【课程】Mathematical Finance

基于格拉斯哥ECON5020的笔记；

General information

Aims

The general aims of this course are to:

• provide an overall introduction to the type of mathematics used extensively in finance;
• provide a necessary foundation for further study of modern finance in the second semester courses.

ILOs

By the end of this course students will be able to:

1. Analyse stochastic processes closely related to finance and apply stochastic calculus
2. Discuss how well various stochastic processes function as models in finance and explain the limitations of these models
3. Deploy techniques of mathematical finance within a problem solving context
4. Demonstrate skills based on data interpretation and numeracy
5. Demonstrate an ability to construct a focused argument based on coherent general principles
6. Explain financial issues, problems and solutions.

Unit 1：Introduction

Two types of Financial markets:

• Exchanges (stock exchange, options exchange, futures exchange, etc.)

• Over-the-counter (OTC) markets.

Exchanges:

• Equity markets: stocks and shares.

• Bond market: zero-coupon bonds (ZCBs), government bonds and corporate bonds.

• Futures market: futures contracts.

• Foreign exchange (FX) market: foreign currencies, options on foreign currencies.

• Options market: equity options and interest rate options.

• Commodity market: oil and metals futures.

OTC Markets:

• Interest rate market (mostly OTC): caps, áoors, swaps and swaptions.

• Exotic options market: barrier options, lookback options, compound options.

• Credit market (mostly OTC): credit derivatives such as CDSs and CDOs.

Bear and Bull Markets:

• In over-supply or under-demand, security prices will generally fall. This is referred to as the bear market.
• In under-supply or over-demand, security prices will generally rise. This is called the bull market.
• Prices that satisfy supply and demand are called to be in equilibrium (at least, in the academic literature).

Perfect Markets:

Assumption 1. Markets are frictionless, i.e., no transaction costs, no taxes or other costs and there are no penalties for short selling.

Assumption 2. Infinitely divisible security prices. It is possible to buy or sell any (also non-integer) quantity of any security.

Assumption 3. All traders have access to the same information. The Efficient Market Hypothesis (EMH) holds:

• Weak-form: Current security prices reflect all past market prices and data. Technical analysis is of no benefit.
• Semi-strong form: Current security prices reflect all publicly available information. Fundamental analysis is of no benefit.
• Strong form: Current security prices reáect all available information. Inside information is of no benefit.

Assumption 4. Markets are assumed to be arbitrage-free:

• Arbitrage is the guarantee of certain future profit without any current investment. Much of the modern financial theory holds under the assumption that efficient financial markets should forbid such an arbitrage opportunity.

• It follows from the no-arbitrage principle that if two financial securities have the same pattern of future cash-flows, they should have the same price today. This property is commonly known as the law of one price.

Standing assumptions 1. to 4. provide a framework in which we can develop mathematical models for primary securities and propose pricing methods for derivative securities.

European Call Option:

$$C_T := max(S_T-K,0) = (S_T-K)^+$$

European Call Option:

$$P_T := max(K-S_T,0) = (K-S_T,0)^+$$

Zero-Coupon Bond:

$$B_t = (1+r)^t\ ,\ B(t,T) = \frac{B_t}{B_T}=(1+r)^{-(T-t)}$$

Unit 2：Elementary Market Model

Stock price movements are more complicated than indicated by the elementary market model. Hence it cannot be claimed that the elementary model gives a realistic picture of the stock price fluctuations.

Notation:

• Initial Wealth： $x$
• Riskless Asset：$B$ (Bank account)
• Risky Asset：$S$ (Stock)
• Shares of the stock：$\phi$
• Sample space：$\Omega = \{\omega_1,\omega_2\}$
• Probability measure：$\mathbb{P}(\omega_1)=p>0$ and $\mathbb{P}(\omega_2)=1-p>0$
• A deterministic interest rate：$r > -1$
• The price of a risky asset at time $t$：$S_t$ ，$S0 > 0$
• $u = \frac{S_1(\omega_1)}{S_0}$ and $d = \frac{S_1(\omega_2)}{S_0}$，$0<d<u$
• Model：$M = (B, S)$

Trading Strategy: $(x,\phi)$

$$V_0(x,\phi):=x=(x-\phi S_0)+\phi S_0$$ $$V_1(x,\phi)(\omega_i):=(x-\phi S_0)(1+r)+\phi S_1(\omega_i)\ ,\ i=1,2$$

Arbitrage:

A trading strategy $(x, \phi)$ in the single-period market model is called an arbitrage opportunity if

1. $x = 0$, that is, no initial investment is required.
2. $V_1(x, \phi) \ge 0$, that is, no risk of losing money.
3. $\mathbb{E}_{\mathbb{P}}\{V_1(x,\phi)\}>0$, that is, a strictly positive expected payoff.
   1. OR There exists an $\omega_i$ such that $V_1(x,\phi)(\omega_i)>0$.

Arbitrage-free: if no arbitrage opportunity exists in the model.

• The elementary market model $M = (B, S)$ is arbitrage free if and only if $d<1+r<u$.

Contingent claim: a trading strategy $(x, \phi)$ obtain $X=h(S_1)=V_1(x,\phi)$.

