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Financial Engineering Course: Lecture 7/14, part 2/2, (Swaptions and Negative Interest Rates)
Financial Engineering Course: Lecture 7/14, part 2/2, (Swaptions and Negative Interest Rates)
The video lecture delves into the intricacies of pricing swaptions under a negative interest rate environment. The instructor introduces the algorithm proposed by Farshid Jamshidian in 1989, which facilitates the transformation of the problem of calculating the maximum of a sum into a sum of specific maximums, given certain conditions. A crucial requirement for this approach is that the function psi_k(x) must be monotone increasing or decreasing in order to achieve accurate calculations. The lecture concludes by assigning homework and providing a Python exercise that focuses on numerical computation techniques.
The speaker emphasizes the significance of determining the value of x_star, which corresponds to the maximum summation of psi equating to zero. Finding this value enables the substitution of the summation size, k, into the equation. The speaker then explores how this condition, along with the use of monotone increasing functions, allows for the elimination of the maximum from the outermost to the innermost part of the equation. Additionally, an exercise is presented that involves computing the expectation of a maximum using both brute force and James' junction streak techniques.
The speaker proceeds to share a personal exercise involving the evaluation of the summation of all psi_i terms for i ranging from 0 to 14. They also touch on the use of Monte Carlo simulation for pricing, employing the Jump Diffusion trick to determine the optimal x value, which significantly influences the summation outcome. The speaker iterates through all the terms for each strike to identify the maximum and subsequently applies the Jamshidian trick by taking the expectation of the maximum and summing the maximum values. However, it is important to recognize certain limitations associated with this technique, such as its inapplicability to high-dimensional factors and the need for careful consideration of underlying assumptions.
Next, the lecture delves into the pricing equation for solutions using the whole white model. This involves defining a zero coupon bond within the whole white model framework, with explicit functions A and B expressed in terms of model parameters. The speaker explains how the function Theta can be represented in terms of zero coupon bonds, which can then be substituted for forward rates. The key takeaway is that, compared to the Black-Scholes equation utilized for pricing swaptions under the annuity measure, it is more advantageous to transition to the measure associated with discounting, which necessitates simulating a short rate process. By employing the Jamshidian trick, it becomes possible to search for R_star and obtain a summation comprising two components: one related to optimization and the other related to zero coupon bonds with specific weights.
The lecture progresses to discuss the pricing of swaptions using Jamshidian's trick, showcasing how this approach facilitates the calculation of implied volatility. The pricing of a swaption can be expressed as a weighted sum of options on zero-coupon bonds, where the weights c_k represent the proportions of options and the zero-coupon bond options are adjusted put options. The pricing of these zero-coupon bond options follows a straightforward process based on previously covered material. The implementation of this approach is relatively straightforward as it involves analyzing monotonic functions during the computation of implied volatility or the pricing of swaptions.
Moving forward, the lecturer explains the sequence of economic events that led to negative interest rates, highlighting the distinction between real and nominal interest rates. They discuss how a lack of trust and deflationary events can impact trading activity and the overall economy. The lecturer acknowledges the interventions carried out by central banks to stimulate the monetary supply and regain trust during the Great Recession, including lowering interest rates to encourage investment and economic activity. However, they also acknowledge the potential drawbacks and unfairness associated with the situation, particularly in terms of purchasing power if inflation exceeds nominal rates.
The lecture delves into the use of negative interest rates as an unconventional measure to incentivize investors to borrow money and invest in the market. The goal is to stimulate the economy by encouraging major financial institutions to purchase assets or engage in market activities. The concept of negative interest rates can work effectively when there is no inflation present. However, if inflation occurs and surpasses the expectations of central banks, the rates may need to be increased to compensate. This can pose a risk to companies and investors with low-rate debts, potentially leading to bankruptcy. These developments highlight the existence of both long economic cycles spanning up to 100 years and shorter-term cycles lasting around 10 years. The lecturer also touches upon the concept of inflation and emphasizes the importance of understanding how the inflation market operates in order to be prepared for any inflation-related phenomena.
Furthermore, the instructor delves into the issue of negative interest rates, which have become more prevalent in the current economic environment. A comparison of European rates between 2008 and 2017 demonstrates that short-term investments now yield negative rates, providing little incentive for saving. The instructor also discusses the challenges posed by negative interest rates when it comes to calculating volatilities and dealing with float rate bonds. Consequently, there is a need for new and alternative models to address these issues effectively. Additionally, the instructor mentions that banks often attempt to mitigate the adverse consequences of negative interest rates by incorporating maximums or waiving coupon payments for clients.
The video lecture proceeds by exploring strategies for dealing with negative interest rates and determining implied volatility for pricing options. This is crucial because, in a scenario where interest rates become negative, trading activities for derivatives can come to a halt. When using the traditional Black-Scholes model to calculate implied volatilities, the output may be "NaN" (not a number). One approach to tackle this challenge is to utilize shifted implied volatilities. This involves incorporating an additional shift parameter in the Black-Scholes model to account for the maximum negative interest rate. However, it is important to monitor this shift parameter closely. If it approaches the negative forward, the issue arises once again.
The speaker further discusses the use of the shifted variant of the LIBOR for pricing swaptions, highlighting how it resolves the problem of negative interest rates. By introducing an extra shift parameter, even if the considered strike is negative, it does not affect the pricing outcome. This is because the shifted model guarantees that rates remain above the negative range, given the log-normal nature of the model. Moreover, it is crucial to associate the shift parameter with the expiry and tenor of the underlying asset. To illustrate these concepts, the speaker provides visual representations of the log-normal distribution and showcases option prices under different shift parameters.
Expanding on the notion of shifting within the Black-Scholes formula, the lecture delves into the impact of shift parameters on volatilities and distribution shapes. A code implementation is presented for pricing, utilizing both Monte Carlo simulation and analytical expressions. The simulation involves generating paths for the shifted Geometric Brownian Motion (GBM) and calculating the average price. The code also adjusts initial points, generates densities for the local model with a shift for theta, and plots log-normal densities for different shift parameters. The importance of keeping the shift parameter as close to zero as possible is emphasized, as higher shift parameters can significantly affect the distribution and volatility.
The professor emphasizes the crucial aspect of accurately accounting for shift parameters when pricing swaptions, highlighting that even a small mistake can lead to significant pricing errors. The lecture consolidates the concepts covered, including the pricing of caplets and floors, interest rate swaps, pricing of swaptions using the Black model, negative interest rates, and the application of Jamshidian's trick in swaption pricing under the Hull-White model. To conclude, the professor assigns homework to students, encouraging them to apply the concepts learned in the lecture to calculate implied volatilities and price options.
In the final section of the video, the speaker discusses how to price an option under the whole line model by combining two blocks together. The objective is to compare the results with Monte Carlo simulation to ensure the code is free from bugs and errors. The lecture concludes with the instructor encouraging students to enjoy their assignments and delve further into the topics covered.
Tthe video lecture provides a comprehensive exploration of negative interest rates, pricing swaptions, and the application of various mathematical techniques and models. It emphasizes the importance of understanding concepts such as Jamshidian's trick, shifted implied volatilities, and the influence of shift parameters on pricing and distribution shapes. By equipping students with these tools and insights, the lecture prepares them to navigate the complexities of the financial world, make informed decisions, and accurately price options and swaptions under challenging market conditions.
Financial Engineering Course: Lecture 8/14, part 1/4, (Mortgages and Prepayments)
Financial Engineering Course: Lecture 8/14, part 1/4, (Mortgages and Prepayments)
In the lecture, the concept of pricing mortgages is thoroughly discussed, highlighting the complex nature of this task from a financial engineering perspective. The main challenge lies in managing the risks associated with client prepayments and additional payments made on top of regular monthly installments. Two types of mortgages are specifically focused on: bullet mortgages and annuity mortgages.
