Category: Learning

Simulating Geometric Brownian Motion (GBM)

Simulating-Geometric-Brownian-Motion-GBMDownload

Here is a realisation of the previous blog, A bit Stochastic Calculus.

Implications

  • 1. Stochastic Integral is different from Ordinary Integral, so as to differentiation. The Stochastic Differential Equation, S.D.E. we normally use to mimic the movement of a stock is as the following,

$$ d S=\mu S d + \sigma S dW$$

, where dW follows a Brownian Motion.

The Brownian Motion also has the following critical properties:

  1. Martingale: E(S_{t+k}|F_t)=E(S_t|F_t) for k>1.
  2. Quadratic Variation: \sum_{i=1}^n (S_i-S_{i-1})^2 \to t
  3. Normality: An increment of S_{t+dt} is normally distributed, with mean zero and variance t_i – t_{i-1}
  • 2. What does Ito’s Lemma tell us?

If the stock price, S_t, follows a stochastic process, then the financial contracts F(S_t), as a function of the underlying stock price, follow another stochastic process.

  • 3. Taylor Expansion helps

See notes about Paul Willmott’s book’s paragraph.

  • 4. Different Forms of Stochastic Processes could be applied.

Example 1: dS=\mu \ dt + \sigma \ dX

Example 2: dS=\mu S \ dt +\sigma S \ dX

For this, dV = \frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial S}dS+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt, which is also called the Geometric Brownian Motion (GBM). If in a specific form of value function V=F(S)=ln(S), then dF = (\mu -\frac{1}{2}\sigma^2)dt + \sigma \ dX. See derivation in attached notes.

Example 3: A mean-reverting random walk. dS=(v-\mu S)dt +\sigma dX

Example 4: Another mean-reverting r.w. dS=(v-\mu S)dt +\sigma S^{1/2}dX

The stochastic term is altered compared with example 3. Now if S ever gets close to zero the randomness decreases,

A bit Stochastic Calculus

A-bit-about-Stochastic-CalculusDownload

See the HTML file for full detals.

Key Takeaways

  • Property of \{W_t\}:
  1. $ W_t – W_s \sim N(0, t-s) $
  2. $(W_t – W_{t-1})$ and $W_{t-i} – W_{t-i+1}$ are uncorrelated. So, $\int_0^t dW_u = \sum^t dW_u = W_t$
  • For,

$$ dS_t = \mu S_t \ dt +\sigma S_t \ dW_t $$

Why the Geometric Brownian Motion of \{S_t\} is designed in that form?

The answer might be,

$$ dS_t /S_t = \mu \ dt +\sigma \ dW_t $$

$$\int_0^t dS_t /S_t = \int_0^t \mu \ dt + \int_0^t \sigma \ dW_t $$

$$ log(S_t) = \mu \ t +\sigma \ W_t $$

Taking the first difference is similar to differentiation. (d(log(S_T)) = log(S_t/S_t-1) = log(1+r_t) \ approx r_t).

$$ r_t = \mu + \sigma \ \Delta W_t $$

The return, \{r_t\}, is equal to a mean, \mu, plus a stochastic term. That is a random walk.

  • We apply a transformation, if S_t follows a Geometric Brownian Motion, then f(S_t) follows another,

In calculating d f(S_t), we would get, (by Taylor Expansion)

$$ df(S_t) = \bigg( \frac{\partial f}{\partial t} + \frac{\partial f}{\partial S_t}\mu S_t +\frac{1}{2}\frac{\partial^2 f}{\partial S_t^2}\sigma^2 S_t^2 \bigg)dt + \frac{\partial f}{\partial S_t}\sigma S_t \ dW_t $$

  • A Special Form of f(\cdot) is f(S) = log(S),

$$ d\ log(S_t) = \bigg( \mu – \frac{1}{2}\sigma^2 \bigg)dt + \sigma \ dW_t $$

Integrate the above equation from time 0 to time t, then we would get,

$$ log(S_t) = log(S_0) + (\mu – \frac{1}{2}\sigma^2)t + \sigma W_t $$

Brownian Motion to Normal Distribution

Codes are shown in the HTML file.

Brownian-Motion-Normal-DistDownload

Markov Property

The Markov Property states that the expected value of the random variable \(S_i\) conditional upon all of the past events only depends on the previous value \(S_{i−1}\). The implication of the Markov Property is that the

Martingale Property

best expectation of future price, \((S_{t+k}\), is the current price, \(S_t\). The current price contains all the information until now.

$$ \mathbb{E}(S_{t+k}|F_t)=S_t, \text{ for k>=0} $$

That is called the Martingale Property.

