FRM-I Valuation Models Notes

Volume

IV

FRM 2010: STUDY NOTES Valuation & Risk Models

Author:

David Harper

Published: March 2010 Source:

www.bionicturtle.com

4. Valuation & Risk Models

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AIM (Applying Instructional Material) Statement Not an assigned AIM but “good to know” This section will give explain a formula/equation or some mathematical portion of the AIM.

Here we share a concept review/idea about the AIM.

With this symbol we describe the symbols used in the AIM.

This is a warning. It usually contains information about what to study.n

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Linda Allen, Chapter 3: Putting VaR to Work ........................................................................ 4 Linda Allen, Chapter 5: Extending the VaR Approach to Operational Risks..................................... 11 Hull, Chapter 11: Binomial Trees ..................................................................................... 19 Hull, Chapter 13: The Black-Scholes-Merton Model ................................................................ 27 Hull, Chapter 17: The Greek Letters ................................................................................. 38 Tuckman, Chapter 1: Bond Prices, Discount Factors, and Arbitrage ............................................ 48 Tuckman, Chapter 2: Bond Prices, Spot Rates, and Forward Rates.............................................. 54 Tuckman, Chapter 3: Yield to Maturity .............................................................................. 62 Tuckman, Chapter 5: One-Factor Measures of Price Sensitivity .................................................. 69 Jorion, Chapter 14: Stress Testing .................................................................................... 83 Narayanan, Chapter 6: The Rating Agencies ........................................................................ 92 Narayanan, Chapter 23: Country Risk Models ....................................................................... 98 de Servigny, Chapter 2: External and Internal Ratings .......................................................... 102 Cornett, Chapter 15 (excluding Appendix 15A): Sovereign Risk ................................................ 114 Ong, Chapter 4: Loan Portfolios and Expected Loss.............................................................. 121 Ong, Chapter 5: Unexpected Loss .................................................................................. 128 Dowd, Chapter 2: Measures of Financial Risk .................................................................... 132 Hull, Chapter 18: Operational Risk ................................................................................. 140 Priniciples for Sound Stress Testing Practices and Supervision ―(Basel Committee on Banking Supervision Publication, Jan 2009). http://www.bis.org/publ/bcbs147.pdf ............................................... 144

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LINDA ALLEN, CHAPTER 3: PUTTING VAR TO WORK  Explain and give examples of linear and non-linear derivatives.  Explain how to calculate VaR for linear derivatives.  Describe the delta-normal approach to calculating VaR for non-linear derivatives.  Describe the limitations of the delta-normal method.  Explain the full revaluation method for computing VaR.  Compare delta-normal and full revaluation approaches. © 2009 by Global Association of Risk

Professionals, Inc.  Explain structural Monte Carlo, stress testing and scenario analysis methods for computing VaR,

identifying strengths and weaknesses of each approach.  Discuss the implications of correlation breakdown for scenario analysis.  Describe worst case scenario analysis.

Explain and give examples of linear and non‐linear derivatives. A linear derivative is when the relationship between the derivative and the underlying pricing factor(s) is linear. It does not need to be one-for-one but the ―transmission parameter‖ (delta) needs to be constant for all levels of the underlying factor. A non-linear derivative has a delta that is not constant.  Linear derivative. Price of derivative = linear function of underlying asset. For example, a futures contract on S&P 500 index is approximately linear. The key is that the transmission parameter (delta) is constant.  Non-linear derivative. Price of derivative = non-linear function of underlying asset. For example, a stock option is non-linear

All assets are locally linear. For example a equity option is the classic example of a non-linear derivative: the option is convex in the value of the underlying. But maybe a better perspective is that its delta is not constant. The option delta is the slope of the tangent line. However, for tiny (infinitesimal) changes in the underling, the delta is approximately constant. So, we consider delta to be an approximation or we say the relationship is locally linear.

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Explain how to calculate VaR for linear derivatives. By definition, the transmission parameter is constant. Therefore, in the case of a linear derivative, VaR scales directly with the underlying risk factor.

VaR Linear Derivative    VaR Underlying Risk Factor

VaR S&P 500 Futures Contract  $250  VaR Index Describe the delta‐normal approach to calculating VaR for non‐linear derivatives. In the delta-normal approach, the linear approximation is assumed (i.e., as if the derivative were linear) and the underlying factor is assumed to follow a normal distribution. We use delta-normal, for example, when relying on both option delta and bond duration to estimate underlying price changes—based respectively on asset price and yield changes (the risk factors). Both are first derivatives (or functions of the first derivative, in the case of duration). In the case of an option, the underlying factor is the stock price and we‘d assume the stock price is normally distributed. In the case of a bond, we‘d assume the yield is normally distributed.

