Bm Fi6051 Wk8 Lecture

Derivative Instruments FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2009

Week 8 – Volatility & Exotics

Lecture Summary 

We will discuss lognormality, what exactly does this mean and what are the implications.

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We will discuss the causes of volatility and the significance of volatility in option pricing

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We will examine implied volatility and how the market forecasts volatility

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We will look at implied volatility shapes, surfaces and implied volatility smiles

Lecture Summary 

This lecture also looks at non-vanilla (exotic) options. Each of the more recognised options are examined in turn. We look at their uses and discuss how they might be valued.

Random Walk 

Underlying the Black-Scholes model is the assumption that the price of stock follows a lognormal random walk, aka, Geometric Brownian Motion (GBM) with drift

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This approach has been found to work very well in industrial practice

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We will look at why

Random Walk 

Changes in one time interval are independent over preceding time intervals

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The size and direction of the change is (in some way) random

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Assumes market efficiency, I.e. the current stock price reflects all available information and future changes reflect future information

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With Drift, refers to the fact that stock prices tend to increase over time

Lognormality 

Lognormality refers to the fact that the natural log of the changes in the stock price are assumed to be normally distributed

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Why do we use the natural log and not just simply the “changes in stock price”?  Non-negative  Recombining  Works well in practice

Volatility 

The relative rate at which the price of a security moves up and down. Volatility is found by calculating the annualized standard deviation of daily change in price. If the price of a stock moves up and down rapidly over short time periods, it has high volatility. If the price almost never changes, it has low volatility†

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What causes volatility?  Trading  News †

Source: InvestorWords.com

Volatility  

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A stock is priced at €50, it’s volatility is 15% In one week, a 1-STD DEV move is

= €1.04

50 0.15 1  52  

A stock is priced at €50, it’s volatility is 45% In one week, a 1-STD DEV move is

= €3.12

50 0.45 1  52  

Volatility  

How to measure volatility? We will examine data for the ISEQ taking daily closing values for last three months  

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11th August 2005 to 11th October 2005 That is, 44 observations

The Volatility of the stock σ, is given by

s σ= τ 

Where s = the standard deviation and τ is the time interval (in years)

Volatility 

All we need to calculate is s … 2 1 n (u − u ) s= ∑ i =1 i n −1

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Where ui is ln(Si/Si-1 )

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And ū is the average of the ui’s

Volatility 

A sample from the ISEQ daily returns… i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Date 10-Aug-05 11-Aug-05 12-Aug-05 15-Aug-05 16-Aug-05 17-Aug-05 18-Aug-05 19-Aug-05 22-Aug-05 23-Aug-05 24-Aug-05 25-Aug-05 26-Aug-05 29-Aug-05 30-Aug-05 31-Aug-05

Close 6,784.93 6,772.67 6,739.24 6,697.64 6,717.50 6,691.91 6,668.44 6,677.70 6,685.23 6,688.59 6,659.78 6,639.90 6,644.13 6,666.68 6,647.73 6,677.28

Si/Si-1

ln(Si/Si-1)

0.998193 0.995064 0.993827 1.002965 0.996191 0.996493 1.001389 1.001128 1.000503 0.995693 0.997015 1.000637 1.003394 0.997158 1.004445

-0.000785 -0.002149 -0.002689 0.001286 -0.001658 -0.001526 0.000603 0.000489 0.000218 -0.001875 -0.001298 0.000277 0.001471 -0.001236 0.001926

Volatility 

From these (44) observations, I calculated the standard deviation of the daily return as

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S = 0.0052659 Or 0.52659% Assuming 260 trading days in the year, the annualised volatility is given as

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σ = 0.0052659 260 = 0.0849101 σ = 8.49101%

Volatility – Use in Black Scholes 

Recall:

c0 = S 0 N ( d1 ) − Ke 

− rT

N(d2 )

Where

d1 =

(

)

ln ( S 0 / K ) + r + σ 2 / 2 T

;

σ T ln ( S 0 / K ) + ( r − σ 2 / 2 )T d2 = = d1 − σ T σ T 

Notice where volatility σ is embedded in the equation

Volatility – Use in Black Scholes 

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If the call option price c0 is known from market data, along with the other variables We can reverse out the volatility implied by the market price. Although it is not possible to calculate σ from the Black Scholes equation, we can get the value by iterative techniques

Implied Volatility 

At the time of writing, Apple Computers (symbol: AAPL) have just released quarterly results which were disappointing. Simultaneously, they announced the release to market of a 60Gbyte iPod video recorder which was well received.

Implied Volatility 

Notice the subsequent erratic stock price and high volumes

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It should be interesting to calculate the implied volatility of AAPL and compare against historical averages

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Maybe we can spot a trading opportunity!

