## The Pricing of Exotic Interest Rate Derivatives

The critical difference between modelling interest-rate derivatives and equitv FX options is that an interest rate derivative is really a derivative of the yield curve and the yield curve is a one-dimensional object whereas the price of a stock or an FX rate is zero-dimensional. One might be tempted to think that as most movements of the yield curve are up and down it is unnecessary to model the one-dimensional behaviour. However, the yield curve can and does change shape over time, and we...

## An example of arbitragefree pricing

Within the context of our assumptions, we now look at a few examples of arbitrage-free pricing. Suppose a company wishes to enter into a contract to exchange a fixed number of dollars, say one million, for yen one year from now. This is called a forward contract. Note that the company has no choice once it has entered the contract - this is not an option. How do we set the exchange rate One could attempt to estimate the exchange rate in a year, and use this to price this contract....

## Fttl ftt2 i0

And let the partition size go to zero. If we again approximate by N up or down moves of size Jt N, then this sum will always be t. Letting N tend to infinity, we deduce that Brownian motion paths always have second (or quadratic) variation of t over an interval 0, T , This actually holds with probability one. This is why we can use measure changes to change the drift but not the volatility of a Brownian motion. There is an immense rigidity in the paths caused by the fact that the quadratic...

## D ogB V rr 08210

Whilst we have reduced the problem to a diffusion equation, we have done so at the cost of making the barrier level non-constant. We therefore look for a different approach to eliminating the first order term which does not involve changing Z. Our approach is to multiply D by a function which we will choose in such a way to eliminate the extra terms. In particular, we try D(Z,t) eaZ+btE(Z,t). (8.2.11) Differentiating and substituting into (8,2,7), we obtain If...

## MinijB82

As we observed in Chapter 9, setting is a necessary and sufficient condition. We now wish to try out various choices of A. The incremental method is the Choleskv decomposition, that is the unique choice of A which is lower triangular. Try the following methods and compare convergences for Asian options, (1) Use spectral decomposition to write C PDP1 with D diagonal and with the diagonal elements decreasing. Try A PD1 < 2 and A PD P1. See 105 for code to carry out diagonal-ization. Here P is...

## D y q

In our analysis of a dividend-paving asset, we hedged the option with contracts involving payment today but delivery at the expiry of the option. We could equally well have hedged with forward contracts, that is with contracts that involve payment and delivery at time T but with the size of the payment fixed today. This has certain advantages and we will return to this point when we have developed more theory, In this chapter we have covered a lot of ground. We introduced the concept of an Ito...

## Ft Atxi Axixn Axn fx t x tAt

Plus an error of order three in Axi and order two in At. If we wish to imitate our arguments in the one-dimensional ease, we have to understand how the terms A.r A.r - behave when xt xf Recall that our definition of the process for implies that Oj X , t wj At - Wtij smallerror 11.3.3 and the Zj are ordinary N 0,1 draws. However, our definition of correlated Brownian motions mean that the variates Zj are correlated to the same extent as the Brownian motions. Thus if we let pjk be the correlation...

## Dttt6 9

The fact that Bt is deterministic makes the final term disappear. Thus we have that dCt Bt d j lBt. 6.9.10 Using the determinism of Bt again, we have that Combining the last equations, we have that dCt dSt dBt, 6.9.12 whieh is precisely the self-financing condition. We have shown that the payoff of an option is replieable by a self-financing portfolio if we are working in a Blaek-Seholes world, and this once again guarantees the arbitrage-free price. In the perfect Blaek-Seholes world we have...

## Contents

The most important assets 19 1.4. Diversifiable Risk 23 1.5. The use of options 24 1.6. Classifying market participants 27 Chapter 2. Pricing Methodologies and Arbitrage 31 2.1. Some possible methodologies 31 2.3. What is Arbitrage 35 2.4. The assumptions of mathematical finance 36 2.5. An example of arbitrage-free pricing 38 2.6. The time value of money 40 2.7. Mathematically defining arbitrage 43 2.8. Using arbitrage to bound option prices 46 Chapter 3. Trees and Option Pricing 61 3.1. A...

## Chapter

The Blaek-Seholes no-arbitrage argument is an example of dynamic replication by continuously trading in the underlying and bonds, we can precisely replicate a vanilla option's pay-off. The cost of setting up such a replicating portfolio is then the unique arbitrage-free price of the option. If we do not allow continuous trading in the underlying, but instead restrict ourselves to portfolios in the underlying and zero coupon bonds set up today, the only derivative we can precisely replicate is...

## Questions

If a dollar is 120 yen and a pound is 1,4 dollars. What can we say about the pound yen exchange rate Question 2,12,2. Each of the following products pays a function of the spot price, S, of a non-dividend paying stock one year from now. If there are no interest rates and spot is 100, give optimal upper and lower bounds on their prices today, 1 The pay-off is one between 110 and 130 and zero otherwise, 3 The pay-off is zero below 80, increases linearly from zero at 80 to 20 at...

## DW2 dt5428

Note that all our Ito processes here are defined by the same Brown-ian motion. Later on we will want to consider stocks driven by different Brownian motions. One important consequence of Ito's lemma is a product rule for Ito processes, if we let fix, y xy, then we have Proposition 5.4.1. If Xt and Yt are Ito processes then d XtYt XtdYt YtdXt d.X , lt . 5.4.29 Note that this generalizes the Leibniz rule from ordinary calculus which does not involve the final term. As this may all have been a bit...

## Info

2. pricing methodologies and arbitrage working with the forward, in that at-the-forward price is where the forward contract's value changes sign, and therefore it is the point where call and put options have equal value. This result has an interesting consequence. It means that our views on where the future value of the asset price is likely to be at expiry, cannot affect the price of a call or put option struck at the forward. For if we believe the spot is more likely to finish in the money...

## Using arbitrage to bound option prices

In this section, we study bounds on option prices which do not involve any assumptions on the way the assets moves. Instead we prove upper and lower bounds on prices, and prove relationships between the prices of differing options. Recall that a European call option on an asset is the right to buy the asset for a fixed price K at some fixed time, T, in the future, A rational investor who owns the option will exercise i.e. use the option if and only if the market price of the asset, S, is more...

## Market Efficiency

Before we can understand why risk is so important we first have to understand the concept of market efficiency. Most of financial mathematics and modern economics is based on the concept of market efficiency, This roughly states that in a free market, all available information about an asset is already included in its price. Therefore there is no such thing as a good buy - the only value an asset has is its market value and it is meaningless to attempt to think otherwise. Is this hypothesis...

## What is risk

It is arguable that risk is the key concept in modern finance. Every transaction can be viewed as the buying or selling of risk. The success of an organization is determined by how much return it can achieve for a given level of risk. Before we can justify these statements, we need to achieve some understanding of what risk is. In a typical pure mathematical ploy, let us start by trying to understand the absence of risk, A riskless asset is an asset which has a precisely determined future...