# Notes on Option Theory

This is a study note on the fundamental theory of the pricing of a `financial derivative`

, whose payoff is defined in terms of an underlying asset. We hereby try to compute a consistent price of the derivative `in relative terms to`

the market price of the underlying asset.

01. Option Pricing Theory

02. Numerical Methods

# Option Pricing Theory ^{↺}

We make our first assumption that the market is `frictionless`

, by which we mean that:

- no transaction cost (commission, bid-ask spread, taxes)
- can hold negative asset (shortting) and there is no margin constraint
- can hold fractional asset
- no market impact from trading

## Arbitrage (Static Portfolio)

We assume that the market lives in a probability space and it includes `tradable assets`

with non-random time- prices and random time- prices:

A `static portfolio`

is a vector of quantities, where each is non-random and constant in time:

Thus the time- value of the static portfolio is;

A static portfolio is an `arbitrage`

if its value satisfies that:

Suppose portfolio `super-replicates`

portfolio , which means that . Then , otherwise arbitrage exists. Same goes if it is a `sub-replication`

. Therefore, if `replicate`

, which menas that , then . This is called `the law of one price`

.

## Assets

### Discount Bond

A `discount bond`

pays at maturity . Given non-random interest rate , the no-arbitrage price of the discount bond is:

### Forward Contract

A `forward contract`

on with non-random delivery price obligates its holder to pay and receive at time . The time- value of the forward contract is .

A `forward price`

is `delivery price`

such that the value of forward contract at time- is zero.

### European Call Option

An `European call option`

gives its holder the right at time to pay and receive . A call has payoff , and it is `in the money`

if at time .

The time- price of a call option satisfies:

For strike :

### European Put Option

An `European put option`

gives its holder the right at time to pay and receive . A put has payoff , and it is `in the money`

if at time .

The time- price of a put option satisfies:

For strike :

In addition,

### Put-Call Parity

# Binomial Tree

We can create a `replicating portfolio`

to calculate the value of a call option under a simple binomial tree:

Where,

And,

Plugging in and :

We can interpret and as probabilities that construct a risk-neutal measure and that:

# The Fundamental Theorem

`The fundamental theorem of asset pricing`

states that:

no arbitrage

if and only if:

there exists a probability measure equivalent to P such that the discounted prices of all tradable assets are martingales w.r.t.

The proof can be summarized as two ideas:

- :
*a martingale is the cumulative P&L from betting on zero-**games, which is always zero no matter how you vary your bet size across games and time. you cannot riskless make something from nothing.* - :
*the**probability of an event is simply the**price**of an asset that pays 1 unit of B iff that event happen*

## Risk-Neutral Measure

The physical probability is not accurate in evaluating a payoff’s true market price. Considering a 50/50 coin flip worth or nothing. Using physical probability the price will be .

However, the actual `market`

price would be different. If the market is risk-adverse, the price would be lower, say . We can view it as this market represents a risk-neutral measure where the down move has higher risk-neutral probabilities than up move.

We can see that the risk-neutral probability `is`

price, that the risk-neutral probability of an event is the price of one-unit payout contingent on the event. Taking a risk-neutral expectation is the same as pricing by replication.

## Radon-Nikodym derivative

In a discrete settimgn with outcomes , the relatioship between the risk-neutral measure and physical measure can be expressed by the `Radon-Nikodym Derivative`

, or liklehood ratio:

The LR is typically larger in bad states than good states, reflecting the price margin on adverse events.

## The Second Fundamental Theorem

A market is said to be `complete`

if every random variable can be replicated by a static portfolio .

`The second fundamental theorem of asset pricing`

states that:

a no arbitrage market is complete

if and only if:

there exists a

unqiuemeasure equivalent to P such that the discounted prices of all tradable assets are martingales w.r.t.

# Trading Strategy

A `filtration`

represents all information revealed at or before time . A stochastic process is `adapted`

to if is -`measurable`

for each , meaning that the value of is determined by the information in .

A `trading strategy`

is a sequence of static strategy adapted to . A trading strategy is `self-financing`

if for all :

This implies that the change in the portfolio value is fully attributable to gains and losses in asset prices:

Therefore,

We define that a trading strategy `replicates`

a time-T payoff if it is `self-financing`

and the value . By the `law of one price`

, at any time , the no-arbitrage price of an asset paying must have the same value of the replicating portfolio.

## Arbitrage (Trading Strategy)

We now expand on the previous definition of arbitrage, that an `arbitrage`

is a self-finance trading strategy whose value satisfies:

# Ito Process

We define an `Ito process`

to be a stochastic process that:

The existence and uniqueness of a solution of can be guaranteed by Lipschitz-type technical condition on and

## Ito’s Rule

The `Ito's rule`

states that give n an Ito process , and a sufficiently smooth function :

With two processes and , and :

In a special case where , the formula becomes:

Note that the Ito’s Rule applies under any probability measure, it is purely math.

## Black-Scholes Model

** Assumptions** Consider two basic assets and in continuous time, where:

And follows GBM dynamics,

** Conclusion** Then by

`no-arbitrage`

and `Ito's rule`

, the time- price of a call option with payoff satisfies the `Black-Scholes PDE`

for We can solve the call price analytically with the `Black-Scholes formula`

:

Here we plotted the BS call price , the intrinsic value and the lower bound against the current underlying price , with paramters , , and

# The Greeks

## Delta

Suppose an asset has a time t value , then its `Delta`

at time is . Delta can be interpreted as:

- the slope of the asset value , plotted as a function of S_t.
- how much the asset value movies per unit move in
- humber of needed to replicate this asset.