A bullet mortgage entails clients paying only the interest rate and outstanding notional at the end of the contract, while an annuity mortgage involves gradual reduction of the notional until no outstanding notional remains at the contract's conclusion. Prepayments, pipeline risks, and the inclusion of people's behavior and incentives in financial contract pricing are also addressed in the lecture.
It is emphasized that risks related to prepayments are minimized for floating rate mortgages since clients have no optimal incentive to make prepayments. The constant prepayment rate is discussed in relation to portfolio management. Assessing the repayment profile of a mortgage portfolio requires considering prepayment risks based on the overall repayment profile rather than individual clients.
The lecture delves into the index amortizing swap and how it can be utilized to match prepayment risks within the portfolio. Furthermore, the behavioral aspect of prepayments is explored, taking into account refinancing incentives and individuals' rational or irrational decision-making when deciding to allocate extra funds towards their mortgage.
The risks faced by banks and other financial institutions are also highlighted, particularly regarding mortgage cash flows and the uncertainty surrounding them. This includes the potential for client defaults and the need for banks to resell houses, sometimes at a loss. The lecture emphasizes the importance of pricing and risk management in issuing mortgages, specifically addressing pipeline risk and prepayment risk. Pipeline risk arises due to the time delay between agreeing to a mortgage and signing the contract, which leaves room for interest rate changes during that period.
The risks associated with mortgages, such as pipeline risk and prepayment risk, are further elaborated upon. Pipeline risk refers to the risk that a client may opt for a lower interest rate, which occurs when a client has the optionality to execute a contract at a lower rate. On the other hand, prepayment risk pertains to a client's desire to modify the contract and the associated risk of prepayments. Financial institutions that enter into contracts with clients face unhatched positions that introduce additional risks in derivative pricing. Mortgages possess an embedded option that enables the mortgagee to pay off their mortgage faster than the agreed schedule, resulting in prepayment risk. The lecture highlights that it is logical for a mortgagee to prioritize paying off their mortgage rather than keeping savings in an account with negative or no interest rates.
While pricing mortgages under the risk-neutral measure is important, the lecture emphasizes that consumer incentives to take out or prepay mortgages may not be solely driven by market circumstances. Factors such as age or financial freedom can influence the incentive to prepay mortgages and avoid monthly payments. The lecture explores the connection between pricing under the risk-neutral measure and the behavioral aspects involved in pricing prepayments. It also delves into two types of amortization schedules: annuity mortgages and bullet mortgages, which ensure borrowers ultimately repay the initial borrowed sum for purchasing the house along with additional amounts representing loan costs.
The video explains the relationship between mortgages, prepayments, and the risks faced by financial institutions. Prepayments made by borrowers, exceeding their scheduled payments, require the bank to adjust their hedge, leading to additional costs. Large prepayments can also decrease the bank's incoming cash flow and contract duration. However, a significant number of sudden prepayments generate prepayment risk that needs to be analyzed and mitigated. To manage these risks, banks create mortgage portfolios and utilize swaps to offset fixed-rate payments.
The lecturer discusses the risks and profits associated with mortgages and prepayments. Mortgages are priced at the portfolio level, with hedges consisting of significantly larger notionals. The profitability for a bank in a mortgage depends on factors such as the notional amount, duration of the loan, and the interest rate. Prepayments, however, pose a potential loss for the bank. Other risks associated with mortgages include pipeline risk, tax risk, default risk, and the risk of a housing market crash. The lecture emphasizes that the amortization plan chosen for a mortgage can impact the amount of interest accrued.
The lecturer provides a detailed exploration of different types of mortgages and their associated amortization schedules. One such type is the bullet mortgage, which involves a single lump sum payment at the end of the mortgage term. While this simplifies payment obligations throughout the term, it carries the risk of a substantial payment due at the end. The lecturer suggests that a bullet mortgage may be suitable for individuals who have alternative investment opportunities, such as a savings account with a higher interest rate than the mortgage. The lecture also offers an overview of monthly payments and accrual periods, providing a comprehensive understanding of mortgage payment structures.
Constant prepayment rates associated with mortgages are discussed in detail. These rates represent fixed amounts that homeowners choose to prepay towards their mortgages. The prepayment rate is typically estimated based on a large portfolio of mortgages, and it affects the notional value over the amortization period. Legal constraints on prepayment amounts are also mentioned. The lecturer calculates the total amount of interest paid on a mortgage using a prepayment rate and emphasizes the importance of considering prepayments in mortgage pricing. Numerical experiments and exercises are presented to illustrate the concepts, and a Python plot and code are used to analyze cash flows and amortization schedules effectively.
The lecture emphasizes the impact of prepayment rates on the amortization of a mortgage over time. An example is provided for a 10-year fixed-rate mortgage at a 3% interest rate, which the bank needs to hedge using a swap. The experiment compares scenarios with and without prepayments, demonstrating how prepayments gradually decrease over time as the outstanding notional decreases. The results highlight that prepayments can significantly reduce the amount of interest paid, but a substantial lump sum payment is still required at the end. The lecturer also notes that in practice, mortgages may be combined with savings accounts or derivatives that offer higher returns, while also minimizing taxation on the outstanding notional.
Furthermore, the lecture dives into the construction of an amortization schedule for a bullet mortgage using Python code. The code allows for the calculation of payment schedules based on given interest rates and prepayment rates. It provides a matrix array that outlines the required payments throughout the mortgage's lifetime. The prepayment rates can be expressed as percentages, making it convenient for analyzing a large portfolio of mortgages. The payment schedule is affected when prepayments are introduced, showcasing the flexibility and usefulness of the Python code for analyzing payment structures.
The speaker explains the columns of a mortgage payment matrix. Time is represented in the first column, followed by the outstanding notion in the second column. Prepayment, repayment, and notional quote are defined in the subsequent columns. The prepayment column indicates the fraction of the notional that will be prepaid and is determined by the constant prepayment rate (CPR). Repayment, in the fourth column, signifies the reduction in the outstanding notion each month with regular payments. The fifth column represents interest payments, while the last column displays the monthly installments required. The lecturer showcases the model using a 30-year bullet mortgage example with no prepayment.
In summary, the lecture provides an extensive exploration of mortgage pricing, prepayment risks, and their impact on financial institutions. It covers various types of mortgages, including bullet mortgages and annuity mortgages, and emphasizes the importance of considering client behavior and incentives in mortgage pricing. The lecture delves into the risks faced by financial institutions, such as pipeline risk and prepayment risk, and discusses strategies for mitigating these risks through portfolio management and the use of financial derivatives like swaps. The lecture also highlights the uncertainty surrounding mortgage cash flows, including the possibility of client defaults and the need for banks to resell houses at a potential loss.
Moreover, the lecture acknowledges that pricing mortgages solely under a risk-neutral measure may not capture the full range of consumer incentives and behaviors. Factors such as age, financial freedom, and personal preferences can significantly influence clients' decisions to prepay or refinance their mortgages. Therefore, the lecture emphasizes the importance of integrating behavioral aspects into mortgage pricing models, considering the motivations and rational/irrational decision-making of borrowers.
The lecturer explores the concept of constant prepayment rates and their relationship to portfolio management. Instead of analyzing prepayment risks on an individual client level, the lecture stresses the need to assess the overall repayment profile of a mortgage portfolio. By considering the aggregate prepayment behavior, banks can better manage the associated risks and use tools like index amortizing swaps to match and hedge prepayment risks effectively.