Quadratic Variation

The Quadratic Variation is defined as,

$$ \sum_{i=1}^k (S_i – S_{i-1})^2 $$

In the coin toss case, each movement must be either “1” or “-1”, so \(S_i – S_{i-1} = \pm 1\). Thus, \((S_i – S_{i-1})^2 =1\) for all “i”. Therefore, the Quadratic Variation equals “k”.

Brownian-Motion-sub-insights-1Download

A Realisation

Here below is the Code Realisation of a Brownian motion – Chapter 5.6 Paul Wilmott Introduces Quantitative Finance.

Codes-for-Chapter-5.6-Brownian-motion-Paul-Wilmott-Introduces-Quantitative-Finance-1Download

ARCH and GARCH

Let’s begin with the ARCH model.

ARCH Model

The ARCH model was initially raised by Engle (1982), and the ARCH model means the Autoregressive Conditional Heteroskedasticity model.

We assume here \(u_t\) is the return.

$$ u_t=\frac{P_t-P_{t-1}}{P_{t-1}} $$

$$u_t\sim N(0,\sigma_t^2)$$

The data-generating process (DGP) is like an AR form, as the name of ARCH. The volatility is autoregressively generated by \(u^2_i\).

$$\sigma_t^2=\delta_0+\sum_{i=1}^{p} \delta_i u_{t-i}^2$$

, where \(p\) is the number of lags, and \(\delta_i\) are a set of parameters. The DGP of that model shows that the volatility of the return is heteroscedastic, correlated with the squared term of the return per se.

For example, an ARCH(1) model is like,

$$ \sigma_t^2=\delta_0+\delta_1 u^2_{t-1} $$

  • Stationarity

Note here we need our time series to be stationary for better forecasting. Thus, \(Var(u_t)=\sigma^2 \)

$$ Var(u_t)=\delta_0+\delta_1 Var(u_{t-1}) $$

$$ \sigma^2=\frac{\delta_0}{1-\delta_1} $$

As the variance has to be positive. We need \(\delta_0 > 0\), and \(\delta_1<1\).

  • Estimation

For this time series data, OLS assumptions are violated, because our series are autoregressive heteroskedasticity.

Instead, the Maximum Likelihood Estimation (MLE) would be a better estimation method by assuming the probability distribution of variables.

MLE allows iterations to find parameters \(\delta\) that can maximise the maximum likelihood function.

GARCH Model

The ‘G’ in the GARCH model means ‘generalised’, and the GARCH model has a set of additional terms, \(\sum \gamma_i \sigma^2_i \). Thus, the DGP of the GARCH(p,q) model is as the following,

$$u_t\sim N(0,\sigma_t^2)$$

$$ \sigma_t^2=\delta_0 + \sum_{i=1}^{p} \delta_i u^2_{t-i} +\sum^q_{j=1} \gamma_j \sigma^2_{t-j} $$

ARMA-GARCH Model

That is a further application, in which the GARCH model is applied to mimic the movement of error terms in the ARMA model.

We initially assume an ARMA(p,q) model,

$$ y_t=\beta_0 +\sum^p_{i=1} \beta_i y_{t-i} + \sum^{q}_{j=1} \theta_j u_{t-j} +u_t$$

Then, we assume the error term here, \(u_t \sim GARCH(m,n)\).

$$ u_t \sim N(0,\sigma_t^2)$$

$$ \sigma_t^2 = \delta_0 +\sum^m_{i=1} \delta_i u_{t-i}^2 +\sum_{j=1}^n \gamma_n \sigma_{t-n}^2 $$

Reference

Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the econometric society, pp.987-1007.

Why do Banks run?

Assumption

Entrepreneurs borrow from banks to invest in long-term projects. Banks themselves borrow from risk-averse households, who receive endowments every period. Households deposit their initial endowment in banks in return for demandable deposit claims. There is no uncertainty initially about the average quality of a bank’s projects in our model, so the bank’s asset side is not the source of the problem. However, there is uncertainty about household endowments (or equivalently, incomes) over time.

Process

Firstly, households deposit their initial endowments and have an unexpectedly high need to withdraw deposits.

Anticipated prosperity, as well as current adversity, can increase current household demand for consumption goods substantially.