If the asset is a call option on and we assumes the `Black-Scholes assumptions`

on , then:

The `Delta`

of a call option is `strictly between 0 and 1`

. As the time-to-maturity decreases, the Delta increases faster the the option becomes more ITM. Here we plotted the BS Delta for equals and against the current underlying price .

## Gamma

For a call option in a B-S model,

In this case, the Gamma can be interpreted as:

- the convextity of w.r.t. S_t
- how much the Delta moves, per unit move in
- how much rebalancing of the replicating portfolio is needed, per unit move in

The `Gamma`

of a call option is `strictly positive`

. As the time-to-maturity decreases, the Gamma increases for ATM options. Here we plotted the BS Delta for equals and against the current underlying price .

## Theta

For a call in B-S model,

The `Theta`

of a call option is `strictly negative`

. As the time-to-maturity decreases, the Theta decreases for ATM options (faster time-decay). Here we plotted the BS Theta for equals and against the current underlying price .

## Discrete Delta Hedge and Gamma Scalping

A discretely `Delta-hedged`

portfolio could buy and short . In this case it is a `Delta neutral`

and `long Gamma`

/`Gamma scalping`

portfolio:

- Delta of the portfolio is
- Gamma of the portfolio is positive
- achieve net profit only if the
`realized volatility`

of is`high`

enough to overcome`time decay`

, otherwise portfolio loss happens. This is the opposite from a`short Gamma`

position, e.g. sell and long Delta

We can visualize the P&L of a long Gamma portfolio in the following graph, where the green area indicate profits and the red area indicate losses. The curved line is the straight line is . As increases, shifts downwards due to time-decay.

In addition, we can show that the P&L of such portfolio does not depend on the drift of the stock:

Continue on L5

# Numerical Methods^{↺}

The `Taylor series`

of a real or complex value function that is differentiable at is:

# Implied Volatility

Given the time- price of a European call option on a non-dividend stock , the time- Black Scholes `implied volatility`

is the unique solution to .

Uniqueness is because is strictly increasing in and Existence is because covers the full range of arbitrage-free prices of the European option

If follows the SDE dynamic , where a non-random function of , then we can first find the implied volatility given call prices with different maturity , and use the equation below to find (not uniquely) the true function :

## Volatility Smile, Skew and Surface

If truely follows GBM with constant volatility , then . However, empirically the is lower when (`volatility smile`

), possibly because

- the market price option using a risk-neutral distribution of log-returns with fatter tails than Normal

Note that is also higher when (`volatility skew`

), possibly due to:

- instantaneous volatility increases as price decreases
- possibility of severe crash fuels demand for downside protection

In addition, the has a term structure and varies for different . The function is call the implied `volatility surface`

# Tree Model

## Binomial Tree

### European Option

Given option price at the -th node , we can induct backward to find :

### American Option - Put

Given option price at the -th node , we can induct backward to find :

### American Option - Call

Given option price at the -th node . If and stock dividend , then it is `never`

optimal to exercise early on an American call option. Therefore

`Argument 1`

At all , the American call is worth more than the exercise payoff :

`Argument 2`

If then construct portfolio . Then V is an arbitrage as and .

## Trinomial Tree

Let and choose to improve accruacy.

# Finite Difference Model

## Explicit Scheme

Inducting backward from to :

Solving for the B-S PDE: where , we get:

Where:

Note that are trinomial tree probabilities.

## Implicit Scheme

Inducting backward from to :

Solving the requires solutions of a system of equation with unknowns.

## Crank-Nicolson Scheme

Inducting backward from to :

If given terminal conditions, then we know ‘s and can solve for .

# Monte Carlo Model

Given be a discounted payoff and the time- price of the payoff . The `Monte Carlo estimator`

of :

By the `strong law of large numbers`

, the sample average converges almost surely to the expected value as . By the `central limit theorem`

:

Often times we need to estimate with sample estimator for the variance of :

The `standard error`

, and a confident interval for is

## Variance Reduction Techniques

### Antithetic Variate

Let . The `antithetic variate estimoator`

:

### Control Variate

A `control variate`

is a random variable, correlated to such that has an explicit formula.

`Example`

Let be the discounted payoff on a call on where . We can choose to be the discounted payoff on a call on where , in which case can be calculated explicitely through B-S formula given constant close to .

The `control variate estimator`

estimates by simulating .

Choose to minimize , we get:

Note that when using sample estimate , the estimated is biased, only when is small.

### Importance Sampling

Suppose are IID draws from density , and . Ordinary Monte Carlo estimator provides:

With importance sampling, find s.t. iff . Then re-draw from density and the `importance sampling estimator`

is:

### Conditional Monte Carlo

Given a random variable :

The `condintional Monte Carlo estimator`

:

# Fourier Transform Model

Given be integrable, meaning . The `Fourier transform`

of is the function defined by:

`Theorem`

If is also integrable, then the `inversion formula`

holds:

## Characteristic Function

The `complex conjugate`

of a complex number is given by . so .

The `characteristic function`

of any random variable is the function defined by:

Therefore if has density , then . A characteristic function `uniquely`

identifies a distribution. For example, , if

- To calculate the
`moments`

of using CF, take the -derivatives of w.r.t. :

- To calculate the
`CDF`

of using CF:

- To calculate
`asset-or-nothing`

call price using CF, given be the asset share price, define the share measure with likelihood ratio .

Therefore for any , the asset-or-nothing call price:

- To calculate a vanilla
`European`

call price on struck at with :

## Heston Model

Provided that:

Where and are BM with correlation , is the rate of mean-reversion, is the long-term mean, and is the volatility of volatility.

We want to find the CF of in order to price options on . The time- conditional `Heston CF`

provides an answer:

Reference:

- FINM 32000 & 33000 Lecture Notes, Roger Lee, the University of Chicago