Replicationg Strategy (or a hedge):

$$\begin{align} (x-\phi S_0)(1+r)+\phi S_1(\omega_1)=h(S_1(\omega_1))\\ (x-\phi S_0)(1+r)+\phi S_1(\omega_2)=h(S_1(\omega_2)) \end{align}$$

Arbitrage Price: If $(x, \phi)$ is a replicating strategy of a contingent claim then $x$ is called the arbitrage price. $x=\pi_0(X)$

Delta hedging formula:

$$\phi = \frac{h(S_1(\omega_1))-h(S_1(\omega_2))}{S_1(\omega_1)-S_1(\omega_2)}=\frac{h(uS_0)-h(dS_0)}{(u-d)S_0}$$ $$\tilde{p}:=\frac{1+r-d}{u-d}\in(0,1)\ ,\ \tilde{\mathbb{P}}(\omega_1)=\tilde{p}\ ,\ \tilde{\mathbb{P}}(\omega_2)=1-\tilde{p}$$ $$x = \frac{1}{1+r}[\tilde{p}h(S_1(\omega_1))+(1-\tilde{p})h(S_1(\omega_2))]$$

Market Completeness:

All contingent claims (that is, all derivative securities) in thethe elementary market model have replicating strategies, the market described by this model is called complete.

Risk-Neutral Probability Measure:

A probability measure $\mathbb{Q}$ on the sample space $\Omega = \{\omega_1,\omega_2\}$ is called a risk-neutral probability measure (or an equivalent martingale measure) for the market model $M = (B, S)$ if $\mathbb{Q}$ is equivalent to $\mathbb{P}$ and the following equality holds

$$\mathbb{E}_{\mathbb{Q}}(\frac{S_1}{1+r})=S_0$$

• The risk-neutral probability measure for the market model $M = (B, S)$ is unique and it satisfies if and only if $d < 1 + r < u$.

Risk-Neutral Valuation Formula:

$$\pi_0(X) = \frac{1}{1+r}\mathbb{E}_{\tilde{\mathbb{P}}}(h(S_1))=\mathbb{E}_{\tilde{\mathbb{P}}}(\frac{X}{1+r})$$

Put-Call Parity:

$$C_T - P_T = S_T - B(t,T)K\ ,\ B(t,T) = (1+r)^{-(T-t)}$$

Generalisation of the Elementary Market Model:

If the sample space $\Omega = \{\omega_1,\omega_2,…,\omega_k\}$ where $k>3$.

• Some (but not all) contingent claims can be replicated (that is, are attainable). Hence the model $M = (B, S)$ is incomplete.

• The risk-neutral probability measure $\mathbb{Q}$ for the model $M$ exists if and only if $S_1(\omega_k ) < S_0(1 + r) < S_1(\omega_1)$. It is not unique.

Unit 3：Single Period Market Model

General Single-Period Market Model:

The main differences between the elementary and general single-period market models are:

• The investor is allowed to invest in several risky securities instead of only one.

• The sample set is bigger, that is, there are more possible states of the world at time $t = 1$.

Trading Strategy: $(x,\phi^1,…,\phi^n)$

Wealth Process:

$$V_1(x,\phi^1,...,\phi^n):=(x-\sum_{j=1}^n\phi^jS_0^j)(1+r)+\sum_{j-1}^n\phi^jS_1^j$$

Gains Process:

$$\Delta S_1^j = S_1^j-S_0^j$$ $$\begin{aligned} G_1(x,\phi^1,...,\phi^n) : &= V_1(x,\phi^1,...,\phi^n)-V_0(x,\phi^1,...,\phi^n)\\ &= (x-\sum_{j=1}^n\phi^jS_0^j)r + \sum_{j=1}^n \phi^j \Delta S_1^j \end{aligned}$$

Discounted stock prices:

$$\hat{S}_0^j := S_0^j = \frac{S_0^j}{B_0},\ \hat{S}_1^j := \frac{S_1^j}{1+r} = \frac{S_1^j}{B_1}$$

Discounted wealth process:

$$\hat{V}_0(x,\phi^1,...,\phi^n) := x$$ $$\hat{V}_0(x,\phi^1,...,\phi^n) = x + \sum_{j=1}^n \phi^j(\hat{S}_1^j-\hat{S}_0^j)$$

Discounted gains process:

$$\Delta S_1^j = S_1^j-S_0^j$$ $$\hat{G_0}(x,\phi^1,...,\phi^n)=0$$ $$\hat{G}_1(x,\phi^1,...,\phi^n) := \sum_{j=1}^n\phi^j\Delta\hat{S}_1^j$$

Arbitrage:

A trading strategy $(x,\phi^1,…,\phi^n)$ in a general single-period market model is called an arbitrage opportunity if

1. $V_0(x,\phi^1,…,\phi^n) = 0$,
2. $V_1(x,\phi^1,…,\phi^n)(\omega_i) \ge 0$, for $ i = 1,2…,k,$
3. $\mathbb{E}_{\mathbb{P}}\{V_1(x,\phi^1,…,\phi^n)\}>0$,
   • That is $\sum_{i=1}^k V_1(x,\phi^1,…,\phi^n)(\omega_i)\mathbb{P}(\omega_i)>0$
   • OR There exists an $\omega_i$ such that $V_1(x,\phi)(\omega_i)>0$.

