Questions and Answers of Introduction Finance Markets

We consider a bond with maturity $T$, priced $P(t, T)=$ $\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s} \mid \mathcal{F}_{t}\right]$ at time $t \in[0, T]$.a) Using the forward
Consider a tenor structure $\left\{T_{1}, T_{2}\right\}$ and a bond with maturity $T_{2}$ and price given at time $t \in\left[0, T_{2}\right]$ by\[P\left(t, T_{2}\right)=\exp
Assume that the price $P(t, T)$ of a zero-coupon bond with maturity $T>0$ is modeled as\[P(t, T)=\mathrm{e}^{-\mu(T-t)+X_{t}^{T}}, \quad t \in[0, T],\]where \(\left(X_{t}^{T}\right)_{t \in[0,
Consider a short rate process $\left(r_{t}\right)_{t \in \mathbb{R}_{+}}$of the form $r_{t}=h(t)+X_{t}$, where $h(t)$ is a deterministic function of time and
a) Given two LIBOR spot rates $L(t, t, T)$ and $L(t, t, S)$, compute the corresponding LIBOR forward rate $L(t, T, S)$.b) Assuming that $L(t, t, T)=2 \%, L(t, t, S)=2.5 \%$ and \(t=0, T=1,
a) Compute the forward rate $f(t, T, S)$ in the Ho-Lee model (17.48) with constant deterministic volatility.In the next questions we take $a=0$.b) Compute the instantaneous forward rate \(f(t,
Consider a floorlet on a three-month LIBOR rate in nine month's time, with a notional principal amount of $\$ 10,000$ per interest rate percentage point. The term structure is flat at $3.95 \%$
Consider a payer swaption giving its holder the right, but not the obligation, to enter into a 3-year annual pay swap in four years, where a fixed rate of $5 \%$ will be paid and the LIBOR rate
Consider a receiver swaption which is giving its holder the right, but not the obligation, to enter into a 2-year annual pay swap in three years, where a fixed rate of $5 \%$ will be received and
Jamshidian's trick (Jamshidian (1989)). Consider a family $\left(P\left(t, T_{l}\right)\right)_{l=i, \ldots, j}$ of bond prices defined from a short rate process \(\left(r_{t}\right)_{t \in
We work in the short rate model\[d r_{t}=\sigma d B_{t}\]where $\left(B_{t}\right)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion under $\mathbb{P}^{*}$, and $\widehat{\mathbb{P}}_{2}$ is
Analysis of user login activity to the DBX digibank app showed that the times elapsed between two logons are independent and exponentially distributed with mean$1 / \lambda$. Find the CDF of the
Consider a standard Poisson process $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$with intensity $\lambda>0$, started at $N_{0}=0$.a) Solve the stochastic differential equation\[d S_{t}=\eta
Consider a standard Poisson process $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$with intensity $\lambda>0$.a) Solve the stochastic differential equation $d X_{t}=\alpha X_{t} d t+\sigma d N_{t}$
Consider a standard Poisson process $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$with intensity $\lambda>0$.a) Solve the stochastic differential equation $d X_{t}=\sigma X_{t^{-}} d N_{t}$ for
Let $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$be a standard Poisson process with intensity $\lambda>0$, started at $N_{0}=0$.a) Is the process $t \mapsto N_{t}-2 \lambda t$ a submartingale,
Affine stochastic differential equation with jumps. Consider a standard Poisson process $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$with intensity $\lambda>0$.a) Solve the stochastic differential
Consider the compound Poisson process $Y_{t}:=\sum_{k=1}^{N_{t}} Z_{k}$, where $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$is a standard Poisson process with intensity $\lambda>0$, and
Show, by direct computation or using the moment generating function (20.10), that the variance of the compound Poisson process $Y_{t}$ with intensity $\lambda>0$
Consider an exponential compound Poisson process of the form\[S_{t}=S_{0} \mathrm{e}^{\mu t+\sigma B_{t}+Y_{t}}, \quad t \geqslant 0\]where $\left(Y_{t}\right)_{t \in \mathbb{R}_{+}}$is a compound
