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Language: English. Brand new Book. As today's financial products have become more complex, quantitative analysts, financial engineers, and others in the financial industry now require robust techniques for numerical analysis. He also covers various filtering techniques and their implementations and gives examples of filtering and parameter veloped from the author's courses at Columbia University and the Courant Institute of New York University, this self-contained text is designed for graduate students in financial engineering and mathematical finance as well as practitioners in the financial industry.
Seller Inventory LHB Condition: NEW. Chapter 4, Section 4. To obtain the formula 6. The proof of this fact relies in a crucial way on the martingale property and it is the purpose of Exercise This follows immediately from Proposition 6. Then, using the same rationale as in the proof of Proposition 6. The formula 6. It is the term in dWt which r t St0 dt, makes the bond riskier. Note that, by solving equation 6. The method is the same as in Chapter 4. Remark 6. Actually, it can be shown cf. This condition, slightly weaker than the hypothesis of Proposition 6.
See Remark 6. For the pricing of options on bonds with coupons, the reader is referred to Jamshidian and El Karoui and Rochet We can also rewrite 6. We know that the discounted price process of a risky asset is a martingale under the risk neutral probability. It follows from the second assertion of Proposition 6. From 6. So, the result follows from Girsanov's theorem. A consequence of Proposition 6. Going back to 6. Before calculating the price of bonds according to this model, let us give some consequences of equation 6.
Chapter 3, Sec- tion 3. It follows that r t can be negative with positive probability, which is not very satisfactory from a practical point of view unless this probability is always very small. Chapter 3, Remark 3. We know cf. Chapter 3 that the process Xtx is Gaussian with continuous paths. This last property is considered as a drawback of the model by prac- titioners. In practice, parameters must be estimated and a value for r must be chosen. Theorem 6. For a proof of this result, the reader is referred to Ikeda and Watanabe , p.
Before investigating the Cox-Ingersol-Ross model, we give some prop- erties of this equation. Denote by Xtx the solution of 6. Ikeda and Watanabe , p. Let us go back to the Cox-Ingersoll-Ross model. L1 L2 Remark 6. Some authors have resorted to a two-dimensional analysis to improve the models in terms of discrepancies between short and long rates; cf. Brennan and Schwartz , Schaefer and Schwartz and Courtadon More recently, Ho and Lee have proposed a discrete-time model describing the behavior of the whole yield curve.
The continuous-time model we present now is based on the same idea and has been introduced by Heath, Jarrow, and Morton see also Morton We also assume joint continuity with respect to t, T. We then have to make sure that this model is compatible with hypothesis H. Morton We see that, for any continous process q t , it is then possible to build a model of the form 6.
This is a consequence of equation 6. This situation is similar to Black-Scholes. In practice, these rates are not directly observable. In words, the interest rate is reduced to min L T, T , K. We have, using Proposition 6. In this model, cap prices are easily derived, as sums of caplet prices.
Notes: We have restricted our presentation to models driven by a single Brow- nian motion. The main results of Section 6. We re- fer to Brigo and Mercurio for an exhaustive account of interest rate modelling and a thorough discussion of practical issues. As mentioned above, the BGM model does not provide simple formulae for swaptions. Jamshidian proposed a model in which swaptions can be priced using Black's for- mula. This model is not compatible with the BGM model.
These techniques can be applied in many situations apart from interest rate modelling. In fact, some of the problems in Chapter 4 can be addressed in this way especially, the problem on the Garman-Kohlagen model and the one on Asian options: see Shreve , Chapter 9. Denote by Lt the density of the restriction of Q to Ft.
We use the same notation as in Remark 6. Explain why the option can be hedged by holding HtT resp. Compute the value of the call option and the hedge ratios. Exercise 37 The aim of this exercise is to prove Proposition 6. Exercise 38 Let d be an integer and let X1 ,. P Exercise 39 Use Proposition 6. Show that the solution of 6. Exercise 42 1. Assume, as in the BGM model, that we have 6. Assume that we have 6. To model this kind of phenomena, we have to introduce discontinuous stochastic processes.
The usual approach to pricing and hedging in this context consists of choosing one of these probability measures and taking it as a pricing measure. In this chapter, we will study the simplest models with jumps. We will then investigate the dynamics of the risky asset, discuss the computation of European option prices and examine hedging strategies that minimize the quadratic risk under the pricing measure. The following proposition gives an explicit expression for the law of Nt for a given t.