$\hat{V}(x,\phi^1,…,\phi^n)$ can take the place of $V(x,\phi^1,…,\phi^n)$

When $V_0(x,\phi^1,…,\phi^n)=0$, $\hat{G}(x,\phi^1,…,\phi^n)$ can take the place of $\hat{V}(x,\phi^1,…,\phi^n)$

Risk-Neutral Probability Measure:

A probability measure $\mathbb{Q}$ on $\Omega$ is called a risk-neutral probability measure for a general single-period market model $M$ if:

• $\mathbb{Q}(\omega_i)>0$ for all $\omega_i \in \Omega$
• $\mathbb{E}_{\mathbb{Q}}(\Delta\hat{S}_1^j) = 0$ for $j = 1,2,…,n$
  • OR $\mathbb{E}_{\mathbb{Q}}(\hat{S}_1^j) = \hat{S}_0^j$
  • OR $\mathbb{E}_{\mathbb{Q}}(S_1^j) = (1+r)S_0^j$

Fundamental Theorem of Asset Pricing (FTAP)

A general single-period market model $M = (B, S^1, . . . , S^n)$ is arbitrage-free if and only if there exists a risk-neutral probability measure for $M$, that is, $\mathbb{M}\ne\empty$.

Proof

Contingent Claim: is a random variable $X$ defined on $\Omega$ and representing a payoff at the maturity date.

Replication and Arbitrage Price:

A trading strategy $(x,\phi^1,…,\phi^n)$ is called a replicating strategy (a hedging strategy) for a claim X when $V_1(x,\phi^1,…,\phi^n)=X$. Then the initial wealth is also denoted as $\pi_0(X)$ and it is called the arbitrage price of $X$.

No-Arbitrage Principle:

Assume that a contingent claim X can be replicated by means of a trading strategy $(x,\phi^1,…,\phi^n)$. Then the unique price of X at 0 consistent with no-arbitrage principle equals $V_0(x,\phi^1,…,\phi^n)=x$.

Stochastic Volatility Model:

$$\Omega = \{\omega_1,\omega_2,\omega_3,\omega_4\}$$

The volatility is defined as

$$v(\omega_i) = \begin{cases} h\ for\ i = 1,4,\\ l\ for\ i = 2,3. \end{cases}$$

We furthermore assume that $0 < l < h < 1$. The stock price $S_1$ is given by

$$S_1(\omega_i) = \begin{cases} (1+v(\omega_i))S_0\ for\ i = 1,2,\\ (1-v(\omega_i))S_0\ for\ i = 3,4. \end{cases}$$

It is easy to check that the model is arbitrage-free whenever $1-h<1+r<1+h$.

We claim that for some contingent claims a replicating strategy does not exist. In that case, we say that a claim is not attainable.

To justify this claim, we consider the digital call option $X$ with the payoff.

$$x = \begin{cases} 1\ if\ S_1>K,\\ 0\ otherwise,\end{cases}$$

We assume that $(1 + l)S_0 < K < (1 + h)S_0$, and ($\Rightarrow$ lead to not attainable)

$$X(\omega_i)=\begin{cases} 1\ for\ i = 1,\\ 0\ otherwise.\end{cases}$$

Suppose that $(x, \phi)$ is a replicating strategy for $X$. Equality $V_1(x, \phi) = X$ becomes

$$(x-\phi S_0)\begin{pmatrix}1+r\\1+r\\1+r\\1+r \end{pmatrix}+\phi \begin{pmatrix}(1+h)S_0\\(1+l)S_0\\(1+h)S_0\\(1+l)S_0 \end{pmatrix}=\begin{pmatrix}1\\0\\0\\0 \end{pmatrix}$$

It is easy to see that the above system of equations has no solution and thus a digital call is not an attainable contingent claim within the framework of the stochastic volatility model.

The heuristic explanation is that the randomness generated by the volatility cannot be replicated, we do not have anough traded assets to replicate volatility, since the volatility itself is not a traded asset in this model.

Attainable Contingent Claim:

A contingent claim $X$ is called to be attainable if there exists a replicating strategy for $X$.

Risk-neutral valuation formula:

Let $X$ be an attainable contingent claim and let $\mathbb{Q}\in\mathbb{M}$ be any risk-neutral probability measure. Then the arbitrage price of $X$ at $t = 0$ equals

$$\pi_0(X)=\frac{1}{1+r}\mathbb{E}_{\mathbb{Q}}(X)$$

which shows that risk-neutral probability measures can be used to price attainable contingent claims.

Pricing Stochastic Volatility Model:

The increments of the discounted stock price $\hat{S}$ are represented in the following

	$\omega_1$	$\omega_2$	$\omega_3$	$\omega_4$
$\Delta\hat{S}_1$	$hS_0$	$lS_0$	$-lS_0$	$-hS_0$

Recall that a vector $(q1, q2, q3, q4)^{\top}$ must satisfy $\sum_{i=1}^4q_i=1$ holds and $q_i > 0$ for $i = 1, 2, 3, 4$.

The class $\mathbb{M}$ of all risk-neutral probability measures in our stochastic volatility model is therefore given by

$$\mathbb{M} = \begin{Bmatrix}\begin{pmatrix}q_1\\q_2\\q_3\\q_4\end{pmatrix}\vline \begin{matrix}q_1>0,q_2>0,q_3>0,\\q_1+q_2+q_3<1,\\l(q_2-q_3)=h(1-(2q_1+q_2+q_3))\end{matrix}\end{Bmatrix}$$

Indeed, it suffices to take $q_1\in(0,\frac{1}{2})$ and to set

$$q_4=q_1,q_2=q_3=\frac{1}{2}-q_1$$ $$\mathbb{E}_{\mathbb{Q}}(X)=q_1·1+q_2·0+q_3·0+q_4·0=q_1$$

Valuation of Non-Attainable Claims: still using the risk-neutral probability measure.

