liam mescall fin engineering 1

8/4/2019 Liam Mescall Fin Engineering 1

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Option Pricing

Included in this report is the pricing mechanisms used fora variety of options and analysis of their associated deltas.

Submission Date: 18/March/2011

Student ID: 0144126

Name: Liam Mescall


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What follows is a three part report detailing the process undertaken to calculate the value of; a va-

nilla equity call and put option, binary call and put options and an exchange call option along with

their individual deltas. This is further developed into an analysis of how the formulae per pricing

models combine to produce these values, the rationale behind the workings, a discussion on the

Black Scholes model and an assessment of the behaviour of associated deltas as a function of asset

price and time.

Part A

Vanilla Options and Black Scholes

An option is a contract which gives the owner the right but not the obligation to enter into another

contract during a particular period of time. The underlying contract can be a stock, bond, financial

asset, currency, commodity or interest rate. Interest rate options give the purchaser the right to pay

or receive an interest rate. For the purposes of this paper we are referring to an equity option. A call

option gives the holder the right to buy the underlying at fixed price while a put option gives the

right to sell the underlying at a fixed price. Also, there are two types of options, European; which

only allow exercise of option right on a specified date and American which can be exercised at any

point up to a specified date. This paper refers to European options. The payoff for these options is:

Call Payoff = max (S-K, 0)

Put Payoff = max (K-S, 0)

Markets for option trading began as far back as the seventeenth century when options were traded

on tulip bulbs in Amsterdam but have since ballooned c. $US65 trillion in size in 2007. Demand

comes from four sources:

- Market makers, who profit from the bid-ask spread.- Hedgers, who use options to offset some specific risk,- Speculators, who gamble on market volatility.- Investment banks, who tailor to the needs of the OTC markets for a premium.

Much of this market growth has come since 1973 when the Black Scholes model was developed as a

means to calculate the fair value of an option. The model for valuing a vanilla European option a

partial differential equation was derived based on a delta hedging argument and is satisfied under

Geometric Brownian Motion (GBM) pricing dynamics. GBM, as described in the question, states that

an underlying stock follows a stochastic differential equation:

= r dt + dW(t) (1)

Where W(t) is a Weiner process i.e. the random element of stock price diffusion. The application of

Itos Lemma to this equation allowed us to alter it to:

d ln S(t) = (r-1/2) dt + dW(t) (2)


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The increments d W(t) are independent and normally distributed with zero mean and variance dt,

implying that log returns over T are independent and identically normally distributed with mean of

(r-1/2) and standard deviation of. Assuming constant volatility also allows us to infer stockprices have a lognormal distribution when following the GBM [1]. Adopting assumptions of

continuous buying/selling and no transaction costs allowed for the assumption of no arbitrage. Inthe absence of this arbitrage the return is then the risk free rate. This allowed for the development

of the following partial differential equation for the price of an option on a stock:

+

+ r S

rV = 0 (3)

In essence the value of an option using this model is the greater of the discounted expectation of the

expiry payoff or zero. The model has three other assumptions:

- No dividends.- Ability to borrow or lend at the risk free rate.- Short term risk free rate is constant.

Assumptions used in the model are the source of much controversy as many do not model the actual

market. Empirical observations of asset returns over the past twenty years highlight that the

frequency of three sigma events are far from the normality assumption i.e. dot com crash, Credit

Crisis, LTCM . Transaction costs do exist and are a factor in purchasing decisions, it is not possible to

trade in arbitrarily small portions of shares and stock prices are not continuous over time. Like any

model, it should be used with a firm understanding of its limitations.

Application of Black Scholes to Code Formulae

Building on the above description of Black Scholes, we will analyse the code in the sequence they

have been constructed:

= (

)

(4)

In equation (4) the log expression is the continuously compounded return over the time period T

with the remainder of that half of the equation equal to the mean of the continuously compounded

return. The bottom half is approximately equal to the standard deviation of the percentage change

in the stock price at time T, which in our case is one year. This means that our uncertainty about the

stock price movement is gauged by standard deviation and by the square root of how far ahead we

are looking, which increases the further ahead you look [1]. The component is the componentminus one standard deviation price movement in time.