Let $\left(N_{t}\right)_{t \in[0, T]}$ be a standard Poisson process started at $N_{0}=0$, with intensity $\lambda>0$ under the probability measure $\mathbb{P}_{\lambda}$, and consider the
Consider a standard Poisson process $\left(N_{t}\right)_{t \in[0, T]}$ with intensity $\lambda>0$ and a standard Brownian motion $\left(B_{t}\right)_{t \in[0, T]}$ independent of
Consider a standard Poisson process $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$with intensity $\lambda>0$ under a probability measure $\mathbb{P}$. Let \(\left(S_{t}\right)_{t \in
Consider a long forward contract with payoff $S_{T}-K$ on a jump diffusion risky asset $\left(S_{t}\right)_{t \in \mathbb{R}_{+}}$given by\[d S_{t}=\mu S_{t} d t+\sigma S_{t} d B_{t}+S_{t^{-}} d
Consider $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$a standard Poisson process with intensity $\lambda>0$ under a probability measure $\mathbb{P}$. Let \(\left(S_{t}\right)_{t \in
Consider a standard Poisson process $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$with intensity $\lambda>0$ under a probability measure $\mathbb{P}$. Let \(\left(S_{t}\right)_{t \in
Pricing by the Esscher transform (Gerber and Shiu (1994)). Consider a compound Poisson process $\left(Y_{t}\right)_{t \in[0, T]}$ with
Show that when the terminal condition is a constant $\phi(T, x)=c>0$ the implicit scheme (22.8) recovers the known solution $\phi(s, x)=c \mathrm{e}^{-r(T-s)}, s \in[0, T]$.
Let $X_{t}$ be the geometric Brownian motion given by the stochastic differential equation\[d X_{t}=r X_{t} d t+\sigma X_{t} d W_{t}\]a) Compute the Euler discretization
Consider a two-step binomial model $\left(S_{k}\right)_{k=0,1,2}$ with interest rate $r=0 \%$ and risk-neutral probabilities $\left(p^{*}, q^{*}\right)$ :a) At time $t=1$, would you exercise
Consider an American butterfly option with the following payoff function.Price the perpetual American butterfly option with $r>0$ in the following cases.a) \(\widehat{K} \leqslant L^{*}
(Barone-Adesi and Whaley (1987)) We approximate the finite expiration American put option price with strike price $K$ as\[f(x, T) \simeq \begin{cases}\operatorname{BS}_{\mathrm{p}}(x,
Consider the process $\left(X_{t}\right)_{t \in \mathbb{R}_{+}}$given by $X_{t}:=t Z, t \in \mathbb{R}_{+}$, where $Z \in\{0,1\}$ is a Bernoulli random variable with
Consider a dividend-paying asset priced as\[S_{t}=S_{0} \mathrm{e}^{(r-\delta) t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where $r>0$ is the risk-free interest rate,
American call options with dividends, see § 9.3 of Wilmott (2006). Consider a dividend-paying asset priced as \(S_{t}=S_{0} \mathrm{e}^{(r-\delta) t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, t
Consider an underlying asset whose price is written as\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where $\left(B_{t}\right)_{t \in \mathbb{R}_{+}}$is a
Perpetual American binary options.a) Compute the price\[C_{b}^{\mathrm{Am}}\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{\tau \geqslant t \\ \tau \text { Stopping time }}}
An American binary (or digital) call (resp. put) option with maturity $T>0$ on an underlying asset process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}=\left(\mathrm{e}^{r t+\sigma B_{t}-\sigma^{2}
American forward contracts. Consider $\left(S_{t}\right)_{t \in \mathbb{R}_{+}}$an asset price process given by\[\frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}\]where $r>0$ and
Consider an underlying asset price process written as\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where \(\left(\widehat{B}_{t}\right)_{t \in
Let $p \geqslant 1$ and consider a power put option with payoff\[\left(\left(\kappa-S_{\tau}\right)^{+}\right)^{p}= \begin{cases}\left(\kappa-S_{\tau}\right)^{p} & \text { if } S_{\tau} \leqslant
