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.

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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 unqiue measure 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

img.png

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 .

img2.png

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 .

img4.png

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 .

img3.png

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.

img5.png

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