10 votes
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

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^{...
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8 votes

Options On Earthquakes

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 ...
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  • 1,662
6 votes
Accepted

How to take the differential of a stochastic integral?

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]...
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  • 1,585
6 votes
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Strike / delta relationship for FX options

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 ...
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  • 20.4k
5 votes

Basic question on Ito integrals

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 ...
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4 votes
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12-month rate calculation for Problem 4.23 in Hull's Options, Futures, and Other Derivatives

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{...
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  • 520
4 votes
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Using crude Monte Carlo

Here's some pseudo code to generate your valuations: ...
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  • 2,406
3 votes

Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

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 ...
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  • 4,961
3 votes

Strike / delta relationship for FX options

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 ...
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3 votes
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Asset Liability Management Test Topic Interpretation

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 ...
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  • 450
3 votes
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Coupon bond pricing problem with reinvestment

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 ...
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  • 2,108
3 votes
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How to define the $f$ function to apply Ito's lemma?

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 $...
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  • 10.9k
2 votes

construct portfolio offering risk free profit

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 ...
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  • 709
2 votes

Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$

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[ ...
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  • 13.9k
2 votes
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Cause of difference in theoretical vs observed value of a (call) option under the Black-Scholes model?

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 ...
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  • 1,346
2 votes
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Calculating value of bond

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 ...
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  • 709
2 votes
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Calculating beta when holding market portfolio

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 ...
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  • 2,926
1 vote

For a market with a bank and risky assets $S_1, S_2$ with different volatility, what should be the short interest rate in this market?

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)...
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  • 264
1 vote
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Self finance conditions - proof check

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 ...
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1 vote
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Monte Carlo European Option Pricing

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 ...
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1 vote
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A forward Monte Carlo method for American Options Pricing

To simply answer this question the author is just multiplying the numbers.
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  • 710
1 vote
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Use random-shift Halton sequence to obtain 40 independent estimates for the price of a European call

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 ...
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1 vote

Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$

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 $\mathcal{F} \left( T_0 \right)$ ...
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1 vote

Is there an efficient method or technique to find an arbitrage between two FX dealers?

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 ...
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  • 682
1 vote

Is there an efficient method or technique to find an arbitrage between two FX dealers?

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 ...
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