Same questions as in Exercise 15.14 , this time for the option with payoff $\kappa-\left(S_{\tau}\right)^{p}$ exercised at time $\tau$, with $p>0$.Data from Exercise 15.14Let $p \geqslant 1$
Let $\left(B_{t}\right)_{t \in \mathbb{R}_{+}}$be a standard Brownian motion started at 0 under the riskneutral probability measure $\mathbb{P}^{*}$. Consider a numéraire \(\left(N_{t}\right)_{t
Consider two zero-coupon bond prices of the form $P(t, T)=F\left(t, r_{t}\right)$ and $P(t, S)=G\left(t, r_{t}\right)$, where $\left(r_{t}\right)_{t \in \mathbb{R}_{+}}$is a short-term interest
Using a change of numéraire argument for the numéraire $N_{t}:=P(t, T), t \in[0, T]$, compute the price at time $t \in[0, T]$ of a forward (or future) contract with payoff $P(T, S)-K$ in a
Consider a price process $\left(S_{t}\right)_{t \in \mathbb{R}_{+}}$given by $d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}$ under the risk-neutral probability measure $\mathbb{P}^{*}$, where \(r
Consider two bonds with maturities $T$ and $S$, with prices $P(t, T)$ and $P(t, S)$ given by\[\frac{d P(t, T)}{P(t, T)}=r_{t} d t+\zeta_{t}^{T} d W_{t}\]and\[\frac{d P(t, S)}{P(t, S)}=r_{t} d
Compute the price $\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\mathbb{1}_{\left\{R_{T} \geqslant \kappa\right\}} \mid \mathcal{F}_{t}\right]$ at time $t \in[0, T]$ of a cashor-nothing "binary"
Assume that $\left(S_{t}\right)_{t \in \mathbb{R}_{+}}$and $\left(N_{t}\right)_{t \in \mathbb{R}_{+}}$satisfy the stochastic differential equations\[d S_{t}=r_{t} S_{t} d t+\sigma_{t}^{S} S_{t}
Consider an asset priced $S_{t}$ at time $t$, with\[d S_{t}=r S_{t} d t+\sigma^{S} S_{t} d W_{t}^{S}\]and an exchange rate $\left(R_{t}\right)_{t \in \mathbb{R}_{+}}$given by\[d
In the mean-reverting Vasicek model (17.47) with $b>0$, compute:i) The asymptotic bond yield, or exponential long rate of interest\[r_{\infty}:=-\lim _{T \rightarrow \infty} \frac{\log P(t,
Consider the Marsh and Rosenfeld (1983) short-term interest rate model\[d r_{t}=\left(\beta r_{t}^{\gamma-1}+\alpha r_{t}\right) d t+\sigma r_{t}^{\gamma / 2} d B_{t}\]where $\alpha \in \mathbb{R}$
Consider the Cox et al. (1985) (CIR) process $\left(r_{t}\right)_{t \in \mathbb{R}_{+}}$solution of\[d r_{t}=-a r_{t} d t+\sigma \sqrt{r_{t}} d B_{t},\]where $a, \sigma>0$ are constants
Convertible bonds. Consider an underlying asset price process $\left(S_{t}\right)_{t \in \mathbb{R}_{+}}$given by\[d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}^{(1)}\]and a short-term interest rate
Write down the bond pricing PDE for the function\[F(t, x)=\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s} \mid r_{t}=x\right]\]and show that in case $\alpha=0$ the corresponding bond price
Consider a zero-coupon bond with prices $P(1,2)=91.74 \%$ and $P(0,2)=$ $83.40 \%$ at times $t=0$ and $t=1$.a) Compute the corresponding yields $y_{0,1}, y_{0,2}$ and $y_{1,2}$ at times
Consider a one-step interest rate model, in which the short-term interest rate $r_{0}$ on $[0,1]$ can turn into two possible values \(r_{1}^{u}=r_{0} \mathrm{e}^{\mu \Delta t+\sigma \sqrt{\Delta
Compute the price\[\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left.\left(\exp \left(\frac{1}{T} \int_{0}^{T} \log S_{u} d u\right)-K\right)^{+} \rightvert\, \mathcal{F}_{t}\right], \quad 0 \leqslant
Pricing Asian options by PDEs. Show that the functions $g(t, z)$ and $h(t, y)$ are linked by the relation\[g(t, z)=h\left(t, \frac{1-\mathrm{e}^{-(T-t) r}}{r T}+\mathrm{e}^{-(T-t) r} z\right),
Given $S_{t}:=S_{0} \mathrm{e}^{\sigma B_{t}+r t-\sigma^{2} t / 2}$ a geometric Brownian motion and letting\[\widetilde{Z}_{t}:=\frac{\mathrm{e}^{-(T-t) r}}{S_{t}}\left(\frac{1}{T} \int_{0}^{t}
Consider an underlying asset price process $\left(S_{t}\right)_{t \in \mathbb{R}_{+}}$modeled as $d S_{t}=(\mu-\delta) S_{t} d t+\sigma S_{t} d B_{t}$, where \(\left(B_{t}\right)_{t \in
