After a market downturn, especially in an uncertain economic environment such as the current state, there can be a relatively long period with a sideways market, where indexes, stocks, etc. We discuss option pricing in such scenarios, in both cases of unattainable as well as attainable boundaries, and obtain closed-form option pricing formulas. Option pricing, channel, reflecting boundaries, Brownian motion, volatility, drift, barriers, mean-reversion, mean-repelling, FX, digital currencies, target zone, sideways market, interest rate, attainable boundaries, unattainable boundaries, arbitrage, stock, put, call, binary, knockout, rebate.
Baxter, M. Beaglehole, D. Working Paper. Bhagavatula, R. Ithaca, NY: Cornell University. Broadie, M. Review of Financial Studies 8 1 : Carr, P. Applied Mathematical Finance 2 3 : Quantitative Finance 17 10 : Geman, H. Mathematical Finance 6 4 : We then get a P 1 and a C 1 option. The P 1 and the C 1 options, as well as every further options in the sequences, are priced in the usual way in the binomial framework.
The price of an American-type option at a given non-final node is the maximum of its intrinsic value and its discounted expected future value. There were 56 realignments in the period — in the EMS, implemented in 17 discrete adjustments. Among these realignments, I analyze the band widening in August , which took place approximately 5 years before the euro was born.
The 15 days average of the daily exchange rates just before the widening was 3. I choose the process of the floating exchange rate so as to fit an exchange rate with future locking.
I assume that the underlying floating exchange rate will be fixed at the same time and at the same rate as the target zone exchange rate. As the option pricing method developed here is based on the discrete binomial model, I have a great flexibility at choosing the process of the floating exchange rate.
The assumed process of the floating exchange rate is a discretized Brownian Bridge process that can be represented by a binomial-tree.
EURUSD tries to crack the next floor/target area
The locking rate and date are set to their values historically developed later. The volatility of the floating exchange rate is calibrated to match the volatility of the target zone exchange rate both before and after the realignment. The model is used to decompose the effect of band widening into the direct effect of realignment and changing uncertainty.
The direct effect captures the changing strike prices, while the other captures the changing volatility. I apply a comparative static analysis. Figure 2 shows the relationships between the floating exchange rate and the exchange rate in the target zone. Its Line 0 demonstrates the relationship before the realignment.
References
As the exchange rate was approximately 3. Line 1 shows the same relationship, but with the post-realignment strike prices. If the floating rate remained unchanged, then the target zone exchange rate should have weakened to 3. According to Line 2, the exchange rate should have weakened further to 3. To sum up, the model attributes almost 6 per cent weakening to the direct effect and further 4 per cent to the growing volatility, while the observed weakening was about 2. The model fails to take into account the limited credibility of the regime, and therefore the deviation of the observed depreciation from the one implied by the model can be interpreted as evidence for the anticipation of a realignment.
In this article, I developed an options-based model of the target zone arrangements. The exchange rate in a target zone system is equivalent to the exchange rate of a currency in an underlying freely floating system adjusted by the price of two options. The advantage of showing the analogy between the problems of option pricing and determining the effect of the target zone is that the option pricing literature offers solution for a wide range of processes of the underlying asset.
Exchange Rate Target Zones
I determined the underlying assets of the options accurately and provided an option pricing method applicable for these options. By applying the options-based approach, I analyzed the band widening of the French franc in I found that the model overestimates the actual depreciation.
One can expect a model to overestimate the change in the exchange rate if it assumes perfect credibility that is violated in practice. Along these lines, the analyses can be thought of as a test of credibility. One potential generalization of the options-based model is to allow the strike prices to change stochastically.
FX Options in Target Zone
By this modification, one could model not only the exchange rates in target zones under imperfect credibility, but also the exchange rates in countries that engage in manage floats. Krugman, P. The Quarterly Journal of Economics 3 : — Article Google Scholar. Copeland, L. London: Pearson Education. Google Scholar.
Understanding the former assumes no knowledge on derivatives. Rangvid, J. European Economic Review —, refer to the same term as the shadow exchange rate. Margrabe, W. Journal of Finance — One can also think of these compound options as exotic knockout options. The knockout condition for both options is that the other option is exercised. Geske, R. Journal of Financial Economics 7: 63— Hull, J. The Economic and Social Review 31 2 : — Pownall, M. Dubuque, IA: Brown Communications. Download references. You can also search for this author in PubMed Google Scholar. Correspondence to Anna Naszodi.
She works for the Central Bank of Hungary as a researcher. Her research interests include international finance, applied econometrics, banking and finance, and asset pricing. She has several working papers that have been presented at prestigious conferences, like the European Winter Meeting of the Econometric Society, the Meeting of the European Economic Association.
The sequence of the created put and call binomial trees are monotonously increasing on each node, but they never exceed the value of the corresponding node of the put and call components of the target zone exchange rate. The proof of the boundedness and the monotony rely on the fact that the value of a put option is monotonously decreasing in its underlying asset, whereas a call option is monotonously increasing in its underlying asset.
First, I check the boundedness of the sequences of the binomial trees of the options, by comparing their underlying assets with the underlying assets of the put and call component of the target zone exchange rate. So the underlying product of the P i is not less than the underlying product of the put component of the target zone exchange rate, consequently, no node value of the binomial tree of P i is greater than the corresponding node value of the binomial tree of the put component.
Similar argument holds for the C i. Therefore, I proved by total induction that for every i , P i and C i are bounded. Second, I prove that the put and call sequences are monotonously increasing on each node by comparing the underlying assets again. According to a convergence theorem see for example: Pownall, 11 p.
The sequence of the put binomial trees converges to the binomial tree of the put component of the target zone exchange rate and the sequence of the call binomial trees converges to the binomial tree of the call component of the target zone exchange rate. In this case, the comparison of the underlying product of the limit of the put options with the underlying product of the put component indicates the following: the binomial tree of the limit of the sequence of the put options exceeds at least at one node the binomial tree of the put component.
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Naszodi, A. The options-based model of target zone and its application to the French franc. J Deriv Hedge Funds 16, — Download citation. Received : 25 February Revised : 25 February Published : 20 October Issue Date : 01 November