Complete and Incomplete Markets:

A financial market, described by a model, is called complete if for any contingent claim $X$ there exists a replicating strategy $(x, \phi) \in \mathbb{R}^{n+1}$ . A market is incomplete when there exists a claim $X$ for which a replicating strategy does not exist.

$M$ is complete if and only if the $k\times(n+1)$ matrix A

$$A=\begin{pmatrix}1+r&S_1^1(\omega_1)&···&S_1^n(\omega_1)\\ 1+r&S_1^1(\omega_2)&···&S_1^n(\omega_3)\\ ···&···&···&···\\1+r&S_1^1(\omega_k)&···&S_1^n(\omega_k)\end{pmatrix}=(A_0,A_1,···,A_n)$$

has a full row rank, that is, $rank(A) = k$.

Unit 4：Modelling Uncertainty

$\sigma$-Field:

can be used to describe the amount of information available at a given moment.

• A collection $\mathcal{F}$ of subsets of $\Omega$ is called a $\sigma$-field (or a $\sigma$-algebra)

• Most of the literature uses $\sigma$-algebra than​ $\sigma$-field.

• Any set $A \in \mathcal{F}$ is interpreted as an observed event.

• Let $\mathbb{N} = \{1, 2, . . .\}$ be the set of all natural numbers.

Probability Measure: A map $\mathbb{P}:\mathcal{F} \to [0,1]$ is called a probability measure if

• $\mathbb{P}(\Omega)=1$
• For any sequence $A_i, i \in \mathbb{N}$ of pairwise disjoint events we have

$$\mathbb{P}(\bigcup_{i\in\mathbb{N}}A_i)=\sum_{i\in\mathbb{N}}\mathbb{P}(A_i)$$

• The triplet $(\Omega, \mathcal{F}, \mathbb{P})$ is called a probability space.

Partition: By a partition of $\Omega$, we mean any collection $P = (A_i)_{i \in l}$of non-empty subsets of $\Omega$.

F-Measurability: A map $X : \Omega \to \mathbb{R}$ is said to be F-measurable (measurable with respect to algebra F) if the function $\omega \to X (\omega)$ is constant on any subset in the partition corresponding to $\mathcal{F}$. Equivalently, for every real number $x$ the subset $\{\omega\in\Omega:X(\omega)=x\}$ is an element of algebra $\mathcal{F}$. If $X$ is $\mathcal{F}$-measurable then $X$ is called a random variable on $(\Omega,\mathcal{F})$.

Filtration: A family $(\mathcal{F}_t)_{0\leq t\leq T}$ of $\sigma$-fields on $\Omega$ is called a filtration if $\mathcal{F}_s \subset \mathcal{F}_t$ whenever $s \leq t$. For brevity, we denote $\mathbb{F}=(\mathcal{F}_t)_{0\leq t\leq T}$.

Stochastic Process: A stochastic process is a real-valued function $S_n (t, \omega) : \{0, 1, …,T\} \times \Omega \to \mathbb{R}$. For each fixed $\omega \to \Omega$ the function $t \to S_n (t, \omega)$ is called sample path. For each fixed $t$ the function $\omega \to S_n (t, \omega)$ is a random variable.

Conditional Expectation:

$$\mathbb{E}(X|A)=\sum_xx\mathbb{P}(X=x|A)$$

• Assume that $\mathcal{G}$ is a $\sigma$-field which is contained in $\mathcal{F}$.

$$\mathbb{E}_{\mathbb{P}}(X|\mathcal{G})(\omega)=\sum_{i\in l}\frac{1}{\mathbb{P}(A_i)}\mathbb{E}_{\mathbb{P}}(X\mathbb{I}_{A_i})\mathbb{I}_{A_i}$$ $$\sum_{\omega\in G}X(\omega)\mathbb{P}(\omega)=\sum_{\omega\in G}\mathbb{E}_{\mathbb{P}}(X|\mathcal{G})(\omega)\mathbb{P}(\omega),\ \forall G \in \mathcal{G}$$

Properties of Conditional Expectation:

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be endowed with sub-$\sigma$-fields $\mathcal{G}$ and $\mathcal{G}_1 \subset \mathcal{G}_2$ of $\mathcal{F}$. Then

• Tower property: If $X : \Omega\to \mathbb{R}$ is an F-measurable random variable, then

$$\mathbb{E}_{\mathbb{P}}(X|\mathcal{G}_1) =\mathbb{E}_{\mathbb{P}}(\mathbb{E}_{\mathbb{P}}(X|\mathcal{G}_2)|\mathcal{G}_1)=\mathbb{E}_{\mathbb{P}}(\mathbb{E}_{\mathbb{P}}(X|\mathcal{G}_1)|\mathcal{G}_2)$$

• Taking out what is known: If $X : \Omega\to\mathbb{R}$ is a $\mathcal{G}$-measurable random variable and $Y : \Omega\to\mathbb{R}$ is an $\mathcal{F}$-measurable random variable, then

$$\mathbb{E}_{\mathbb{P}}(XY|\mathcal{G})=X\mathbb{E}_{\mathbb{P}}(Y|\mathcal{G})$$

• Trivial conditioning: If $X : \Omega\to\mathbb{R}$ is an $\mathcal{F}$-measurable random variable independent of $\mathcal{G}$, then

$$\mathbb{E}_{\mathbb{P}}(X|\mathcal{G}) = \mathbb{E}_{\mathbb{P}}(X)$$

Martingales: are stochastic processes representing fair games.

