Geometric Brownian Motion

Having defined a Brownian motion, the next important process to examine is the closely related geometric Brownian motion. We state that a stochastic process 𝑉(𝑡) is a geometric Brownian motion if 𝑙𝑜𝑔 𝑉(𝑡) is a Brownian motion with initial value 𝑙𝑜𝑔 𝑉(0). This simply means that a geometric Brownian motion is an exponenti-ated Brownian motion.

This process is probably the most used process to model or simulate the evolution of asset prices. A desirable feature of the geometric Brownian motion is that values are always positive because of the exponential function. This makes the process attractive in modeling asset prices compared to the ordinary Brownian motion, which also can take negative values. Another fundamental feature of the geometric Brownian motion is that the percentage changes

𝑉 𝑡₂ − 𝑉(𝑡₁)

𝑉(𝑡₁) ,𝑉 𝑡₃ − 𝑉(𝑡₂)

𝑉(𝑡₂) , … ,𝑉 𝑡_𝑛 − 𝑉(𝑡_𝑛−1) 𝑉(𝑡_𝑛−1)

are independent for 𝑡₁ <𝑡₂ <⋯<𝑡_𝑛, instead of absolute changes in value 𝑉 𝑡_𝑘+1 − 𝑉 𝑡_𝑘 . These properties all make the geometric Brownian motion partic-ularly useful for modeling asset prices.

We assume that the W is a standard Brownian motion, and that the variable X satisfies the stochastic differential equation

𝑑𝑋 𝑡 =𝜇𝑑𝑡+𝜎𝑑𝑊(𝑡)

We then write the evolution of the variable, V as 𝑉 𝑡 =𝑉 0 𝑒^𝑋(𝑡) ≡ 𝑓 𝑋 𝑡

Then, by applying Itô’s Lemma⁶, we have that

6 For the scope of this thesis, we will not go into Itô’s Lemma. For a derivation of the Lemma, the reader is referred to Hull (2012, Ch. 13, App.)

𝑑𝑉 𝑡 =𝑓^′ 𝑋 𝑡 𝑑𝑋 𝑡 +1

2𝜎²𝑓^′′ 𝑋 𝑡 𝑑𝑡

=𝑉 0 𝑒^{𝑋 𝑡} 𝜇𝑑𝑡+𝜎𝑑𝑊 𝑡 +1

2𝜎²𝑉 0 𝑒^{𝑋 𝑡}𝑑𝑡

=𝑉 𝑡 𝜇+1

2𝜎² 𝑑𝑡+𝑉 𝑡 𝜎𝑑𝑊(𝑡)

We now arrive at an expression that we can use to describe the evolution of the price of an asset. Usually, we write the geometric Brownian motion as

𝑑𝑉 𝑡

𝑉(𝑡) =𝜇𝑑𝑡+𝜎𝑑𝑊(𝑡)

where 𝜇 denotes drift term, and W is a standard Brownian motion. This is how the assets of a bank are usually in the literature. This is also the approach we will take in developing our model.

Risk-Neutral Dynamics

Having introduced the concept of risk-neutral pricing and stochastic processes, let us briefly touch upon the dynamics of a stochastic process under the risk-neutral probability measure. Consider an asset, V, those value follows a geometric Brown-ian motion. Assume that there exists a risk-free asset that pays a constant contin-uously compounded interest rate r. Since the expected return on any asset under the risk-neutral probability measure is the risk-free rate, the drift term of the geo-metric Brownian must be 𝜇=𝑟. Thus, under this measure, the process can be written as

𝑑𝑉 𝑡

𝑉(𝑡) =𝑟𝑑𝑡+𝜎𝑑𝑊(𝑡)

We will use this assumption when developing our model.

The Basics of Bond Pricing

Since our goal is to build a bond pricing model, we will briefly discuss the most basic theory on this topic. We know that the price of an asset is equal to the present value of the expected cash flows from the asset. Thus, to price an instrument, one

the discount rate is simply the risk-free rate. Thus, for pricing we only need to compute the expected cash flows. For a bond, the cash flows consist of coupon payments that are paid periodically until the date of maturity, and the face value, or principal, of the bond, which is paid at maturity.

Since the price of a bond at time zero is given by the present value of all cash flows, it is equal to the sum of the present value of coupon payments and the principal payment. Thus, under the risk-neutral probability measure, the price at time zero of a bond with coupon rate c and face value B is

𝐵₀ = 𝑐𝐵

1 +𝑟 + 𝑐𝐵

1 +𝑟 ²+⋯ 𝑐𝐵

1 +𝑟 ^𝑛 + 𝐵

1 +𝑟 ^𝑛 or

𝐵₀ = 𝑐𝐵

1 +𝑟 ^𝑡+ 𝐵 1 +𝑟 ^𝑛

𝑛

𝑡=1

assuming that the bond does not default, the bond pays annual coupons, the annual risk-free interest rate is given by r, and the bond has a maturity of n. In this case, the yield on the bond will be equal to the risk-free rate, because the bond does not have any risk. In general, the yield on a bond, y, is given by the rate that satisfies

𝐵₀ = 𝐶𝐹_𝑡 1 +𝑦 ^𝑡

𝑛

𝑡=1

that is, the rate that makes the present value of the expected cash flows equal the market price. The spread on a bond is given by the yield of the bond minus the risk-free rate: 𝑠=𝑦 − 𝑟.