Call Option Price

First-second derivative: Delta-gamma, duration-convexity

European call option price Delta

$6 $5 $4 $3 $2 $1 $$-

$5

$10

$15

$20

Stock Price Taylor Series Approximation

f (x)  f (x0 )  f (x0 )(x  x0 )  1 2 f (x0 )(x  x0 )2

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Six Inputs 1 Stock (S) 2 Strike (K) 3 Volatility Variance 4 Riskfree rate (r) 5 Term (T) 6 Div Yield

Inputs: Initial Adjust $50.00 $51.00 $50.00 $50.00 30.0% 32.0% 9.00% 10.24% 4.00% 4.00% 1.00 1.00 0.00% 0.00%

Pricing both calls (initial & adjusted) d1 0.28 0.35 N(d1) 0.61 0.64 d2 -0.02 0.03 N(d2) 0.49 0.51 Call Price $6.88 $7.88

Greeks (approximations) N'(d1) Delta Gamma Vega

0.383 0.612 0.0255 19.16

0.376 0.636 0.0230 19.16

Major steps: 1. Give six inputs into the Black-Scholes model 2. Imagine stock price jumps +$1 and volatility jumps +2%; i.e., we ―shock" two risk factors. The value of the derivative (the call option) has a non-linear relationship with the underlying "risk factors"

3. The second column (―Adjust‖) is a full re-pricing. The first column is the option value under initial assumptions. The second column uses Black-Scholes to re-price the option under the "shocked" assumptions

4. Compare to a Taylor Series approximation where we use the derivatives. Gamma is second-order approximation (this is the convexity - it is essentially the same thing as convexity in the bond price/yield curve). Vega is sensitivity to change in volatility (what is the small change in call option value given small change in volatility)

Re-price with Greek approximations Let's change stock price and volatility: Stock price change $1.00 New Stock Price $51.00 Volatility change New Volatility

2.00% 32.00%

Estimate change in price with delta/gamma (& vega) approximation Change in Stock 1.00 5. Instead of a (full) re-pricing, the Taylor series Delta 0.61 approximation says that the estimated price change is Gamma 0.03 $1.01. Change in Volatility 0.02 Note this gives a new option price ($7.884) that is nearly Vega 19.16 the same as the full re-pricing. Truncated Taylor Series: 1.01 Instead of re-pricing the option, we “merely” estimated Estimated new call price: $7.884 the new price based on delta and vega.

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Describe the limitations of the delta‐normal method.  Although the delta-normal method analytically tractable, it is only an approximation  It is not good for ―derivatives with extreme nonlinearities‖ (e.g., MBS). The Taylor approximation is not helpful when the derivative exhibits extreme non-linearities. This includes mortgagebacked securities (MBS) and fixed income securities with embedded options  In the case of ―delta-normal,‖ we are assuming the underyling risk factors are normally distributed.

Explain the full revaluation method for computing VaR. Full revaluation is the full re-pricing of the portfolio under the assumption that the underlying risk factor(s) are ―shocked‖ to experience a loss. Effectively, full revaluation shocks the risk factors according to VaR; i.e., what is the worst expected change in the risk factor, given some confidence and time horizon. Then, full revaluation prices the portfolio under the changed risk factors. Full revaluation considers portfolio value for a wide range of price levels. New values can be generated by: 

Historical simulation,



Bootstrap (simulation), or



Monte Carlo simulation

dV  V (S1 )  V (S0 ) Compare delta‐normal and full revaluation approaches.  Full Revaluation — Every security in the portfolio is re-priced. Full revaluation is accurate but computationally burdensome.  Delta-Normal — A linear approximation is created. This linear approximation is an imperfect proxy for the portfolio. This approach is computationally easy but may be less accurate. The delta-normal approach (generally) does not work for portfolios of nonlinear securities.