An $\mathbb{F}$-adapted process $X = (X_t)_{0\leq t\leq T}$ on a finite probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is called a martingale whenever for all $s < t$

$$\mathbb{E}_{\mathbb{P}}(X_t|\mathcal{F}_s) = X_s$$

• supermartingale:

$$\mathbb{E}_{\mathbb{P}}(X_t|\mathcal{F}_s) \leq X_s$$

• submartingale:

$$\mathbb{E}_{\mathbb{P}}(X_t|\mathcal{F}_s) \ge X_s$$

Unit 5：Multi-Period Market Model

Trading Strategy:

• $\phi_t=(\phi_t^0,\phi_t^1,…,\phi_t^n)$ for $t=0,1,…,T$
• $\phi_t^0$ is the number of ‘shares’ of the money market account $B$ held at time $t$.
• $\phi_t^j$ is the number of shares of the $j$th stock held at time $t$.

Wealth process:

$$V_t(\phi) = \phi_t^0B_t + \sum_{j=1}^n\phi_t^jS_t^j$$

Self-Financing Trading Strategy:

$$\phi_t^0B_{t+1}+\sum_{j=1}^n\phi_t^jS_{t+1}^j= \phi_{t+1}^0B_{t+1}+\sum_{j=1}^n\phi_{t+1}^jS_{t+1}^j$$

Gains Process:

$$G_t(\phi):=V_t(\phi)-V_0(\phi)$$ $$G_t(\phi) = \sum_{u=0}^{t-1}\phi_u^0B_{u+1} + \sum_{u=0}^{t-1}\sum_{j=1}^n\phi_u^jS_{u+1}^j$$

Increment Processes:

$$\Delta S_{t+1}^j := S_{t+1}^j-S_t^j$$ $$\Delta B_{t+1} := B_{t+1}-B_t=(1+r)^tr=B_tr$$

Discounted stock prices:

$$\hat{S}_t^j := \frac{S_t^j}{B_t}$$

Discounted wealth process:

$$\hat{V}_t(\phi) := \frac{V_t(\phi)}{B_t}$$ $$\hat{V}_t(\phi) = \hat{V}_0(\phi)+\sum_{u=0}^{t-1}\sum_{j=1}^n\phi_u^j\Delta \hat{S}_{u+1}^j$$

Discounted gains process:

$$\hat{G}_t(\phi):=\hat{V}_t(\phi)-\hat{V}_0(\phi)$$ $$\hat{G}_t(\phi) = \sum_{u=0}^{t-1}\sum_{j=1}^n\phi_u^j\Delta \hat{S}_{u+1}^j$$

Hence for every $t = 0, . . . ,T-1 $

$$\Delta\hat{V}_{t+1}(\phi)=\Delta\hat{G}_{t+1}(\phi)=\sum_{j=1}^n\phi_u^j\Delta \hat{S}_{u+1}^j$$

Arbitrage:

A trading strategy $\phi \in \Phi$ is an arbitrage opportunity if

• $V_0(\phi)=0,$
• $V_T(\phi)(\omega)\ge 0$ for all $\omega \in \Omega$,
• $V_T(\phi)(\omega)> 0$ for some $\omega \in \Omega$,
  • OR $\mathbb{E}_{\mathbb{P}}(V_T(\phi))>0$

Risk-Neutral Probability Measure:

A probability measure $\mathbb{Q}$ on $\Omega$ is called a risk-neutral probability measure for a multi-period market model $M = (B, S^1, . . . , S^n)$ whenever

• $\mathbb{Q}(\omega)>0$ for all $\omega \in \Omega$
• $\mathbb{E}_{\mathbb{Q}}(\Delta\hat{S}_{t+1}^j|\mathcal{F}_t) = 0$ for $j = 1,2,…,n$ and $t=0,…,T-1$.
  • OR $\mathbb{E}_{\mathbb{Q}}(\hat{S}_{t+1}^j|\mathcal{F}_t) = \hat{S}_t^j$

Fundamental Theorem of Asset Pricing:

$$\mathrm{Class\ } \mathbb{Q} \mathrm{\ is\ non-empty } \Leftrightarrow \mathrm{Market\ model}\ M\ \mathrm{is\ arbitrage-free}$$

Replicating Strategy:

A replicating strategy (or a hedging strategy) for a contingent claim $X$ is a trading strategy $\phi\in\Phi$ such that $V_T(\phi) = X$, that is, the terminal wealth of the trading strategy matches the claim’s payoff for all $\omega$.

Attainable Contingent Claim: A contingent claim $X$ is called to be attainable if there exists a trading strategy $\phi\in\Phi$, which replicates $X$, i.e., $V_T (\phi) = X$.

Completeness: A multi period market model is said to be complete if and only if all contingent claims have replicating strategies.

Risk-neutral valuation formula:

$$\pi_t(X)=B_t\mathbb{E}_{\mathbb{Q}}(\frac{X}{B_T}|\mathcal{F}_t)$$

Completeness: Assume that a multi-period market model $M = (B, S^1, . . . , S^n)$ is arbitrage-free. Then $M$ is complete if and only if there is only one risk-neutral probability measure, that is, $\mathbb{M} = \{\tilde{\mathbb{P}}\}$ is a singleton.

Unit 6：Binomial Asset Pricing Model

If $d < 1 + r < u$ then the CRR market model $M = (B, S)$ is arbitrage-free and complete.

