[SOLVED] Delta Hedging and Option Pricing

Delta Hedging and Option Pricing
In this topic, we will study one of the most important subjects in financial mathematics the pricing and hedging of options.
So far, the finance in this course has focused on optimal investment, but that is not the area of finance where mathematics has had the biggest impact.
The entire industry of derivatives trading relies upon financial mathematics and did not really exist until financial mathematics made it possible to trade in derivatives with low risk.

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Most people think that the aim of financial mathematics is to somehow beat the stock market, but that is a mistaken view. Financial mathematics is for the most part about creating financial products people want to buy and making a profit by selling them these products at a little over cost price.
The financial product we will focus on as our main example is a Call Option.
Recall that a European call option on a stock with strike K and maturity T is a financial derivative that pays
out max{ST K, 0} at the maturity T where ST is the stock price at maturity.
It is called a derivative because it is a financial product whose payouts are derived from some formula
involving other underlying financial products. In this case the underlying is the stock.
A European put option with strike K and maturity T is a financial derivative that pays out max{K ST , 0}
Anyone can make up their own derivative products, just choose whatever formulae you want to determine the derivatives payoff from information in the underlying.
Example: An Asian call pays out max{S K, 0} where S is the mean stock price from time 0 (when the contract was created) to time T .
Example: A digital call option with strike K pays out according to the formula: {
digital call payoff =
0 otherwise
You can be as creative as you like! However, if you invent a new kind of derivative you might find it difficult to find anyone who wants to buy it off you. It is also far from obvious how much any given derivatives contract should cost.
In a major paper, that essentially started the subject of financial mathematics as well as the entire industry of derivatives trading, Black and Scholes gives a simple model for stock prices where you can work out the price for any derivatives contract.
Intuitively, you would probably expect that the price of a derivative depends upon your view about the future, but in Black and Scholes model everyone agrees on the price of all derivatives.
The reason for this is that Black and Scholes show how to replicate the payoff of a derivative by following a particular trading strategy in the stock and a risk-free bank account only. To start the trading strategy you must charge a fixed amount that their theory shows you how to calculate: this is simply the price. You then follow their trading strategy up to the maturity of the derivative and make the payouts determined in the derivatives contract. You will then exactly break even.

Structure of the week
This week we will
Discuss the concept of replication in more detail and state Black and Scholes main result.
Look in detail at the stock price model used by Black and Scholes: continuous time geometric Brow-
nian motion.
Test with a numerical simulation whether their trading strategy works
Give a sketch proof that their trading strategy works.
(1938-1995)
(1941-) for economics 1997

Problem: How to buy and sell options without taking undue risks Key Paper: The Pricing of Options and Corporate Liabilities
Keywords: Option, Delta Hedging, Replication
1 Pricing by replication
The idea of pricing by replication is not in itself new. It has been used for centuries by bookmakers. A bookmaker is someone who is willing to let you gamble with them and who will give you prices for different bets. They are different from a gambler themselves because they are willing to take both sides on the same bet.
Example: The big fight.
Suppose there is a boxing match between Dr John Armstrong and.

Everyone is excited and so a lot of people want to bet. Suppose that the current odds at the bookmakers mean that if you bet $1 on me winning and I win, then you will receive $1000 plus your $1 stake. If I you lose you will lose your $1 stake.
In the language that bookmakers use, this means the odds on me winning are 1000 to 1: you win $1000 on a stake of $1.
What happens if you want to bet on instead? You have two choices: you could either go to the bookmaker and find out what the current odds are on or you could find people who wanted to bet that I will win and bet with them directly.
If you go down the latter route, you could offer to pay someone $1000 (plus their $1 stake) if I win, but you will get to keep their $1 stake if I lose. You would then be replicating a bet on winning by selling bets on me winning instead.
This bet is equivalent to you betting $1000 that will win, you receive $1 if he wins (plus your $1000 stake) and lose your $1000 stake.
What this shows is that by replication you can get odds on of 1 to 1000 (Meaning you win $1 on a stake of $1000).
This means that the bookmaker cant expect to sell any bets on winning unless the odds they are offering are 1 to 1000, or at least very close.
Perhaps you will be willing to accept slightly worse odds in exchange for not having to bother to find 4