As households withdraw deposits to satisfy consumption needs, banks will have to call in loans to long gestation projects in order to generate the resources to pay them. The real interest rate will rise to equate the household demand for consumption goods and the supply of these goods from terminated projects.

Results

Thus greater consumption demand will lead to higher real rates and more projects being terminated, as well as lower bank net worth. This last effect is because the bank’s loans pay off only in the long run, and thus fall in value as real interest rates rise, while the bank’s liabilities, that is demandable deposits, do not fall in value.

$$Asset = Liability + Equity$$

in the balance sheet, so as to banks. However, the difference is that banks’ assets are loans and liabilities are deposits from households. If the real interest rate increases, which conveys the increase in the discount rate, then the value of assets for banks would decrease (,by the present value of future cash flows). Liability (debts) keeps constant, then the equity of banks is destroyed.

Eventually, if rates rise enough, the bank may have negative net worth and experience runs, which are destructive of value because all manner of projects, including those viable at prevailing interest rates, are terminated.

Solution

How can this tendency towards banking sector fragility be mitigated?

  1. Capital Structure of Banks

One possibility is to alter the structure of banks. Long-term loans’ value is more volatile if the real interest rate fluctuates.

If banks financed themselves with long-term liabilities (in part我国政策行if the bank finances through long-term loans, that means A=D+E, `D is also volatile to the real interest rate changes, and moves in the similar direction as Asset) that fell in value as real interest rates rose, banks would be doubly stable. The bank hedge itself, hedging the assets by bank debts.

Deposits from households do not make banks stable, compared with financing through bank loans, because deposits could be withdrawn.

The authors stated that competition that banks strive for efficiency determines the capital structure of banks. I personally do not understand that idea, so I will leave it here.

P.S.

Diamond and Rajan (2001) 中指出，银行，作为金融中介，的功能是有human capital能量化或者保证depositors withdraw时 borrower能提供足够的liquidity还给lender (depositor)的问题。

  • 2. Government Intervention

The government may have to intervene to pull the economy or consumption back into place. A typical way of doing so is through lower the interest rate.

The paper states that, reducing interest rates drastically when the financial sector is in trouble, but not raising them quickly as the sector recovers could create incentives for banks to seek out more illiquidity than good for the system. Such incentives may have to be offset by raising rates in normal times more than strictly warranted by macroeconomic conditions.

Put differently, reduce in interest rates could encourage banks to increase leverage or fund even more illiquid projects up front. This could make all parties worse off.

Reference

Diamond, D. and Rajan, R. (2009) (w15197) Illiquidity and Interest Rate Policy. Cambridge, MA: National Bureau of Economic Research DOI: 10.3386/w15197.

Diamond and Rajan’s Study about Financial Crisis 2008

The authors noted the financial crisis of 2008 was caused by mainly three reasons.

  1. U.S. financial sectors misallocated resources to real estate.
  2. Commercial and Investment banks had a large proportion of their instruments in their Balance Sheet.
  3. Investments were largely financed with short-term debts.

The following will illustrate why those facts happen.

1. Misallocation of Investment

Step 1. World Crisis pushed up risks.

The financial crisis in emerging markets, East Asia Econ Collapsed, `Russia Defaulted, South America, etc made investors circumspect.

Step 2. Capital Controls made CA surplus.

To react to those unexpected events and prevent domestic industries from the incumbents, governments started to conduct capital controls. Also, investors were unwilling to invest (they cut down investments and even consumptions) or charge a high-level risk premium. A number of countries became net exporters.

Step 3. “dot-com” bubble derived another global crisis.

Those exporters then had a current accounts surplus and transferred the CA surplus into “savings” (investment). Those savings were invested into the high-return business, the IT industry. However, another nightmare happened that is the “dot-com” bubble collapsed around the 2000s.

Step 4. CB QE and US financial innovations made a housing bubble

Central Banks QE, lowered the interest rate, which ignited demand for housing. The house price spiked. In the U.S., financial innovation (securitization) drew more marginal-credit-quality buyers into the market. The crisis manifested itself.

Step 5. Asymmetric information enforced the bubble.

Because rating agencies were at a distance from the homeowner, they could process only hard information. Asymmetric information enforced the bubble. Housing prices surged to prevent “default”.

Step 6. Securitization Iterate itself.

The slicing and dicing through repeated securitization of the original package of mortgages created very complicated securities. The problems in valuing these securities were not obvious when house prices were rising and defaults were few.

