9

In general, quantitative finance requires mathematics, finance, and numerical programming. The mix of the three and the areas of focus within the three will depend on the particular area you intend to work in. For example, option pricing, risk, and asset management are all related but derivative modeling would draw more on stochastic processes and ...


6

It is correct that $$ \mathbf{P}(t^{-1/2}W(t) \in[a,b])=Φ(b)−Φ(a), \forall t\in(0,\infty) $$ due to the stationary increments property of the Wiener process and the fact that you normalized the random variable by dividing by its standard deviation. $\mathbf{P}$ is a probability measure on an abstract space, not a random variable. Hence, you probably mean ...


5

You seem to use the term "volatility" to describe two very different quantities: (1) the diffusion coefficient of your SDE and (2) the standard deviation of the log-returns under your modelling assumptions. While the first may be negative, the second may not. [Interpretation 1] Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a standard ...


3

It is my understanding that a replicating portfolio for a put involves short selling stock and lending money. You cannot statically replicate an option. So this is not true in general, you'll need to re-balance your replicating portfolio (underlying + cash) dynamically if you want to replicate the option. This will imply sometimes buying stock and borrowing ...


3

In the black scholes model, today's stock price, risk free rate and stock volatility are considered independent variables. They are inputs to the model. Hence the cross partial derivatives are zero. You could invent a model where you tried to explain the current stock price in terms of risk free rate and volatility. Then indeed the partial derivatives ...


3

It is a very badly worded question in my humble opinion. There are three "prices" to contend with. (1) If you want to buy a stock and pay for it now, you pay the current stock price S. (2) If you want to buy a stock and not have to pay for it until a future delivery date T, then you enter into a "forward" or (in the United States) a "futures contract" ...


3

The standard starting point with modelling a stock price process is to use the Black-Scholes model for the stock price. This simply asserts that the changes in the stock price are described by the following stochastic differential equation (SDE) $$\dfrac{\textrm{d}S}{S} = \mu\:\textrm{d}t + \sigma\:\textrm{d}W_t$$ where $W_t$ is a standard Brownian motion (...


3

Your formula for $p$ is $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$ where $\alpha$ is not expected return on stock but continuous risk free rate, i.e. 1%. If you use $\alpha$ as 1%, you will get $p=0.009125828 $ which is within $[0,1]$ EDIT: With the information given in the question, it must satisfy following equality: $$S_0e^{\alpha - \delta}=...


3

It costs 0.03 dollars for the option to (sell 1 pound/buy 1.5 dollars. Now divide everything by 1.5: It costs 0.02 dollars for the option to (sell 2/3 pound / buy 1 dollar). Now convert to pounds at spot rate: It costs 0.0133 pounds for the option to (sell 2/3 pound / buy 1 dollar). Done


3

The 1 you are referring to is a vector of ones The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as: s = (ones(n,k) + R)' \ ones(n, 1) where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets). 1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02)....


3

I think you mixed several things up. I will try to help you out. Everything started with your claim that $\Bbb E \bigl[W(T) \mid \mathcal F_t \bigr] = 0$ which is wrong! if $W$ is a Brownian notion, then $$ \Bbb E \bigl[W(T) \mid \mathcal F_t \bigr] = W(t), \quad t\leq T. $$ This follows from the fact that Brownian motions are martingales. Here and in ...


3

The risk neutral measure is used to price assets (e.g. derivatives) and not to base your investment decisions on. In the first part of you question your simulation gives you the Risk-Neutral expectation of the stock at time $T$. If you want the expectation at time $t$, then why don't you just simulate from time 0 up to time $t$? (I might have misunderstood ...


3

Imho that's more of a probability question than finance really. If you take $N$ attempts, then the probability of at least one (or more) failures is the complementary probability of never failing on those $N$ attempts: $p=1-p_{pass}^N$ Solving for $p=50\%$, you need to evaluate $p_{pass}^{N}=0.5$, where $p_{pass}=0.99$, getting that $N\geq 69$.


2

If you're lucky enough that the payment schedules (start/end dates, frequency, day count, business day adjustment etc.) are the same between the fixed leg of the interest rate swap and the "spread" leg of the basis swap, then you can simply use: OIS rate (%) = IRS Rate (%) - 0.01 * (basis spread (bps)) Otherwise, to do it accurately, you'll need to do a ...


2

What a difficult problem. The first line gave $165 e^{-rt} -3 S e^{-dt} = 15$ [since 50+55+60 = 165]. In the second line we want to evaluate $110 e^{-rt} -2 S e^{-dt} $. We notice that this is exactly two thirds of the left side of the above, because 110 is two thirds of 165 and 2 is two thirds of 3. So we take two thirds of the right hand side of the first ...


2

First of all I’ll work with column vectors because I find it easier than with row vectors as you did. I guess it’s a little bit easier if we modify your first equation a little bit. Notice that is really the first order condition of the following lagrangian: $$L(w, \lambda, \delta)= \frac{1}{2}{\bf w^TCw} - \lambda({\bf w^Tm} - \mu_v) - \delta({\bf w^Tu} - 1)...


