Questions and Answers of Introduction Finance Markets

Consider an asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$given by the stochastic differential equation$$\begin{equation*}d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}
(Carr and Lee (2008)) Consider an underlying asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$given by $d S_{t}=r S_{t} d t+\sigma_{t} S_{t} d B_{t}$, where $\left(B_{t}ight)_{t \in
Compute the moment $\mathbb{E}^{*}\left[R_{0, T}^{4}ight]$ from Lemma 8.2.
Consider the Black-Scholes call pricing formula$$C(T-t, x, K)=K f\left(T-t, \frac{x}{K}ight)$$written using the function$$f(\tau, z):=z \Phi\left(\frac{\left(r+\sigma^{2} / 2ight) \tau+\log
The prices of call options in a certain local volatility model of the form $d S_{t}=S_{t} \sigma\left(t, S_{t}ight) d B_{t}$ with risk-free rate $\underline{r=0}$ are given by$$C\left(S_{0}, K,
Let $\sigma_{\mathrm{imp}}(K)$ denote the implied volatility of a call option with strike price $K$, defined from the relation$$M_{C}(K, S, r, \tau)=C\left(K, S, \sigma_{\mathrm{imp}}(K), r,
(Hagan et al. (2002)) Consider the European option priced as $e^{-r T} \mathbb{E}^{*}\left[\left(S_{T}-ight.ight.$ $\left.K)^{+}ight]$in a local volatility model $d S_{t}=\sigma_{\text {loc
Show that the result of Proposition 9.4 can be recovered from Lemma 8.2 and Relation (9.18).
Find an expression for $\mathbb{E}^{*}\left[R_{0, T}^{4}ight]$ using call and put pricing functions.
(Henry-Labordère (2009), § 3.5).a) Using the gamma probability density function and integration by parts or Laplace transform inversion, prove the formula$$\int_{0}^{\infty} \frac{\mathrm{e}^{-u
Let $\left(W_{t}ight)_{t \in \mathbb{R}_{+}}$be standard Brownian motion, and let $a>W_{0}=0$.a) Using the equality (10.2), find the probability density function $\varphi_{\tau_{a}}$ of the first
a) Compute the mean value$$\mathbb{E}\left[\operatorname{Max}_{t \in[0, T]} \widetilde{W}_{t}ight]=\mathbb{E}\left[\operatorname{Max}_{t \in[0, T]}\left(\sigma W_{t}+\mu tight)ight]$$of the maximum
Exercise 10.3 Consider a risky asset whose price $S_{t}$ is given by$$\begin{equation*}d S_{t}=\sigma S_{t} d W_{t}+\frac{\sigma^{2}}{2} S_{t} d t \tag{10.25}\end{equation*}$$where
a) Compute the "optimal call option" prices $\mathbb{E}\left[\left(M_{0}^{T}-Kight)^{+}ight]$estimated by optimally exercising at the maximum value $M_{0}^{T}$ of $\left(S_{t}ight)_{t \in[0, T]}$
Consider an asset price $S_{t}$ given by $S_{t}=$ $S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, t \geqslant 0$, where $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion,
Recall that the maximum $X_{0}^{t}:=\operatorname{Max}_{s \in[0, t]} W_{s}$ over $[0, t]$ of standard Brownian motion $\left(W_{s}ight)_{s \in[0, t]}$ has the probability density
Using From Relation (10.11) in Proposition 10.3 and the Jacobian change of variable formula, assuming $S_{0}>0$, compute the joint probability density function of geometric Brownian motion
a) Compute the hedging strategy of the up-and-out barrier call option on the underlying asset price $S_{t}$ with exercise date $T$, strike price $K$ and barrier level $B$, with $B \geqslant K$.b)
Pricing Category ' $\mathrm{R}$ ' CBBC rebates. Given $\tau>0$, consider an asset price $\left(S_{t}ight)_{t \in[\tau, \infty)}$, given by$$S_{\tau+t}=S_{\tau} \mathrm{e}^{r t+\sigma W_{t}-\sigma^{2}
Compute the Vega of the down-and-out and down-and-in barrier call option prices, i.e. compute the sensitivity of down-and-out and down-and-in barrier option prices with respect to the volatility
Price the up-and-out binary barrier option with payoff$$C:=\mathbb{1}_{\left\{S_{T}>Kight\}} \mathbb{1}_{\left\{M_{0}^{T}K \text { and } M_{0}^{T} \leqslant Bight\}}$$at time $t=0$, with $K \leqslant