someone to bet with. On the other hand, they can offer you odds of 1 to 1000 by using the replication strategy. We deduce that the odds offered by a bookmaker on winning will be close to 1 to 1000 if the odds on me winning are 1000 to 1.
1.1 Making a profit by offering a service
Suppose that the bookmaker offers odds of exactly 1000 to 1 on me winning and 1 to 1000 on. If they have one customer willing to bet $1 on me at 1000 to 1 and one customer willing to bet $1000 on at 1 to 1000 then they will exactly break even whatever happens in the fight. In practice, the bookmaker will offer very slightly different odds to that they make a small profit whatever happens in the fight.
They will only get customers if their profit is rather small as people could decide to replicate the bets for themselves. So if we assume that there are a lot of bookmakers competing with each other, we can expect that, due to market forces, the bookmakers will only be able to make a small profit.
Since I could, in principle, bet with other people myself and not give the bookmaker a profit, it might seem that it is irrational ever to use a bookmakers at all. However, weve not included the costs involved in finding someone to bet with in this calculation. In practice, people are normally happy to let the bookmaker make a small profit for the convenience of not having to advertise for someone to bet with. Thus the bookmaker is able to make a profit by offering a service.
When analysing this mathematically, we usually make the assumption that the bookmaker will exactly break even. This is a mathematical model that is close to reality, but not exactly the same. The advantage of making this assumption is we can then state clearly that if you know the odds on me winning a boxing match you can calculate the odds of my oponent winning the boxing match.
Here are the key points:
A bookmaker does not attempt to predict the outcome of a boxing match at all. They do not have a view on who wins, they just know how much people are willing to pay to place a bet.
A bookmaker does not gamble themselves! They make sure they guarantee to make a small profit. For convience it is easier to model a bookmaker as exactly breaking even.
Bookmakers use a replication strategy to ensure that they will break even. Financial Services
When you think about how to make money from the financial markets, most people start thinking about how they might predict future stock price movements and invest to make their fortune. But this is not how most people in finance make money, instead they offer some form of service:
Accountants provide advice on tax efficient investments
Pensions companies manage funds on behalf of pension investors
Financial advisors give advice on financial planning
Insurance companies help people manage risks
Bankskeepyourmoneysafefromthievesandmakeiteasyandconvenienttopayforgoodswithyour
bank card.
In the first three examples, the strategies used guarantee a profit. If you provide financial advice and charge 1000 for your advice, you are guaranteed to make money. Although insurance companies in principle

take some risk, they make sure they have so many customers that they can be pretty confident of what will happen on average and so they too are attempting to get a guaranteed profit.
By contrast, speculators invest in the hope of making large profits, but run the risk of making large losses. In financial mathematics we usually focus on financial services rather than speculation.
1.3 Derivatives Traders
The derivatives traders who work in banks are providing a service which is very similar to the service provided by a bookmaker.
The customers of a derivatives trader wish to buy and sell derivative contracts. One of the groupwork exercises considers why anyone might want to buy or sell a derivatives contract, but for now just assume that people really do want to buy and sell such contracts.
The job of the derivatives trader is to sell their customers the derivatives they want to buy, just as the job of a bookmaker is to let people place the bets they want. In fact, a derivatives contract is simply a bet on the financial markets, so there really is very little difference between the job of a bookmaker and the job of a derivatives trader. A bookmaker deals in bets on boxing matches, football matches and horses, a derivatives trader deals in bets on the financial markets, but otherwise they are very similar.
So the job of a derivatives trader is to replicate the derivatives their customers want to buy. They will sell them to their customers at just a little more than the cost of replication. In this way, a derivatives trader should guarantee that they will make a profit.
Since in principle their customers could replicate the derivatives for themseleves, a derivatives trader will only be able to charge slightly more than the cost of replication: they are charging for the service of repli- cating the derivative on the customers behalf.
We will use the mathematical model that a derivatives trader exactly breaks even. This isnt a perfectly accurate model, but it is close to reality. We will then be able to say that the price of a derivative is equal to the cost of replicating that derivative.
In reality, as we know already, there is no such thing as a price. There is the ask price you need to pay to buy something and the bid price you need to pay to tell something and in fact the bid and ask prices quoted on the market are only the price for buying and selling small quantitis. In general, the price that you have to pay for something will also depend upon the quantity you wish to buy or sell. Nevertheless it is convenient mathematically to pretend financial products have a single price, and this is a good approximation for many purposes.
In reality, therefore, a derivatives trader will charge different prices for buying and selling the same derivative and this will be one way in which they make a small profit. Thus, ignoring the profit made by a derivatives trader is an essentially similar assumption to ignoring the difference between the buy and sell prices for a stock. So while it is a bit of a simplification, it is makes the analysis so much easier it is worth making this simplification.
Always remember that a good model has to balance realism with simplicity. For most purposes, ignoring the profit of a derivatives trader is a good approximation.

