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.

# Option Pricing Theory↺

We make our first assumption that the market is frictionless, by which we mean that:

• 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 $\mathbb{P}$ and it includes $N$ tradable assets with non-random time-$0$ prices and random time-$T$ prices:

A static portfolio is a vector of quantities, where each $\theta$ is non-random and constant in time:

Thus the time-$t$ value of the static portfolio $\Theta$ is;

A static portfolio $\Theta$ is an arbitrage if its value $V_t$ satisfies that:

Suppose portfolio $\Theta^a$ super-replicates portfolio $\Theta^b$, which means that $\mathbb{P}[V^a_T > V^b_T] = 1$. Then $V^a_0 \geq V^b_0$, otherwise arbitrage exists. Same goes if it is a sub-replication. Therefore, if $\Theta^a$ replicate $\Theta^b$, which menas that $\mathbb{P}[V^a_T = V^b_T] = 1$, then $V^a_0 = V^b_0$. This is called the law of one price.

## Assets

### Discount Bond

A discount bond $Z$ pays $1$ at maturity $T$. Given non-random interest rate $r_t$, the no-arbitrage price of the discount bond is:

### Forward Contract

A forward contract on $S_T$ with non-random delivery price $K$ obligates its holder to pay $K$ and receive $S_T$ at time $T$. The time-$0$ value of the forward contract is $S_0 - KZ_0$.

A forward price $F_0$ is delivery price such that the value of forward contract at time-$0$ is zero.

### European Call Option

An European call option gives its holder the right at time $T$ to pay $K$ and receive $S_T$. A call has payoff $(S_T-K)^+$, and it is in the money if $S_t>K$ at time $t\leq T$.

The time-$0$ price $C_0$ of a call option satisfies:

For strike $K_1:

### European Put Option

An European put option gives its holder the right at time $T$ to pay $S_T$ and receive $K$. A put has payoff $(K-S_T)^+$, and it is in the money if $S_t at time $t\leq T$.

The time-$0$ price $P_0$ of a put option satisfies:

For strike $K_1:

# 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 $\alpha$ and $\beta$:

We can interpret $p_u$ and $p_d$ as probabilities that construct a risk-neutal measure $\mathbb{P}$ and that:

# The Fundamental Theorem

The fundamental theorem of asset pricing states that:

no arbitrage

if and only if:

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

The proof can be summarized as two ideas:

• $\exists\ M.G. \mathbb{P} \rightarrow \text{no arb}$:

a martingale is the cumulative P&L from betting on zero-$\mathbb{E}$ games, which is always zero no matter how you vary your bet size across games and time. you cannot riskless make something from nothing.

• $\text{no arb} \rightarrow \exists\ M.G. \mathbb{P}$:

the $\mathbb{P}$ 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 $1M$ or nothing. Using physical probability the price will be $500K$.

However, the actual market price would be different. If the market is risk-adverse, the price would be lower, say $300K$. 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.

In a discrete settimgn with outcomes $\{\omega_1\,\dots ,\omega_n\}$, the relatioship between the risk-neutral measure and physical measure $P$ 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 $Y_T$ can be replicated by a static portfolio $\Theta$.

The second fundamental theorem of asset pricing states that:

a no arbitrage market is complete

if and only if:

there exists a unqiue measure $\mathbb{P}$ equivalent to P such that the discounted prices of all tradable assets are martingales w.r.t. $\mathbb{P}$

A filtration $\{\mathcal{F}_t\}$ represents all information revealed at or before time $t$. A stochastic process $X$ is adapted to $\{\mathcal{F}_t\}$ if $X_t$ is $\mathcal{F}_t$-measurable for each $t$, meaning that the value of $X_t$ is determined by the information in $\mathcal{F}_t$.

A trading strategy is a sequence of static strategy $\Theta_t$ adapted to $\mathcal{F}_t$. A trading strategy is self-financing if for all $t>0$:

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 $\Theta$ replicates a time-T payoff $X_T$ if it is self-financing and the value $V_T=X_T$. By the law of one price, at any time $t$, the no-arbitrage price of an asset paying $X_T$ must have the same value of the replicating portfolio.