Check that the function $g(t, x)$ in (11.26) satisfies the boundary conditions$$\left\{\begin{array}{l}g(t, B)=0, \quad t \in[0, T] \\g(T, x)=0, \quad x \leqslant K
Consider a market made of a riskless asset priced $A_{t}=A_{0}$ with zero interest rate, $t \geqslant 0$, and a risky asset whose price modeled by a standard Brownian motion as \(S_{t}=B_{t}, t
Given the price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$defined as the geometric Brownian motion\[S_{t}:=S_{0} \mathrm{e}^{\sigma B_{t}+\left(r-\sigma^{2} / 2ight) t}, \quad t \geqslant
Consider an asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$given by the stochastic differential equation\[\begin{equation*}d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}
Consider an asset price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$which is a martingale under the risk-neutral probability measure $\mathbb{P}^{*}$ in a market with interest rate $r=0$,
Consider an underlying asset price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$under a risk-neutral measure $\mathbb{P}^{*}$ with risk-free interest rate $r$.a) Does the European call
Consider an underlying asset price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$under a risk-neutral measure $\mathbb{P}^{*}$ with risk-free interest rate $r$.a) Show that the price at time
The following two graphs describe the payoff functions $\phi$ of bull spread and bear spread options with payoff $\phi\left(S_{N}ight)$ on an underlying asset priced $S_{N}$ at maturity time
Given two strike prices $K_{1}- One long call with strike price \(K_{1}$ and payoff function $\left(x-K_{1}ight)^{+}$,- One short put with strike price $K_{1}$ and payoff function
Exercise 7.9 Butterfly options. A long call butterfly option is designed to deliver a limited payoff when the future volatility of the underlying asset is expected to be low. The payoff function of a
Forward contracts revisited. Consider a risky asset whose price $S_{t}$ is given by $S_{t}=S_{0} \mathrm{e}^{\sigma B_{t}+r t-\sigma^{2} t / 2}, t \geqslant 0$, where \(\left(B_{t}ight)_{t \in
Option pricing with dividends (Exercise 6.3 continued). Consider an underlying asset price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$paying dividends at the continuous-time rate
Forward start options (Rubinstein (1991)). A forward start European call option is an option whose holder receives at time $T_{1}$ (e.g. your birthday) the value of a standard European call option
Cliquet option. Let \(0=T_{0}
Consider the price process $\left(S_{t}ight)_{t \in[0, T]}$ given by\[\frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}\]and a riskless asset valued $A_{t}=A_{0} \mathrm{e}^{r t}, t \in[0, T]$, with
Consider the price process $\left(S_{t}ight)_{t \in[0, T]}$ given by\[\frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}\]and a riskless asset valued $A_{t}=A_{0} \mathrm{e}^{r t}, t \in[0, T]$, with
a) Consider the solution $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$of the stochastic differential equation\[d S_{t}=\alpha S_{t} d t+\sigma d B_{t}\]For which value $\alpha_{M}$ of $\alpha$ is
Compute the arbitrage-free price\[C\left(t, S_{t}ight)=\mathrm{e}^{-(T-t) r} \mathbb{E}_{\alpha}\left[\left(S_{T}ight)^{2} \mid \mathcal{F}_{t}ight]\]at time $t \in[0, T]$ of the power option with
Consider two assets whose prices $S_{t}^{(1)}, S_{t}^{(2)}$ at time $t \in[0, T]$ follow the geometric Brownian dynamics\[d S_{t}^{(1)}=r S_{t}^{(1)} d t+\sigma_{1} S_{t}^{(1)} d W_{t}^{(1)}
Exercise 7.20 Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$be a standard Brownian motion generating a filtration $\left(\mathcal{F}_{t}ight)_{t \in \mathbb{R}_{+}}$. Recall that for \(f \in
Consider a price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$given by\[\frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}, \quad S_{0}=1\]under the risk-neutral probability measure $\mathbb{P}^{*}$.
