Is This A Viable Alternative Options Pricing Method?
i’m currently a high school student who hasn’t gone past Algebra II, and thus I have minimal Calculus knowledge. I know the basics of Integration and Derivation (drop the coefficient, raise to the coefficient), Infinite series, and some basics of random walks (ex. t^0.5 = sigma).
I am unable to solve the Black Scholes equation due to it’s use of SDEs. I am however familiar with probability theory and have attempted to approximate the CDF of a Gaussian distribution using a Monte Carlo Method. I plugged in the standard deviation and strike price as (strike/sigma) then multiplied by the payout as a percent and added the risk free rate (10 year treasury) times days till expiration divided by 365 (365 days in a year duh). This gave me a value almost equal to the price on the open market. I assume that the inaccuracy is due to loss of precision when retrieving values such as the risk free rate or the asset price.
What I want to know is whether or not this method of pricing options is viable, or was the semi-successful calculation due to chance? Basically I am seeking peer review since I don’t know anyone working in finance who can verify whether I messed up or not.
My proposed method:
- a function that generates a random number using the Gaussian distribution and if the number is greater than the strike price it adds 0.0001 to an increment variable, otherwise it does not add anything. Repeat 10k times and this approximates the probability of hitting the strike where each random number is one basis point significance.
- Multiply this probability by the payout in order to find the price where the expected return of the option is zero.
- 加上期權在未平倉合約期間的無風險利率應計利息，因為出於某種原因，這包含在夏普比率和布萊克斯科爾斯模型之類的東西中。
- 該值是通過這種方法計算的期權價格，似乎與交易所交易的期權價格接近。
您正試圖通過蒙地卡羅模擬為期權定價。假設 Black-Scholes 擴散框架，它應該如何工作。
在 Black-Scholes 模型的假設下，風險資產的價值[數學處理錯誤] $ S $ 當時 $ t=T $ 是一個隨機變數，它讀取
[數學處理錯誤]$$ S_T = S_0 e^{\left(\mu-\frac{\sigma^2}{2}\right)T + \sigma \sqrt{T} Z}\tag{1} $$ 和
- [數學處理錯誤] $ S_0 $ , 資產的初始值 $ t=0 $
- [Math Processing Error] $ \mu $ , 資產的風險中性漂移[Math Processing Error] $ \color{red}{(*)} $
- [Math Processing Error] $ \sigma $ , 資產波動率
- [Math Processing Error] $ Z $ ，一個標準的高斯隨機變數， $ Z \sim N(0,1) $
因為對數[Math Processing Error] $ S_T $ 是正態分佈的
[Math Processing Error]$$ \begin{align} \ln(S_T) &= \ln(S_0) + \left(\mu-\frac{\sigma^2}{2}\right)T + \sigma\sqrt{T}Z \ &\sim N\left(\ln(S_0) + \left(\mu-\frac{\sigma^2}{2}\right)T, \sigma^2 T\right) \end{align} $$ [Math Processing Error] $ S_T $ 遵循我們所說的對數正態分佈。[Math Processing Error] $ \color{red}{(**)} $ 現在，假設一個人可以寫一個期權價格作為其未來支付的貼現期望 $ \phi(S_T) = \max(S_T-K,0) $ （這個斷言背後隱藏著很多理論）我們在哪裡使用過[Math Processing Error] $ K $ 表示期權的行使價或行使價，即
[Math Processing Error]$$ V_0 = e^{-rT} \mathbb{E} \left[ \phi(S_T) \right] = \mathbb{E} \left[ \max(S_T-K,0) \right] $$ 然後可以使用蒙特卡羅方法來計算 RHS 中的期望值。為此，只需
- 模擬隨機變數的獨立繪製[Math Processing Error] $ S_T $ （代表終端資產價值的不同場景）。這應該很容易，因為我們剛剛看到只需要模擬標準高斯的獨立繪製[Math Processing Error] $ Z $ 並使用公式 $ (1) $ 上面給出的。
- 計算相關的支出 $ \phi(S_T) = \max(S_T-K,0) $ 每次抽獎
- 平均所有採樣的支付值以獲得他們的期望。
- 最後應用折扣因子[Math Processing Error] $ e^{-rT} $ 獲得期望的期權價格
在數學上，蒙地卡羅估計[數學處理錯誤] [數學處理錯誤] $ \hat{V}_0 $ 的真實期權價格 $ V_0 $ 因此由下式給出
[Math Processing Error]$$ \begin{align} \textbf{[STEP 1] }&\ \ \ S_T^{(m)} = S_0 \exp\left( (r-\frac{1}{2}\sigma^2)T + \sigma \sqrt{T} Z^{(m)} \right),\ \ \forall m=1,…,M \ \textbf{[STEPS 2-3-4] }&\ \ \ \hat{V}0 = e^{-rT} \left( \frac{1}{M} \sum{m=1}^M \phi\left(S_T^{(m)}\right) \right) \end{align} $$
和 $ (Z^{(m)})_{m=1,…,M} $ 代表[Math Processing Error] $ M $ iid 樣本超出標準正態分佈和[Math Processing Error] $ M $ 模擬的總數。
[Math Processing Error] $ \hat{V}_0 $ 是真實溢價的無偏估計[Math Processing Error] $ V_0 $ 其變異數與[Math Processing Error] $ M^{-1/2} $ （這是中心極限定理的直接結果，該結果背後還有一些理論）。
[Math Processing Error] $ \color{red}{(*)} $ $ \mu = r_{FOR}-r_{DOM} $ 為一個[Math Processing Error] $ DOM/FOR $ 外匯匯率和風險——“自由利率”應改為 $ r = r_{FOR} $
[數學處理錯誤] [數學處理錯誤] [數學處理錯誤] $ \color{red}{(**)} $ 您可以按照您的建議使用高斯模型（Bachelier 模型），其中 $ S_T = \mu + \sigma_N \sqrt{T} Z \sim N(\mu, \sigma_N^2T) $ ，這不會改變討論的其餘部分。注意 $ \sigma_N $ 被稱為正常（或 Bachelier）波動率，而 $ \sigma = \sigma_{LN} $ 我們上面使用的現在稱為 LogNormal（或 Black-Scholes）波動率。