# Tag Info

1

What you seem to be missing is $$\sum_{i,j} w_i w_j \sigma_i \sigma_j = \left(\sum_i w_i\sigma_i\right)^2$$ Now apply Jensen's inequality to get $$\left(\sum_i w_i\sigma_i\right)^2 \leq \sum_i w_i\sigma_i^2$$ QED (note that nonnegative weights is a crucial assumption here)

0

There are two mistakes in the code: 1) In the line vt[t] = np.abs(vt[t-1] + kappa*(theta-np.abs(vt[t-1]))*dt + xi*np.sqrt(np.abs(vt[t-1]))*W_v[t]) you forgot to multiply W_v[t] by np.sqrt(dt). This is the reason the volatility increases so much. 2) The line St[t] = St[t-1]*np.exp((mu - 0.5*vt[t])*dt + np.sqrt(vt[t]*dt)*W_S[t]) should be St[t] = St[t-...

1

Ok, I found the problem. I was not discounting the value according to interest rate. The following modified code works fine for non-zero interest rate. When dividends rate are non zero, it is probably better to hedge using a forward contract instead of the asset itself. I will skip this for now. def hedging_portfolio(path, strike, r, d, vol, T): t = ...

1

"Whose price function $V$ fluctuates according to the actual market price of that derivative"—this is not true. The reason being that we are 'modeling' the derivative price (where a model is a simplified version of reality). So $V$ tells us what the derivative price would be under our model—and since this model doesn't use the actual derivative price as an ...

3

In one sentence, time value has to do with the probability of crossing the strike before expiration (whether from below or above). Doesn’t matter whether the crossing results in the option being in the money or not.

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The answer given is mostly wrong: @msitt uses a convoluted way without explicitly mentioning it (put-call symmetry) to actually give the price of a USD Put, not of a USD Call as requested. Here is a more direct and correct approach. I will consider, as mentioned by @FinanceGuyThatCantCode, that the volatility convention is ACT/365, which is standard. I ...

1

The "square-root rule" for time-to-expiration only (roughly) applies when the spot price = strike price. Even in that case there is a second-order term that is a function of the risk-free rate and implied volatility. This can be seen in the Black-Scholes pricing formula: the time-to-expiration is included in a term that also varies with log(spot/strike), ...

2

Let's say the company was bankrupt (ie, stock price is 0). A put option effectively becomes a bond with face value equal to the strike and maturity equal to the expiration. With positive interest rates, zero coupon bonds generally become more valuable as time passes. In this extreme case, an American option is worth more because you could early ...

1

In its simplest form, an option is a combination of two binary options. The buyer of a call option is long of an "asset-or-nothing" binary call. I.E. if Spot>Strike, it is worth Spot; else 0. To fund that, he is selling a "cash-or-nothing" call: worth Strike if Spot>Strike, else 0. The positive value of the option obviously derives from the fact that as ...

2

The Black Scholes (1973) model assumes that $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t$. Thus, $$S_t=S_0\exp\left(\left(r-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right).$$ Please note the factor $-\frac{1}{2}\sigma^2t$ in the exponential. If you incorporate dividends, replace $r$ by $r-q$. You do not need an extra term $\sqrt{t}$ in front of the ...

0

Given the variation, ATM vol = alpha * F ^(beta-1), if your stochastic process for forward price dF= alphaF^beta dW, that means your effective beta, CEV, is 1. This gives horizontal backbone of the vol surface. I think it all depends on whether this is what you expect to see - the vol surface is stickey under shocked price scenarios.

1

This is easy to answer with the meta theorem given in the same chapter. Here you have two sources of randomness (W and N), and one risky asset. Q1: Arbitrage generally happens when you have more assets than the number of random sources, but here it is the other way around, so the answer is yes. Q2: You have one risky asset so you can delta hedge one ...

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A free course to learn quant finance is: Python for Trading A very basic and free course on Python programming. It includes everything a beginner needs to know: data structures, expressions, functions and various libraries used in financial markets. This is a detailed and comprehensive course to build a strong foundation in Python. The less pricey quant ...

1

BlackRock has the best commercially available prepayment model and Yield Book is basically the industry standard for trading and is decent.

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Your function returning (minus) the log-likelihood seems weird to me, I would go with function y = findGARCH_LLy(params,S,rf) % Finds log-likelihood for the GARCH option pricing model. alpha0 = params(1); alpha1 = params(2); beta1 = params(3); lambda = params(4); N = length(S); % Define the returns (pad first return with zero) r = [0, diff(log(S))]; % ...

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