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Explain structural Monte Carlo, stress testing and scenario analysis methods for computing VaR, identifying strengths and weaknesses of each approach. Structured Monte Carlo The main advantage of the use of structured Monte Carlo (SMC) simulation is that we can generate correlated scenarios based on a statistical distribution. Structured Monte Carlo

Advantage Able to generate correlated scenarios based on a statistical distribution By design, models multiple risk factors

Disadvantage Generated scenarios may not be relevant going forward

Scenario analysis (3.2.3) and Stress Testing (3.2.3.3) Please note that Jorion has scenario analysis as a sub-class of stress testing; i.e., stress testing includes scenario analysis as one tool. But Linda Allen, on the other hand, essentially classifies stress testing as a type of scenario analysis. The key advantage of scenario analysis is that it gives us a means to explicitly incorporate scenarios (e.g., correlations spiking to one during a crisis) that would not necessarily be accessible by historical or simulated means.

Stress Testing

Advantage Can illuminate riskiness of portfolio to risk factors By design, models multiple risk factors Can specifically focus on the tails (extreme losses) Complements VaR

Disadvantage May generate unwarranted red flags Highly subjective (can be hard to imagine catastrophes)

The snapshot on the next page (from a learning spreadsheet) illustrates a structured Monte Carlo simulation.

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1a. INPUT: Time horizon & confidence level 1b. INPUT: Returns: Expected excess return; Volatilities; Factor Exposure 1c. INPUT: Correlation Matrix 2. [Next sheet] Cholesky Decomposition returns correlated matrix 3. Per INVERSE TRANSFORM, random standard normals generated: 5 factors, 100 Trials 4a. Random standard normals multiplied by matrix (A') returns: correlated volatilities 4b. Add expected excess return = CORRELATED RETURNS 5. Portfolio Return = Sum of [Factor Expore]*Return 6. VaR 1a. Time horizon (T days) 1a. Confidence Level

10 99% Five Risk Factors 1 2 4.0% 5.0% 20.0% 30.0% 0.75 0.50 0.16% 0.20%

1b. Returns Expected Excess Return (Annual) Risk Factor Volatilty (Annual) Risk Factor Exposure Expected Excess T-day Return

1c. Correlation Matrix (Unitless)

2. Cholesky Decomposition (A') Scaled to T days i.e., this represents a correlated, time-scaled matrix (Sigma) that can be multiplied by the normal Zs 6. Monte Carlo Value at Risk (VaR) i.e., loss of %

3 2.0% 15.0% 0.25 0.08%

4 0.0% 10.0% 0.10 0.00%

5 -1.0% 40.0% -0.05 -0.04%

1

2

3

4

5

1

1.0

0.8

0.5

0.3

(0.1)

2

0.8

1.0

0.3

0.4

(0.3)

3

0.5

0.3

1.0

0.5

(0.5)

4

0.3

0.4

0.5

1.0

0.1

5

(0.1)

(0.3)

(0.5)

0.1

1.0

1 2 3 4 5

1 0.0400 0.0000 0.0000 0.0000 0.0000

2 0.0450 0.0397 0.0000 0.0000 0.0000

3 0.0150 -0.0057 0.0254 0.0000 0.0000

4 0.0050 0.0049 0.0100 0.0159 0.0000

5 -0.0040 -0.0257 -0.0507 0.0462 0.0319

11.79%

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Discuss the implications of correlation breakdown for scenario analysis.  The problem with the SMC approach is that the covariance matrix is meant to be ―typical‖  But severe stress events wreak havoc on the correlation matrix. That‘s correlation breakdown.  Scenarios can attempt to incorporate correlation breakdowns. One approach is to stress test (simulate) the correlation matrix. This is easier said than done; e.g., the variance-covariance matrix needs to be invertible.

Describe worst case scenario analysis  The worst case scenario measure asks, what is the worst loss that can happen over a period of time? — Compare this to VAR, which asks, what is the worst expected loss with 95% or 99% confidence? The probability of a ―worst loss‖ is certain (100%); the issue is its location  As an extension to VAR, there are three points regarding the WCS: — The WCS assumes the firm increases its level of investment when gains are realized; i.e., that the firm is ―capital efficient.‖ — The effects of time-varying volatility are ignored — There is still the extreme tail issue: it is still possible to underestimate the likelihood of extreme left-tail losses

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LINDA ALLEN, CHAPTER 5: EXTENDING THE VAR APPROACH TO OPERATIONAL RISKS  Describe the following top-down approaches to measuring operational risks: 

Multifactor models



Income based models



Expense based models



Operating leverage models



Scenario analysis models



Risk profiling models

 Describe the following bottom-up approaches to measuring operational risk: 



Process approaches 

Causal networks and scorecards



Connectivity models



Reliability models

Actuarial approaches 

Empirical loss distributions



Parametric loss distributions



Extreme value theory

 Compare and contrast top-down and bottom-up approaches to measuring operational risk.  Describe ways to hedge against catastrophic operational losses.  Describe the characteristics of catastrophe options and catastrophe bonds.  Describe various methods of hedging operational risks and discuss the limitations of hedging

operational risk.