Consider an underlying asset whose price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$ is given by a stochastic differential equation of the form\[d S_{t}=r S_{t} d t+\sigma\left(S_{t}ight) d
Chooser options. In this problem we denote by $C\left(t, S_{t}, K, Tight)$, resp. $P\left(t, S_{t}, K, Tight)$, the price at time $t$ of the European call, resp. put, option with strike price
Consider a risky asset priced\[S_{t}=S_{0} \mathrm{e}^{\sigma B_{t}+\mu t-\sigma^{2} t / 2}, \quad \text { i.e. } \quad d S_{t}=\mu S_{t} d t+\sigma S_{t} d B_{t}, \quad t \geqslant 0\]a riskless
The Capital Asset Pricing Model (CAPM) of W.F. Sharpe (1990 Nobel Prize in Economics) is based on a linear decomposition\[\frac{d S_{t}}{S_{t}}=(r+\alpha) d t+\beta \times\left(\frac{d
Quantile hedging (Föllmer and Leukert (1999), $\S 6.2$ of Mel'nikov et al. (2002)). Recall that given two probability measures $\mathbb{P}$ and $\mathbb{Q}$, the Radon-Nikodym density
Bachelier (1900) model. Consider a market made of a riskless asset valued $A_{t}=A_{0}$ with zero interest rate, $t \geqslant 0$, and a risky asset whose price $S_{t}$ is modeled by a standard
Consider a risky asset price $\left(S_{t}ight)_{t \in \mathbb{R}}$ modeled in the Cox et al. (1985) (CIR) model as\[\begin{equation*}d S_{t}=\beta\left(\alpha-S_{t}ight) d t+\sigma \sqrt{S_{t}} d
Black-Scholes PDE with dividends. Consider a riskless asset with price $A_{t}=$ $A_{0} \mathrm{e}^{r t}, t \geqslant 0$, and an underlying asset price process \(\left(S_{t}ight)_{t \in
a) Check that the Black-Scholes formula for European call options\[ g_{\mathrm{c}}(t, x)=x \Phi\left(d_{+}(T-t)ight)-K \mathrm{e}^{-(T-t) r} \Phi\left(d_{-}(T-t)ight), \]satisfies the following
Power option Power options can be used for the pricing of realized variance and volatility swaps. Let $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$denote a geometric Brownian motion solution of\[ d
On December 18, 2007, a call warrant has been issued by Fortis Bank on the stock price $S$ of the MTR Corporation with maturity $T=23 / 12 / 2008$, strike price $K=$ HK\$ 36.08 and entitlement
Forward contracts. Recall that the price $\pi_{t}(C)$ of a claim payoff $C=h\left(S_{T}ight)$ of maturity $T$ can be written as $\pi_{t}(C)=g\left(t, S_{t}ight)$, where the function \(g(t,
a) Solve the Black-Scholes PDE\[\begin{equation*}r g(t, x)=\frac{\partial g}{\partial t}(t, x)+r x \frac{\partial g}{\partial x}(t, x)+\frac{\sigma^{2}}{2} x^{2} \frac{\partial^{2} g}{\partial
Log contracts can be used for the pricing and hedging of realized variance swaps.a) Solve the PDE \[\begin{equation*}0=\frac{\partial g}{\partial t}(x, t)+r x \frac{\partial g}{\partial x}(x,
Binary options. Consider a price process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$given by\[ \frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}, \quad S_{0}=1 \]under the risk-neutral probability measure
a) Bachelier (1900) model. Solve the stochastic differential equation\[\begin{equation*}d S_{t}=\alpha S_{t} d t+\sigma d B_{t} \tag{6.41}\end{equation*}\]in terms of \(\alpha, \sigma \in
a) Show that for every fixed value of $S$, the function\[ d \longmapsto h(S, d):=S \Phi(d+|\sigma| \sqrt{T})-K \mathrm{e}^{-r T} \Phi(d), \]reaches its maximum at \(d_{*}(S):=\frac{\log (S /
a) Compute the Black-Scholes call and put Vega by differentiation of the Black-Scholes function\[g_{\mathrm{c}}(t, x)=\mathrm{Bl}(x, K, \sigma, r, T-t)=x \Phi\left(d_{+}(T-t)ight)-K