Furthermore, the lecture delves into the risks faced by financial institutions due to mortgages and prepayments. When borrowers make significant prepayments, it necessitates adjustments to the bank's hedging strategy, resulting in additional costs and potential disruptions to cash flow and contract duration. The sudden prepayment of a significant number of clients creates prepayment risk, which must be carefully analyzed and hedged to mitigate its impact on the bank's portfolio. The lecturer highlights that banks create mortgage portfolios and utilize swaps to offset fixed-rate payments, thereby reducing risks.
The lecture concludes with a discussion on the valuation of mortgage securities, noting that it depends on market observable quantities. Although this aspect is briefly mentioned, the lecture implies that a more in-depth exploration of these quantities will be covered in subsequent parts of the course.
The lecture provides a comprehensive understanding of mortgage pricing, prepayment risks, and their implications for financial institutions. It addresses various types of mortgages, behavioral aspects, portfolio management techniques, and risk mitigation strategies. By considering the complex dynamics of mortgage cash flows, prepayments, and client behavior, the lecture equips viewers with the knowledge and tools necessary to navigate the challenges of pricing and managing mortgage portfolios effectively.
Financial Engineering Course: Lecture 8/14, part 2/4, (Mortgages and Prepayments)
Financial Engineering Course: Lecture 8/14, part 2/4, (Mortgages and Prepayments)
In addition to the topics covered so far, the lecture further explores the concept of annuity mortgages and their essential characteristics. An annuity mortgage is a type of mortgage where the outstanding notion gradually decreases over time due to regular repayments. The monthly payments for annuity mortgages comprise two components: interest rate payments and contractual repayment schedules denoted by "q." These repayments are structured in a way that the outstanding notional is reduced with each payment until the final payment covers the remaining balance.
The instructor explains that annuity mortgages have fixed installment payments throughout the contract's duration, ensuring a balance between the interest rate and principal portions. This balance results in a constant sum on each payment date. As the outstanding notional decreases, both the repayments and interest rate payments follow opposite trends. The interest compounded on the remaining notional diminishes over time. To calculate the correct installment amount, the discounted future cash flows of the mortgage must be equal to the value of the outstanding notional. Any prepayments made should adjust the constant payment amount accordingly.
The lecture delves into the calculation of constant payments or annuities. The value of an annuity is determined by summing all future cash flows discounted to the present day. By utilizing the formula for geometric sums, one can derive an analytical expression for the annuity. However, if prepayments are made, the constant payment amount will change, necessitating a recalculation. The lecturer also explains how to calculate interest rate payments and principal payments, as well as how to adjust the outstanding notional after prepayments are made.
Furthermore, the lecturer emphasizes the notion of time and its impact on mortgages, repayments, and prepayments. As repayments and prepayments are made, the outstanding notional of a mortgage decreases, leading to a corresponding decrease in the monthly payments. The prepayment rate can be seen as a reformulation of the interest rate payment and is included in the interest rate component. When a borrower decides to prepay an installment, the remaining payment schedule is adjusted to reflect the updated outstanding notional. Graphs are presented to illustrate the impact of varying prepayment levels on the constantly reducing notional, considering scenarios with zero percent and 12 percent prepayment rates. The lecture concludes that larger prepayment rates can hinder the reduction in the outstanding notional.
The lecture also delves into the structure of annuity mortgages and their repayment mechanism. An annuity mortgage consists of fixed monthly payments that encompass both repayment and interest rate components. These fixed payments ensure a balanced repayment structure over the lifetime of the mortgage. The lecturer explores the impact of prepayments on monthly payments and explains how the constant payment amount (c) needs to be recalculated when prepayments are made. Additionally, the notional amount of the mortgage gradually decreases until there is no outstanding notional remaining. By the end of the mortgage period, all payments reach zero, facilitating a smooth transition in the presence of prepayment rates. The lecturer provides Python code for the repayment schedule and explains its significance.
Furthermore, the lecture discusses the steps involved in calculating the new notional after a repayment or prepayment takes place in a mortgage. This process is iterative and considers the previous notional, repayment, prepayment rates, and interest rate payments over the lifetime of the contract. If the prepayment is time-dependent or stochastic, adjustments need to be made in the calculations. Additionally, the lecture highlights that prepayments reduce monthly costs, while a zero prepayment rate leads to constant installments throughout the mortgage's lifetime. It is explained that if prepayment occurs only on a specific date, the installments will remain constant until that date, after which everything will be recomputed.
The lecturer then moves on to explain how prepayment rates for mortgages are estimated from a portfolio management perspective. The prepayment rate, represented by the lambda coefficient, is a crucial factor in portfolio management as it affects the performance and risk of the portfolio. Estimating the prepayment rate involves considering historical data and analyzing various factors that influence a borrower's decision to prepay their mortgage. These factors may include interest rates, individuals' financial goals, and market conditions. The lecture explores how market observable quantities impact the prepayment rate and discusses methods for estimating it based on a portfolio of mortgages.
Next, the lecture delves into the concept of refinancing incentive and its relationship to prepayment models for mortgages. Borrowers are more likely to prepay their mortgage when they observe a lower interest rate compared to the rate of their current mortgage. This refinancing incentive is a key driver in any prepayment model and is closely linked to market rates. Additionally, the type of mortgage, its maturity, and the collateral associated with it can affect mortgage rates. The lecturer emphasizes that the attractiveness of the collateral influences the interest rate offered by banks. Other factors that can impact prepayment rates include the age of the mortgage, the month of the year, tax considerations, and burnout.
The lecture discusses factors that affect prepayment rates, considering both the market situation and individual client profiles. The interest rate incentive is identified as the most significant factor influencing prepayment rates. Determining the prepayment incentive involves evaluating market observable quantities. The lecture suggests that the most reasonable benchmark for pricing a mortgage is a swap rate, which banks use to derive the mortgage rate for new clients. The liquidity risk factor determines an additional spread for the mortgage rate. Prepayments are viewed as a cost for banks as they reduce the hedging position, and determining the mortgage rate involves assessing associated risks and profits.
The focus then shifts to the incentive function of mortgage prepayments. The swap rate is dependent on prepayment amounts, which are directly related to the initial mortgage rate of a fixed-rate mortgage and the rate associated with refinancing. The liquidity risk coefficient and the bank's profit margin further contribute to determining the new mortgage rate. However, the lecture acknowledges that people do not always behave logically or rationally when deciding to prepay their mortgage. For example, individuals may choose to prepay when it is not necessarily optimal, such as when they come into extra money. The incentive function is defined as the difference between the current mortgage rate and the new mortgage rate, and it is used to assess whether it makes sense to refinance or prepay a mortgage.
The instructor emphasizes the importance of understanding the shape of the incentive function in different market circumstances. The graph representing the incentive function exhibits breakpoints and a sigmoid shape, which reflects both the incentive function and the non-rational behavior of borrowers. The lecture highlights the significance of considering small details when implementing incentive functions, as even subtle variations can have a crucial impact.
The lecture concludes with the speaker discussing the concept of prepayments on mortgages. As the swap rate decreases or reaches zero, the incentive for prepayment diminishes. In cases where swap rates become negative, the incentive may reach its maximum level. The shape of the incentive function graph is further explored, with particular attention given to the difference between the old mortgage rate and swap values. It is underscored that although the shape is generally decreasing, it is essential to pay attention to small details when implementing incentive functions.
The lecture provides a comprehensive understanding of annuity mortgages, their repayment mechanisms, the calculation of constant payments, the impact of prepayments, estimation of prepayment rates, refinancing incentives, and the factors influencing prepayment behavior. By considering these aspects, individuals can make informed decisions regarding their mortgages and understand the dynamics of the mortgage market.