Describe the following top‐down approaches to measuring operational risks:      

Multifactor models Income based models Expense based models Operating leverage models Scenario analysis models Risk profiling models Page 11 of 146

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… Multifactor Models Stock returns (as the dependent variable) are regressed against multiple factors. This is a multiple regression where Iit are the external risk factors and the betas are the sensitivity (of each firm) to the external risk factors:

Rit  it  1i I1t  1i I1t 

  it

The risk factors are external to the firm; e.g., interest rates, GDP. Also, note the multi-factor model cannot help model low-frequency, high-severity loss (LFHS) events. Please note the characteristics of a multi-factor model. The mult-factor model based on a multiple regression has several applications in the FRM. It has an intercept (). After the intercept, it has several terms. Each term contains an external risk factor (I) and a sensitivity to the risk factor (). Finally, it has an error term or residual ().

… Income Based Models These are also called Earning at Risk (EaR) models. Income or revenue (as the dependent variable) is regressed against credit risk factor(s) and market risk factor(s). The residual, or unexplained, volatility component is deemed to be the measure of operational risk.  Extract market & credit risk from historical income volatility  Residual volatility (volatility of ε) is operational risk measure

Eit  it  1tC1t  2t M2t   it

C1  Credit Risk M1  Market Risk  it  Residual. Volatility of residual is operational risk

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…Expense based and operating leverage models  Expense based — Operation risk ~ Fluctuations in historical expenses — Simplest — Ignore all non-expense events; e.g., reputational risks Operational risk is measured as fluctuations in historical expenses. This is the easiest approach but ignores operational risks that are unrelated to expenses; further, a risk-reducing initiative that happened to increase expenses (because it involved a cost) would be mischaracterized.  Operating leverage — Joins income- and expense-based models — Also does not measure some events This is a model that measures the relationship between variable costs and total assets. Operating leverage is the change in variable costs for a given change in total assets.

…Scenario analysis models  Management imagines catastrophic shock → estimates impact on firm value  May not incorporate LFHS events (if they haven‘t happened!)  Subjective In this context, this is a generic label referring to an attempt to ―imagine‖ various scenarios that contain catastrophic shocks. By definition, scenario analysis attempts to anticipate low frequency high severity (LFHS) risk events – but doing this generally is a subjective exercise.

… Risk profiling models  Performance indicators track operational efficiency; e.g., number of failed trades, system downtime, percentage of staff vacancies.  Control indicators track the effectiveness of controls; e.g., number of audit exceptions. Refers to a system that directly monitors either performance indicators and/or control indicators. Performance indicators track operational efficiency; e.g., number of failed trades, system downtime, percentage of staff vacancies. Control indicators track the effectiveness of controls; e.g., number of audit exceptions.

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Describe the following bottom‐up approaches to measuring operational risk

Process approaches The process approach attempts to identify root causes of risk; because it seeks to understand cause-and-effect, in should be able to help diagnose and prevent operational losses. — Causal networks and scorecards The scorecard breaks down complex processes (or systems) into component parts. Data are matched to the component steps in order to identify lapses or breakdowns. Scorecards are process-intensive and require deep knowledge of the business processes. The outcome is a process map. — Connectivity models Connectivity models are similar to scorecards but they focus on cause-and-effect. Examples of connectivity models include fishbone analysis and fault tree analysis. It is subjective; does not attach probabilities to risk events. — Reliability models Reliability models emphasize statistical techniques rather than root causes. They focus on the likelihood that a risk event will occur. The typical metric is the event failure rate, which is the time between events.