Proposition 7. Bouleau , Chapter VI, Section 7. Remark 7. The independence of the increments is a consequence of this property of exponential laws. Bouleau , Chapter III; for the second one, cf. The dynamics of Xt , the price of the risky asset at time t, can now be described in the following manner. The process Zt is called a compound Poisson process. Lemma 7. Let Ws resp.
Going back to 7. We keep the hypotheses and notations of Lemma 7. Doob's inequality, Chapter 3, The- L2 orem 3. We also have N. We want to price and hedge European options with maturity T in this model. For technical reasons, we will also assume that the random variables Uj are square- integrable.
Since the process Xt itself is right-continuous, this means, intuitively, that one can react to the jumps only after their occurrence. This condition is the counterpart of the condition of predictability that is found in the discrete models cf. The following proposition is the counterpart of Propo- sition 4. This results from the expression in Proposition 7.
He sells the option at a price V0 at time 0 and then follows an admissible strategy between times 0 and T. According to Proposition 7. If this quantity is non-negative, the writer of the option loses money, otherwise he earns some. Each term of this series can be computed numerically if we know how to simulate the law of the Uj 's.
For some laws, the mathematical expectation in the formula can be calculated explicitly cf. For such a strategy, equality 7. To do so, we need the following proposition. From Proposition 7. We deduce easily exercise from formula 7. From Exercise 19 of Chapter 3, it can be written as a stochastic integral. But, when there are jumps, the minimal risk is generally positive cf. Exercise 48 and Chateau As for the determination of the volatility in the Black-Scholes model, we can distinguish two approaches: 1 a statistical approach, from historical data, and 2 an implied approach, from market data, in other words, from the prices of options quoted on an organised mar- ket.
Notes: Financial models with jumps were introduced by Merton Assume N and V1 are square-integrable. Exercise 45 The hypotheses and notations are those in Exercise Suppose, with the notations of Section 7. Write the price formula 7. Hint: use Exercise Exercise 48 The objective of this exercise is to show that there is no perfect hedging of calls and puts for the models with jumps we studied in this chapter. Using Proposition 7. Chapter 8 Credit risk models In the last few years, the market of credit derivative instruments has de- velopped dramatically.
Credit risk is associated with the risk of default of a counterparty. In the second section, we intro- duce intensity models, which consider the default time as an exogenous ran- dom time, characterized by its hazard rate. We then describe the valuation of credit default swaps CDS. The last section is devoted to the concept of cop- ula, which is very useful in models involving several default times. For more information on credit risk models, we refer the reader to the recent second edition of Brigo and Mercurio and, for mathematical developments, to Bielecki and Rutkowski We will limit our presentation to Merton's model see Merton , which appears as the pioneering model of this approach.
The constant r is the instantaneous interest rate and the constant k is an expenditure rate. In Merton's model, default may occur at the deterministic time T only. Within this model, the computation of the price of a defaultable zero-coupon bond is similar to the pricing of a barrier option see Exercise Remark 8. This is in contrast to the structural approach.
In the intensity approach, the default time appears as an exogenous variable and default may occur as a surprise.
On the other hand, at a given date t, investors know if default has occurred or not. Now, consider a defaultable zero-coupon bond, with maturity T , which, at time T , pays one unit of currency if default has not occurred and nothing in case of default before or at time T. The following proposition relates the computation of conditional expectations given Gt to conditional expectations given Ft. Proposition 8. We have proved the following result. This assumption is often used in practice.
An investor A, who wants to be protected against default, agrees with a bank B on the following. At dates T1 ,. On the other hand, in case of default before time Tn , B will make a payment to A. The number N is called the nominal of the swap, and s is the spread of the swap. The idea is that A holds a bond with nominal N issued by a company C that may default and that, in case of default, the holder of the bond recovers N R, instead of N.
See Brigo and Mercurio or Overhaus et al. Since credit default swaps are the most liquid credit deriva- tives, they are often used for calibration purposes. In practice, the spreads of credit default swaps with various maturities are used to derive implied default probabilities. Denote by sj the spread of a swap with payment dates T1 ,.
In fact, 8. COPULAS The following result, known as Sklar's theorem, shows that the law of a vector can be characterised in terms of its marginal distributions and a copula. Theorem 8. There exists a copula C such that, for all x1 ,. Moreover, if the functions Fi are continuous, the copula C is unique. For the proof of Sklar's theorem, we will use the following lemma.
