Derivatives pricing pdf

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R Understanding Cash Flow Statements. The model features hump-shaped, level dependent, and unspanned stochastic volatility, and accommodates a … Expand. View 1 excerpt, cites background. We discussed utility-based pricing of defaultable bonds where their recovery values are unpredictable. By considering an optimal investment problem for bond holders, we have derived … Expand.

Risk Pricing Models: Applications. This chapter provides a number of examples and problems applying the pricing models set in Chaps. These models are assuming essentially that risk models are complete in the sense that states … Expand. Approximations and control variates for pricing portfolio credit derivatives. Computer Science, Mathematics. Portfolio credit derivatives that depend on default correlation are increasingly widespread in the credit market. Valuing such products often entails Monte Carlo simulation.

However, for large … Expand. View 2 excerpts, cites methods and background. Interaction of credit and liquidity risks: Modelling and valuation. In this paper we discuss the interaction of default risk and liquidity risk on pricing financial contracts. We show that two risks are almost indistinguishable if the underlying contract has … Expand. Related Papers. Abstract Citations Related Papers. As we are getting used to, the move to a continuous process extracts'a formal technical penalty.

In this case, the IQ-martingale's volatility must be positive with probability 1 - but otherwise our chapter two result has carried across unchanged. Drlftlessness We need just one more tool. Thrown into the discussion of martingales was the intuitive description of a martingale as neither drifting up or drifting down.

We have, though, a technical definition of drift via our stochastic differential formulation. An obvious question springs to mind: are stochastic processes with no drift term always martingales, and vice versa can martingales always be represented asjust C1't dWt for some F-previsible volatility process C1't? One way round we can do for ourselves with the martingale representation theorem. If the technical condition fails, a driftless process may not be a martingale.

Such processes are called local martingales. Exponential martingales The technical constraint can be tiresome. In the simplest models, Black-Scholes for example, we'll have a market consisting of one random security and a riskless cash account bond; and with this comes the idea of a portfolio.

The processes can take positive or negative values we'll allow unlimited shortselling of the stock or bond. There is an intuitive way to think about previsibility. Self-financing strategies With the idea of a portfolio comes the idea of a strategy.

And one particularly interesting set of strategies or portfolios are those that are flllancially self-contained or selffinancing. A portfolio is self-flllancing if and only if the change in its value only depends on the change of the asset prices.

In the discrete framework this was captured via a difference equation, and in the continuous case it is equivalent to an SDE. What SDE? At the next time instant, two things happen: the old portfolio changes value because St and B t have changed 80 3. If the cost of the adjustment is perfectly matched by the profIts or losses made by the portfolio then no extra money is required from outside - the portfolio is self-fmancing.

What kind of portfolios are self-fmancing? The second example should convince us that being self-financing is not an automatic property of a portfolio. Every time we claim a portfolio is selffinancing we have to turn the handle on Ito's formula to check the SDE. Trading strategies Now we can define a replicating strategy for a claim: Replicating strategy Suppose we are in a market of a riskless bond B and a risky security S with volatility C1't, and a claim X on events up to time T. For the same reason as we wanted them in the discrete market models.

The claim X gives the value of some derivative which we need to payoff at time T. We want a price if there is one, as of now, given a model for Sand B. The profit created by the mismatch at time t can be banked there and then without risk. And, as usual with arbitrage, one unit could have been many; no risk means no fear.

Replicating strategies, if they exist, tie down the price of the claim X not just at payoff but everywhere. We can layout a battle plan. We define a market model with a stock price process complex enough to satisfy our need for realism. Then, using whatever tools we have to hand we find replicating strategies for all useful claims X. And if we can, we can price derivatives in the model. The rest of the book consists of upping the stakes in complexity of models and of claims.

We have the tools and we've seen the overall approach at the end of chapter two. So taking the stock model of section 3. Ito will oil the works. There are no transaction costs and both instruments are freely and instantaneously tradable either long or short at the price quoted.

We need a model for the behaviour of the stock - simple enough that we 83 Continuous processes actually can find replicating strategies but not so simple that we can't bring ourselves to believe in it as a model of the real world. Following in Black and Scholes' footsteps, our market will consist of a riskless constant-interest rate cash bond and a risky tradable stock following an exponential Brownian motion. As we've seen in section 3. And as we shall see here, it is quite hard enough to start with.

Zero interest rates If there's one parameter that throws up a smokescreen around a first run at an analysis of the Black-Scholes model, it's the interest rate r.

The problems it causes are more tedious than fatal- as we'll see soon, the tools we have are powerful enough to cope. But we'll temporarily simplify things, and set r to be zero. So now we begin. For an arbitrary claim X, knowable by some horizon time T, we want to see if we can fmd a replicating strategy Pt, 'lj;t.