These are then input into the following equations:

C = . N ( . N ( (5)P= . N ( . N ( (6)


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Both and in equations (5) and (6) represent the normal cumulative distribution function. Inequation (5), as the stock price increases, N ( and N ( approach a value of 1, increasing the callvalue as the likelihood of it being ITM and therefore exercised increases. Similarly the value of N

( and N ( will decrease with an increasing and with it decrease the value of the putoption. An intuitive understanding of the above equations is that the stock price times N (

is the

present value of receiving the stock if and only if the option finishes ITM. The discounted exercise

payment times N (is the present value of paying the exercise price in that event. We can see inthis that N ( is the risk adjusted probability that the option will be exercised. N ( is morecomplicated as it represents the expected value, computed using risk adjusted probabilities, of

receiving the stock at expiration of the option, contingent upon the option finishing in the money

[2].

The option deltas were calculated using the following formulae:

Call Delta =

. N(

(7)

Put Delta =- . N( (8)The elements of which have been discussed above.

Vanilla Option Deltas

The Black Scholes delta describes the rate of change of the option price with respect to changes in

the underlying stock. It is the first derivative [3]. Taking thirty different stock prices, adjusting the

strike price to 65 and holding all other variables constant, the blsdelta function in MatLab was used

to calculate the below profiles illustrating the change in delta as stock price moves (refer tab 1 of

Data workbook attached).

As the above graph illustrates, with the increase in underlying price, a corresponding increase in call

delta value is noted and decrease in put delta value. As both approach the strike price (65) a

steepening in the curves is noted highlighting the increased sensitivity of the delta variables at this

point. As the OTM option approaches being ATM and ITM, the probability of the option taking on

intrinsic value increases as does the probability the option will be exercised. The call delta increases

towards a value of 1 the higher the stock rises telling us that for every movement in the stock, an


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almost corresponding movement is noted in the option value. An opposite value is taken on for the

put option.

I have also graphed the deltas as a function of time using the same blsdelta function and holding

everything constant except the maturity (refer tab 1 of Data workbook attached). Below shows a

deep OTM call option delta and a deep ITM put delta as a function of changing time.

Clearly we can see that the put delta starts at .78 but rapidly decreases for the next 15 periods as it

gets further away from maturity as the probability of finishing ITM decreases. Similarly we can see

the call value rapidly increase in the first ten periods when it is removed from maturity as with the

volatility at 35% there is a greater chance the underlying will move to bring the asset ITM. After ten

periods it is

Code Development and Functioning

Please refer Code1 mfile attached for relevant code. To develop the vanilla option code:

- Initially the arguments outlined in the instructions were typed out and with correspondingvalues.

- Noting the hints in instructions and help function, a list of functions required to completethe formulae was drawn up i.e. normcdf, exp etc. These were verified in the help function.

- These were then put into sequence with those being called first, coded first i.e. the d1 and d2formulae preceded the call and put formulae.

- Having previous experience of coding and the maths involved, writing out the equations wasstraight forward.

- The benchmark code blsprice and blsdelta was drawn up after consulting MatLab help forargument input.

- The code was saved to the MatLab folder and ran. No differences noted to benchmark.


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Part B

A binary option is one with a discontinuous payoff which will pay a predetermined. These are quite

straight forward as if an underlying asset is above or below a certain point at a certain point in time

(depending on whether European or American) then the option holder is entitled to the greater of

the predetermined amount or asset or nothing at all. There are four types of binary options which

have associated payoffs:

- Call binary: Pays one unit if S>K or zero otherwise- Put binary: Pays one unit if K>S or zero otherwise- Cash-or-nothing: K for a call and K for a put- Asset-or-nothing: S(t) for a call and S(t) for a put

These are considered exotic options i.e. other than plain vanilla as described above. For the

purposes of this report we will focus on the cash or nothing option. The valuation formulae for the

cash or nothing calls and puts adopts principals from the Black Scholes model also. The call formula

is the product of the option payoff discounted at the risk free rate (assets grow at risk free rate per

Black Scholes) and the risk adjusted probability the option will be exercised. The put formula is the

value of the call cash or nothing subtracted from the discounted payoff. This makes sense as the

higher the call option goes the higher the probability of it finishing and paying out which will

increase its value. Subtracting this from the discounted payoff will reduce the put value i.e. as the

underlying grows the value of the put will decrease.

When calculating the rate of change of the options value relative to stock price the normal

probability density function is introduced in place of the normal cumulative distribution function.

