Option Pricing

Much of the apparent complexity of current approaches to real-options analysis arises from the attempt to fit financial-option formulae to real-world problems. Usually this does not work since real-world options are often quite different from financial options. Option-pricing formulae are treated as a pro-crustean bed by academics Either the real world is simplified beyond recognition or unwarranted assumptions are added to make the facts fit the theory. Neither approach satisfies managers. 

Merton, R. C. 1973. Theory of rational option pricing. Bell Journal of Economics, 4 141-183. 

Applications of option-pricing theory twenty-five years later. American Economic Review, 88 323-349. 

Roman Introduction to the Mathematics of Finance From Risk Management to Options Pricing. 

Lajbcygier, P. R. Connor, J. T. (1997). Improved option pricing using artificial neural networks and bootstrap methods. Int J Neural Syst 8,457-71. 

Edwards, S. F. (1988). Option prices for groundwater protection. Journal of Environmental Economics and Management, 15,475-87. 

We overcome this inconsistency, by deriving a unified framework that directly leads to consistent cap/floor and swaption prices. Thus, in general we start from a HJM-like framework. This framework includes the traditional HIM model as well as an extended approach, where the forward rates are driven by multiple Random Fields. Furthermore, even in the case of a multifactor unspanned stochastic volatility (USV) model we are able to compute the bond option prices very accurately. First, we make an exponential affine guess for the solution of an expectation, which is comparable to the solu-... 

In chapter (6), we extend the traditional HJM approach, by assuming that the sources of uncertainty are driven by Random Fields. For that reason, we introduce a non-differentiable Random Field (RF) and an equivalent T-differentiable counterpart. Given the particular Random Field, we derive the corresponding short rate model and show in contrast to Santa-Clara and Sor-nette [67] and Goldstein [33] that only a T-differentiable RF leads to admissible well-defined short rate dynamics". Santa-Clara and Sornette [67] argue that there is no empirical evidence for a T-differentiable RF. We conclude that the existence of some pre-defined short rate dynamics enforces the usage of a r-differentiable RF. Furthermore, we compute bond option prices when... 

Now, applying the same approach as in section (2.1) we derive the theoretical option pricing formula for the price of a swaption based on the Fourier inversion of the new transform... 

Starting from the payoff function of a European option on a coupon bearing bond we can write the option price at the exercise date Tq as follows... 

Recapitulating, we have derived theoretically a unified setup for the computation of bond option prices in a generalized multi-factor framework. In general, the option price can be computed by the use of exponential affine solutions of the transforms t z)i for z C applying a FRFT and S,(n), for n G N performing an lEE. 

In option pricing theory, we are usually not interested in a method for the approximation of an unknown pdf. Typically, there is a need for a practicable method to eompute the (exereise) probabilities... 

In option pricing theory, we often assume that the dynamics of the underlying is driven by a lognormal-distributed source of uncertainty. Therefore, in this section we show that our integrated version of the generalized Edge-... 

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

This formula has been derived by applying our general option pricing framework, based on exponential affine solutions of the transform t(z). Later on, when we compute option prices coming from more sophisticated models there typically exists no closed-form solution for the characteristic functions. Then, the FRET- or the lEE-approach can be applied to compute the option prices. 

Following the last section, we introduce the FRFT technique to derive the price of a zero-coupon bond option. In doing so, we are able to compare the option price coming from the FRFT approach with the appropriate closed-form solution (5.27). 

Note that the FRFT technique is not only very accurate, but also very efficient. The computation of option prices for 512 different strikes takes less than 0.1 seconds. Furthermore, it has to be pointed out that the FRFT is a very powerful method, as long as the characteristic functions are available at least in semi-closed form. This appealing feature is directly linked to the high efficiency running a FRFT without wasting any computational time. On the other hand, the Fourier inversion approach is widely useless, if no semi-closed solution of the characteristic functions are available. 

Fig. 5.1 Approximation error running a FRFT to compute the bond option price ZBOi t, 1,2) with j3 = 0.6 and 5 = 0.2...

As in section (5.2.2), we introduce the lEE technique by starting from a simpler model such that we obtain a closed-form solution for the option price. Then, the numerical approximations are directly comparable with the equivalent findings for the closed-form solution. Therefore, we derive the option pricing formula of a receiver swaption with only one (u = l) payment date in T. ... 

Now, we extend this analysis to the computation of an option price on a coupon hearing bond, with multiple payment dates Te 7i, Again,... 

Overall, we find a difference between the simulated and approximated swaption prices of up to 1.3% for out-of-the-money options (see figure ( 5.3.SI)). Nevertheless, based on our results of the last section, where the lEE performed up to 10 times more accurate than the corresponding MC simulation study, we expect that the difference between the simulated and the approximated option prices is mainly linked to the impreciseness of the MC approach (figure (5.3.SI)). The lEE approach performs very efficient and accurate, even if we compute the price of a 1x20 swaption (see table 5.1). Again, all 21 single probabilities TlJ iEEm for i = 0,. ..,20 are in between the 97.5% confidence coming from the simulation approach. ... 

In the following, we compute the price of bond options assuming these two types of Random Fields as correlated sources of uncertainty, while dZ(t T) leads to anon-differential and dU (t, T) to a T-differential type of term structure model. Note that the computation of the particular option price differs only in the proposed type of correlation function. 

None the less, following Kennedy [50], [51], Goldstein [33], Longstaff, Santa-Clara and Schwartz [57], Collin-Dufresne and Goldstein [20] and Santa-Clara and Somette [67] we compute bond option prices assuming the non-differential RF dZ t,T), as well as the T-differential counterpart dU t,T), keeping in mind that only the T-differential RF model leads to a well-defined short rate dynamics. 

Equivalent to section (5.1), we first have to change the measure of the (log) bond price process, before we are able to compute the option prices. Therefore, we transform the (log) bond price dynamics... 

Now we are able to analyze the impact of the correlation structure on the option price. First of all, we find that the option price is increasing with a decrease in the dependency structure (figure (6.1)). Hence, we obtain the lowest option price given perfect correlation (7 = 0). Note that this special solution reflects the price for a bond option in a traditional HIM world. 

The other way round, we obtain the highest option price by postulating completely uncorrelated Random Fields (7 = < ). These findings are directly linked to the fact that the variance A (t) is a decreasing function of the correlation structure leading to lower option prices. 

Now, we immediately see that the volatility of the underlying is a decreasing function of the correlation structure. In reverse, this implies that the volatility increases as the correlation decreases leading to higher option prices. 

Fig. 6.2 Impact of the type of Random Field on the bond option price given /3 = 0.1, 5 = 0.05, 7b= land7i =2...

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