3.) Black Scholes Merton Model
唔知大家有冇聽過Black, Scholes同Merton 呢三個人名呢
呢三個人可以話係Modern finance發展得咁快嘅功臣之一
基本上任何一個financial market背後都可以見到佢地嘅蹤影
相信有唔少人都買過call/put (認購證/認沽證) 呢類option
大家有冇好奇過究竟呢啲derivative嘅價錢係點計出黎?
仲有 眼利嘅朋友喺買呢啲option嘅時候可能會留意到
有一樣好奇怪嘅嘢叫"Implied volatility"
Volatility大家都明 但係加個implied係頭又係點解
大家唔洗心急 一切嘅問題都會係下面慢慢迎刃而解
而BSM model強大之處就在於佢可以幫大家搵到一個derivative嘅價錢嘅"close form"出黎
(記唔記得我講過SDE其實好難有close form?
)
當然佢用途唔止咁 慢慢大家就會知道點解呢個model咁重要
(background) No arbitrage pricing
喺講任何嘢之前 我地仲有一個非常非常重要嘅concept要講清楚
呢個就係no arbitrage pricing啦
Arbitrage (套利) 通常就係指risk-less profit
呢到"risk-less"嘅意思就係照字面解 要完全零風險
無論個market點樣郁 郁幾多都好
你都可以依照一套方法 係任何情況下都有profit 而呢一舊profit係
必然會賺到返黎
咁嘅話我地就會話個market有arbitrage opportunity
(
大家口中所講嘅risk-free rate
係現實其實未必等同於risk-less
因為擺deposit係銀行收risk-free rate根本就唔係risk-less 2008年已經有好例子比大家睇
)
而通常我地都會consider一個比較簡單嘅case先
假設依家個market得一隻stock 一隻derivative 一個risk-free rate
咁點check有冇arbitrage opportunity? 我地就要靠上面三樣嘢去砌一個portfolio出黎
(e.g. short sell 1 share of stock, long 1 share of call, deposit the money I get from short selling the stock in bank to receive risk-free rate )
然後我地就要睇下會唔會有一個portfolio係符合下面是但一個case
Case I:
一開頭(t=0) payoff必然>0 然後到maturity (t=T) payoff必然>= 0
Case II:
一開頭(t=0) payoff必然=0 然後到maturity (t=T) payoff必然>0
如果比你搵個一個咁嘅portfolio 咁你就可以話呢個market有arbitrage
咁大家可能又會問: 點解個market無端端會有arbitrage?
我地首先去返比較簡單嘅case先 (1 stock, 1 derivative, 1 risk-free rate)
interest rate其實係depends on money demand同money supply
stock price就係depends on本身間公司嘅underlying value
我地再假設stock price係一個fair estimate of a company's underlying value 同埋 interest rate已經比money demand同supply determine咗出黎
咁令到呢個market有arbitrage嘅元凶只有一個 --- 就係嗰隻derivative
因為derivative嘅價錢set錯咗(not its fair price)
所以就令到呢個simple market有risk-less profit
咁當然個market都唔係白痴 所有人都會意識到呢一點
而當derivative嘅價錢本身定得太低 咁就會愈黎愈多人買 價錢慢慢就會被推高
而如果定得太高 咁就會愈黎愈多人賣 價錢慢慢被推低
總之根據demand&supply 慢慢derivative嘅價錢就會去返fair price
冇人再可以有任何方法賺到risk-less profit => Arbitrage opportunity does not exist
之但係price derivative嘅人都唔係白痴
佢地當然知咩係arbitrage 亦都知就算一開頭有arbitrage都好 最終靠個market都可以achieve到fair price
個問題就係佢唔會咁傻仔比我地賺一開頭嗰啲risk-less profit呀嘛
而呢個就係所謂嘅no arbitrage pricing --- 一開頭就將derivative嘅價錢set做fair price
(其實唔單止一開頭 任何time point都唔應該有arbitrage)
咁就唔會有任何arbitrage出現
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(a) Black Scholes Equation
講咗咁耐arbitrage 我地終於要入戲肉
首先講下Black Scholes Model嘅setting同assumption先
大家都知道任何model都冇可能同現實100%一樣 (就算強如particle physics嘅standard model都唔係)
所以適量嘅assumption係必須嘅 同時亦都可以令我地之後輕鬆啲
基本assumptions如下:
1.) Market consists of at least one stock and one risk-less asset (i.e. money market, cash, or bond)
2.) The rate of return on the risk-less asset is constant and thus called the risk-free interest rate
3.) Stock price follows Geometric Brownian motion with constant μ and σ
4.) The stock does not pay a dividend (We can generalize it later)
5.) There is no arbitrage opportunity in the market
6.) It is possible to borrow and lend any amount, even fractional, of cash at the risk-less rate
7.) It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling)
8.) No transaction cost (i.e. friction-less market) (We can generalize it later)
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