In this section, we provide some graphs to illustrate the results obtained from our analytical pricing formula as well as reveal some interesting features of a Parisian up-and-in call.
It should be noted that the calculation procedure for an American-style Parisian up-and- in call option is similar to that for a European-style Parisian up-and-out call as presented in [83], except that we have replaced the value of the European vanilla option by the numerical value of its American counterpart, which can be obtained by using the highly efficient integral equation method ([54]). Once the value of the American vanilla option is found, the integrals in our analytical formula can be computed by using quadrature rules (Gauss-Laguerre, Gauss- Legendre, Gauss-Jacobi rules) in a very similar way as that in [83]. Therefore, the computation cost of our formula should not be too much different with that in [83].
Figures 4.1(a) and 4.1(b) present comparisons of the values of Parisian up-and-in calls for various J values with those of their embedded vanilla calls. The parameters used in our calculations are E = $10, ¯S = $18, T − t = 0.8 (year), ¯J = 0.2 (year), σ = 30%, r = 5%, D = 10%. As can be seen clearly from both the figures that the values of the Parisian options
8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 15 Asset price Option price Parisian: J = 0 Parisian: J = 0.05 Parisian: J = 0.1 Parisian: J = 0.15 Parisian: J = 0.18 Payoff function Vanilla
(a) American-style options
8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 Asset price Option price Parisian: J = 0 Parisian: J = 0.05 Parisian: J = 0.1 Parisian: J = 0.15 Parisian: J = 0.18 Vanilla (b) European-style options
Figure 4.1: Comparison between the prices of a Parisian up-and-in call at various J with that of its embedded vanilla option; parameters are: E = $10, ¯S = $18, T − t = 0.8 (year),
¯
J = 0.2 (year), σ = 30%, r = 5%, D = 10%.
are always less than those of their embedded vanilla options. This is indeed expected as the holders of the Parisian calls have to wait until the knock-in feature is activated to obtain the same exercise right as the holders of the embedded vanilla options. This waiting period, with the risk that the knock-in feature may never be activated, would definitely devalue the Parisian calls, in comparison with their embedded vanilla counterparts.
Figures 4.1(a) and 4.1(b) also reveal some interesting properties of the Parisian up-and-in calls with respect to changes in S and J. One can observe that when J is fixed, the Parisian call prices are increasing functions of asset price. In fact, when the asset price increases,
the knock-in feature is more likely to be activated and thus the values of the Parisian calls increase and finally approach the values of their embedded vanilla options. Similarly, with a fixed value of S, the knock-in feature is more likely to be activated when J gets closer to ¯J. As a result, the Parisian option prices increase when J increases.
14 16 18 20 22 24 26 28 0 0.5 1 1.5 2 Asset price Price difference Parisian: J = 0 Parisian: J = 0.05 Parisian: J = 0.1 Parisian: J = 0.15 Parisian: J = 0.18
Figure 4.2: Differences between the prices of American-style and European-style Parisian up-and-in calls at various J, with parameters: E = $10, T − t = 0.8 (year), ¯S = $18,
¯
J = 0.2 (year), σ = 30%, r = 5%, D = 10%.
It is also expected that the price of an American-style Parisian up-and-in call should be higher than that of its European-style option counterpart because upon the activation of the knock-in feature, the holder of the former can buy the underlying asset before and up to expiry, while that of the latter can only buy the underlying asset at expiry. This expectation is clearly illustrated in Figure 4.2, which shows the differences between the prices of the former at various J and those of the latter. It is clear from this figure that the differences between the two sets of prices become larger when S becomes larger or J gets closer to ¯J. This is because the knock-in feature is then more likely to be activated and the values of the Parisian options approaches those of their embedded options. As a result, the differences between the values of the Parisian options approach those of their embedded vanilla options, which become larger when S increases.
4.5 Conclusion
In this chapter, we have derived a simple analytical formula for Parisian up-and-in calls by using the “moving window” technique proposed in [83]. Unlike the “knock-out” cases, the valuation of American-style Parisian up-and-in calls is very similar to that of its European counterpart and both can be handled with the same solution procedure. As a result, we are able to derive a pricing formula that can be used to evaluate both American-style and European-style Parisian up-and-in calls. We have also provided examples to illustrate some interesting features of a Parisian up-and-in call.
Pricing American-style Parisian
down-and-in options
5.1 Introduction
Continuing on the topic of pricing Parisian knock-in options, this chapter discusses the pricing problem of another type of these options, the down-type options. The difference between the up-type and down-type options lies on the knock-in condition, i.e., the condition activates the knock-in feature. More specifically, the knock-in feature of a Parisian down-and-in option is activated only if the underlying asset price has continually stayed below the barrier ¯S for a prescribed time period ¯J. This knock-in condition of the down-type option clearly contrasts with that of its up-type option counterpart.
It should be noted that there is no clear relation between the pricing formulas of Parisian up-and-in options and Parisian down-and-in options. In other words, knowing an analytical pricing formula of the former does not allow us to straightforwardly obtain that of the latter. Finding an analytical formula for the latter is therefore not a trivial task, even though that of the former is already derived in Chapter 4. In this chapter, we show that the “moving window” technique can be used to derive an analytical solution for American-style Parisian down-and-in calls.
This chapter is organized as follows. In Section 5.2, we introduce the PDE systems govern- ing the price of an American-style Parisian down-and-in call option. The solution procedure is presented in Section 5.3, while Section 5.4 provides some selected graphs to illustrate the
implementation of our formulas. This chapter ends with some concluding remarks given in Section 5.5.