We now expand on the previous definition of arbitrage, that an arbitrage is a self-finance trading strategy $\Theta_t$ whose value $V_t$ satisfies:

# Ito Process

We define an Ito process to be a stochastic process $X$ that:

The existence and uniqueness of a solution of $X$ can be guaranteed by Lipschitz-type technical condition on $\mu_t$ and $\sigma_t$

## Ito’s Rule

The Ito's rule states that give n an Ito process $X_t$, and a sufficiently smooth function $f(X_t)$:

With two processes $X_t$ and $Y_t$, and $f(X_t, Y_t)$:

In a special case where $Y_t = t$, 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 $B_t$ and $S_t$ in continuous time, where:

And $S_t$ follows GBM dynamics,

Conclusion Then by no-arbitrage and Ito's rule, the time-$t$ price $C_t$ of a call option with payoff $(S_T-K)^{+}$ satisfies the Black-Scholes PDE for $(S, t)\in [0, \infty]\times (0, T)$

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

Here we plotted the BS call price $C^{BS}$, the intrinsic value $(S_t - K)^{+}$ and the lower bound $(S_t - Ke^{-r(T-t)})^{+}$ against the current underlying price $S_t$, with paramters $K=100$, $T-t=1$, $\sigma=0.2$ and $r=0.05$

# The Greeks

## Delta

Suppose an asset has a time t value $V_t(S_t, t)$, then its Delta at time $t$ is $\partial V_t(S_t, t)/\partial S_t$. Delta can be interpreted as:

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

If the asset is a call option on $S_t$ and we assumes the Black-Scholes assumptions on $S_t$, 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 $T-t$ equals $1$ and $0.25$ against the current underlying price $S_t$.

## Gamma

For a call option in a B-S model,

In this case, the Gamma can be interpreted as:

• the convextity of $C^{BS}$ w.r.t. S_t
• how much the Delta moves, per unit move in $S_t$
• how much rebalancing of the replicating portfolio is needed, per unit move in $S_t$

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 $T-t$ equals $1$ and $0.25$ against the current underlying price $S_t$.

## 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 $T-t$ equals $1$ and $0.25$ against the current underlying price $S_t$.

## Discrete Delta Hedge and Gamma Scalping

A discretely Delta-hedged portfolio could buy $C$ and short $\text{Delta} \cdot S$. In this case it is a Delta neutral and long Gamma/Gamma scalping portfolio:

• Delta of the portfolio is $0$
• Gamma of the portfolio is positive
• achieve net profit only if the realized volatility of $S$ is high enough to overcome time decay, otherwise portfolio loss happens. This is the opposite from a short Gamma position, e.g. sell $C$ and long Delta $S$

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 $C_{t+\Delta t}$ the straight line is $\text{Delta}_t \cdot S_{t+\Delta t}$. As $\Delta t$ increases, $C_{t+\Delta t}$ shifts downwards due to time-decay.

In addition, we can show that the P&L of such portfolio $dV = dC - C_S dS$ does not depend on the drift $\mu$ of the stock:

Continue on L5

# Numerical Methods↺

The Taylor series of a real or complex value function $f(x)$ that is differentiable at $a$ is:

# Implied Volatility

Given the time-$t$ price of a European call option on a non-dividend stock $S$, the time-$t$ Black Scholes implied volatility $\sigma(t)$ is the unique solution to $C_t = C^{BS}(\sigma(t))$.

Uniqueness is because $C^{BS}$ is strictly increasing in $\sigma$ and Existence is because $C^{BS}$ covers the full range of arbitrage-free prices of the European option $[S_0-Ke^{-rT}, S_0]$

If $S_t$ follows the SDE dynamic $dS_t = rS_tdt + \sigma(t)S_tdW_t$, where $\sigma(t)$ a non-random function of $t$, then we can first find the implied volatility $\bar{\sigma}_T$ given call prices with different maturity $T$, and use the equation below to find (not uniquely) the true function $\sigma(t)$:

## Volatility Smile, Skew and Surface

If $S_t$ truely follows GBM with constant volatility $\sigma$, then $\sigma_{imp}(K, T) = \sigma, \;\forall\;K, T$. However, empirically the $\sigma_{imp}$ is lower when $K \approx S_t$ (volatility smile), possibly because

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

Note that $\sigma_{imp}$ is also higher when $K < S_t$ (volatility skew), possibly due to:

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

In addition, the $\sigma_{imp}$ has a term structure and varies for different $T$. The function $\sigma_{imp}(K, T)$ is call the implied volatility surface

# Tree Model

## Binomial Tree

### European Option

Given option price at the $j$-th node $C_T^j = f(S_T^j)$, we can induct backward to find $C_t^j$:

### American Option - Put

Given option price at the $j$-th node $C_T^j = (K - S_T^j)^+$, we can induct backward to find $C_t^j$:

### American Option - Call

Given option price at the $j$-th node $C_T^j = (S_T^j - K)^+$. If $r>0$ and stock dividend $\delta=0$, then it is never optimal to exercise early on an American call option. Therefore $C^{American} = C^{European}$

Argument 1 At all $t>0$, the American call is worth more than the exercise payoff $S_t - K$:

Argument 2 If $C^{American} > C^{European}$ then construct portfolio $V = [-C^{American}, C^{European}]$. Then V is an arbitrage as $V_0 > 0$ and $V_T \geq 0$.

## Trinomial Tree

Let $\Delta t:= T/N$ and choose $\Delta x \approx \sigma\sqrt{3\Delta t}$ to improve accruacy.