Actuarial approaches — Empirical loss distributions Internal and external data on operational losses are plotted in a histogram in order to draw the empirical loss distribution. Basically, it is assumed that the historical distribution will apply going forward. As such, no specification or model is required (i.e., Monte Carlo simulation can fill in the gaps).This approaches attempts to describe the operational loss distribution with a parametric distribution. A common distribution for operational risk events is a Poisson distribution. This approach is not mutually exclusive to the empirical and Page 14 of 146

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parametric approaches. EVT conducts additional analyses on the extreme tail of the operational loss distribution. For LFHS events, a common distribution is the Generalized Pareto Distribution (GPD). Extreme value theory (EVT) implies the use of a distribution that has fattails (leptokurtosis or kurtosis > 3) relative to the normal distribution. — Parametric loss distributions This approaches attempts to describe the operational loss distribution with a parametric distribution. A common distribution for operational risk events is a Poisson distribution. — Extreme value theory This approach is not mutually exclusive to the empirical and parametric approaches. EVT conducts additional analyses on the extreme tail of the operational loss distribution. For LFHS events, a common distribution is the Generalized Pareto Distribution (GPD). Extreme value theory (EVT) implies the use of a distribution that has fat-tails (leptokurtosis or kurtosis > 3) relative to the normal distribution. — Proprietary Operational Risk Models Proprietary models include, for example, OpVar offered by OpVantage. A proprietary model implies the vendor has their own database of event losses that can be used to help fit distributions.

Compare and contrast top‐down and bottom‐up approaches to measuring operational risk Top-down approaches   

Assesses overall, firm-wide cost of operational risk Is typically a function of a target (macro) variable or variance in target variable; e.g., revenue, earnings Does not distinguish between HFLS and low frequency high severity (LFHS) operational risk events Advantages SIMPLE LOW DATA requirements (not data-intensive)

Disadvantage Little help or utility with regard to designing/modifying procedures that mitigate risk; does not really help with prevention Because top-down approach both (i) aggregates risk (note: may ―over-aggregate‖) and (ii) is backwardlooking, it is a poor diagnostic tool. Tendency to overaggregate.

Bottom-up approaches   

Analyzes risk from the perspective of individual business activities Maps processes (activities) and procedures to risk factors & loss events to generate future scenario outcomes Distinguishes between HFLS and LFHS

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Advantages DIAGNOSTIC: Useful because it can help employees correct weaknesses PROSPECTIVE: Forward-looking DIFFERENTIATES between HFLS and LFHS

Disadvantage High data requirements By overly disaggregating risk from different business units/segments, may omit interdependencies and therefore correlations. Note the tendency to under-aggregate (or overly disaggregate)

Describe ways to hedge against catastrophic operational losses Three ways a firm can hedge against catastrophic operational loss include: 1. Insurance 2. Self-insurance 3. Derivatives

Insurance Includes fidelity insurance (covers against employee fraud); electronic computer crime insurance, professional indemnity (liabilities to third parties caused by employee negligence); directors‘ and officers‘ insurance (D&O, covers lawsuits against Board and executives related to bread of their fiduciary duty to shareholders); legal expense insurance; and stockbrokers‘ indemnity (covers stockbroker losses arising from the regular course of business). Insurance contracts transfer risk to the insurance company, which can absorb the company-specific (aka, non-systemic or unique) risk because they can diversify this risk among of pool of customers. In theory, diversification can minimize/eliminate the company-specific risk. However, insurance contracts create a moral hazard problem: the policy creates an incentive for the policy-holder to engage in risky behavior. Two other disadvantages: insurance policies limit coverage (―limitation of policy coverage‖) and Hoffman claims ―only 10 to 30% of possible operational losses are covered‖ by insurance policies. In short, they rarely provide total coverage and never provide coverage of all possible operational losses. Insurance is a critical component of risk management. Remember the positives (risk transfer to the insurance company; insurance company absorbs company-specific risk via portfolio diversification) and the negatives (moral hazard, limitation of policy coverage, lack of coverage for all operational loss events)

Self-insurance This is when the company holds (its own) capital as a buffer against operational losses. Holding capital is expensive—firm can use liquid assets or line of credit. Some firms self-insure through a wholly-owned insurance subsidiary.

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Derivatives Swaps, forward and options can all transfer operational risk. But derivatives do not necessarily hedge against operational risk. It depends. Specifically, derivatives hedge if and when the derivative hedges a risk (e.g., credit or market risk) that itself is correlated to operational risk.