Financial Engineering Course: Lecture 8/14, part 3/4, (Mortgages and Prepayments)
Financial Engineering Course: Lecture 8/14, part 3/4, (Mortgages and Prepayments)
In today's lecture, we aim to establish a strong connection between refinancing incentives, prepayments, and various types of mortgages. We begin by examining the concept of a constant payment rate and its relationship to mortgages as amortizing swaps without uncertainty. Building upon this foundation, we introduce the concept of an index amortizing swap, which incorporates clients' willingness to prepay or refinance based on market conditions. This further leads us to link refinancing incentives and the benchmark swap rate in derivative pricing, specifically applied to a mortgage portfolio that amortizes over time.
To better understand the dynamics involved, we explore both deterministic and stochastic functions of amortization schedules. While a deterministic function suffices in simpler cases, the more advanced scenario introduces stochasticity, primarily driven by the swap rate. This stochasticity captures the irrational behavior of clients, which is important to consider when observing market rates and incorporating them into the pricing of an amortizing swap. However, pricing a stochastic notion poses challenges, and a standard approach may not suffice, necessitating the involvement of advanced counterparties to create such derivatives.
We delve into the impact of stochastic factors, such as the swap rate and volatility, on mortgage pricing and prepayment risk. Employing Ito's lemma becomes essential to ascertain whether observed quantities adhere to martingale properties, particularly when the factor being observed is a function of Libor. It is noteworthy that prepayment risk only exists in fixed-rate mortgages, as floating-rate mortgages lack the incentive for prepayment. By understanding the principles behind index amortizing swaps, we can effectively manage prepayment risk and reduce interest rate risk.
Expanding our knowledge, we introduce the concept of an index amortizing swap—an over-the-counter interest rate swap that combines a plain vanilla swap with partial absorption. Typically designed for sophisticated investors due to its large notionals, this exotic derivative is not commonly included in XVA evaluations. Nevertheless, exploring the pricing of mortgages and their connection to prepayment behavior, refinancing incentives, and market observations holds significant value. Deterministic amortization schemes serve as commonly traded instruments, facilitating their processing and integration into the framework of an index amortizing swap, which inherently carries embedded optionality.
Our focus now shifts to the modeling of the notional of an index amortizing swap, which encapsulates the possibility of stochastic amortization via a complex function tied to the type of mortgage. The prepayment rate, in turn, becomes a function dependent on the swap rate, while the refinancing incentive relies on historical estimations derived from various factors such as age, income, wealth, and taxes. Estimating the coefficients involved in these prepayment models requires historical data and detailed analysis. As each bank's portfolio of clients differs, determining these coefficients becomes an extensive study unique to each institution.
In the lecture, the speaker also discusses the estimation of coefficients used in mortgage prepayment models, emphasizing that they are not market-driven but solely based on historical behavior estimations. Moreover, the concept of an index amortizing swap is defined, highlighting its utilization of refinancing incentives and prepayment rates, which are determined based on historical data, to establish mortgage notional values. By evaluating these expectations, one can ascertain the overall value of a mortgage portfolio and make necessary adjustments according to market conditions.
The instructor further elaborates on the complexities involved in the decomposition of notionals, explaining that they cannot be further divided as they depend on the swap rate, which, in turn, is not independent of the Libor swap rate. While assuming independence is possible, it is not recommended without careful study of the correlation's impact. Instead, employing Monte Carlo simulation is advisable. This entire process entails several steps, including pricing a swap rate, estimating the refinancing function, constructing a function based on the mortgage type, and adjusting notionals. The upcoming block of the lecture will focus on simulating the north node, which provides insights into how notionals behave over time based on the type of mortgage. It is crucial to approach this process with meticulous attention to detail and careful consideration of each step involved.
In summary, today's lecture has emphasized the interplay between refinancing incentives, prepayments, and different types of mortgages. We have explored the concept of amortizing swaps, both with and without uncertainty, and introduced the index amortizing swap, which incorporates market-driven prepayment behavior. By linking refinancing incentives, benchmark swap rates, and derivative pricing, we can effectively manage a mortgage portfolio's amortization over time.
Stochastic factors such as the swap rate and volatility play a significant role in pricing and assessing prepayment risk. The use of Ito's lemma becomes essential to evaluate observed quantities' martingale properties accurately. It is also important to differentiate between fixed-rate and floating-rate mortgages when considering prepayment risk.
We have delved into the intricacies of the index amortizing swap, an exotic derivative that combines a plain vanilla swap with partial absorption. Although typically designed for sophisticated investors, it offers valuable insights into mortgage pricing, prepayment behavior, and market observations. Deterministic amortization schemes align well with this type of swap, simplifying its processing and incorporating embedded optionality.
The lecture has emphasized the modeling of the notional of an index amortizing swap, considering stochastic amortization and the intricate function tied to the mortgage type. The estimation of coefficients for prepayment models requires historical data and detailed analysis, varying among banks based on their unique client portfolios.
Furthermore, we have discussed the challenges associated with decomposing notionals and the importance of understanding the correlation between swap rates and Libor rates. Employing Monte Carlo simulation is recommended for pricing derivatives with stochastic notions, offering a comprehensive approach to handle the complexity of the process.
This lecture has shed light on the connection between refinancing incentives, prepayments, and various mortgage types. By incorporating market observations, historical data, and advanced modeling techniques, we can effectively manage prepayment risk and navigate the complexities of pricing mortgage portfolios.
Financial Engineering Course: Lecture 8/14, part 4/4, (Mortgages and Prepayments)
Financial Engineering Course: Lecture 8/14, part 4/4, (Mortgages and Prepayments)
In the lecture, the pricing of mortgages takes center stage, and the instructor demonstrates a Python experiment that combines the knowledge of pricing annuities and mortgages, including refinancing incentives, to simulate the stochasticity in notional values. The lecture covers various aspects such as Swaps, pricing models, and the associated risks, including pipeline options, that banks face.
A significant part of the lecture focuses on the behavior of the notional profile for bullet and annuity mortgages and how they can be simulated. It is highlighted that the randomness of simulated paths has a substantial influence on the notional profile. Prepayments are shown to have a significant impact on the notional value, especially for bullet mortgages, while annuity mortgages are comparatively less affected. The lecturer presents Python codes that are extended to make the constant prepayment rate time-dependent, requiring inputs such as the zero coupon bond curve, swap rate, and stochastic paths at each time step.
The speaker delves into the prepayment rate for mortgages and its influence on the outstanding notional and incentive function, which is dependent on market factors like the swap rate. Two mortgage payment profiles, bullet and annuity, are presented, and their indexing for time and prepayment behavior is explained. The lecture introduces two incentive functions, sigmoid and logistic, and emphasizes that the yield curve used for market simulation is fixed at five percent. The Monte Carlo paths generated for interest rate parts serve as the basis for evaluating the incentive functions.
The instructor further discusses the simulation of swap rates, considering the client's perspective and their outstanding mortgage. They define the incentive function based on the client's mortgage and iterate over time steps to create notional schedules. The incentive function is evaluated for the mortgage profile at each time step, and this information is stored in metrics, resulting in a stochastic notional that depends on the incentive function, stochastic interest rates, and the type of mortgage. The lecture includes plotted results, showcasing the paths with and without prepayment options.