Implied Volatility 

I’ve selected a list of near-term, at-the-money call and put options from CBOE

Implied Volatility 

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Now, using MatLab’s in-built Black-Scholes function: [CALL,PUT] = BLSPRICE(SO,X,R,T,σ,Q) 

SO= Current Stock Price

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X = Strike Price R = Risk Free Rate T = Time to Maturity σ = Volatility Q = Asset dividend rate

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We can calculate the theoretical value of the call options

Implied Volatility 

[CALL,PUT] = BLSPRICE(49.25,45,0.0375,(37/365),0.25,0)

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Call Option = 4.6385

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[CALL,PUT] = BLSPRICE(49.25,50,0.0375,(37/365),0.25,0)

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Call Option = 1.3081

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[CALL,PUT] = BLSPRICE(49.25,55,0.0375,(37/365),0.25,0)

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Call Option = 0.1729

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Compare these results with the actual market: 5.00, 2.30 and 0.75 respectively

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The volatility we used (25%) is lower than that implied by the market prices

Implied Volatility 

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Now, using another of MatLab’s in-built functions, we can calculate the implied volatility: ImpVol = BLSIMPV(SO,X,R,T,CallPx) 

CallPx = The actual call option value

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ImpVol = BLSIMPV(49.25,45,0.0375,(37/365),5.0)

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ImpVol = 34.55%

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ImpVol = BLSIMPV(49.25,50,0.0375,(37/365),2.3)

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ImpVol = 40.89%

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ImpVol = BLSIMPV(49.25,55,0.0375,(37/365),0.75)

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ImpVol = 40.14%

Implied Volatility 

In other words, the options prices tells us that “the market” believes volatility of Apple Shares to be about 40%

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IF, we believed that volatility was in fact lower, how could we exploit our belief and make money?

Volatility Smile 

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Notice that the implied volatility was “about” 40%. The underlying stock is the same and so the implied volatilities should be the same, right?

Volatility Surface 

Extend the previous graph to a 3D model with implied volatility, moneyness and maturity

Compound Options 

These are options on options    

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A A A A

call put call put

(option) on a call (option) on a put on a put on a call

Used to hedge for a certain period. E.g. Buy a 3month compound call option on oil futures during a particularly volatile period. Leverage on leverage But if both options are exercised, then the premium for the compound option will be larger than that for a single option

Barrier Options 

Barrier is an umbrella name for a two different option types 

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Knock-Out Calls and Puts  Down and Out  Up and Out Knock-In Calls and Puts  Down and In  Up and In

A knock-Out, causes the option to terminate if the underlying breaches a barrier A knock-In, causes the option to activate if the underlying breaches a barrier

Barrier Options 

The following diagram shows a Knock-In

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If the barrier was set below 90 at initiation, these would become “Down and Ins” The strike of the option should be judiciously set

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Barrier Options 

An airline might buy a Up-And-In Call option on aviation fuel costs, cheaper than a standard call

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The premium for a barrier is lower than that of a normal option

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A down-and-out call plus a down-and-in call equals? A standard call!

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AKA 

kick-outs, kick-ins, ins, outs, exploding options, extinguishing options and trigger options

Barrier Options 

Barrier Options are path-dependent. I.e. at expiration, it matters how the underlying price arrived at the final value

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Barrier options are typically priced using MonteCarlo techniques, these will be discussed in some detail next semester in “Financial Engineering”

Binary Options 

Binary Options

Lookback Options   

Also known as Hindsight Options There are two (call) types Fixed Strike 

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Floating Strike 

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Payoff is dependent on the Smax -X Payoff is dependent on the ST-Smin

The equivalent put options have 

Fixed Strike Payoff is dependent on the X-Smin

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Floating Strike Payoff is dependent on the Smax -ST

Lookback Options 

Consider the payback on each option using the following graph SMAX ST

X

SMIN

Shout Options 

Shout options are very similar to lookback options but the option holder “shouts” (decides) during the life of the option when she believes that the market has peaked or bottomed.

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These are cheaper than lookback options

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But no guarantee that you can pick the market bottom or top!

Asian Options 

Asian Options have a payoff that depends on the average asset price during at least some part of the option life

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Consider a pension fund manager with a predominantly equity fund. The fund is held for 5 years. The manager should consider an Asian put option on the basket, this will protect the fund from sudden drops in the fund value in the last days of the fund

Asian Options  

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There are two types of Asian Options Average Rate Options  The payoff is dependent on the average underlying asset price for a given period Average Strike Options  The strike (and therefore payoff) is dependent on the average underlying asset price for a given period Both forms can be either calls or puts Asian options are path-dependent

Asian Options 

Note how the payoff on a regular call option (ST) is less than that of the Asian payoff (STAVG ) described by the diagram below

STAVG ST

Averaging Period

Basket Options  

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Also known as Rainbow Options The payoff is dependent on the value of a portfolio of assets (e.g., stocks) The volatility of the basket is dependent on the price correlation of the constituent stocks Lower correlation implies higher variability (volatility) Typical baskets belong to fund managers or index of shares (FTSE, ISEQ, etc)

Other Exotic Options 

An exotic option is a non-standard option

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We’ve looked at the more common exotic options

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An option can be contrived to suit a particular clients needs

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The pricing of these options is a financial engineering skill. The construction/engineering of the options is a much prized skill used in “structured products” desks in investment banks

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2009, 7th Ed. 

Chapter 13, 18, 24

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2005, 6th Ed. 

Chapter 13