10

First, let's check if these models are abritrage free. The first fundamental theorem of asset pricing says that if there exists an equivalent probability measure under which $\frac{S_t}{\beta_t} = e^{-t}S_t$ is a martingale, then the market is arbitrage free, so we will check whether such an equivalent martingale measure exists. This is where we will use ...


8

The heart of option pricing is the ability to replicate. If you can make a mango from apple and orange, the price of the mango is determined by the cost of an apple and an orange. People may value the mango less or more than that, but the market price is already constrained and there is no scope for pricing in these (real world) preferences. No replication ...


6

You can rewrite $X_t = e^{-kt}Z_t$ and define $Z_t:=\int_{0}^{t}e^{ks}dW_s$. There is a theory (Lemma 4.15 in Björk if you use his book) which states that $$\text{Var}\left[\int_{0}^{t}f(u)dW_s\right]=\int_{0}^{t}(f(u))^2ds$$ You can use that. furthermore, You can use Ito to compute $dX_t$. By standard stochastic calculus theory the dynamics of $Z_t $ is $...


6

In FX world, the ATM strike is the delta-neutral strike, that is, the absolute delta values of a call and the corresponding put are the same. Moreover, the delta can be premium adjusted or not depending on the particular currency pair. See the linked paper as mentioned by @AntoineConze. For AUD/USD, the delta is not premium adjusted, and then the delta-...


4

The rate is the return on your investment. Since you'll receive 100\$ after 12 months, $\frac{100 - P}{P} = \frac{100 - 89.0}{89.0} = \frac{11}{89} = 12.36 \%$. Same for the 6-month T-Bill: $\frac{100 - P}{P} = \frac{100 - 94.0}{94.0} = \frac{6}{94} = 6.38 \%$.


4

Here's some pseudo code to generate your valuations: Asset asset(); //some asset object that holds all the data. vector<double> timeline; //this is your vector of times. size_t nTimes = timeline.size(); vector<double> dts(nTimes); //vector of fwd prices on the same timeline. vector<double> fwds(nTimes); //vector of fwd prices on the same ...


4

To verify @AntoineConze's suggestion, the variance should be: $$\int_0^4 (2_{[0,1]}(t)+3_{(1,3]}(t)-5_{(3,4]}(t))^2\,dt.$$ Since the supporting domains are disjoint, the product of any two of the terms $2_{[0,1]}(t), 3_{(1,3]}(t), 5_{(3,4]}(t)$ is identically 0, so the integral is just $$\int_0^4 2^2_{[0,1]}(t)+3^2_{(1,3]}(t)+5^2_{(3,4]}(t)\,dt=4(1-0)+9(3-1)+...


3

I assume all three models are stated under the money-market measure: then there is no arbitrage if the discounted pay-off is a martingale under the money-market Numeraire. Therefore to show no arbitrage for all three models, we would want to show that: $$\mathbb{E}\left[\frac{S_t}{\beta_t}|\mathcal{F_0}\right]=\frac{S_0}{\beta_0}$$ Model a: $$\frac{S_0}{\...


3

There are specific quotation conventions for specifying ATM and deltas for FX options quotes (unadjusted deltas, premium adjusted deltas, etc.) and converting deltas to strikes. These conventions vary across currency pairs. See this paper https://ideas.repec.org/p/zbw/cpqfwp/20.html for details.


3

Portfolios for some kind of investors effectively balance asset investments with liabilities incurred. Think about a pension account, where the future liability of the pension payment represents the liability and the currently invested monies are the assets. I am sure you can think of other similar situations but I will illustrate regarding pensions below. ...


3

In part (a) use discount rate $e^.07 -1 = .072508181$ to get the right answer. For part (b) I am just giving you hint: Calculate bond price at the end of 1st year and 2nd year in the same way as you did in part (a). Use the above calculated price to buy bond from the dividend at the end of first and second year. You may assume bond can be purchased in ...


3

In fact, the variable $Z_t$ is a function of $W_t$, which is the stochastic variable. Therefore, you can see $Z_t$ as $f(W_t) = \exp(aW_t)$. The rest is a trivial application of Ito's lemma to find $dZ_t=df(W_t)$.


2

The question is asking if there is a way to create arbitrage by borrowing in one currency, exchanging at the current spot rate, lending in another currency and converting the future payments back to the original currency at the forward exchange rate. Specifically, given the assumption above, if there were no arbitrage the inequality above could not hold. ...