The lecturer emphasizes the significance of incentive functions and stochasticity in the context of mortgages and prepayments. Various examples of notional profiles are shown, illustrating their behavior under different scenarios, including rational and irrational behavior using the sigmoid function. The impact of increasing uncertainty and volatility is discussed, emphasizing the role of the incentive function in risk exposure and the need for buying or selling index amortizing swaps or swoptions. The number of steps in the simulation is shown to impact the notion profile, and practical adjustments are highlighted.
An in-depth discussion is held on annuity mortgages in the rational setting, with a graph depicting how prepayment incentives work and how clients determine their maximum prepayment. Limitations such as legal restrictions or penalties may exist, influencing the client's choices. A comparison between bullet mortgages and annuity mortgages reveals that uncertainty strongly depends on the schedule, with a reduction in notional leading to lower uncertainty. Decomposing a complex order portfolio into linear and non-linear parts is explained, with financial engineering offering a possibility for financing without necessarily resorting to index amortizing swaps.
The calculation of payments and the notional value of a mortgage are explained using a simplified case of a two-period mortgage. The notional value is split into two parts: n-up and the difference between n-up and n-low. The latter part handles mortgage prepayment and is only positive if the strike is greater than L-K, similar to a call option's nonlinear effect. The calculation for the second payment involves a summation of two payments, with the first payment being deterministic and the second payment being discounted based on possible outcomes of n-up and n-low.
The lecture redefines the index amortizing swap as a combination of a deterministic amortizing swap and a nonlinear floorlet. The lecturer highlights that purchasing a mortgage can be seen as entering into a long position in a swap, with prepayments reducing the mortgage's notion, which is akin to an option to enter a swap. The composition of an index amortizing swap can be optimized to replicate its risk profile, and advanced exotic derivatives like this can be hedged or replicated using simplified liquid instruments available in the market. The lecture consistently emphasizes the prepayment risks and their impact on the notion of the mortgage portfolio.
Another topic discussed in the video is the additional risk associated with European mortgages or Dutch mortgages, specifically related to the client's ability to choose the fixing rate of the mortgage. The lecture highlights two critical dates: t0, the quotation day, and t1, the time when the client signs a contract with the bank. The risk for the bank is that the client may choose the lower rate, leading to substantial losses. This risk is referred to as pipeline risk, and it is crucial to manage it effectively to protect the bank's profits.
The discussion revolves around pricing pipeline risk for mortgages and prepayments. Hedging pipeline risk poses challenges as it requires the use of swaptions, necessitating continuous recalculation of values and associated profiles. This process is not a one-time occurrence for a single client; it applies to each individual client. Furthermore, risks are accumulated in a portfolio, necessitating bundling of mortgages into a larger portfolio that needs to be aged. The lecture concludes by focusing on pricing pipeline risk, incorporating optionality for clients to choose the rate at the quotation date or settlement date, depending on which rate is smaller.
The lecturer explains the decomposition of the index amortizing swap into a linear product and the remaining swaption part. This decomposition strategy is common in finance when dealing with structures involving optionality. To handle the associated risk, Black's formula is introduced as a straightforward approach, requiring only volatility for the swaption of those configurations. The lecture emphasizes the importance of considering client behavior and incentives, along with pricing in the risk-neutral world when working with mortgages.
In addition, the speaker compares bullet mortgages and annuity mortgages, highlighting that annuity mortgages involve regular repayments over time instead of a lump sum payment at the contract's end. The lecture explores the factors that lead to client prepayments, such as refinancing incentives, and presents numerical experiments on notional simulation based on market and incentive functions of mortgages. The discussion also covers the risks associated with transitioning from an index amortizing swap to stochastic prepayment and options.
Towards the end of the lecture, exercises are provided for students to simulate notionals and price mortgage contracts. The focus shifts to the concept of convexity and its impact on expectations in finance. Students are tasked with determining the side of a function that yields equality when compared to a library with a martingale payment measure, using analytical or numerical methods. The lecture introduces the concept of convexity collection and explores its effects on expectations. Students are also encouraged to modify code to ensure that prepayments occur only a few times during the lifetime of the mortgage contract, further developing their programming skills in Python.
Overall, the lecture provides a comprehensive understanding of mortgage pricing, covering various complexities such as prepayment risks, incentive functions, stochasticity, pipeline risk, and the decomposition of index amortizing swaps. It equips students with the necessary knowledge and practical skills to analyze and simulate mortgage portfolios while considering market factors and client behavior.
Financial Engineering Course: Lecture 9/14, part 1/2, (Hybrid Models and Stochastic Interest Rates)
Financial Engineering Course: Lecture 9/14, part 1/2, (Hybrid Models and Stochastic Interest Rates)
In the lecture, the focus is on hybrid models and their significance within the portfolios of financial institutions. These models are utilized to simulate future scenarios for various asset classes, including interest rate swaps, foreign exchange contracts, and stocks. The lecturer begins by discussing the importance of employing hybrid models for xVA (valuation adjustments) and VaR (value at risk) calculations. They introduce the Black-Scholes hybrid model, which establishes a connection between stocks and interest rates and can be easily extended to forex pricing. This model serves as a foundation for further discussions on stochastic volatility models.
The lecture is divided into blocks, with the second block centering on stochastic volatility models. The Heston-Hull-White model is discussed, which involves incorporating stochastic volatility into the hybrid model framework. The lecturer provides an overview of the model's dynamics and highlights its application in simulating potential future values of portfolios. The aim is to evaluate risks and assess the value of portfolios that encompass multiple asset classes, such as interest rates, stocks, foreign exchange, commodities, credit, and inflation. The speaker emphasizes the correlation between different asset classes and the need to account for their interdependencies.
The lecture also emphasizes the calibration of multi-dimensional stochastic differential equations (SDEs) to market quotes, particularly for simulating correlated processes from different asset classes. Hybrid models are particularly useful for hybrid payoffs and were initially popular for pricing exotic derivatives. However, due to cost considerations and regulatory restrictions, they have found more efficiency in the xVA and hVAR (hybrid value at risk) framework. The concept of netting effect, which considers the offset values of different asset classes due to their correlations, is highlighted as an important factor in portfolio evaluation and exposure calculation.
While hybrid models offer benefits in evaluating call options and potential future exposures, the lecture acknowledges the challenges associated with these models. The instructor suggests keeping the models as simple as possible to facilitate fast evaluations, as speed is crucial in pricing derivative products. Calibration to market data and considering correlations between different stochastic differential equations are essential. Some approximations may be necessary when dealing with non-zero correlations. The lecture suggests Monte Carlo simulations or partial differential equations (PDEs) as methods to evaluate hybrid models.
The limitations of using PDEs for valuing portfolios with assets from different classes are discussed due to the high dimensionality involved. The lecture advocates for the use of Monte Carlo simulations, which provide a more practical approach. Efficient valuation and calibration are highlighted as crucial for portfolio evaluation, as thousands of evaluations are typically required. The lecturer mentions the extension of the Black-Scholes model with Hull-White for interest rates, emphasizing the role of stochasticity and time dependence in hybrid models. The remaining mechanics of the model remain similar to the standard Black-Scholes model.
The lecturer also delves into the concept of changing the measure from risk-neutral to the T forward measure to leverage the advantages of hybrid models in dealing with stochastic discounting. They discuss the calculation of expectations for European payoff types based on time and underlying variables, using integral forms and the Radon-Nikodym derivatives from measure transformations. The dynamics of stock and discounted stock are explained, emphasizing the need for them to be martingale processes. The concept of forward stock price is introduced to simplify the process.
Further explanations are provided on the derivation of the forward stock price stochastic differential equation (SDE) and the importance of performing log transformations to make it linear in state variables. The lecturer applies Ito's lemma to the forward stock price SDE and addresses the measure transformation required for the process. The resulting driftless SDE features two separate Brownian motions, corresponding to the stock and interest rates, with correlation between them. The factorization of the two Brownian motions is discussed in terms of their distributional properties.