1.4 The BlackScholes Model
Black and Scholes breakthough was to give a strategy to replicate a put option or a call option by investing only in the underlying stock and a risk-free bank account. This strategy describes exactly how much a derivatives trader should charge and what they need to do if they want to exactly break even. Of course, in practice they will charge a little more so they make a profit.
Black and Scholes had to make various assumptions, most of which are unrealistic but which do give a good first approximation to a real market.
They assume that there is a stock whose price follows geometric Brownian motion: that is it satisfies the SDE
dSt = St(dt+dWt)
for some constants and and a Brownian motion Wt. They assume that this stock does not pay
any dividends.
They assume that you are able to buy and sell stock in arbitrary quantities at the price St
They assume that you can buy and sell stock in continuous time.
Theyassumethatanymoneyyoumayhave,oranymoneythatyouowe,growsatarisk-freeinterest
rate of r (continuously compounded).
These assumptions give us the simplest possible model for investing in a risky stock and a risk-free bank
account in continuous time. Ill justify this claim in the next notebook.
You might think that it is more complex to work in continuous time than discrete time as we have done previously, but actually it makes life simpler because we can use Itos Lemma. There is a downside, you have to explain rather carefully what you mean by statements such as you can buy and sell stock in continuous time. It also means that your model can only ever be a mathematical approximation to reality
Definition: The Black-Scholes price of a European call option with strike K and maturity T in the market described above is given by the formulae:
Of course this looks insanely complicated and you will quite rightly wonder where this formula comes from. I will explain! But first I want to state the main result of Black and Scholes
Theorem: (Replicating a call option) Suppose that at time 0 you receive the amount V (S0, 0, K, T, r, ). Suppose that at every moment in time from 0 to maturity T you ensure that you are holding exactly
:= V St
units of the stock and you put any remaining cash into a risk-free bank account. At time T you liquidate your position, selling any stock you hold and withdrawing any money from the risk-free account. You will thenhaveexactlymax{ST K,0}.
V (St,t,K,T,r,) := StN(d1) er(Tt)KN(d2) d1:= 1 (log(St)+(r+12)(Tt))
d2:= 1 (log(St)+(r12)(Tt))
and N is the cumulative distribution function of the standard normal distribution.