2

The valuation formula for a contingent claim delivering a payoff at $T$, as seen of today $t$ knowing that the underlying is currently worth $s$ reads $$ \Pi(t,s) = e^{-r(T-t)} \Bbb{E}^\Bbb{Q} \left[ f(S_T) \mid \mathcal{F}_t \right] $$ where $f(S_T)$ is the $\mathcal{F}_T$-measureable payoff of your contingent claim. In the exercise you mention, we have $t ...


2

A curious piece of homework, but let’s just consider the information at hand. You are given a somewhat odd process $S_t = S_0e^{-W_t+t}$ and the rest of the question pointing you to the “implied probability measure” looks like an indication that the stated process is under the physical measure $P$. The SDE followed by $S$ under $P$ must be: $\frac{dS_t}{...


2

You question is saying that you have 14 payments coming starting in 6 years. This implies that the formula is as you have it, but replace both $N_1$ and $N_2$ with $N_2-N_1$ and discount that whole cashflow from T=6 to today. Accordingly, this is the equation you're looking for. $$P=(1+y)^{-{N_1}} * [\frac{CPN}{y}(1-\frac{1}{(1+y)^{N_2-N_1}})+\frac{FV}{(1+y)...


2

The phrase "The CAPM holds" refers to the assumption, that any asset return $r_i$ fulfills the pricing relation $r_i=r_f+\beta_i(r_m-r_f)$, where $r_m$ denotes the market return, $r_f$ the risk-free rate and $\beta_i$ the beta-factor of the asset. The CAPM is an economic theory, but be aware that plenty of empirical research does not support the CAPM, it is ...


1

Thanks to @Antoine Conze, here's my answer. Using self-finance condition, straight forward calculation shows, with some sloppy notation, $$dV=(a\mu_1S_1+b\mu_2 S_2+crG)dt+(a\sigma_1 S_1+b\sigma_2S_2)dB =r(aS_1+bS_2+cG)dt\\ \Rightarrow a(\mu_1-r)S_1+b(\mu_2 -r)S_2=0,\ a\sigma_1S_1 +b \sigma_2 S_2=0 \\ \Rightarrow r=\frac{\mu_1\sigma_2-\mu_2\sigma_1}{\sigma_2 ...


1

Let me define $B_t=A_t=e^{rt}$ $-$ to avoid confusing it with the geometric average $1/t\int S_u\text{d}u$. Your portfolio value is: $$ V_t =\psi_tB_t+\phi_tS_t $$ To be self-financing we need to enforce one of the following equivalent conditions: $$\begin{align} & \text{[1]} \quad \text{d}V_t =\psi_t\text{d}B_t+\phi_t\text{d}S_t \\[3pt] & \text{[2]}...


1

A few suggestions: As your underlying follows a geometric Brownian motion and you are solely interested in pricing European options, there is no need to simulate intermediate steps. Since your solution is exact, you can directly sample $S_T$ as \begin{equation} S_T = S_0 \exp \left\{ \left( r - \frac{1}{2} \sigma^2 \right) T + \sigma \sqrt{T} Z \right\},...


1

To simply answer this question the author is just multiplying the numbers.


1

Actual Question I suppose this is homework, so I will only outline the steps. The way I understand this question is as follows: Build a function that simulates different 10,000 sample paths of your underlying asset with 10 equidistant time steps each. For each of these paths, compute the terminal European call option payoff. Their discounted sample mean is ...


1

The spot price process is driven by a constant coefficient geometric Brownian motion. Thus, the ratio $S \left( T_1 \right) / S \left( T_0 \right)$ is independent of $\mathcal{F} \left( T_0 \right)$ and its distribution only depends on the length of the time interval $T_1 - T_0$. It follows that \begin{equation} S \left( T_1 \right) / S \left( T_0 \right) ...


1

As far as I know, the answer is yes and people do it all the time. There's something to add to the textbook example though. First the bid/ask spread on FX spot market is usually much tighter, meaning the room for taking advantage of the such arbitrage is smaller than you think (or you would need huge capital to leverage this kind of trade). For some currency ...


1

Can't wait to see you implement it in real life... You will experience so many uncontrolled variables and scenarios... One common scenario is: you see what you think is a good price... Then you aggress on it... By the time you are filled, your entire great arbitrage formula is gone. (It is not enough for you to be fast - in nanoseconds - you would also ...


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