The dynamics of the forward stock are explored in the lecture using a hybrid model with two stochastic differential equations. It is emphasized that the volatility of the forward stock is no longer constant but influenced by the volatility of interest rates. The speaker discusses the calculation of implied volatilities within the context of stochastic interest rates. They suggest using prices to determine implied volatilities and highlight the importance of switching between risk-neutral and T-forward measures to exclude stochastic discounting from payoffs. This section underscores the complexities involved in working with stochastic interest rates in financial engineering.
The lecture introduces a stochastic interest rate model with a one-dimensional process and a time-dependent volatility function reminiscent of the Black-Scholes equation without interest rates. The discounting component is factored outside of the expectation, and the pricing process for European options involves only the constant value of the integral of the time-dependent function. The speaker also presents the cost method for pricing, leveraging the affinity of the Black-Scholes model, and provides insights into how stochastic discounting is handled within this approach.
In the subsequent segment, the speaker discusses the integration process required to obtain the expression for the constant "c" and its relevance in pricing with a stochastic interest rate. They explain that the Black-Scholes model with a stochastic interest rate can represent European option prices as a modified Black-Scholes equation with adjusted volatility. However, it is noted that even with a two-dimensional stochastic differential equation for the interest rate, there is no impact on implied volatility for stock options. The inclusion of interest rates only results in a time-dependent volatility for stocks, without additional stochasticity, leading to a flat volatility across different strike prices. The speaker conducts an experiment to illustrate the influence of different parameters on the term structure of implied volatility.
The lecture further delves into the utilization of forward values in option price implied volatility calibration using actual data. The impact of the speed of mean reversion (lambda) on the implied volatility term structure of stocks is discussed, along with the volatility of interest rates. The speaker highlights that fixing one of these parameters can result in a similar shape of implied volatilities, simplifying the calibration process. Moreover, the effect of correlation on implied volatilities is addressed, where the positivity or negativity of the overall variance of sigma_f impacts the implied volatilities accordingly.
The lecture emphasizes the importance of hybrid models in financial institutions' portfolios, particularly for xVA and VaR calculations. It explores the dynamics and complexities of stochastic volatility models, discusses the calibration of multi-dimensional stochastic differential equations, and highlights the correlations between different asset classes. The lecture also covers the application of measure transformations, the derivation of forward stock price SDEs, and the challenges and considerations related to stochastic interest rates. The calibration of implied volatilities and the impact of various parameters on the term structure of implied volatility are also addressed.
Financial Engineering Course: Lecture 9/14, part 2/2, (Hybrid Models and Stochastic Interest Rates)
Financial Engineering Course: Lecture 9/14, part 2/2, (Hybrid Models and Stochastic Interest Rates)
In this lecture, the focus is on advanced hybrid models, particularly stochastic volatility hybrid models like the Scholes-Black, Heston, and Shobel-Zoo full white models. The lecturer demonstrates the impact of different correlation coefficients on the hybrid payoff of a basket consisting of a stock and a bond. Efficient simulation techniques for these hybrid models using Monte Carlo simulation are also discussed.
The lecture delves into the Shobel-Zoo full white model, which extends the Black-Scholes model by introducing a normally distributed process for volatility. However, it has limitations due to its structural model. The lecturer discusses the constraints and limitations of the Schobel-Zhu model compared to the Heston model. The volatility structure of the Schobel-Zhu model is less flexible, resulting in a more limited range of implied volatility skew and smiles compared to the Heston model.
Another model discussed is the Shwartz-Zhao model, which introduces an additional process for sigma squared and extends the set of state variables. However, solving the characteristic function analytically becomes computationally expensive due to the complex set of Riccati equations involved. The lecturer shows the shapes of implied volatilities and skews for different parameters and compares them to the Heston model.
The impact of correlations on the pricing of hybrid payoffs is explored. An experiment is conducted to evaluate the derivative's value for different correlations between stock and interest rate motions. The importance of calibrating correlations to market data before calibrating other model parameters is emphasized. The lecture briefly mentions more advanced discretization methods for hybrid models that will be discussed later.
The lecture focuses on extending the flexibility and calibration of the Heston model with stochastic interest rates. Introducing an extra dimension for interest rates creates challenges with instantaneous covariance metrics. Approximations are used to find the connector function and solve the correlation problem. The importance of maintaining the correlation between stock and interest rates for evaluating the characteristic function and calibrating the model to market data is highlighted.
Approximation methods, such as the delta method and Taylor series expansion, are discussed to simplify the evaluation of variance and characteristic functions. The lecturer provides formulas and techniques for approximating variances and discusses the limitations of these approximations.
The time-dependent function of stock volatility and the mapping of the function over time are explained, along with the Euler discretization method of simulation. The lecturer mentions that later on, they will compare the estimates of the simulation against Monte Carlo brute force and Fourier transformation. The iterative step of the Euler discretization method for approximating the integral is also covered.
The lecture addresses the issue of zero attainability by the volatility paths in the CIR model and provides fixes for Euler discretization. The importance of keeping the variances of hybrid models as independent as possible for better simulation results is emphasized. The process for x(t) is discussed, including its correlation matrix and Cholesky decomposition, highlighting the need to maintain independence from the variance.
The challenges of dealing with non-positive definite matrices in financial engineering are discussed, and the importance of adjusting correlations to satisfy the condition for positive terms under the square root is emphasized. The lecture also covers the generic form of discretization and important steps for modeling stochastic interest rates.
The lecturer introduces the trick and representation for almost exact simulation of the Heston model, applicable to the Heston-Hull-White model as well. The simplification achieved through special cases for the variance process and the evaluation of integrals using Euler discretization and non-central chi-squared distributions is explained. The concept of almost exact simulation is discussed, emphasizing the importance of the variance process in determining accuracy. The lecturer highlights the need to use a whole vector of samples for v life and establishes the order of simulation as first sampling the variance process, followed by the short rate.
The lecturer provides an overview of a simulation performed on the Heston for White model and compares it with other methods. Euler discretization, almost exact simulation, and the COS (Characteristic Function-Based Option Pricing Method) method are compared. The results demonstrate that all methods yield good results. The lecturer shares the code for the simulation, including the configuration for the Heston for White model and the three-dimensional discretization of the hybrid model using the Euler method. Adjustments are made to ensure that the realizations for the variance are capped and floored from zero. The COS method for the Heston for White model is also discussed, and the approximation for the characteristic function is derived and coded.
The focus shifts to comparing different methods for hybrid models and stochastic interest rates. The Monte Carlo simulation results show good accuracy with 10,000 samples, but a larger number of Monte Carlo paths is recommended for improved accuracy. Various hybrid models such as Black-Scholes, Heston, and Schulz-Zucchi models are covered. The lecture also touches upon the application of hybrid models in pricing different asset classes within a single evaluation and their use in xVA calculations. Two exercises are assigned to students, one on advanced models like Heston CIR and the other on developing a Monte Carlo simulation.
In the final part of the lecture, the speaker discusses the development of a Monte Carlo simulation using a white model for stochastic interest rates. It is suggested to derive the corresponding ordinary differential equations to achieve faster Monte Carlo simulations that allow for larger steps. This approach will be compared to the Euler discretization method. The speaker concludes the lecture and expresses anticipation for the students' presence in the next session.