2

If $PV_{t, T}(\text{Divs}) \ge K\big(1-e^{-r(T-t)}\big)$, since $P_{Eur}(S_t, K, T-t) >0$, the identity \begin{align*} C_{Eur}(S_t, K, T-t) = P_{Eur}(S_t, K, T-t) + (S_t-K) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big), \end{align*} implies that \begin{align*} C_{Eur}(S_t, K, T-t) > (S_t-K). \end{align*} That is, it is not rationale to exercise the ...


2

The best explanation I came across so far is the one in Gravelle and Rees (2003) chapter 17. I could exactly write here what they state, but that would be copying.


2

I think they have expanded the third term on the LHS and simplified.


2

For general mathematical finance, you may start with the book Stochastic Calculus for Finance, and then the books Martingale Methods in Financial Modelling and Mathematical Methods for Financial Markets. After those preparations, you can start with some books in specialized areas such as the books Interest Rate Models by Brigo and Mercurio, Credit Risk by ...


2

Yes, your steps are valid This is a wrong use of the term "quantile". Here you need to compute a probability (through the normal cdf) and not a quantile (i.e. the value of a random variable corresponding to a given level of the cdf, e.g. the quantile 0.5 (or percentile 50%) is the median)


2

No, you can have $$ \frac{1}{2n}\sum_{i=1}^{2n} C(S^i_T,K,T) = 0 $$ First off, there's the obvious case where $n=1$ and $u_1 = 0.5$ More generally, for options way out of the money it is common to have $$ \frac{1}{n}\sum_{i=1}^{n} C(S^i_T,K,T) = 0 $$ even for very large $n$. Antithetic sampling does not change that.


2

Your computation of $\Delta$ is correct. However, your computation of the cash amount is wrong. You choose the cash amount $\beta$ that you need to initially lend or borrow such that in the up state, the following holds \begin{equation} P_u = \Delta S_u e^{\delta h} + \beta e^{r h}. \end{equation} We get \begin{eqnarray} \beta & = & \left( P_u - \...


2

you got a typo. It should be 40.886 in your last equation. Then $\sigma$ should match. Also, If $\alpha$ means annualized log return, it should be $\mu\,t = \alpha - \frac 1 2\sigma^2\,t$ So in your last two equations, the first term should be $\frac \alpha {250} - \frac 1 2 \sigma^2$


2

Under the Black-Scholes' framework, the underlying stock price price process $\{S_t, \, t\ge 0\}$ satisfies an SDE of the from $$ dS_t = S_t (\alpha dt + \sigma dW_t),$$ where $\{W_t, \, t\ge 0\}$ is a standard Brownian motion, $\alpha$ is the continuously compounded stock return, and $\sigma$ is the constant volatility. Then $$S_t = S_0e^{(\alpha-\frac{1}{2}...


2

Almost, indeed The volatility of an asset over a horizon $[t,T]$ indeed refers to the standard deviation of the log-return observed over that period: $$ \sqrt{\text{Var}\left(\ln\left(\frac{X_T}{X_t}\right)\right)} = \sqrt{\text{Var}\left( \ln X_T - \ln X_t \right)} = \sqrt{\text{Var}\left( \ln X_T \right)} $$ because $X_t$ is a known constant at $t$ The ...


2

You already associated the valuation function ins A, B, C and E with the corresponding products. In order to explicitly exclude D, you don't have to compute all the derivatives but just note that \begin{eqnarray} C_S & = & e^{-0.02 (T - t)} \mathcal{N}' \left( d_1 \right) \frac{\partial d_1}{\partial S}\\ C_t & = & 0.02 \underbrace{e^{-0.02 (...


2

Assume that under the real world measure $$ dS_t/S_t = (\alpha-\delta) dt + \sigma dZ_t^\Bbb{P} \tag{1} $$ Under the EMM $\Bbb{Q}$ one then needs to have (fundamental theorem of asset pricing: in the absence of arbitrage the discounted value of any self-financing portfolio should be a martingale): $$ dS_t/S_t = (r-\delta) dt + \sigma dZ_t^\Bbb{Q} \tag{2} $$ ...


2

I think you nearly got there but made a few mistakes in the application of l'Hopital's rule. First Limit In the first case, you got \begin{eqnarray} \lim_{S_0 \rightarrow \infty} \Omega & = & \lim_{S_0 \rightarrow \infty} \frac{\Gamma_{\text{call}} S_0 + \Delta_{\text{call}}}{\Delta_{\text{call}}}\\ & = & \lim_{S_0 \rightarrow \infty} \...


2

Assuming standard BS dynamics for $S_t$, you have $\frac{Z_t}{Z_0}\equiv(\frac{S_t}{S_0})^p = \exp{((rt-\frac{\sigma^2}{2}t+\sigma W_t)(1-\frac{2r}{\sigma^2}))}$ Now, this is a lognormally distributed r.v. and the Gaussian inside the exponential has mean and variance $m=2rt - \frac{\sigma^2}{2}t -\frac{2r^2}{\sigma^2}t$ $v = (\sigma-\frac{2r}{\sigma})^2 ...


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