But as the house prices stopped rising and defaults started increasing, the valuation of these securities became very complicated.

2. Why Did Bank hold those instruments?

The key answer is bankers thought those securities were worthwhile investments, despite their risks. Risks were vague and unable to be evaluated.

it is very hard, especially in the case of new products, to tell whether a financial manager is generating true excess returns adjusting for risk, or whether the current returns are simply compensation for a risk that has not yet shown itself but that will eventually materialize.

Several facts manifested the problem.

  • 1. Incentive at the Top

CEOs’ performance is evaluated based in part on the earnings they generate relative to their peers. Peer Pressure, which came from holding financial instruments to increase returns, mutually increased the willingness to hold those financial instruments.

  • 2. Flawed Internal Compensation and Control

The top management wants to maximise the long-term bank value and goals. However, many compensation schemes are paid for short-term risk-adjusted performance. The divergency gave managers an incentive to take risks in the short term.

It is not said that the Risk management team is unaware of such incentives. However, they may be unable to fully control them, because tail risks, by the nature, are hard to quantify before they occur.

  • 3. Short-term Debt

Given the complexity of bank risk-taking, and the potential breakdown in internal control processes, investors would have demanded a very high premium for financing the bank long term. By contrast, they would have been far more willing to hold short-term claims on the bank, since that would give them the option to exit — or get a higher premium — if the bank appeared to be getting into trouble.

In good times, short-term debt seems relatively cheap compared to long-term capital and the costs of illiquidity remote. Markets seem to favor a bank capital structure that is heavy on short-term leverage. In bad times, though, the costs of illiquidity seem to be more salient, while risk-averse (and burnt) bankers are unlikely to take on excessive risk. The markets then encourage a capital structure that is heavy on capital.

  • 4. The Crisis Unfolds

Housing Price decreased, => MBS fall in value and becaome hard to price. Balance sheet destorted, and debt level held, and equity shrinked.

Every parties sold out, drived price down again and again.

Panic (no confidence) spreaded worldwide.

Interbank lendings were forzen as inadequate credits.

  • 5. The `Credit Crunch

Banks were reluctant to lend due to two reasons. One possibility is that they worry about borrower credit risks. A second is that they may worry about having enough liquidity of their own, if their creditor demands funds.

  • Dealing with the Crunch

Banks still fear threats from illiquidity. Illiquid assets still compose significant portions of banks and non-banl balance sheets. The price of those illiquid assets fluctuated largely, because liquidty asset could be easily exchanged or sold out for cash, but illiquid assets were unable to do so so that price shrinked and damaged the balance sheet. Debts held constant, but assets shrinked, resulting in shrinkage of equity, and increase in leverage and financial burden.

Coins have two sides. Low prices mean not only insolvent, but also tremendous buying opportunity. The pandic manified the expectation of insolvency, plus illiquid market condition made the fact that less money was availab to buy at the price. Selling iterated itself.

CB standed out, provided liquidty to financial institutes.

However, an interesting thing happened. CB’s intervention to lend against all manner of collateral may not be a unmitigated bless, because it may allow weak entities to continue holding illiquid assets.

Possible ways to reduce the overhand

1. Authorities offer to buy illiquid assets through auctions. `This can reverse a freeze in the market caused by distressed entities. Fair value from the aution can be higher than the prevailing market price. 2. government ensures the stability of financial system that holds illiquid assets through the recapitalization of entities that have a realistic possibility of survival. （我国，纳入国有）.

Reference

Diamond, Douglas W. and Rajan, Raghuram G., The Credit Crisis: Conjectures About Causes and Remedies (February 2009). NBER Working Paper No. w14739, Available at SSRN: https://ssrn.com/abstract=1347262

A Great Introduction to the Nobel Prize Econ 2022

Here below is a great article introducing the Nobel Prize in Econ in 2022.

A-Great-Article-about-Nobel-Econ-2022-1Download

Later, I will start a series of studies about the journal articles from those Nobel Prize winners. Hopefully, that would help us understand the current crisis.

Reference

Bernanke, B., Gertler, M. and Gilchrist, S. The Financial Accelerator in a Quantitative Business Cycle Framework.

Diamond, D.W. and Dybvig, P.H. ‘Bank Runs, Deposit Insurance, and Liquidity’. JOURNAL OF POLITICAL ECONOMY, p. 19.