This lecture covers various advanced hybrid models, their limitations, calibration techniques, impact of correlations on pricing, approximation methods, simulation techniques, and comparisons between different methods. The focus is on understanding the intricacies of these models and their practical applications in financial engineering.
and an attempt to extend the model using a new variable is not successful. Instead, the approach is to use approximations to find the connector C function to solve the problem of correlation between stock and interest rates. Historically, the correlation between short-term interest rates and the stock market is not strong, but it varies depending on economic circumstances and the market overall.
Financial Engineering Course: Lecture 10/14, part 1/3, (Foreign Exchange (FX) and Inflation)
Financial Engineering Course: Lecture 10/14, part 1/3, (Foreign Exchange (FX) and Inflation)
The instructor delves into the realm of financial engineering, focusing on two crucial asset classes: foreign exchange and inflation. He provides a comprehensive understanding of the modeling process for each asset class and demonstrates how options can be priced accordingly. Additionally, the instructor delves into the inclusion of stochastic volatility and stochastic interest rates in the evaluation of these assets.
The lecture begins by exploring the history of foreign exchange, highlighting its significant growth in recent years attributed to globalization. The instructor discusses the impact of the gold standard, which limited private ownership of currency, and how the Bretton Woods system established the current framework of multiple currencies backed by gold. The lecture concludes with the assignment of homework tasks to reinforce the covered material.
Furthermore, the video delves into the historical aspect of currencies and the role of gold within them. Specifically, it outlines the transition that occurred in 1971 when the United States ceased using gold as the standard for determining the value of its currency. This pivotal shift led to the current worldwide system where currencies are exchanged based on their relative strength rather than being backed by gold.
Risk assessment is another significant topic addressed in the video. It explores the various risks investors may encounter when engaging with bonds, foreign exchange, and inflation. The lecture elucidates the intricate relationships and complexities associated with these risk factors. The determination of foreign exchange rates through supply and demand dynamics is also thoroughly discussed. The video emphasizes how central banks manipulate these rates through the utilization of reserves. Moreover, it dispels the notion that gold is an investment and clarifies that owning gold is not a necessity for investors.
Financial engineering concepts take the spotlight, with the video showcasing the replication of a forward FX contract. An example is provided to illustrate the initiation of a forward FX contract and how the exchange rate between the original currencies and the new currency is determined. The application of financial engineering in pricing forward foreign exchange contracts is also examined. The video demonstrates the calculation of the forward rate, which is derived by multiplying the spot rate by the effect rate.
The lecture further delves into the concept of financial engineering, exploring its application in pricing assets and liabilities. The equivalence of two pricing approaches is demonstrated, enabling the calculation of a forward rate using these approaches.
Managing exposure to foreign currencies and inflation through derivatives is a significant aspect of financial engineering. The lecture highlights the determination of a forward rate, which depends on the exchange rate at which a country will trade its currency for another. Additionally, the basis spread adjusts for the difference in demand and supply of various currencies.
The intricacies of foreign exchange (FX) and inflation are explained, with the lecture emphasizing that different rules apply depending on the specific type of FX swap contract being executed.
Valuing a foreign exchange contract while considering the effects of foreign exchange rates and discounting is thoroughly discussed. The instructor demonstrates the calculation process, including the utilization of a forward FX contract for the same purpose.
Finally, the lecture explores how foreign exchange (FX) and inflation impact swaps. It delves into the calculation of the swap's value in domestic and foreign currencies while accounting for exchange rate fluctuations.
Financial Engineering Course: Lecture 10/14, part 2/3, (Foreign Exchange (FX) and Inflation)
Financial Engineering Course: Lecture 10/14, part 2/3, (Foreign Exchange (FX) and Inflation)
The instructor's focus is on pricing options related to foreign exchange or off options, utilizing the Black-Scholes framework as a starting point. The lecture extensively covers the derivation of differential equations for domestic risk-neutral measures and their impact on the dynamics of stochastic differential equations. To illustrate these concepts, Python experiments are conducted, comparing the Western Corridor model in two currencies using both Monte Carlo simulation and Fourier transformation with the COS method. The section also delves into the dynamics of the foreign exchange process and the establishment of martingales as market quantities and their corresponding value.
Moving forward, the lecture addresses the dynamics of foreign exchange (FX) and inflation. It begins by defining a generic effects process and then focuses on pricing, transitioning to the risk-neutral domestic measure for FX. The lecture explains the utilization of the high function to manage foreign money savings accounts, which are subsequently exchanged in domestic amounts and discounted using the domestic money savings account. By applying the Ethos lemma and simplifying the equation, the lecture concludes that the dynamics of FX and inflation do not represent a marked yield under this measure. However, valuable insights are provided that can be applied effectively.
A significant topic covered by the speaker is the process of measure transformation from E to Q, creating a new process used for option pricing evaluation. The derived process represents the FX process under the risk-neutral measure of domestic risk information, ensuring that when foreign money savings accounts are exchanged for local currency, the quantity is marked. This enables the pricing of European options using Black-Scholes equations, with the only differences being the discounting of options under the risk-neutral measure and the inclusion of the drift term rd-rf. The FX market model is an extension of a standard log-normal model, and European options can be priced using the same methodology of changing measures and identifying martingales.
Expanding on the foreign exchange market, the lecture focuses on augmenting the Black-Scholes model with stochastic volatility and stochastic interest rates. While previous lectures discussed deterministic interest rates, introducing stochasticity becomes essential for XVA calculations and VAR simulations. Additionally, the correlation between different stochastic factors is emphasized, highlighting the potential pitfalls of relying solely on deterministic interest rates. The foreign exchange market's complexity arises from its non-tradable nature and the necessity to exchange assets across different columns to enforce martingale conditions. Furthermore, the effects world introduces an additional term in the stochastic differential equations that requires careful analysis and calibration to the market.
The speaker delves into the calibration of various asset classes, including stocks from small companies and interest rate products, one of the largest asset classes globally. It is noted that attempting to calibrate all parameters simultaneously can be challenging, leading to the recommendation of calibrating individual parameters and incorporating them into the stock dynamics. The lecture also explores the evaluation of European options through Fourier transformation, discussing the approximations employed. Furthermore, the importance of defining measures for interest rates in the foreign market and transforming them into the risk-neutral measure under domestic markets is addressed.
Affine models for zero coupon bonds and binary savings accounts are discussed, with a focus on their dynamics and the calibration of options, caps, and tablets. The use of stochastic differential equations to derive models for effects and leverage calibrated parameters for each individual process is proposed. The lecture delves into the complexities of pricing derivatives with intricate drift terms, emphasizing the accurate handling of this additional term. The primary driver of option pricing is the volatility corresponding to the FX process, with higher-order returns influencing interest rate volatility.
Volatility's significance in foreign exchange is emphasized by the speaker, particularly due to the non-linear nature of the process, including the presence of the square root of a term. The challenges associated with drift handling and the necessity of employing a stochastic interest rate are discussed. Two stochastic differential equations corresponding to the foreign zero coupon and couple with domestic measures are explained, emphasizing the requirement for them to be martingales under specific conditions. The importance of correlation between foreign markets and FX is highlighted, emphasizing that it cannot be assumed to be zero. Finally, the speaker derives the pricing equation for European options for FX, incorporating all the discussed concepts.
The professor introduces the payoff of a European call option with a maximum value of yt minus k, involving a discounting process with the domestic money savings account. To address stochastic interest rates, the first step is to transition from a measure flow to the t-forward measure associated with the bond maturity capital t. As the dynamics of FX exhibit no drift, the professor only needs to incorporate volatilities into the diameter. Applying the Ethos lemma to this quantity, the professor includes three different elements in the dynamics, including the previously discussed zero components and the dynamics of yt in the FX process.
