 3 respectively.  After this first tick the averaging process starts.  Most examples assume that the averaging process takes place after the first tick so leave this unchecked.

Finally, leave American Option unchecked (i.e., we are valuing a European option) and leave volatility unchecked because we have specified the up- and down-tick directly.  Alternatively, this can be implied by estimating the volatility of the underlying asset process and entering this estimate directly.

Now click on the Draw Tree button with maturity set equal to the number of partitions (= 2).  So compounding assumption is 1% continuously compounded each period for 2 periods.

Binomial Tree for Stock Prices

With path dependency there are four possible paths.  You should observe that the set of possible terminal stock prices are {40, 10, 10, 2.5}.

Step 2:  To value an Asian option first calculate the average stock price at every node.  Recall that we assume the average price is calculated only after the first tick is realized (i.e., starting after the end of period 1).  Thus, if there is an uptick in period 1, then in the derived average price process under this starting assumption is S'u

= 20 and S'uu = (20 + 40)/2 where now S' refers to the average stock price process.

You can compute the average prices after making the above simplifying assumptions.  The average price, S', at the lowest node {down-tick, down-tick (i.e., D,D)} at the end of period 2 is (5+2.5)/2 = 3.75.

Step 3:  Cash flows from the options at end of partition 2.  Consider a call option.  With a strike price equal to 15 if the sequence of realized ticks is {up-tick,up-tick (i.e., U,U)},  the last price for the underlying (average process) is (20+40)/2 = 30.  Therefore, the call option with strike price equal to 15, finishes in the money at 15 on this highest path.

You can see this by clicking on Draw Tree, and selecting Call Option and Asian option set.  Leave all boxes un-ticked (i.e., American, Volatility, Average from Node 1 etc.,).  Follow along the highest path and the top number is 15 (the call option value at maturity if this path is realized) and the below 15 is 40 (the stock price value at maturity if this path is realized).  The call option has a strike price equal to 15 and is defined on the average stock price over the last partition ((20+40)/2=30).

If the sequence is U, D then the last period's average is:  (20 + 10)/2 = 15.  The call option is at-the-money equal to zero.  Similarly, for D, U and D, D both result in the call option expiring worthless.

Step 4:  Option value at the end of partition 1.  From step 3 you saw that at the end of partition 2 the call option was worth 15 if an up up sequence is realized and 0 otherwise.  As a result, if a down tick is realized at the end of partition 1 the value of the call option is clearly 0.

You can see this in the tree diagram by following the down-tick to the node 3.  The pair of numbers is (0, 5).  The top number 0 is the option value and the bottom number 5 is the stock price at this node.  The call is worth zero because irrespective of whether an additional up- or down-tick is realized the call option terminates out of the money at 0.

But suppose an up-tick is realized so the at node 2 you see the pair of numbers (5.05, 20).  The bottom number is the stock price at this node (i.e., 20 = 10*2).  How is the top number arrived at?

The answer is that this is the expected present value of the option prices at the terminal node discounted back at 1% continuously compounded.  Furthermore, the expectation is taken with respect to risk neutral probabilities, not the true probability, of an up and down tick.

How is the risk neutral probability calculated?

You can see that working through a multi-period tree is equivalent to completing a sequence of one-period binomial option prices.  The risk neutral probability for an up-tick in a one period binomial option price equals (exp(risk free rate) - down-tick)/(up-tick - down-tick).  For the current problem this equals:  (1.01005 - 0.5)/(2 - 0.5) = 0.340033.  The risk neutral probability of a down tick is 1 - 0.340033 = 0.659967.

As a result, the value of the call option at node 1 (up-tick realized at end of first partition) = 0.340033*15*(exp(-0.01) + 0.659967*0*(exp(-0.01) = 5.049751 (or 5.050 to three decimal places).

Step 5:  What is the price for the Asian option today (at the beginning node 1)?  From the tree diagram you can see that this equals 1.700 which is computed as follows:

From step 4 you computed the value of the Asian option if an up-tick is realized (5.049751) and 0 if a down-tick is realized.  At node 1 the value of the option is given as 1.699999 = 0.340033*5.049751*exp(-0.01) + 0.659967*0*exp(-0.01).

Exercise:  By working through the above steps verify that the Asian put option is valued at 6.453.