In our model, the only risk factor we will consider is the credit risk. This is the risk that the issuer of the bond, e.g. a bank, will not satisfy its obligations by not making the required coupon payments or the repayment of the principal (Fabozzi, 2013).

This form of credit risk is called default risk, and causes bonds to carry a positive credit spread.

In this simple valuation framework, credit spread can be viewed as compensation for default risk, which is driven by two factors: the probability of default (hence-forth PD) and the loss given default (hence(hence-forth LGD). Bonds in the same capital structure share the same PD but differ in terms of LGD depending on seniority.

Thus, the recovery rate, i.e. the fraction of the principal that the bondholder re-ceives in the case of default, is higher the more senior it is. This creates a clear relationship between spreads in the capital structure anchored by relative LGD expectations (Barclays, 2016).

Monte Carlo Methods

We will use Monte Carlo simulation techniques when we price bonds in section 9.

Therefore, we will briefly introduce these methods in the following.

Monte Carlo methods refer to techniques for approximating parameters by the sample mean of independent samples of simulated random variables. The methods come from physics, where they are most often used to determine model values for which there is no analytical solution. In finance, we use the methods similarly: when we wish to price an asset whose price cannot be determined analytically, for exam-ple because the payoff of the asset depends on the path that the asset follows.

These methods are based on the relationship between probability and volume. In the mathematics of measurement, the probability of an event is defined as its vol-ume relative to the volvol-ume of a universe of possible outcomes. Monte Carlo methods use the opposite idea, calculating the volume of a set by taking the volume as a probability. Thus, by sampling random outcomes, we can Monte Carlo techniques to obtain an estimate of a given set’s volume by taking the fraction of random draws that fall in that set (Glasserman, 2003). By increasing the number of draws, we know from the law of large numbers that our estimate will converge towards the true value.

As an example of Monte Carlo, let us consider the problem of calculating the inte-gral of a function f over the interval 0,1

𝜇= ¹𝑓 𝑥 𝑑𝑥

0

The integral above may be represented as an expectation 𝐸 𝑓(𝑉), where the vari-able V is uniformly distributed on the interval 0,1. If we are able to draw random points, 𝑉₁,𝑉₂,𝑉₃, …, 𝑉_𝑛, uniformly and independently from the interval 0,1, we can then obtain a Monte Carlo estimate of 𝜇 by evaluating the function f at each of these n random points and averaging the results. Thus,

𝜇_𝑛 = 1

𝑛 ^𝑛 𝑓(𝑉_𝑖)

𝑖=1

If the function f is integrable over 0,1, then we have from the law of large numbers that 𝜇_𝑛⟶ as 𝑛 ⟶ ∞.

This method is convenient in finance because in most cases, we can write the price of an asset as an expectation. Thus, the pricing problem can be reduced to calcu-lating an expectation. By having the evolution of an asset’s value as a stochastic process, we can use Monte Carlo methods to sample a large number of random paths for the asset in a risk-neutral world. Then, by calculating the asset’s payoff on each path and averaging over all simulations, we can obtain an estimate of the expected payoff in a risk-neutral world. Finally, we can discount this expected pay-off to obtain an estimate of the value at time zero. This method is highly powerful and can be used to find the value of complex and path-depend assets for which no analytical solutions are available. A further advantage of the method is that any stochastic process for the asset can be accommodated.

We have chosen to use Monte Carlo simulation as it provides a simple and flexible method of modeling the asset path of the bank, which in our setting incorporates a path-dependent feature. As Boyle (1977) argues, the method has a distinctive ad-vantage when modeling specialized situations, such as when the assets follow a jump process.

Summary

In this chapter, we have investigated how corporate debt instruments are priced in the current literature, and developed the theoretical foundation for our model.

There are several approaches to modelling credit risk. One is structural models, as pioneered by Merton (1974), where bonds and stocks are viewed as contingent claims on the firm’s assets and bonds are priced using option pricing theory. This approach has been extended by for example Leland (1994). Another approach is the intensity-based models which use quantitative techniques to estimate statistical hazard rates. Because these classical models might be difficult to apply for banks because of the complexity of both instruments and capital regulation, we looked towards the literature on contingent convertible capital instruments, CoCos. We found that models such as those developed by Pennacchi (2011) and Glasserman and Nouri (2012) are somewhat better to capture the bank related complexities than the classical models.

But as we wish to incorporate additional features that these models do not, we looked towards the literature on Collateralized Loan Obligations, which provided an intuitive way of looking at a bank: a portfolio of loans on the asset side, and different tranches of debt on the liability side. This idea is useful in a valuation setting because it clearly defines the cash flow waterfall and the priority of pay-ments according to the tranche hierarchy.

Finally, before developing a bond pricing model, we needed to understand some of the most fundamental theoretical concepts. We found, that we under the assump-tion of no arbitrage are able to price assets as if agents were risk-neutral. This means, that we can use the risk-free interest rate as discount rate in our valuation, and that the expected return of all assets is the risk-free rate. Furthermore, we found that we are able to model the evolution of asset prices by a process called a geometric Brownian motion, and that the price of a bond can be seen as the present value of expected future cash flows. Finally, we found that we are able to combine all these concepts in a pricing framework by applying Monte Carlo methods, where expectations can be approximated by sample means.