Finding a replicating strategy We shall follow a three-step process outlined in this box here. Three steps to replication Q under which St is a martingale. We see that the bond option price formula merely changes the discount factor representing the value now of a dollar at time t. Under constant interest rates this was e rt , and under variable interest rates it is just the price of at-bond P O, t.

Otherwise, as long as the other variables are expressed in terms of forward prices and term volatilities, the formula is the same. We can buy or sell the bond before time Tn, transferring the ownership of future but not past coupons along with it. Caps and floors Suppose we are borrowing at a floating rate and want to insure against interest payments going too high.

An individual payment at a particular time Ti is called a caplet, and if we can price caplets, we can price the cap. The option price formula and put-call parity will then price the caplet. A floor works similarly, but inversely, in that we receive a premium for agreeing to never pay less than rate k at each time T i.

That is, we pay an extra amount at time T i. Buying a floor and selling a cap at the same strike k is exactly equivalent to receiving fixed at rate k on a swap. Swaptions A 5waption is an option to enter into a swap on a future date at a given rate. Suppose we have an option to receive fixed on a swap starting at date To. That is not entirely a coincidence Interest rates as a swap is just a coupon bond less a floating bond which always has par value. If you receive fixed on a swap, you have a long position in the bond market; a swap option looks like a bond option.

A simple case is given in Heath-JarrowMorton's original paper. It is an extension of Ho and Lee's model to two factors. Here the W1 Brownian motion provides 'shocks' which are felt equally by points of all maturities on the yield curve, whereas W2 gives short-term shocks which have little effect on the long-term end of the curve.

This model is HJM consistent, so we can read off information about it from that structure. The HJM completeness conditions reduce, in this case, to there being two F-previsible processes ' '1 t and ' '2 t such that the drift a is So the range of available drifts has two degrees of functional freedom away from the martingale measure drift. Like Ho and Lee, this model has normally distributed forward rates - which does allow them to go negative. Nevertheless the model does have the 5. However in a multi-factor setting, the short rate loses its dominant role as the carrier of all information about the bond prices.

We can use the results of section 6. This BlackScholes type of formula allows us to price caps and floors as well as options on the discount T -bonds. However, in the multi-factor setting, the trick we used before to price options on coupon-bearing bonds does not work, making it more involved to price them and the associated swaptions.

We take the instance of the completely general n-factor model, where each volatility surface Ji t, T can be written as a product where Xi and Yi are deterministic functions. For the market to be complete, we need two conditions on the functions 0: and Yi to hold.

Firstly, there should be n F-previsible processes ,1, Consequently the bond 5. We shall see later that this enables us to price caps and swaptions easily.

To price, we only need to know the function " rather than the whole volatility structure. While the, function represents the correlation at time t between changes in the LIBOR rates at different forward dates T, in practice, is calibrated by comparing the model's prices with the market. This valuation has the familiar BlackScholes form because under the forward measure lP'Ti , L Ti-l is log-normal and the calculation proceeds as usual. We can even approximately price swaptions.

We also define 5. It assumes that there is only a single stock in the market. And it assumes that the cash bond is deterministic with zero volatility. None of these assumptions is necessary. The subsequent sections tackle these restrictions one by one and show how a more general model can still price and hedge derivatives. Also we will reveal the underlying framework which governs all these models from behind the scenes.

This is not to say that all models, no matter how complex or bizarre, will always give good prices. But if a model is driven by Brownian motions, and has no transaction costs, it is analysable in this framework. Here r is the constant interest rate, a is the constant stock volatility and J. The process W is lP'-Brownian motion.

Our most general stochastic process can have variable drift and volatility. Not only can they vary with time, but they can depend on movements of the 6. We could replace the constant a by a function of the stock price a St , or even a function of both the stock price and time a St, t.

Even this is not fully general. For instance the volatility at time t might depend on the maximum value achieved by the stock price up to time t. As before, we aim to make the discounted stock price Zt B t-1St Into a martingale. This is achieved by adding a drift It to W. Now the market price of risk depends on the time t and the sample path up to that time. It will, however, continue to be independent of the instrument considered.

It should also be checked that Z is a proper martingale. For instance, it is enough that lEiQI exp! Replicating strategies If X is the derivative to be priced, with maturity at time T, then the procedure is not much different from the basic Black-Scholes technique. Then the martingale representation theorem section 3.

Note that we need at never to be zero. Let us take cPt to be our stock portfolio holding at time t. There is no general expression which will provide a more explicit answer for the option value Vi. To make specific calculations, one needs to know the discount rate Tt, the volatility of the stock Jt - though not its drift - and the derivative itself. Implementation In practice, if the model is much more complex than Black-Scholes, these expectations cannot be performed analytically.