Risk-Neutral Probability Measure:

Assume that $d < 1 + r < u$. Then a probability measure $\tilde{\mathbb{P}}$ on $(\Omega, \mathcal{F}_T )$ is a risk-neutral probability measure for the CRR model $M = (B, S)$ with parameters $p, u, d,r$ and time horizon $T$ if and only if:

• $X_1,X_2,X_3,…,X_T$ are independent under the probability measure $\tilde{\mathbb{P}}$
• $0<\tilde{p}:=\tilde{\mathbb{P}}$(X_t=1)<1$ for all $t = 1,…,T$
• $\tilde{p}u+(1-\tilde{p})d=(1+r)$

where $X$ is the Bernoulli process governing the stock price $S$.

CRR Call Option Pricing Formula

$$C_0=S_0\sum_{k=\hat{k}}^T\begin{pmatrix}T\\k\end{pmatrix}\hat{p}^k(1-\hat{p})^{T-k}-\frac{K}{(1+r)^T}\sum_{k=\hat{k}}^T\begin{pmatrix}T\\k\end{pmatrix}\tilde{p}^k(1-\tilde{p})^{T-k}$$

where

$$\tilde{p}=\frac{1+r-d}{u-d},\ \ \hat{p}=\frac{\tilde{p}u}{1+r}$$

and $\hat{k}$ is the smallest integer k such that

$$klog(\frac{u}{d})>log(\frac{K}{S_0d^T})$$

Put-Call Parity

Set $\mathbb{P}(S_T=S_0u^kd^{T-k})=\begin{pmatrix}T\\k\end{pmatrix}p^k(1-p)^{T-k}$

$$C_t:=S_t\hat{\mathbb{P}}(D|\mathcal{F}_t)-KB(t,T)\tilde{\mathbb{P}}(D|\mathcal{F}_t)$$ $$P_t:=-S_t\hat{\mathbb{P}}(\bar{D}|\mathcal{F}_t)+KB(t,T)\tilde{\mathbb{P}}(\bar{D}|\mathcal{F}_t)$$

where $D = \{\omega\in\Omega:S_T(\omega)>K\}, \bar{D}=\{\omega\in\Omega:S_T(\omega)<K\}$

Therefore:

$$\begin{align} C_t-P_t &=S_t(\hat{\mathbb{P}}(D|\mathcal{F}_t)+\hat{\mathbb{P}}(\bar{D}|\mathcal{F}_t))-KB(t,T)(\tilde{\mathbb{P}}(D|\mathcal{F}_t)+\tilde{\mathbb{P}}(\bar{D}|\mathcal{F}_t))\\ &=S_t-KB(t,T) \end{align}$$

where $B(t,T)=(1+r)^{-(T-t)}$

American Call Option

By an arbitrage free price of the American call we mean a price process $C_t^a, t \leq T$, such that the extended financial market model - that is, a market with trading in riskless bonds, stocks and the American call option - remains arbitrage-free.

$$C_t^a=\max_\tau \mathbb{E}_{\tilde{\mathbb{P}}}((1+r)^{-(\tau-t)}(S_\tau-K)^+|\mathcal{F}_t),\ \forall t\leq T$$ $$\tau^*_t = min\{u\ge t|(S_u-K)^+\ge C_u^a \}$$ $$C_t^a\ge C_t\ge S_t-\frac{K}{(1+r)^{T-t}}>S_t-K$$

The result only holds for non-dividend-paying stock.

American Put Option

$$P_t^a=\max_\tau \mathbb{E}_{\tilde{\mathbb{P}}}((1+r)^{-(\tau-t)}(K-S_\tau)^+|\mathcal{F}_t),\ \forall t\leq T$$ $$\tau^*_t = min\{u\ge t|(K-S_u)^+\ge P_u^a \}$$

The stopping time $\tau^*_t$ is called the rational exercise time of an American put option that is assumed to be still alive at time $t$.

Dynamic Programming Recursion:

In the CRR model, the arbitrage pricing of the American call/put option reduces to the following recursive recipe, for $t \leq T-1$,

$$C_t^a=max\{(S_t-K)^+,(1+r)^{-1}(\tilde{p}C_{t+1}^{au}+(1-\tilde{p})C_{t+1}^{ad})\}$$ $$P_t^a=max\{(K-S_t)^+,(1+r)^{-1}(\tilde{p}P_{t+1}^{au}+(1-\tilde{p})P_{t+1}^{ad})\}$$

with the terminal condition

$$C_T^a = (S_T-K)^+\ ,\ P_T^a = (K-S_T)^+$$

The quantities $C_{t+1}^{au}/P_{t+1}^{au}$ and $C_{t+1}^{ad}P/_{t+1}^{ad}$ represent the values of the American call/put in the next step corresponding to the upward and downward movements of the stock price starting from a given node on the CRR lattice.

