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1

Given your code, the following will yield what you are after: t,S = generateGBM(TIME_HORIZON/365, DRIFT, ANNUALIZED_VOL, INITIAL_PRICE, 1/365/24/3) As all inputs are annualized, you must also think in units of year fractions: The time horizon is 30 days over 365 days, and the time step size, being 20 minutes, is one year over 365 * 24 * 3 (there are three ...


2

For an option with delayed cash settlement, expiry time $T$ and settlement time $T_p(\geq T)$, paying $(S_T-K)^+$ at $T_p$, the present value of this payment at $T$ is: $$ E_T\left[\beta_T \beta_{T_p}^{-1} (S_T-K)^+ \right] = P(T,T_p)(S_T-K)^+,$$ with $\beta_t = \exp \left(\int_0^t r_u du \right) $, $r$ risk-free interest rate, $P$ associated zero-coupon ...


2

Just to be clear we are talking about an option that pays $max(0,S_1-K)$ paid at time $t=2$. Then the only difference between this and a standard option is the extra discounting from $ t=1$ to $t=2$ . So the price $P$ must satisfy $$ P=BS/(1+r)$$ where BS is the regular Black Scholes price and $r$ is the forward risk free rate from $t=1$ to $t=2$. The ...


1

Consider stock price process (Geometric Brownian Motion): $$S_t=S_0exp((\mu-0.5\sigma^2)t+\sigma W_t) \tag{1}$$ where $W_t$ is a Wiener process and $\mu$ is a drift - or average return. If you are not familiar with Wiener process you can see this equation as: $$S_t=S_0exp((\mu-0.5\sigma^2)t+\sigma \sqrt t Z) \tag{2}$$ where $Z$ is standard normal random ...


3

You are correct, in that you need not (explicitly) specify real world dynamics to calculate option prices. Indeed in many rates derivatives models, you simply assume a unique risk neutral measure exists (completeness), specify the dynamics under the risk neutral measure (risk neutral probabilities) and price your options. At the same time, it is important to ...


2

Everything (warning: I have not checked 3rd order greeks) that is not delta is in terms of ccy2 in the standard Garman Kohlhagen model. Gamma is not in CCY1 by default either (some vendors like Bloomberg display it like that to be consistent with Delta). Why Let's start by not looking at FX but equity to help build intuition. The actual price of an option is ...


3

The best way is to start with definitions (instantaneous and their finite difference versions) of Greeks. For a currency pair $(FOR,DOM)$ with FX rate $S$, the number of $[DOM]$ (domestic, numeraire, right-side) units needed to buy one $[FOR]$ (foreign, asset, left-side) unit, let $V(S)$ be an option's price in $[DOM]$ units. Note that the unit of $S$ is: $$ ...


1

The shape you observe is really only due to spot being higher for ITM calls & OTM puts. The plots are definitely correct. You can quickly check by using standard closed form volga, which is $$vega*\frac{d1*d2}{\sigma}$$. Changing risk free rate and dividends just shifts the whole graph slightly left or right respectively. Volga is $$vega*\frac{d1*d2}{\...


0

I think that your approach is exact. Let the market prices $P^M(0,T)$ of zero bonds be given for some maturities $T_1,...,T_m$. Let $P^M(0,T_0)=1$ for $T_0=0$. The market prices of zero bonds should be calculate for $t \in [T_i,T_{i+1}]$ and $0 \leq i \leq m$ using log linear interpolation $$ln P^M(0,t)=lnP^M(0,T_i)+\frac{t-T_i}{T_{i+1}-T_i}*(lnP^M(0,T_{i+1})...


2

For simplicity, let's say that your time $0.003$ equals 1 day, and your second pillar (probably $0.083$ instead of $0.00833$) equals 1 week. What you do: Approximate the short rate with the 1-day interest rate. What they do: Employ additional information about the shape of the yield curve at the short end, i.e. extrapolating from the first two available ...


0

Since you write under BS model, it is a tautology in this model. Assume you use Black, which you can without loss of generality, as one can easily be transformed into the other (FX is easier to show with covered interest rate parity in my opinion). Looking at $$d1 = ( log(F/K) + 0.5*σ^2*t ) / (σ*sqrt(t))$$ $$ d2 = d1 - σ*sqrt(t)$$ it is not immediately ...


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