Lemma 8. If X is a real-valued random variable with a continuous distribution function F , the random variable F X is uniformly distributed on [0, 1]. We only prove the result in the case of continuous marginals. See Sklar for the general case. We know from Lemma 8. Note that the uniqueness of C follows from the fact that any vector u1 ,. The so-called collateralized debt obligations CDO produce i. For practical purposes, it is important to have parametric families of copulas that can be adjusted to market data.
They can be character- ized by the covariance matrix of the Gaussian vector. Note that the compo- nents of the vector can be assumed to be standard normal variables, since the copula is invariant under increasing transformations of the coordinates. So the diagonal entries of the matrix are equal to 1. Exercise 52 In the framework of Section 8. Show that the intensity process is deterministic and explain formula 8. Exercise 54 Let U1 ,. Prove that C u1 ,. Prove Pm that C u1 ,. Hint: observe that for any events A1 ,. When we can write the option price as the ex- pectation of a random variable that can be simulated, Monte-Carlo methods can be used.
Consider a random variable X and assume we are able to generate a sequence of independent trials, X1 ,. Remark 9. This happens, for instance, in the simulation of a Poisson random variable see page Convergence rate of Monte-Carlo methods Let X1 ,. Theorem 9. Note that it is impossible to bound the error, using this theorem, since the support of any Gaussian random variable is R. It is easy to do this by using the same samples as for the expectation. Let X be a random variable and X1 ,. With a probability close to 0.
The possibility to give an error estimate with a small numerical cost is an extremely useful feature of Monte-Carlo methods. A simple and very common method is to use a linear congruential gen- erator. Then it is possible to create random number generators with an arbitrary long period by increasing m. We recall here some elementary methods used to simulate each of these distributions. Thus N1 has the same distribu- tion as the variable X we want to simulate. For the simulation of distributions not mentioned above, or for other meth- ods of simulation of the previous distributions, one may refer to Rubinstein Simulation of Gaussian vectors Multidimensional models will generally involve Gaussian processes with values in Rn.
The problem of simulating Gaussian vectors see Section A.
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We now give a method of simulation for these kinds of random variables. It is easily checked that this vector is a Gaussian vector with zero-mean. This implies that the coordinates of Z are n independent standard normal variables. This method of calculation of the square root is called Cholesky's method for a complete algorithm, see Ciarlet or Press et al.
Sometimes we need to know how to simulate the whole path of a process for example, when we are studying the dynamics through time of the value of a portfolio of options; see Exercise This section suggests some simple tricks to simulate paths of processes. Two approaches are available. Simulation of models with jumps We have investigated in Chapter 7 an extension of the Black-Scholes model with jumps; we now describe a method for simulating this process.
We take the notations and the hypothesis of Chapter 7, Section 7. All these variables are assumed to be independent. Then, from equation 8. An obvious con- sequence of this fact is that one always has interest to rewrite the quantity to compute as the expectation of a random variable that has a smaller variance: this is the basic idea of variance reduction techniques. Suppose that we want to evaluate E X.
A lot of techniques are known in order to implement this idea. We give an introduction to some standard methods. For the Black-Scholes model, explicit formulae for the vari- ance of the put and the call options can be obtained. The variance of the put option is often but not always smaller than the variance of the call. In these cases, one should compute put option prices even when one needs call prices. Basket options. A very similar idea can be used for pricing basket or index options. The simplest model used to price basket options is the multi- dimensional Black-Scholes model.
It is obtained by changing the sampling distribution. We introduce this method in a very simple context. This fact can lead to a very large error in a standard Monte-Carlo method. In order to increase the probability of exercise, we can use the equality 9. We consider here the multidimensional Black-Scholes model already intro- duced on page In view of 9. The reader is referred to Glasserman et al. We now extend these techniques to the case of path-dependent options. We use the Girsanov theorem 4. We start by considering an elementary importance sampling technique. It is a straightforward extension of the technique used in the preceding example.
According to Girsanov theorem see Theorem 4. We begin by considering a very simple but instructive example. We will now compare the variances of I2n and I2n0. Observe that in doing this we assume that most of the numerical work relies on the evaluation of f and the time devoted to the simulation of the random variables is negligible.
This is often a realistic assumption. One can prove that if f is a monotonic function, this is always true and thus the Monte-Carlo method using antithetic variables is better than the standard one. Suppose in addition that Z is square- integrable. A basic example. We want to simulate a random variable X Ru with distribution f x dx. Deduce a method of simulation of the distribution of X. We recall see Section 1. Describe a method to sample the vector S0 ,. Note that the price of options does not depend on the value of p see Chapter 1.
