#options
➡️ Quantitative Finance from Scratch, part 35: how market makers make money. The first source is the spread. Like an exchange booth, a market maker buys a little below the fair price at the bid and sells a little above it at the ask, so every trade earns half the gap whatever the stock does next. After selling a call, it buys delta shares, so small moves no longer matter. The second source comes from the other Greeks: theta is the income, because the call it owes loses value every day, and gamma is the cost, because after every big move it must buy shares high and sell them low to keep the hedge. If the stock moves exactly as much as the implied volatility in the price, income and cost cancel. If it stays calmer, the market maker earns, and if it moves more, it loses. Vega is the risk to this income, because a jump in implied volatility makes the call it owes more expensive. If a customer sells it the call instead, everything flips. The code (marketmaker.py, 16 lines) sells our call from part 33 at the ask (fair $10.45, ask $10.55, bid $10.35) and hedges it every day on 2,000 simulated years with an implied volatility of 20 %. With a real volatility of 16 % it earns $1.59 per call on average and every path ends in profit; at 20 % it earns $0.10, about the spread; at 24 % it loses $1.41 and only 0.3 % of paths end in profit. Honest notes: a simplified model with one option, daily hedging, Black Scholes prices, constant volatility, no trading costs, no dividends and no other customers. Real market makers manage many options at once and also face fees, inventory limits and jumps. #quantfinance #options #marketmaking #greeks #finance
QuantFinanceToGo102,600 次观看 • 15 天前
➡️ Quantitative Finance from Scratch, part 33: delta and delta hedging. Delta is how much the price of an option moves when the stock moves by one dollar. For a call it lies between zero and one, and in Black Scholes it is N(d1), the same term as in the price formula from part 30. In our test run with the stock and the strike at $100, a 5 % rate, one year and 20 % volatility, the call costs $10.45 and has a delta of 0.64, so a $1 rise lifts the call by about 64 cents. Whoever sold the call holds 0.64 shares against it, and for small moves the loss on the call and the gain on the shares cancel out, which is called delta neutral. Because delta changes whenever the stock moves, the shares have to be rebalanced again and again. The code (delta.py, 16 lines, seed 33) also simulates 2,000 years of a stock with daily rebalancing: the hedged result has a spread of $0.44, against $15.39 without the hedge. Honest notes: the model assumes constant volatility, no trading costs and no dividends; real hedging adds costs and jumps the model does not see. Education only, not financial advice. #quantfinance #options #finance #delta #hedging
QuantFinanceToGo90,400 次观看 • 19 天前
➡️ Part 31 of Quantitative Finance From Scratch: implied volatility. The Black-Scholes formula from part 30 needs five inputs. Four of them you can look up (stock price, strike, rate and time), but the fifth, the volatility over the life of the option, lies in the future. Traders form their own views, buy and sell options, and supply and demand set the price. Put that market price into the formula and solve it backwards for the volatility, and you get the implied volatility. Our test run (impliedvol.py, 16 lines of plain Python): stock 100, strike 100, rate 5 %, 1 year, and a call that costs 12 dollars on the market. The volatility sits in several places of the formula, so there is no way to solve for it by hand, but the price always rises with the volatility. A bisection search tries the middle of the range (50 %, 25 %, 12.5 % and so on) and halves the range each time. After 10 steps the price fits to half a cent, and the implied volatility is 24.1 %. Every strike and every expiry date has its own implied volatility. Together all these values form the implied volatility surface. Historical volatility, measured from past returns, serves as a comparison, and as the estimate when no options trade. Honest notes: the smile, the expiry curve and the 3D surface in the video are an illustration in the typical shape for index options, anchored at our test run, and contain no market data. The price paths and the past returns are simulated. Implied volatility shows how the market prices the expected swing, and the real swing can turn out larger or smaller. #quantfinance #impliedvolatility #options #blackscholes #finance
QuantFinanceToGo51,900 次观看 • 21 天前
Almost more important than winning any fantasy football league is having bragging rights to the best team name. The boys discuss potential options for their team name this season. #henryandjimmyv #fantasyfootball #teamname #options #ideas @Colston Loveland @Jesser @Caleb Williams
The Maley Way196,600 次观看 • 2 个月前
➡️ Quantitative Finance from Scratch, part 37: Binomial model option pricing The stock moves in steps, and each step goes up or down by a fixed factor that comes from the volatility. In one step of one year, a stock at $100 goes to $122.14 or $81.87, and a call with strike $100 pays $22.14 or nothing. To get one price today, we average the two payoffs with weights that price the stock correctly: on average it must grow like money in the bank at 5 %, which gives 58 % for up and 42 % for down. These weights are the risk neutral probability. They also price the option, because a mix of stock and cash can copy it exactly. 58 % of $22.14, discounted at 5 %, gives $12.16. With many small steps the branches join again, and walking backwards through the tree gives the price, which zigzags toward the Black-Scholes value of $10.45. The code (binomial.py, 16 lines) builds the Cox-Ross-Rubinstein tree for S = K = 100, r = 5 %, volatility 20 %, one year: $12.16 with one step, $9.54 with two, $11.04 with three, $10.25 with ten and $10.45 with 1,000 steps, against Black-Scholes $10.45. With the same numbers, a put is worth $5.57 as a European option and $6.09 as an American option that can be exercised early (1,000 steps). Source: Cox, Ross and Rubinstein, "Option pricing: A simplified approach", Journal of Financial Economics, 1979. Honest notes: the model assumes a constant volatility and interest rate, no fees and trading at every step; real prices do not move in fixed steps. #quantfinance #options #trading #blackscholes #finance Education only, not financial advice.
QuantFinanceToGo26,700 次观看 • 13 天前
#phoenix_AI our second run for our free trail. Soon includes target $NVDA, $MU, $TSLA, $AAPL, $AMZN, $GME and many more Like+Retweet+Comment to enter the trail for random 100 members. #GAMMA #Options #Stocks #Elite_Indicators #Phoenix_Gamma الان الوجبة الثانية للفترة التجريبية المجانية ل #فينيكس_AI مع الاهداف والوقف لايك وتعليق وريتويت لتكن من ضمن المجموعة الجديدة
Phoenix Stocks & Options939,782 次观看 • 2 年前
➡️ Part 32 of Quantitative Finance From Scratch: put call parity. A call gives you the right to buy a stock at the strike on a fixed date, and a put gives you the right to sell it at that price. If you buy a call and sell a put with the same strike and the same date, you end up buying the stock at the strike whatever happens, because above the strike you use your call, and below it the buyer of your put sells you the stock. That sure deal is worth the stock price today minus the present value of the strike, so the call minus the put equals the stock minus the present value of the strike. Our test run (putcall.py, 16 lines of plain Python) prices both options with Black Scholes for a stock and a strike of 100 dollars, a rate of 5 %, one year and a volatility of 20 %. The call costs 10.45 dollars and the put 5.57 dollars, and their difference of 4.88 dollars equals 100 minus 95.12, the present value of the strike. Six end prices drawn with seed 32 show that the call minus the put always pays the end price minus the strike. The rule needs no model for the stock price (Merton 1973), so traders use it to check quotes and to turn a call price into a put price. It holds exactly for European options, which can only be used at expiry, on stocks without dividends. #quantfinance #putcallparity #options #blackscholes #python
QuantFinanceToGo23,300 次观看 • 20 天前