#blackscholes
➡️ 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 просмотров • 21 дней назад
➡️ 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 просмотров • 13 дней назад
➡️ 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 просмотров • 20 дней назад
![➡️ Part 30 of Quantitative Finance From Scratch: the Black-Scholes formula. A call option is the right to buy a stock at a fixed price (the strike) on a fixed date. Fischer Black and Myron Scholes published the formula in 1973 (Journal of Political Economy), Robert Merton derived it another way, and in 1997 Scholes and Merton received the Nobel Memorial Prize in Economic Sciences. Black had died in 1995. C = S N(d1) - K e^(-rt) N(d2), with d1 = [ln(S/K) + (r + σ²/2) t] / (σ √t) and d2 = d1 - σ √t. S is today's stock price, K the strike, r the risk free rate, t the time to expiry in years, σ the volatility, and N the cumulative standard normal distribution. The stock's expected return does not appear, because stock and cash can copy the option. Our test run (blackscholes.py, 16 lines of numpy, seed 30): stock 100, strike 100, 1 year, rate 4 %, volatility 20 %. Then d1 = 0.3, d2 = 0.1, N(d1) = 0.618, N(d2) = 0.540, and the price is 61.79 - 51.87 = 9.93. One million risk neutral Monte Carlo prices give an average discounted payoff of 9.92 (standard error 0.014). With the stock at 100 the same option costs 4.49 with 3 months left and 15.08 with 2 years left. Honest notes: the model assumes a constant volatility and price paths without jumps. If you solve the formula backwards from real option prices, the volatility changes with the strike (the smile), so in practice each option gets its own implied volatility for the stock. #quantfinance #blackscholes #options #finance #python](https://ttanon.com/i/c/7699456957379398944-720.webp)
➡️ Part 30 of Quantitative Finance From Scratch: the Black-Scholes formula. A call option is the right to buy a stock at a fixed price (the strike) on a fixed date. Fischer Black and Myron Scholes published the formula in 1973 (Journal of Political Economy), Robert Merton derived it another way, and in 1997 Scholes and Merton received the Nobel Memorial Prize in Economic Sciences. Black had died in 1995. C = S N(d1) - K e^(-rt) N(d2), with d1 = [ln(S/K) + (r + σ²/2) t] / (σ √t) and d2 = d1 - σ √t. S is today's stock price, K the strike, r the risk free rate, t the time to expiry in years, σ the volatility, and N the cumulative standard normal distribution. The stock's expected return does not appear, because stock and cash can copy the option. Our test run (blackscholes.py, 16 lines of numpy, seed 30): stock 100, strike 100, 1 year, rate 4 %, volatility 20 %. Then d1 = 0.3, d2 = 0.1, N(d1) = 0.618, N(d2) = 0.540, and the price is 61.79 - 51.87 = 9.93. One million risk neutral Monte Carlo prices give an average discounted payoff of 9.92 (standard error 0.014). With the stock at 100 the same option costs 4.49 with 3 months left and 15.08 with 2 years left. Honest notes: the model assumes a constant volatility and price paths without jumps. If you solve the formula backwards from real option prices, the volatility changes with the strike (the smile), so in practice each option gets its own implied volatility for the stock. #quantfinance #blackscholes #options #finance #python
QuantFinanceToGo24,700 просмотров • 23 дней назад
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