This theorem shows that you can exactly replicate a call option for the Black-Scholes price. The replication strategy is called delta hedging because V is usually denoted by delta. The word hedging comes from
gambling, where the process of offsetting different bets to give a guaranteed outcome is called hedging
your bets.
It follows from this replication theorem that, under our simplifying assumption that derivatives traders ex- actly break even, they can do so by charging the Black-Scholes price. This is why it is called a price!
2 Exercises
These exercises should help you become more familiar with the Black-Scholes formula, which might seem rather intimidating at first. They also give more simple examples of replication arguments.
2.1 Exercise
WehavedefinedaEuropeancalloptiontobeaderivativecontractthatpaysoffmax{ST K,0}attimeT. A European call option with strike K and maturity T is often described as the right (but not the obligation) to buy the stock at time T at a pre-determined price K. Explain why having such a right is equivalent to receivingapaymentofmax{ST K,0}.Notethatitisarightbutnotanobligation,soifyoulosemoney you would not take up this right. This is why it is called an option, you have the option of whether or not to exercise your right at time T .
2.2 Exercise
Plot a graph of the payoff of a European call option with strike K and maturity T against ST , the final stock price. Take K = 100 to be concrete.
2.3 Exercise
Suppose K1 < K2. Explain why a European call option with strike K1 and maturity T is worth more than a European call option with strike K2 and maturity T .2.4 ExercisePlot a graph of the payoff of a European put option against ST2.5 ExerciseConsider a portfolio consisting of: +1 units of a European call option with strike K and maturity T 1 units of a European put option with strike K and maturity TPlot a graph of the portfolio payoff against ST . Deduce that such a portfolio can be replicated by purchasing1 unit of stock and putting erT K into a risk-free bank account at time 0. Deduce the following:Priceofcallattime0Priceofputattime0=S0 erTK 82.6 ExercisePlot a graphs of the Black-Scholes price of an option against the stock price St for a European call optionwith strike K = 100, maturity T = 1 in a model where r = 0.02, = 0.2 and S0 = 110. You should plotgraphsforvaluest{0, 1 , 2 ,…, 9 }. 1010 102.7 ExerciseLet di(S, t, K, T, r, ) be as in the Black-Scholes formula. Suppose that S > K. Show that lim d1(S,t,K,T,r,) =
Deduce that
if S > K. What is the value of
if $ S < K $?lim d2(S,t,K,T,r,) = . tTlim V (S, t, K, T, r, ) = S K tTlim V (S, t, K, T, r, ) = S K tT3 Geometric Brownian MotionWe want to test the theory of Black and Scholes through a simulation. We will therefore need to simulate the stock price.According to the assumptions of the Black-Scholes model is that the stock follows geometric Brownian motion. This means it satisfies the SDE:dSt = St(dt+dWt)for some constants and .3.1 Option 1: Use simulate using the Euler schemeThe most obvious way to simulate St is to approximate it on a grid using the Euler scheme.This means we should simulate St using the difference equation. If we were to simulate a Brownian motionWt we would have:where Wt = Wt+t Wt.St+t St +St(t+Wt)As usual with the Euler scheme, we arent interested in the Brownian motion Wt itself, so we can simply this to St+tSt+St(t+ tt)where we simulate the t as independent, identically distributed standard normals. Notice that this is only an approximate simulation. Only in the limit as t 0 does this become exact.3.2 Option 2: Solve the SDE analyticallyFor the very specific SDE of geometric Brownian motion, we can actually solve the SDE and write down an expression for St in terms of Wt.We do this using Itos Lemma. Take Zt = log(St). Our original equation is dSt = Stdt+StdWtBy Itos Lemmat2St2 dt+ (111)1( d log(S )1 d2 log(S ) )S St2S22St2 dt+S StdWt = 212 dt+dWt 3.3 Solving Geometric Brownian MotionWe can solve the equationbecause it is a constant coefficient SDE. SoTaking exponentials we have proved: Lemma: If St satisfiesdZt =(122)dt+dWt Z t = Z 0 + ( 12 2 ) t + W t .dSt = Stdt+StdWt (1)St=S0exp (22)+Wt3.4 Simulating Geometric Brownian MotionSuppose we want to simulate Zt on a discrete time grid t, 2t, 3t, . . .. Because we have the equation Z t = Z 0 + ( 12 2 ) t + W t ,we can simulate Zt on a discrete time grid using the equationZ t + t = Z t + ( 12 2 ) t + W t ,Notice that this isnt an approximation, it is an exact equation.Again, we dont need to compute Wt itself, we can simulate Zt using the difference equationZ t + t = Z t + ( 12 2 ) t + t t where the t are independent standard normal random variables.We may then compute St = exp(Zt).3.5 Relationship with discrete time Geometric Brownian MotionI introduced discrete-time Geometric Brownian motion earlier in the course.Our equations for simulating continuous-time geometric Brownian motion on a grid are exactly the oneswe used for simulating discrete-time Geometric Brownian motion.So discrete-time Geometric Brownian motion is exactly the same thing a CS: assignmentchef QQ: 1823890830 Email: [email protected]