Moving forward, the speaker delves into the dynamics of the FX forward and variance processes in the short-rate model, where the volatility parameter remains constant. However, the volatility contribution from FX is time-dependent and not constant, resulting in a reduction of dimensionality from four to two. The speaker also mentions the additional quantum correction that arises when switching measures from risk-neutral to domestic t-forward measure, which poses challenges when using small time steps. The section concludes with a discussion on numerical experiments and approximations employed for the characteristic function.
The speaker emphasizes the importance of carefully selecting model parameters as they significantly impact pricing and hedging decisions. The Heston model is discussed, and the characteristic function is defined, enabling the pricing and calculation of FX impact volatilities. A comparison is made between Monte Carlo simulation and Fourier approximation, involving 20 different Monte Carlo runs with 1000 paths per run. The results demonstrate alignment between Monte Carlo option pricing and the Fourier approximation, with satisfactory differences for calibration to implied volatility market data. However, it is noted that the quality of results can vary depending on the specified model parameters.
The professor proceeds to discuss the Python code for the COS method and analyzes its accuracy. The code encompasses specifications for 500 expansion terms and incorporates different model parameters and configurations for domestic and foreign markets, as well as comprehensive metric collections. The professor emphasizes the significance of random samples in Monte Carlo simulations and suggests changing the random seeds to improve results. A Monte Carlo simulation with multiple runs is performed, evaluating option prices using the payoff evaluation method. The average of all runs is calculated, along with the expectation and standard deviation, allowing for error monitoring arising from changes in the random seed.
Lastly, the lecturer highlights the importance of accurate model parameter selection, as it greatly influences pricing and hedging decisions. The characteristic function for the Heston model is defined, enabling the pricing and calculation of FX impact volatilities. A comparison between Monte Carlo simulation and Fourier approximation is conducted, involving 20 Monte Carlo runs with 1000 paths per run. The results demonstrate satisfactory alignment between Monte Carlo option pricing and the Fourier approximation, providing calibration to implied volatility market data. However, the speaker emphasizes the influence of specified model parameters on result quality.
Financial Engineering Course: Lecture 10/14, part 3/3, (Foreign Exchange (FX) and Inflation)
Financial Engineering Course: Lecture 10/14, part 3/3, (Foreign Exchange (FX) and Inflation)
The lecturer delves into the topic of inflation, tracing its development over the past century. Initially, inflation was associated with monetary policy and the increase in money supply, but its definition has now shifted to encompass changes in price levels. The importance of inflation derivatives for hedging inflation risks, particularly for banks and pension funds, is highlighted. The pricing of these derivatives is closely linked to foreign exchange pricing, adding to their significance in the financial market. The section provides a concise overview of inflation and its relevance in the financial sector.
Moving forward, the lecturer examines the variations in inflation measures used across countries, with a specific focus on the European Harmonized Consumer Price Index (HICP) and the US Consumer Price Index (CPI). Comparing these measures is not always straightforward, as they may not accurately reflect actual price increases. However, they are still used to price derivatives contracts, with derivatives often linked to CPI index values. To illustrate historical inflation trends in the US, the lecturer presents a graph showcasing the fluctuation of CPI figures over time, using a reference date from 2000-2015.
In the subsequent part of the lecture, the instructor explores the non-linear nature of inflation and its evolution over different periods. A graph is presented, highlighting the impact of market crashes on deflation and the potential deflationary effects of globalization. The lecturer also delves into the concepts of sticky and transitory inflation, explaining their implications for prices and the economy. It is emphasized that due to its dynamic nature, inflation cannot be easily described by simple economic models. Various factors, such as demographics and the global economy, influence inflation, making it a complex phenomenon to analyze. Furthermore, changes in the composition of price measurement baskets over time can affect inflation figures significantly.
Continuing the discussion, the lecturer explains that comparing inflation over time is challenging due to the changing definitions associated with different goods and services. The lecture also sheds light on the composition of elements used in calculating the CPI index and the techniques employed to adjust and smooth out results. These techniques include the hedonic effect, which factors in the utility of a product when considering price increases, and substitution, where consumers switch to cheaper goods to mitigate rising prices.
Housing's impact on inflation and inflation measures is then examined. In the US, housing prices are not included in CPI or inflation measures because housing is seen as a capital investment. However, CPI measures do incorporate a "shelter impact," which estimates the cost of living in a rented house. The lecture emphasizes that the basket of products used for inflation calculations changes over time, leading to potentially unreliable inflation figures. While the CPI index is considered a lagging indicator of inflation, it serves as an underlying observable quantity for derivative pricing. Pension funds, insurance companies, and banks dealing with inflation-dependent derivatives are the primary users of inflation products, as inflation can significantly affect their payments. The break-even inflation rate is determined by the spread between legal and inflation-linked bonds.
Shifting the focus, the lecturer explains the distinction between nominal and real instruments in relation to inflation. Nominal instruments do not account for inflation and are considered nominal prices that do not protect against inflationary forces. Inflation swaps and inflation forwards are products that expose individuals to the difference between the real and nominal economies. The basic contract discussed is an inflation swap, where the performance is based on the CPI index at a given time, exchanging the floating and fixed parts. The lecturer highlights the importance of considering delays when modeling inflation, as inflation data is released with a lag and is based on past months.
The lecture goes on to discuss how commodities can provide a better representation of inflation compared to inflation figures, as commodity prices are immediately observable in daily markets, while inflation figures have a few months of lag. Forward inflation is defined as inflation observed at a particular time, and if forward inflation is available in the market and the yield curve for nominal zero coupon bonds is known, the real zero coupon bond can be calculated. The lecture also covers the pricing of inflation swaps using similar methodologies as foreign exchange and interest rate swaps. Additionally, the lecturer touches on pricing options using inflation processes and the possibility of defining and extending hybrid models for inflation with stochastic interest rates.
Expanding on the similarities and differences between foreign exchange and inflation, the professor explains the relationship between nominal and real rates. The transfer of funds between nominal and real economies creates a connection term that influences the risk-neutral measure. The lecture also delves into derivative options such as call options and explores year-on-year inflation, which measures the performance of inflation over a specific period of time. Furthermore, the professor examines the distribution of inflation in the log-normal case and how this ratio is affected in the Black-Scholes framework. The lecture encompasses various processes related to foreign exchange and inflation, including risk-neutral measures, derivative options, and inflation performance over time.
The professor further elaborates on the connection between inflation and foreign exchange in pricing inflation products and cross-currency swaps. The derivation of the characteristic function for the distribution of the log of forward inflation rates is explained using Fourier transformations and pricing techniques. The importance of pricing options is emphasized as it aids in calibrating volatility parameters to market instruments, enabling the evaluation of future portfolio exposures and the application of risk measures such as VAR calculations.
Shifting the focus to the foreign exchange (FX) market and inflation, the lecture covers the evaluation of FX rates, determining the fair value of FX contracts, and deriving the fair value of cross-currency. Pricing FX options is discussed, extending the pricing methodology to incorporate stochastic volatility and interest rates. Additionally, the lecture explores the definition of inflation forwards and the pricing of inflation swaps. The lecture concludes by presenting three exercises for students to apply their knowledge, including deriving the question function for year-on-year inflation within the Black-Scholes framework and using simulations to find the expectations of a function.
Lastly, the instructor presents an exercise centered around the Stochastic Differential Equation for Foreign Exchange. The objective of the exercise is to simplify the equation, factorize the Brownian motions to obtain Sigma hat, and subsequently determine the Sigma and Sigma Sigma hat terms. The instructor concludes the lecture by bidding farewell to the students and expressing hopes that they have enjoyed the course and the exercises.