A comprehensive guide to analytical solutions for option pricing, stochastic calculus foundations, and the Greeks.
A general 1-dimensional SDE is described as:
\[dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dW_t\]Where:
For a stochastic process $X_t$ following the above SDE, the differential of a function $f(t, X_t)$ is:
\[df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma \frac{\partial f}{\partial x} dW_t\]Special case $X_t = W_t$ ($\mu=0, \sigma=1$): \(df(t, W_t) = \left( \frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \right) dt + \frac{\partial f}{\partial x} dW_t\)
Standard model for stock prices $S_t$:
\[dS_t = \mu S_t dt + \sigma S_t dW_t\]Applying Ito’s Lemma to $f(S_t) = \ln S_t$:
\[S_T = S_t \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right)(T-t) + \sigma (W_T - W_t) \right)\]Properties:
| $E[S_T | S_t] = S_t e^{\mu(T-t)}$ |
In an arbitrage-free market, there exists a measure $\mathbb{Q}$ under which the discounted asset price is a martingale. Stock price dynamics under $\mathbb{Q}$ (with continuous dividend yield $q$):
\[dS_t = (r - q) S_t dt + \sigma S_t dW_t^\mathbb{Q}\]The value $V_t$ of a derivative with payoff $V_T$ is the discounted expected value under $\mathbb{Q}$:
\[V_t = e^{-r(T-t)} E^\mathbb{Q} [ V_T | \mathcal{F}_t ]\]The conditional expectation $V(t, x) = E^\mathbb{Q} [ e^{-r(T-t)} \psi(S_T) | S_t = x ]$ solves the Partial Differential Equation (PDE):
\(\frac{\partial V}{\partial t} + (r-q) x \frac{\partial V}{\partial x} + \frac{1}{2} \sigma^2 x^2 \frac{\partial^2 V}{\partial x^2} - rV = 0\) Boundary Condition: $V(T, x) = \psi(x)$
\(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\partial V}{\partial S} - rV = 0\)
Notation:
Standard Variables: \(d_1 = \frac{\ln(S/K) + (r - q + \frac{1}{2}\sigma^2)\tau}{\sigma\sqrt{\tau}}\) \(d_2 = d_1 - \sigma\sqrt{\tau} = \frac{\ln(S/K) + (r - q - \frac{1}{2}\sigma^2)\tau}{\sigma\sqrt{\tau}}\)
These are options that pay a fixed amount depending on the position of the underlying price at maturity.
| Measure | European Call | European Put | Digital Call (Cash-or-Nothing) | Digital Put (Cash-or-Nothing) |
|---|---|---|---|---|
| Price ($V$) | $S e^{-q\tau} N(d_1) - K e^{-r\tau} N(d_2)$ | $K e^{-r\tau} N(-d_2) - S e^{-q\tau} N(-d_1)$ | $e^{-r\tau} N(d_2)$ | $e^{-r\tau} N(-d_2)$ |
| Delta ($\Delta = \frac{\partial V}{\partial S}$) | $e^{-q\tau} N(d_1)$ | $-e^{-q\tau} N(-d_1)$ | $\frac{e^{-r\tau} N’(d_2)}{S\sigma\sqrt{\tau}}$ | $-\frac{e^{-r\tau} N’(d_2)}{S\sigma\sqrt{\tau}}$ |
| Gamma ($\Gamma = \frac{\partial^2 V}{\partial S^2}$) | $\frac{e^{-q\tau} N’(d_1)}{S \sigma \sqrt{\tau}}$ | $\frac{e^{-q\tau} N’(d_1)}{S \sigma \sqrt{\tau}}$ | $-\frac{e^{-r\tau} d_1 N’(d_2)}{S^2 \sigma^2 \tau}$ | $\frac{e^{-r\tau} d_1 N’(d_2)}{S^2 \sigma^2 \tau}$ |