Derivation of $u$ and $d$ from $r$ and $\sigma$

The Cox-Ross-Rubinstein (CRR) parametrisation:

$$u = e^{\sigma\sqrt{\Delta t}}\ and\ d=\frac{1}{u}$$

Assume that $B_{k\Delta t}=(1+r\Delta t)^k$ for every $k = 0, 1, . . . , n$. Then the risk-neutral probability measure $\tilde{\mathbb{P}}$ satisfies

$$\tilde{\mathbb{P}}(S_{t+\Delta t}=S_tu|S_t)=\frac{1+r\Delta t-d}{u-d}=\frac{1}{2}+\frac{r-\frac{\sigma^2}{2}}{2\sigma}\sqrt{\Delta t}+o(\sqrt{\Delta t})$$

BS-PDE是这个 CRR 求的

$\tilde{\mathbb{P}}$ 部分可以用泰勒展开约到 $o(\Delta t)$求

The Jarrow-Rudd (JR) parameterisation:

$$u=e^{(r-\frac{\sigma^2}{2})\Delta t+\sigma\sqrt{\Delta t}}\ and \ d=e^{(r-\frac{\sigma^2}{2})\Delta t-\sigma\sqrt{\Delta t}}$$

Let $B_{k\Delta t}=(1+r\Delta t)^k$ for every $k = 0, 1, . . . , n$. Then the risk-neutral probability measure $\tilde{\mathbb{P}}$ satisfies

$$\tilde{\mathbb{P}}(S_{t+\Delta t}=S_tu|S_t)=\frac{1}{2}+o(\sqrt{\Delta t})$$

MatlabCode

r = 0.008333333; u = 1.095583494; d = 0.912755628; 
strike = 53; S0 = 50; T = 5; N = T;
SS = zeros(N,T);
SS(1,1) = S0;
for t = 2:T
    for n = 1:t
        SS(n,t) = S0*u^(t-n)*d^(n-1);
    end
end
payoff = max((strike-SS(:,T)),0); % put
rnp = (1+r-d)/(u-d);
PayOffMatr = zeros(N,T);
PayOffMatr(:,T) = payoff;
for j=T-1:-1:1
    for k = 1:N-1
        PayOffMatr(k,j) = max(((PayOffMatr(k,j+1)*rnp+ PayOffMatr(k+1,j+1)*(1-rnp))/(1+r)),(strike-SS(k,j))); 
        % American put
        % PayOffMatr(k,j) = (PayOffMatr(k,j+1)*rnp + PayOffMatr(k+1,j+1)*(1-rnp))/(1+r); 
        % European put
    end
end

Unit 7：Black-Scholes Formula

Wiener Process:

A stochastic process $W = (W_t, t \to \mathbb{R}_+)$ on $\Omega$ is called the Wiener process if the following conditions hold:

• W(0) = 0
• Sample paths of $W$ are continuous functions.
• For any $0 \leq s < t, W (t) - W(s) \sim N(0,t-s)$ where $N(\mu,\sigma^2)$ denotes the normal distribution with expected value $\mu$ and variance $\sigma^2$.
• The process $W_t$ has independent increments, are mutually independent.

Let $N$ denote the normal distribution. Probability distribution function (pdf) or a normally distributed variable is

$$f(x,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

For any $t>0,W_t \sim N(0,t)$ and thus $(\sqrt{t})^{-1}W_t \sim N(0,1)$. The random variable $W_t$ has the pdf given by

$$p(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$$

Hence for any real numbers $a\leq b$

$$\mathbb{P}(W_t\in[a,b])=\int_a^b\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}dx = N(\frac{b}{\sqrt{t}})-N(\frac{a}{\sqrt{t}})$$

Markov Property

$$\mathbb{P}(W_t\leq x|W_{t_1}=x_1,...,W_{t_n}=x_n)=\mathbb{P}(W_t\leq x|W_{t_n}=x_n)$$ $$\mathbb{P}(W_t\leq y|W_s=x) = \int_{-\infty}^yp(t-s,z-x)dz$$ $$p(t-s,z-x)=\frac{1}{\sqrt{2\pi (t-s)}}exp(-\frac{(z-x)^2}{2(t-s)})$$ $$\mathbb{E}_{\mathbb{P}}(W_t|W_s)=W_s$$

Stochastic Differential Equation:

$$dS_t = rS_tdt + \sigma S_tdW_t$$

Geometric Brownian motion:

$$S_t = S_0 exp(\sigma W_t+(r-\frac{1}{2}\sigma^2)t)$$

Random Walk Approximation:

Let $Y_t^h$ for $t = 0,\Delta t, . . . , $ be a random walk starting at 0.

$$\mathbb{P}(Y_{t+\Delta t}^h=y+h|Y_t^h=y)=\mathbb{P}(Y_{t+\Delta t}^h=y-h|Y_t^h=y)=0.5$$ $$W_t = \lim_{h\to0}Y_t^h$$

Approximation of the Stock Price:

$$S_t = S_s e^{(r-\frac{1}{2}\sigma^2)(t-s)+\sigma (W_t-W_s)}$$ $$W_t-W_{t-\Delta t} \approx Y_t-Y_{t-\Delta t}=\pm h = \pm\sqrt{\Delta t}$$

Black-Scholes market model

Assumptions:

• There are no arbitrage opportunities in the class of trading strategies.
• It is possible to borrow or lend any amount of cash at a constant interest rate $r \ge 0$.
• The stock price dynamics are governed by a geometric Brownian motion.
• It is possible to purchase any amount of a stock and short-selling is allowed.
• The market is frictionless: there are no transaction costs (or any other costs).
• The underlying stock does not pay any dividends.