The log-normal cases of section 6. Instead numerical methods must be used. The common feature of models where this happens is that the asset prices are log-normally distributed under the martingale measure Q. The forward price to purchase F at time T is And the value at time zero of an option to buy ST for a strike price of k is Log-normal asset prices When prices, under the martingale measure, are log-normal, there are great advantages. This holds for the Black-Scholes model itself, for some currency and equity models, and also for simple interest rate models.

Explicitly, suppose the stock ST and the cash bond BT are known to be jointly log-normally distributed under the martingale measure Q. The discount factor By. In many cases, this assumption does little harm. If we write an option on, say, General Motors stock, having modelled its behaviour adequately, we are unaffected by the movements of other securities.

However, more complex equity products, such as quantos, depend on the behaviour of at least two separate securities. Even more so in the bond market, where a swap's current value is affected by the movements of a large number of bonds of varying maturities. For instance, our quanto contract of section 4. These two processes have some degree of co-dependence.

In particular, large movements in one may be linked with corresponding movements in the other. Such changes would suggest that the two securities are correlated. Instead of just one lP'-Brownian motion, we will have, in the n-factor case, n independent Brownian motions wi, That means that each Wl behaves as a Brownian motion, and the behaviour of anyone of them is completely uninfluenced by the movements of the others.

Their filtration F t is now the total of all the histories of the n Brownian motions. In other words, FT is the history of the n-dimensional vector Wi, This leads to an enhanced definition of a stochastic process see box. The drift term is unchanged from the original one-factor definition, but there is now a volatility process ai t for each factor.

We must remember that in a multi-factor setting volatility is no longer a scalar, but strictly is now a vector. There is also an n-factor version of Ito's formula and the product rule. If X t and yt are both adapted to the same Brownian motion Wt, then this rule agrees with the first case. Thus the term L:ai t Pi t in the n-factor product rule will be identically zero, agreeing with the second case in section 3.

Finally, we recall from section 5. Generally speaking, if there are more securities than factors there might be arbitrage, and if there are fewer we will not be able to hedge. The situation is not quite as simple as that the bond market, for instance, has an unlimited number of different maturity bonds , but we shall start with the canonical case. Our model then, will contain a cash bond B t as usual, and n different..

Th elr. This vector equation mayor may not have a solution It for any particular t. In fact, not only can the numeraire have volatility, it can be any of the tradable instruments available. We have seen in the foreign exchange context that there can be a choice of which currency's cash bond to use. But no matter which numeraire is chosen, the price of the derivative will always be the same. It is because the choice of numeraire doesn't matter, that we usually pick the stolid cash bond.

When we proved the self-financing condition in chapter three, we assumed that the numeraire had no volatility. This is not actually necessary. But we do have to check that the self-financing equations will still work. We do this with two applications of the product rule. The resulting equation is the self-financing equation.

This also holds for n-factor models with multiple stocks. If we choose B t to be our numeraire, we need to find a measure Q equivalent to the original measure under which B t-1Sti. Then the value at time t of a derivative payoff X at time Tis Suppose however that we choose Ct to be our numeraire instead.

Then we would have a different measure QC under which i Ct-1St. We can actually find out what QC is, or at least what its Radon-Nikodym derivative with respect to Q is. We recall Radon-Nikodym fact ii from section 3. Example -forward measures in the interest-rate market In interest-rate models, it is often popular to use a bond maturing at date T the T-bond with price P t, T as the numeraire.

The new numeraire is the T -bond normalised to have unit value at time zero. Now the forward price set at time t for purchasing X at date T is its current value Vi scaled up by the return on aT-bond, namely Ft Bigger models I F t.

Calculating the forward price for X is now only a matter of taking its expectation under the forward measure. This gives an alternative expression for pricing interest-rate derivatives. If X is a payoff at date T, then its value at time t is So the value of X at time t is just the lP'T-expectation of X up to time t the forward price of X discounted by the T -bond time value of money up to date T. LIBOR rate 6. We have looked at the interest rate market chapter five.

But we have not yet studied an interest rate market of another currency. Now we will. For definiteness, we will imagine ourselves to be a dollar investor operating in both the dollar and sterling interest-rate markets. Our variables will be Table 6. What we have here are two separate interest-rate markets the dollar denominated and the sterling denominated , plus a currency market linking them. The multi-factor model approach is needed to reflect varying degrees of correlation between various securities in the three markets.

Our plan, much as ever, is to follow the three steps to replication. The first thing to do is to find a change of measure under which Xt, yt and Zt are all martingales. J; 6. Bigger models As long as this measure Q is unique, we will be able to hedge. And uniqueness will follow if the volatility vectors of any n of the dollar tradable securities make an invertible matrix. A derivative X paid in dollars at date T will have value at time t The sterling investor The sterling investor is on the other side of the mirror.