Consider the backward induction relation, i.e.\[\widetilde{v}(t, x)=\left(1-p_{N}^{*}ight) \widetilde{v}\left(t+1, x\left(1+a_{N}ight)ight)+p_{N}^{*} \widetilde{v}\left(t+1,
(Leung and Sircar (2015)) ProShares Ultra S\&BP500 and ProShares UltraShort SESP500 are leveraged investment funds that seek daily investment results, before fees and expenses, that correspond to
Today I went to the Furong Peak mall. After exiting the Poon Way MTR station, I was met by a friendly investment consultant from NTRC Input, who recommended that I subscribe to the following
Today I went to the East mall. After exiting the Bukit Kecil MTR station, I was approached by a friendly investment consultant from Avenda Insurance, who recommended me to subscribe to the following
A lump sum of $\$ 100,000$ is to be paid through constant yearly payments $m$ at the end of each year over 10 years.a) Find the value of $m$.b) Assume that the amount remaining at the beginning
Today I received an SMS from Jack, and I opted for the $3 \mathrm{~K}$ loan over 12 months. a) Compute the monthly interest rate earned by Jack using the below graph of the function r+ (1 (1+r)-12)
Consider a two-step trinomial (or ternary) market model $\left(S_{t}ight)_{t=0,1,2}$ with $r=0$ and three possible return rates $R_{t} \in\{-1,0,1\}$. Show that the probability measure
We consider a riskless asset valued $S_{k}^{(0)}=S_{0}^{(0)}, k=0,1, \ldots, N$, where the risk-free interest rate is $r=0$, and a risky asset $S^{(1)}$ whose returns
We consider the discrete-time Cox-Ross-Rubinstein model with $N+1$ time instants $t=0,1, \ldots, N$, with a riskless asset whose price $A_{t}$ evolves as $A_{t}=A_{0}(1+r)^{t}$, \(t=0,1,
We consider the discrete-time Cox-Ross-Rubinstein model on $N+1$ time instants $t=0,1, \ldots, N$, with a riskless asset whose price $A_{t}$ evolves as $A_{t}=A_{0}(1+r)^{t}$ with \(r
Consider a two-step trinomial market model $\left(S_{t}^{(1)}ight)_{t=0,1,2}$ with $r=0$ and three possible return rates $R_{t}=-1,0,1$, and the risk-neutral probability measure
Consider a two-step binomial market model $\left(S_{t}ight)_{t=0,1,2}$ with $S_{0}=1$ and stock return rates $a=0, b=1$, and a riskless account priced $A_{t}=(1+r)^{t}$ at times $t=0,1,2$,
In a two-step trinomial market model $\left(S_{t}ight)_{t=0,1,2}$ with interest rate $r=0$ and three return rates $R_{t}=-0.5,0,1$, we consider a down-an-out barrier call option with exercise
Consider a two-step binomial random asset model $\left(S_{k}ight)_{k=0,1,2}$ with possible returns $a=0$ and $b=200 \%$, and a riskless asset $A_{k}=A_{0}(1+r)^{k}, k=0,1,2$ with interest
We consider a two-step binomial market model $\left(S_{t}ight)_{t=0,1,2}$ with $S_{0}=1$ and return rates $R_{t}=\left(S_{t}-S_{t-1}ight) / S_{t-1}, t=1,2$, taking the values $a=0, b=1$, and
Consider a discrete-time market model made of a riskless asset priced $A_{k}=$ $(1+r)^{k}$ and a risky asset with price $S_{k}, k \geqslant 0$, such that the discounted asset price process
Call-put parity.a) Show that the relation $(x-K)^{+}=x-K+(K-x)^{+}$holds for any $K, x \in \mathbb{R}$.b) From part (a), find a relation between the prices of call and put options with strike
Consider a two-step binomial random asset model $\left(S_{k}ight)_{k=0,1,2}$ with possible returns $a=-50 \%$ and $b=150 \%$, and a riskless asset $A_{k}=A_{0}(1+r)^{k}, k=0,1,2$ with
Analysis of a binary option trading website.a) In a one-step model with risky asset prices $S_{0}, S_{1}$ at times $t=0$ and $t=1$, compute the price at time $t=0$ of the binary call option