By a time shift argument write a function Price n,N,K,R,up,down,x that computes the price at time n, when the asset value is x, at this time. Write a recursive algorithm to compute u 0, x, 0. What is the complex- ity of this algorithm with respect to N? Can you improve the complexity of this algorithm? Implement this hedging formula and check, using simulation, that it gives a perfect hedging strategy. Explain how to sample a sequence of independent normal random vari- ables with mean 0 and variance 1.
This is obviously impossible in practice. Implement the Black-Scholes formula for the price of a call option. The residual risk is identically equal to 0 when using a perfect hedge. Draw its histogram and compute its mean and its variance. We compare the following strategies: a We do not hedge: we sell the option, get the premium, we wait for three months, we take into account the exercise of the option sold and we evaluate the portfolio. Are there arbitrage opportunities? Use the same hedging strategies for a combination of put and call op- tions. Sampling the zero-coupon bond dynamics We denote by P t, T the price at time t of a zero-coupon with maturity date T.
We want to implement a hedging strategy for this option. Show, using Proposition 6. Plot a histogram of the residual risk and study the values of its mean and its variance when h decreases to 0. Write a function that samples a vector of independent normal random variables with mean 0 and variance 1. Draw the histogram of the vector and compare it with the exact distri- bution of a normal random variable with mean 0 and variance 1.
We will now use the random variable ST as a control variate. Write a program using ST as a control variate. Compare the precision of this method with the previous one using various values for K and S0. How is this method related to the call-put parity formula? We want to compute the price of a call option with strike K where S0 is small compared to K.
Empirically check that the variance is reduced by using simulation. Let Wt1 ,. How can you use IT as a control variate? Relate this method to the call-put arbitrage relation. Appendix A. The components X1 ,. However, if X1 , X2 ,. It is well known that if the random variables X1 ,. Theorem A. The random variables X1 ,. The reader should consult Bouleau , Chapter VI, p. Remark A. The importance of normal random variables in modelling comes partly from the Central Limit Theorem cf.
The simulation of normal and multivariate normal distributions is discussed in Chapter 8. The events B1 ,. They are called atoms of B. Jacod and Protter , Chapter The def- inition of the conditional expectation is based on the following theorem see Jacod and Protter , Chapter The almost surely uniquely determined random variable Y is called the con- ditional expectation of X given B and is denoted by E X B.
The converse property is not true, but we have the following result. Proposition A. Given Property 8 above, we just need to prove that 1 implies that X is independent of B. Thus, the conditional expectation of X given B appears as the least-square best B -measurable predictor of X. Assume that X is B -measurable and that Y is independent of B. Denote by PY the law of Y.
In the Gaussian case, the computation of a conditional expectation is particularly simple. Indeed, if Y, X1 , X2 ,. This means that the function of the Xi 's that approximates Y in the least-square sense is linear. One can compute Z by projecting the random variable Y in L2 on the linear subspace generated by the constant 1 and the Xi 's cf.
Bouleau , Chapter 8, Section 5. For more details, the diligent reader can refer to Dudley Let C be a closed convex set that does not contain the origin. The vector x0 is nothing but the projection of the origin on the closed convex set C. This completes the proof. Consider a compact convex set K and a vector subspace V of Rn.
Therefore, the subspace V is included in a hypherplane that does not intersect K. By Theorem A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publica- tions Inc. Reprint of the edition. Amin and A. Bouhari Arouna. Adaptative Monte Carlo method, a variance reduction tech- nique. Artzner and F. Term structure of interest rates: The martingale approach.
Barone-Adesi and R. On the theory of option pricing. Bensoussan and J. Dunod, Paris, Alain Bensoussan and Jacques-Louis Lions. Applications of variational in- equalities in stochastic control, volume 12 of Studies in Mathematics and its Applications. North-Holland Publishing Co. Trans- lated from the French. Tomasz R. Bielecki and Marek Rutkowski.
Credit risk: modelling, valuation and hedging. Springer Finance. Springer-Verlag, Berlin, Black and J. Black and M. The pricing of options and corporate liabilities. Hermann, Paris, Processus Stochastiques et Applications. Bouleau and D. Residual risks and hedging strategies in marko- vian markets. Brace, D. Gatarek, and M. The market model of interest rate dynamics.
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