Discounted Stock Price:

$$\mathbb{E}_{\tilde{\mathbb{P}}}(\hat{S}_t|\hat{S}_s,s\leq t)=\hat{S}_t$$

Recall $W_t-W_s = Z\sqrt{t-s}$ where $Z \sim N(0,1)$, and thus $e^{z} \sim$ Log-Normal Distribution

$$\mathbb{E}_{\tilde{\mathbb{P}}}(e^{aZ})=e^{a^2/2}$$ $$\mathbb{E}_{\tilde{\mathbb{P}}}(e^{\sigma Z\sqrt{t-s}})=e^{\frac{1}{2}\sigma^2(t-s)}$$

The Black-Scholes Formula

$$C_t(S_t)=S_tN(d_+(S_t,T-t))-Ke^{-r(T-t)}N(d_{\_}(S_t,T-t))$$ $$P_t(S_t)=Ke^{-r(T-t)}N(-d_{\_}(S_t,T-t))-S_tN(-d_+(S_t,T-t))$$ $$C_t-P_t = S_T-Ke^{-r(T-t)}$$

where

$$d_\pm(S_t,T-t)=\frac{ln\frac{S_t}{K}+(r\pm\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{(T-t)}}$$

推导过程

The Black-Scholes PDE

Let the price of the contingent claim at t given the current stock price $S_t = x$ be denoted by $V (t, x)$. Then $V (t, x)$ is the solution of the Black-Scholes partial differential equation.

$$\frac{\partial}{\partial t}V(t,x)+\frac{\sigma^2x^2}{2}\frac{\partial^2}{\partial x^2}V(t,x)+rx\frac{\partial}{\partial x}V(t,x)-rV(t,x)=0$$

with the terminal condition $V (T, x) = h(x)$.

推导过程

To summarise, Black-Scholes equation is written in the form

$$\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}+bS\frac{\partial C}{\partial S}-rC=0$$

where $b$ is the cost-to-carry:

Asset with no dividend	$b = r$
Asset with dividend $D$	$b = r - D$
foreign currency with return $r_f$	$b = r - r_f$
Commodity with storage cost $q$	$b = r + q$
Futures	$b = 0$

Unit 8：Greeks, Hedging and limits of Black-Scholes

Suppose that the stock price satisfies the Black-Scholes assumption:

$$\mathbb{E}_{\mathbb{Q}}\{S_t\} = S_0e^{rt}$$ $$Var(S_t) = S_0^2e^{2rt}(e^{\sigma^2t}-1)$$

Two ideas to estimate $σ$ in the Black-Scholes model:

1. Collect some historical stock prices to evaluate the variance $ν$ and then apply equation:

$$v \approx S_0^2e^{2rt}(e^{\sigma^2t}-1)$$

1. Collect some historical stock prices per $\Delta t$, and define the return $R$. Then compute the variance of $R$ and

$$\sigma^2 \approx \frac{Var(R)}{\Delta t}$$

The leads to Idea 2:

Since $dlnS_t \approx lnS_{t+\Delta t}-lnS_t$ and

$$dlnS_t = \frac{dS_t}{S_t} \approx \frac{S_{t+\Delta t}-S_t}{S_t} = R_t$$

we have

$$R_t \approx lnS_{t+\Delta t}-lnS_t = (r - \frac{\sigma^2}{2})\Delta t +\sigma(W^*(t+\Delta t)- W^*(t)).$$

Then $Var(R) = \sigma^2 \Delta t$

Numerical methods are important in mathematical finance:

• The Monte Carlo scheme

• Bellman equations: The finite difference methods, the Markov chain approach, or the projection methods.

Greeks

Delta	Gamma	Rho	Theta	Vega
$\Delta$	$\Gamma$	$\rho$	$\Theta$	$vega$
$\frac{\partial C}{\partial S}$	$\frac{\partial^2 C}{\partial S^2}$	$\frac{\partial C}{\partial r}$	$\frac{\partial C}{\partial t}$	$\frac{\partial C}{\partial \sigma}$

$$\Phi'(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$$ $$S\Phi'(d_1)-e^{-r(T-t)}K\Phi'(d_2)=0$$

Call option

Delta: always positive : An increase in the asset price increases the likely payoff and therefore the price of the option.

$$\Delta = \Phi(d_1)$$

Gamma:

$$\Gamma = \frac{\Phi'(d_1)}{S\sigma\sqrt{T-t}}$$

Rho: always positive : An increase in the interest rate is equivalent to lowering the strike price $K$ ( the present price of this future payment decreases with an increase of the interest rate ). This again makes payoff more likely which in e§ect increases the price of the option.

$$\rho = (T-t)Ke^{-r(T-t)}\Phi(d_2)$$

Theta: always negative : The option price is decreasing in time, given that everything else is fixed. This fact can be deduced from a no arbitrage argument.

$$\Theta = \frac{-S\sigma}{2\sqrt{T-t}}\Phi'(d_1)-rKe^{-r(T-t)}\Phi(d_2)$$

Vega: always positive : This can be understood by considering that an increase in volatility leads to a wider spread of asset prices. However the payoff from a European call option is bounded from below, so that this spread has a more positive effect then a negative effect and is therefore increasing the payoff. The increase in payoff leads to an increase of the price of the option.

$$vega = S\sqrt{T-t}\Phi'(d_1)$$

Implied Volatility:

The solution of equation is called an implied volatility.

$$C(t,T, \sigma_{impl},K) = C^{obs}(t)$$

We can therefore compute its zeros numerically by applying the Newton method.

$$\sigma_{k+1} = \sigma_k - \frac{C(t,T, \sigma_k,K)-C^{obs}(t)}{\frac{\partial}{\partial \sigma}C(t,T, \sigma_k,K)}$$

one can cure some of these problems by using either local volatility models ( nowadays quite famous among financial institutions ) or stochastic volatility models ( for example the Heston model ).