In addition, the concept of complete and incomplete markets is discussed. Elementary probability is briefly revised and discrete-time discrete-space stochastic processes used in financial modelling are considered.
The third part introduces the Wiener process, Ito integrals and stochastic differential equations, but its main focus is the famous Black—Scholes formula for pricing European options. Some guidance for further study within this exciting and rapidly changing field is given in the concluding chapter. There are approximately exercises interspersed throughout the book, and solutions for most problems are provided in the appen. It covers a broad range of foundation topics related to financial modeling, including probability, discrete and continuous time and space valuation, stochastic processes, equivalent martingales, option pricing, and term structure models, along with related valuation and hedging techniques.
The joint effort of two authors with a combined 70 years of academic and practitioner experience, Risk Neutral Pricing and Financial Mathematics takes a reader from learning the basics of beginning probability, with a refresher on differential calculus, all the way to Doob-Meyer, Ito, Girsanov, and SDEs. Includes more subjects than other books, including probability, discrete and continuous time and space valuation, stochastic processes, equivalent martingales, option pricing, term structure models, valuation, and hedging techniques Emphasizes introductory financial engineering, financial modeling, and financial mathematics Suited for corporate training programs and professional association certification programs.
Score: 5. Its scope is limited to the general discrete setting of models for which the set of possible states is finite and so is the set of possible trading times--this includes the popular binomial tree model. This setting has the advantage of being fairly general while not requiring a sophisticated understanding of analysis at the graduate level. Topics include understanding the several variants of "arbitrage," the fundamental theorems of asset pricing in terms of martingale measures, and applications to forwards and futures.
The authors' motivation is to present the material in a way that clarifies as much as possible why the often confusing basic facts are true. Therefore the ideas are organized from a mathematical point of view with the emphasis on understanding exactly what is under the hood and how it works. Every effort is made to include complete explanations and proofs, and the reader is encouraged to work through the exercises throughout the book. The intended audience is students and other readers who have an undergraduate background in mathematics, including exposure to linear algebra, some advanced calculus, and basic probability.
The book has been used in earlier forms with students in the MS program in Financial Mathematics at Florida State University, and is a suitable text for students at that level. Students who seek a second look at these topics may also find this book useful. Popular Books. Discover other editions. Buy Now. Related Books. If a perpetuity-due is used to value the perpetuity, the valuation date is at time 5 years.
The present value of the outflow at time n years is: 4, The present value of the income at time n years is: If this were a multiple-choice exam, we would start with one of the answer choices as our first guess. We could accumulate to time 14 years and then use an annuity-immediate or accumulate to time 15 years and then use an annuity-due. We will use the former approach. The accumulated value that we just calculated is used as the present value for the year annuity-immediate.
We set up the equation of value at time 14 years and solve for the unknown X : , However, any financial calculator should also be able to work this question. Inserting this value for P into formula 7. We have 2. Using the equivalence principle, we have 0.
Figure S7. Note that this uses the assumed, rather than the actual, expenses. In particular, the change in the mortality basis changes 16 p44 from 0. Hence, the benefit on survival to the end of the term is the most significant contribution to the EPV of the benefits.
Since t V repre- sents the value of the investments the insurer should be holding at duration t, adopting the suggestion of a proportionate paid-up sum insured would give the insurer a small profit for each policy becoming paid-up, assuming that experience exactly follows the assumptions. It is generally considered reasonable for the insurer to retain a small profit, on average, as the policy- holder has adjusted the terms of the contract.
Using these values, we have S. The values of S. This is not surprising. As t increases from 0, the time until the sum insured is likely to be paid decreases and so the present value of the loss increases. This will increase the standard deviation of the present value of the loss provided there is still considerable uncertainty about when the policyholder is likely to die, as will be the case for the range of values of t being considered here.
An excerpt from the resulting table of values is shown in part c below. Then, equating policy values before and after the alteration, we have 1 3 Since this term is zero, it has been omitted. For time 0, the explanation is the same except that the amount we accumu- late for one year is the premium rather than the policy value.
An excerpt from the table of values is shown below. The left-hand side of the formula, t V i , is the EPV of all future net cash flows from the insurer from time t, given that the life is in state i at time t. Solutions for Chapter 8 85 8. We can re- consider the model in Figure S8. The life is alive when the process is in state 0 or state 2. The two state model is the same as the three state model under which the two alive states are merged.