# Finite Difference Model

## Explicit Scheme

Inducting backward from $t=T$ to $0$:

Solving for the B-S PDE: $rC = C_t + vC_S + 0.5\sigma^2C_{SS}$ where $v = (r-\sigma^2/2)$, we get:

Where:

Note that $p_u, p_m, p_d$ are trinomial tree probabilities.

## Implicit Scheme

Inducting backward from $t=T$ to $0$:

Solving the $LHS$ requires solutions of a system of $2J-1$ equation with $2J-1$ unknowns.

## Crank-Nicolson Scheme

Inducting backward from $t=T$ to $0$:

If given terminal conditions, then we know $C_{t+1}$‘s and can solve for $C_t$.

# Monte Carlo Model

Given $Y$ be a discounted payoff and the time-$0$ price of the payoff $C=\mathbb{E}Y$. The Monte Carlo estimator $\hat{C}_M$ of $C$:

By the strong law of large numbers, the sample average $\hat{C}_M$ converges almost surely to the expected value $C$ as $M \rightarrow \infty$. By the central limit theorem:

Often times we need to estimate $\sigma$ with sample estimator for the variance of $Y$:

The standard error $SE = \hat{\sigma}_M/\sqrt{M}$, and a $95\%$ confident interval for $C$ is $\hat{C}_M \pm 1.96SE$

## Variance Reduction Techniques

### Antithetic Variate

Let $\tilde{Y}:=Y_{Z = -z\;}$. The antithetic variate estimoator $\hat{C}^{av}_M$:

### Control Variate

A control variate $Y^\ast$ is a random variable, correlated to $Y$ such that $C^\ast := \mathbb{E}Y^\ast$ has an explicit formula.

Example Let $Y$ be the discounted payoff on a call on $S_t$ where $dS_t = \sigma(t)S_tdW_t$. We can choose $Y^\ast$ to be the discounted payoff on a call on $S_t^{\ast}$ where $dS_t^{\ast} = \sigma S^{\ast}_tdW_t$, in which case $C^{\ast}$ can be calculated explicitely through B-S formula given constant $\sigma$ close to $\sigma(t)$.

The control variate estimator $\hat{C}^{cv, \beta}_M$ estimates $C$ by simulating $Y - \beta Y^{\ast}$.

Choose $\beta$ to minimize $Var(\hat{C}^{cv}_M)$, we get:

Note that when using sample estimate $\hat{\beta^{\ast}\;}$, the estimated $\hat{C}^{cv, \hat{\beta}^{\ast}\;}_M$ is biased, only when $M$ is small.

### Importance Sampling

Suppose $X$ are IID draws from density $f$, and $C := \mathbb{E}h(X)$. Ordinary Monte Carlo estimator provides:

With importance sampling, find $g$ s.t. $g(x) > 0$ iff $f(x)g(x) \neq 0$. Then re-draw $X$ from density $g$ and the importance sampling estimator $\hat{C}^{is}_M$ is:

### Conditional Monte Carlo

Given a random variable $X$:

The condintional Monte Carlo estimator:

# Fourier Transform Model

Given $f:\mathbb{R}\rightarrow\mathbb{R}$ be integrable, meaning $\int|f(x)|dx < \infty$. The Fourier transform of $f$ is the function $\hat{f}: \mathbb{R}\rightarrow\mathbb{C}$ defined by:

Theorem If $\hat{f}$ is also integrable, then the inversion formula holds:

## Characteristic Function

The complex conjugate of a complex number $z = x + yi$ is given by $\bar{z} = x - yi$. so $\text{Re}(z) = \text{Re}(\bar{z})$.

The characteristic function of any random variable $X$ is the function $F_X(z)$ defined by:

Therefore if $X$ has density $f$, then $F_X(z) = \hat{f}(z)$. A characteristic function uniquely identifies a distribution. For example, $F_X(z) = e^{-z^2/2}$, if $X\sim\mathcal{N}(0,1)$

• To calculate the moments of $X$ using CF, take the $n$-derivatives of $F_X(z)$ w.r.t. $z$:
• To calculate the CDF of $X$ using CF:
• To calculate asset-or-nothing call price using CF, given $e^X$ be the asset share price, define the share measure $\mathbb{P}^{\ast}$ with likelihood ratio $e^X/\mathbb{E}e^X$.

Therefore for any $k\in\mathbb{R}$, the asset-or-nothing call price:

• To calculate a vanilla European call price on $e^X$ struck at $K$ with $k := log\;K$:

## Heston Model

Provided that:

Where $W^S$ and $W^V$ are $\mathbb{P}$ BM with correlation $\rho$, $\kappa$ is the rate of mean-reversion, $\theta$ is the long-term mean, and $\eta$ is the volatility of volatility.

We want to find the CF of $X$ in order to price options on $S_T$. The time-$t$ conditional Heston CF provides an answer:

Reference:

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