This reflects that his numeraire is the sterling cash bond D t rather than the dollar cash bond. As explained in section 6.

Firstly, the C-M-G theorem is used to make the discounted price processes into martingales under a new measure. Then the 6. The repeated recurrence of this program suggests that there might be a more general result underpinning it.

And there is. Before stating this canonical theorem, it is worth carefully laying out some concepts we have already brushed up against. A market is arbitrage-free if there is no way of making riskless profits. An arbitrage opportunity would be a self-financing trading strategy which started with zero value and terminated at some definite date T with a positive value.

A market is arbitrage-free if there are absolutely no such arbitrage opportunities. A market is said to be complete if any possible derivative claim can be hedged by trading with a self-financing portfolio of securities. Suppose we have a market of securities and a numeraire cash bond under a measure lP'. This is just a more precise name for what we call the martingale measure.

Already we have examples of the binomial trees and the continuous-time Black-Scholes model. Both of these are complete markets with an EMM. We have not found an arbitrage opportunity, but neither are we sure that one might not exist. In both the binomial tree and Black-Scholes models we found there was one and only one EMM, and we were able to hedge claims.

Even more so in the multiple stock models section 6. And it was exactly that invertibility which lets us hedge. Arbitrage-free and completeness theorem Harrison and Pliska Suppose we have a market of securities and a numeraire bond.

Then 1 the market is arbitrage-free if and only if there is at least one EMM Q; and 2 in which case, the market is complete if and only if there is exactly one such EMM Q and no other. In the HJM bond-market model, these conditions were also visible. We now see that this is to make sure that the model is arbitrage-free. The other key HJM condition is that the volatility matrix is non-singular for all sequences of dates Tl A martingale is really the essence of a lack of arbitrage.

The governing rule for a Q-martingale M t is that In other words, its future expectation, given the history up to time s, is just its current value at time s. The martingale is not 'expected' to be either higher or lower than its present value.

An arbitrage opportunity, on the other hand, is a one-way bet which is certain to end up higher than it started. Assuming for simplicity a two security market of stock St and bond Bd Then its value at time t is 6. Can this really be an arbitrage opportunity? Crucially, E t is a Q-martingale because Zt is. From which it is clear that VT is zero as well.

Any strategy can make no more than nothing from nothing. A martingale is essentially a 'fair game' and any strategy which involves only playing fair games cannot guarantee a profit. Or in our language, if an EMM exists, there are no arbitrage opportunities. Hedging means unique prices If we can hedge, then there can only be at most one EMM. The indicator function fA takes the value 1 if the event A happens, and zero otherwise.

This is a valid derivative, so it must be hedgeable. We assumed that we could hedge all claims. So also must E t be. The two measures Q and Q' which were trying to be different actually give the same likelihood for the event A. Harrison and Pliska We have only proved each result in one direction. We showed that if there was an EMM there was no arbitrage, but did not show that if there is no arbitrage then there actually is an EMM.

Also we proved that hedging can only happen with a unique EMM, but not that the uniqueness of the EMM forced hedging to be possible. The full and rigorous proofs of all these results in the discrete-time case are in the paper 'Martingales and stochastic integrals in the theory of continuous trading' by Michael Harrison and Stanley Pliska, in Stochastic Processes and their Applications see appendix 1 for more details.

For the continuous case and more advanced models, there has been other work, notably by Delbaen and Schachermayer.

But the increasing technicality of this should not stand in the way of an appreciation of the remarkable insight of Harrison and Pliska. The lists below have been kept short, in the hope that in this case less choice is more. Ross is an introduction to the basic static probabilistic ideas of events, likelihood, distribution and expectation.

Grimmett and Stirzaker contain that material in their first half, as well as the development of random processes including some basic material on martingales and Brownian motion. Probability with martingales not only lays the groundwork for integration, conditional expectation and measures, but also is an excellent introduc- Appendices tion to martingales themselves. There is also a chapter containing a simple representation theorem and a discrete-time version of Black-Scholes.

Both Revuz and Yor, and Rogers and Williams provide a detailed technical coverage of stochastic calculus. They both contain all our tools; stochastic differentials, Ito's formula, Cameron-Martin-Girsanov change of measure, and the representation theorem.

Although dense with material, a reader with background knowledge will find them invaluable and definitive on questions of stochastic analysis. A number of models are discussed, and numerical procedures for implementation are also included. The chapter-by-chapter bibliographies are another useful feature. Duffle is a much more mathematically rigorous text, but still accessible.