This introduces the relative likelihood of a variable occurring at a given point. Cumulative describes

all the values less than or equal to a certain point. As described in equation (4) above the bottom

half is approximately equal to the standard deviation of the percentage change in the stock price at

time T and is now multiplied by the share price to calculate the delta. The put delta is calculated in

the same manner. Their movement as a function of increasing underlying value and time is

illustrated below (strike price changed to 65 for illustrative purposes):


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As is evident from the graph, the rate of change of the options value is a decreasing function of the

share price increases. Both approach one as the stock price goes from being OTM to ITM. The deltas

behaviour closer to maturity is much more pronounced and best illustrated in the following diagram

(please refer tab 2 of Data sheet attached, all values calculated from MatLab):

We can see here as time to maturity increases, over the first five years there is rapid decline in delta

value which then plateaus and approaches zero. This is understandable as the nature of the option

is to payoff M or zero, as the maturity approaches an increase in underlying could bring the option

from OTM to ITM and from a zero payoff to M. The contrast in values from ATM to ITM is the reason

for the delta greater than 1.


Please refer Code2 mfile attached for code. To develop the binary option code:

- The list of arguments was input, noting the option payoff was to be included.- Again, the help function was consulted for an understanding of the normpdf function in

calculating the binary delta.

- There was only one equation to write out for each price as d1 and d2 had been previouslycalculated.

- The normpdf function was used in the calculation of the deltas.- Variables and functions were put in order they would be called.- To benchmark the code, I reviewed code used in week 2 lab and adjusted it accordingly.- The code was saved to the MatLab folder and ran. No differences noted to benchmark.


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Part C

An exchange option gives the holder the right but not the obligation to exchange one asset for

another at a predetermined date (if American then at any date until expiration). The option derives

its value from the difference between the prices of two or more assets and a two asset payoff is

described by:

Max ( - , 0)To price these options we must first assume the underlying assets follow a GBM, then adopt the

principals previously outlined from Black Scholes to derive. The pricing equation is very familiar from

the vanilla pricing with a few adjustments:

- is calculated by getting the log returns of one asset relative to the other.- The volatility element is based on the correlation between the two assets.- The dividend yield of the second asset is subtracted from the first as opposed to subtracting

the risk free rate from the first asset.

- The second asset is substituted into the pricing equation in place of the strike price- There is also a difference noted in the delta calculation. The second asset uses the

probability density function as it is trying to identify the probability that a particular level will

be hit that would give the option intrinsic value.

The delta of this two asset portfolio will be shaped largely by the spread size and approximation

to maturity as illustrated below (please refer tab 3 of Data attachment for figures, all values

calculated from MatLab):

We can see that the widening of the spread initially has a steep increase in the delta value which

then plateaus when a range of c. $16 is established. The second delta is marginally higher as it

started from an underlying value of $40 compared to the $50 in delta one. The higher the spread the

further the delta values will approach 1 as the probability of a pay-out is heightened. This is then


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reflected in the option price. The second asset is affected by the change in the first as it uses inthe calculation of , both of which increase as the first share price log returns increase.The deltas are also affected by time:

As the time to maturity increases, the delta values decrease highlighting the sensitivity of options to

underlying movement as maturity approaches. The second delta decreases at a slower rate as it is

starting from a lower underlying price of $40 as opposed to the higher delta 1underlying value of

$50.


Please refer Code3 mfile attached for code. The exchange option code was developed by:

- Updating the inputs for the two asset model, noting the inclusion of two assets withseparate prices, volatilities and a correlation variable.

- No new functions were needed.- and required recalculation as equations were altered.- Variables and functions were put in order they would be called.-

The code was saved and ran and prices observed. No benchmarking completed as no inbuiltMatLab function present to do so.


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Bibliography

[1] Alexander, C. 2007Pricing, Hedging and Trading Financial Instruments. London:John Wiley&

Sons Ltd

[2] Nielsen, L.Understanding N ( and N (: Risk Adjusted Probabilities in the Black ScholesModel. [online]. Available at: http://www.ltnielsen.com/wp-content/uploads/Understanding.pdf.Accessed on 17-3-2011.

[3] Hull, J. 2007,Fundamentals of Futures and Options Markets. Toronto:Pearson.
http://www.ltnielsen.com/wp-content/uploads/Understanding.pdfhttp://www.ltnielsen.com/wp-content/uploads/Understanding.pdfhttp://www.ltnielsen.com/wp-content/uploads/Understanding.pdf

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