Describe the characteristics of catastrophe options and catastrophe bonds Catastrophe Options Catastrophe options (―cat options‖) were introduced by the Chicago Board of Trade (CBOT). The CBOT cat option is linked to the Property and Claims Service Office (PCS) national index of catastrophic loss; it trades like a call spread (i.e., a long call is combined with a short call at a higher strike price). The ―cat option‖ has a payoff linked to an index of underwriting losses written on a pool of insurance policies. Technically, it is a spread option. But unlike a typical option, the payoff does not have unlimited upside. Cat options are useful because they have essentially no correlation to the S&P. The weather derivative is a particular type of cat option. Its value derives from a weather-based index. The most common are daily heating degree day (HDD) and cooling degree day (CDD).

Cat Bonds These are bonds with embedded options, where the embedded option is triggered by a catastrophe; e.g., hurricane. The borrower pays a higher rate (i.e., the cost of the embedded catastrophic risk hedge) in exchange for some type of debt relief; the most common is relief of both debt and principal. There are three types of catastrophe bonds:  Indemnified notes: triggered by events inside the firm; i.e., debt relief is granted if the internal event happens. But these require detailed analysis and are especially subject to the moral hazard problem  Indexed notes: triggered by industry-wide losses are reflected by an independent index (external to the company). There has been a trend away from indemnified and toward indexed notes because indexed notes are not subject to moral hazards (i.e., they link to external, index-based loss events not internal, company-specific loss events)  Parametric notes: like an indexed note, linked to an external event. However, cash flow is based on a predetermined formula; e.g., some multiple of Richter scale for earthquakes

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Describe various methods of hedging operational risks and discuss the limitations of hedging operational risk Operational risk is embedded in the firm—assessing it is subjective. It is very difficult to quantify crosscorrelations. Additionally, influences like the incentive scheme produce subtle, complex outcomes. There are at least four limitations to operational risk hedging:  It can be difficult to identify and define the specific operational risk  Measurement of operational risks is often subjective  It is difficult to foresee unanticipated correlations between/among various operational risks  Data is often not available and/or reliable Two normal means of benchmark are peer comparisons and extrapolation from history into the future. However, both of these can be problematic when applied to the measurement of operational risk. As firm cultures vary, peer comparisons may be misleading. Further, catastrophic events are ―once in a lifetime,‖ and therefore do not lend themselves to extrapolation. Both benchmarking approaches (“peer and seer”) problematic: 1. Firm cultures vary peer and 2. Catastrophic events are “once in a lifetime”

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HULL, CHAPTER 11: BINOMIAL TREES  Calculate the value of a European call or put option using the one-step and two-step binomial model.  Calculate the value of an American call or put option using a two-step binomial model.  Discuss how the binomial model value converges as time periods are added.  Describe the impact dividends have on the binomial model.  Discuss how volatility is captured in the binomial model.

Two basic approaches to option valuation The two basic approaches to option valuation are Black-Scholes (analytical or closed-form) and Binomial (simulation or ―open‖ lattice)

c  S0 N (d1 )  Ke rT N (d 2 )

Binomial (discrete time) (lattice)

Black-Scholes (continuous time) (closed form)

Calculate the value of a European call or put option using the one‐step and two‐step binomial model. We need the following notation:

f  price/value of option S0  stock price   number of shares of stock

u = proportional "up" jump (u  1) d  proportional "down" jump (d  1) f u  option payoff if stock jumps up f d = option payoff if stock jumps down

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The probability of an ―up jump‖ (or up movement) is denoted by (p) and given by:

erT  d p ud This probability (p) then plugs into the equation that solves for the option price:

f  e rT [ pf u  (1  p) f d ] Risk neutral Valuation In a risk–neutral world all individuals are indifferent to risk, and investors would require no compensation for risk. The expected return on a stock would be the risk free rate:

E(ST )  S0erT The principle of risk–neutral valuation says that we can generalize: when pricing an option under the risk–neutral assumption, our result will be accurate in the “real world” (i.e., where individuals are not indifferent to risk). Keep in mind there are two basic steps in the binomial pricing model: (i) building the paths forward and (ii) backward induction

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Two step Binomial Trees Here is the two-step binomial for a European call option on a stock index (Asset = $800, Strike = $800, Time = 0.25 years, Volatility = 20%, Riskless rate = 5%, and Dividend Yield = 2%) Asset Strike Time (yrs) Volatility Riskless Div Yield

$810.00 $800.00 0.25 20% 5.0% 2.0%

Stock Option

Solved: u d a p 1-p

1.1052 0.9048 1.0075 0.5126 0.4874