The integral expression for the probability can be evaluated numerically to give 0. We will use that figure to reference the states involved in the annuity payments. Let superscript f denote the female sur- vival model, and superscript m denote the male survival model. The equation of value for the premium is 0. Hence, the first term is likely to be much larger than the second term. Hence, the formula holds whatever values the random variables take.
This will be useful in part d. Then, for example, 0. Solutions for Chapter 9 9. Dividing X by the salary earned between ages 59 and 60 gives the replacement ratio as If the member survives to age 26, the value at that time of the accrued benefits is 0. As there are no exits other than by death and there is no death benefit , the funding equation gives the contribution as this amount, and hence the contri- bution is 9.
The amount of the pension does not depend on the financial performance of the underlying assets. If, for example, the underlying assets provide a lower level of accumulation than expected, the replacement ratio would be reduced under a defined contribution plan, but not under a defined benefit plan. For example, if the underlying assets do not perform strongly, the employer may have to increase its contributions to the fund to ensure that benefits can be paid. The pension is payable from age 65 without actuarial reduction, or at age x with the reduction factor applied.
Her salary in the year prior to retirement is s On retirement at age exact 62, the pension is 0. On retirement at age exact 65, the pension is 0. The normal contribution pays for the increase in benefit arising solely from the impact of additional service.
Under the TUC method, future salary increases are not included in the actuarial liability, and the normal contribution must fund both the impact of the additional service and the impact of salary increases on the whole ac- crued benefit.
Because the PUC pre-funds the projected salary increases, the actuarial liability is always higher than for the TUC, but the values must con- verge at the retirement age.
We see that, in this case, the actuarial liability for Giles is much lower under the TUC method than under the PUC method, because he is a long way from retirement.
To build up the higher early actuarial liability, the contributions under the PUC method start out higher than under the TUC method, as the benefit funded under the PUC method is based on the higher, projected final salary.
Later, the contribution under the TUC method becomes higher; the TUC con- tributions pay for the additional year of accrued benefit each year, and also pay for the entire past accrued benefit to be adjusted for the projected one year salary increase. Solutions for Chapter 10 The change in the interest rate basis has resulted in a 2. Solutions for Chapter 10 b Consider a policy with sum insured S. For the risk to be fully diversifiable we require! Let Z denote the present value of the benefit from an individual policy.
E[Z] 0. As this is the interest rate used to calculate P in part a , the EPV of the net future loss is 0. The significant uncertainty about the profitability of the contract is demon- strated by the variance of the present value of the net future loss. See f below. Thus, the answer to part e above is expected to be 2. To simulate values of the interest rate, we can simulate from the N 0.
Suppose a simulated value is y. For each simulated value of I the calculation of the simulated value of future loss proceeds exactly as in part a , using the same future lifetimes as in part a. Generate a simulated future lifetime, as in part a , t j , say.
Generate a simulated N 0. Note that allowing for interest rate uncertainty, in addition to mortality un- certainty, does not increase the standard deviation of the future loss by very much. This is because term insurance is not very sensitive to the interest rate assumption. However, in this question there is no uncertainty about the parameters of the survival model, so that the mortality risk is diversi- fiable whereas the interest rate risk is non-diversifiable.
Hence, for a large portfolio of identical policies the interest rate risk will be relatively more significant. To illustrate that it is not a diversifiable risk, consider a portfolio consisting of n one-year term insurance policies, each with sum insured S, issued to independent lives who are the same age. Let X denote the total amount of claims in the portfolio in the next year. To asses the impact of pandemic risk, the most suitable approach would be to simulate the losses assuming an appropriate model for the incidence of pandemic risk for each year that the portfolio is in force.
The simulations could be used to re-evaluate the moments of the loss distribution, to quantify the impact of pandemic risk on this portfolio. Stress testing involves assuming a deterministic path which is adverse, to assess the most important vulnerabilities of a portfolio. Solutions for Chapter 11 We show here an excerpt from the profit test table, and explain the entries in more detail below. All other rows accumulate the cash flows to the end of each month i. For each row, the cash flows are determined assuming that the policy is in 1 force at the start of the month i.
Column 2 shows the premium income at the start of each month. Column 3 shows the expenses incurred; the initial expenses in the first row include all first month expenses, which is why there are no further expenses in the second row which shows cash flows during the first month of the contract. Column 4 shows the interest earned during the month on the beginning-month cash flows, i. Column 5 shows the expected benefit outgo for the month for a policy in force at the beginning of the month, accumulated to the end of the month.
We assume that the insurer does not establish a reserve for the policy until the end of the first year, so that the initial expenses represent the only outgo at inception.
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