A put spread collar option requires its holder to sell an asset at the price $f(S)$ when its market price is at the level $S$, where $f(S)$ is the function plotted in Figure 3.10, with
A call spread collar option requires its holder to buy an asset at the price $f(S)$ when its market price is at the level $S$, where $f(S)$ is the function plotted in Figure 3.10, with
Consider an asset price $\left(S_{n}ight)_{n=0,1, \ldots, N}$ which is a martingale under the riskneutral probability measure $\mathbb{P}^{*}$, with respect to the filtration
a) We consider a forward contract on $S_{N}$ with strike price $K$ and payoff\[ C:=S_{N}-K \]Find a portfolio allocation $\left(\eta_{N}, \xi_{N}ight)$ with value\[ V_{N}=\eta_{N}
Power option. Let $\left(S_{n}ight)_{n \in \mathbb{N}}$ denote a binomial price process with returns $-50 \%$ and $+50 \%$, and let the riskless asset be valued \(A_{k}=\$ 1, k \in
Consider the discrete-time Cox-Ross-Rubinstein model with $N+1$ time instants $t=0,1, \ldots, N$. The price $S_{t}^{0}$ of the riskless asset evolves as $S_{t}^{0}=\pi^{0}(1+r)^{t}$, \(t=0,1,
We consider the discrete-time Cox-Ross-Rubinstein model with $N+1$ time instants $t=0,1, \ldots, N$.The price $\pi_{t}$ of the riskless asset evolves as \(\pi_{t}=\pi_{0}(1+r)^{t}, t=0,1,
CRR model with transaction costs (Boyle and Vorst (1992), Mel'nikov and Petrachenko (2005)). Stock broker income is generated by commissions or transaction costs representing the difference between
CRR model with dividends (1). Consider a two-step binomial model for a stock paying a dividend at the rate $\alpha \in(0,1)$ at times $k=1$ and $k=2$, and the following recombining tree
CRR model with dividends (2). We consider a riskless asset priced as\[ S_{k}^{(0)}=S_{0}^{(0)}(1+r)^{k}, \quad k=0,1, \ldots, N \]with $r>-1$, and a risky asset $S^{(1)}$ whose return is given
We consider a ternary tree (or trinomial) model with $N+1$ time instants $k=0,1, \ldots, N$ and $d=1$ risky asset. The price $S_{k}^{(0)}$ of the riskless asset evolves as\[
Compute $\mathbb{E}\left[B_{t} B_{s}ight]$ in terms of $s, t \geqslant 0$.
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion. Let $c>0$. Among the following processes, tell which is a standard Brownian motion and which is not. Justify your
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion. Compute the stochastic integrals\[ \int_{0}^{T} 2 d B_{t} \quad \text { and } \quad \int_{0}^{T}\left(2 \times
Determine the probability distribution (including mean and variance) of the stochastic integral $\int_{0}^{2 \pi} \sin (t) d B_{t}$.
Let $T>0$. Show that for $f:[0, T] \mapsto \mathbb{R}$ a differentiable function such that $f(T)=0$, we have\[\int_{0}^{T} f(t) d B_{t}=-\int_{0}^{T} f^{\prime}(t) B_{t} d t\]Apply Itô's
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion.a) Find the probability distribution of the stochastic integral $\int_{0}^{1} t^{2} d B_{t}$.b) Find the probability
Given $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$a standard Brownian motion and $n \geqslant 1$, let the random variable $X_{n}$ be defined as\[X_{n}:=\int_{0}^{2 \pi} \sin (n t) d B_{t}, \quad n
Apply the Itô formula to the process $X_{t}:=\sin ^{2}\left(B_{t}ight), t \geqslant 0$.
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion.a) Using the Itô isometry and the known relations\[ B_{T}=\int_{0}^{T} d B_{t} \quad \text { and } \quad
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion. Given $T>0$, find the stochastic integral decomposition of $\left(B_{T}ight)^{3}$

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