Compute the first and second moments of the time integral $\int_{\tau}^{T} S_{t} d t$ for $\tau \in[0, T)$, where $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$is the geometric Brownian motion
Consider the short rate process $r_{t}=\sigma B_{t}$, where $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion.a) Find the probability distribution of the time integral
Asian call option with a negative strike price. Consider the asset price process\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where \(\left(B_{t}ight)_{t \in
Consider the Asian forward contract with payoff\[\begin{equation*}\frac{1}{T} \int_{0}^{T} S_{u} d u-K \tag{13.36}\end{equation*}\]where \(S_{u}=S_{0} \mathrm{e}^{\sigma B_{u}+r u-\sigma^{2} u / 2},
Hedging Asian options (Yang et al. (2011)).a) Compute the Asian option price f(t,St,Λt)f(t,St,Λt) when Λt/T⩾KΛt/T⩾K.b) Compute the hedging portfolio allocation (ξt,ηt)(ξt,ηt) when
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$be a standard Brownian motion started at 0 , i.e. $B_{0}=0$.a) Is the process $t \longmapsto\left(2-B_{t}ight)^{+}$a submartingale, a martingale or
Stopping times. Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$be a standard Brownian motion started at 0 .a) Consider the random time $u$ defined by\[u:=\inf \left\{t \in \mathbb{R}_{+}:
Consider a standard Brownian motion $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$started at $B_{0}=0$, and let\[\tau_{L}=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=Light\}\]denote the first hitting time
Consider $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$a Brownian motion started at $B_{0}=x \in[a, b]$ with \(a
Consider a standard Brownian motion $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$started at $B_{0}=0$, and let\[\tau:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=\alpha+\beta tight\}\]denote the first
(Doob-Meyer decomposition in discrete time). Let $\left(M_{n}ight)_{n \in \mathbb{N}}$ be a discretetime submartingale with respect to a filtration \(\left(\mathcal{F}_{n}ight)_{n \in
a) Give the probability density function of the maximum of drifted Brownian motion $\operatorname{Max}_{t \in[0,1]}\left(B_{t}+\sigma t / 2ight)$.b) Taking \(S_{t}:=\mathrm{e}^{\sigma
Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion.a) Compute the expected value\[\mathbb{E}\left[\operatorname{Max}_{t \in[0,1]}
Consider a risky asset whose price $S_{t}$ is given by\[d S_{t}=\sigma S_{t} d B_{t}+\sigma^{2} S_{t} d t / 2\]where $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion.a)
The digital drawdown call option with qualifying period pays a unit amount when the drawdown period reaches one unit of time, if this happens before fixed maturity $T$, but only if the size of
a) Check explicitly that the boundary conditions (12.3a)-(12.3c) are satisfied.b) Check explicitly that the boundary conditions (12.14a)-(12.14b) are satisfied.
Compute the expected realized variance on the time interval [0, T] in the Heston model, with$$d v_{t}=-\lambda\left(v_{t}-might) d t+\eta \sqrt{v_{t}} d B_{t}, \quad 0 \leqslant t \leqslant T$$
Compute the variance swap rate$$\mathrm{VS}_{T}:=\frac{1}{T} \mathbb{E}\left[\lim _{n ightarrow \infty} \sum_{k=1}^{n}\left(\frac{S_{k T / n}-S_{(k-1) T / n}}{S_{(k-1) T /
Convexity adjustment (§ 2.3 of Broadie and Jain (2008)).a) Using Taylor's formula$$\sqrt{x}=\sqrt{x_{0}}+\frac{x-x_{0}}{2 \sqrt{x_{0}}}-\frac{\left(x-x_{0}ight)^{2}}{8 x_{0}^{3 /
Consider an asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$with log-return dynamics$$d \log S_{t}=\mu d t+Z_{N_{t^{-}}} d N_{t}, \quad t \geqslant 0$$i.e. $S_{t}:=S_{0} \mathrm{e}^{\mu
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}

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