| Vega ($\mathcal{V} = \frac{\partial V}{\partial \sigma}$) | $S e^{-q\tau} \sqrt{\tau} N’(d_1)$ | $S e^{-q\tau} \sqrt{\tau} N’(d_1)$ | $-\frac{e^{-r\tau} d_1 N’(d_2)}{\sigma}$ | $\frac{e^{-r\tau} d_1 N’(d_2)}{\sigma}$ |
| Theta ($\Theta = \frac{\partial V}{\partial t}$) | $-\frac{S e^{-q\tau} N’(d_1)\sigma}{2\sqrt{\tau}} - r K e^{-r\tau} N(d_2) + q S e^{-q\tau} N(d_1)$ | $-\frac{S e^{-q\tau} N’(d_1)\sigma}{2\sqrt{\tau}} + r K e^{-r\tau} N(-d_2) - q S e^{-q\tau} N(-d_1)$ | see note* | see note* |
| Rho ($\rho = \frac{\partial V}{\partial r}$) | $K \tau e^{-r\tau} N(d_2)$ | $-K \tau e^{-r\tau} N(-d_2)$ | $-\tau e^{-r\tau} N(d_2) + \frac{\sqrt{\tau}}{\sigma} e^{-r\tau} N’(d_2)$ | $-\tau e^{-r\tau} N(-d_2) - \frac{\sqrt{\tau}}{\sigma} e^{-r\tau} N’(d_2)$ |
Note on Digital Theta: \(\Theta_{DigCall} = r e^{-r\tau} N(d_2) + \frac{e^{-r\tau} N'(d_2)}{\sigma \sqrt{\tau}}\left(\frac{\ln(S/K)}{\tau} - (r - q - \frac{\sigma^2}{2})\right)\) (Often simplified using relationships between Greeks).
Notice that a European Call is essentially: \(\text{Call} = (\text{Asset-or-Nothing Call}) - K \times (\text{Cash-or-Nothing Call})\) Thus:
Options that become active (Knock-in) or null (Knock-out) if the asset price $S_t$ crosses a barrier $H$. Solutions provided here are for Single Barrier Options (Reiner-Rubinstein).
Standard Terms ($A$ - $F$ common notation): (Assuming payout is Vanilla Call/Put if valid, no separate rebate for simplicity)
Payoffs depend on price falling to $H$.
| Option Type | Condition | Formula |
|---|---|---|
| Down-and-In Call | $K > H$ | $(H/S)^{2\lambda} (S^c(y) - S^c(x_1))$ (Often simplified: CD-like terms) |
| Down-and-Out Call | $S_0 > H$ | Standard: $A - C$ (where $A$=Vanilla Call, $C$=Reflection) $C_{do} = S N(x_1) - K e^{-r\tau} N(x_2) - (H/S)^{2\lambda} [ S N(y_1) - K e^{-r\tau} N(y_2) ]$ (Note: Validity strictly for $K > H$. If $K < H$, formula adjusts). |
| Down-and-In Put | Any | $P_{di} = S e^{-q\tau} (H/S)^{2\lambda+2} N(y_1) - K e^{-r\tau} (H/S)^{2\lambda} N(y_2)$ |
| Down-and-Out Put | $K > H$ | $A - B + C - D$ (Complex interactions if $K$ vs $H$ varies). Simplified: $P_{do} = P_{vanilla} - P_{di}$ |
| Down-and-In Call (Full) | $S (H/S)^{2\lambda} N(y_1) - K e^{-r\tau} (H/S)^{2\lambda - 2} N(y_2)$ |
To be rigorous, here is the full set of 8 formulas relative to Vanilla Price ($V_{BS}$):
Let $\eta = 1$ for In, $-1$ for Out. Use indicator functions for $K$ relative to $H$. The breakdown is complex so standard texts define terms $A, B, C, D$.
Analytical Formula Summary (Haug representation):
Call Options ($S > H$, Down)
Put Options ($S > H$, Down)
Up Options ($S < H$, Up) (Symmetric to Down options by swapping $S \leftrightarrow K$, etc., but specific formulas exist).
Definitions:
Payoff: $C_T = S_T - m_T$ (Buy at min, sell at spot) \(C = S e^{-q\tau} N(a_1) - S e^{-q\tau} \frac{\sigma^2}{2b} N(-a_1) - m_t e^{-r\tau} N(a_2) + S e^{-r\tau} \frac{\sigma^2}{2b} e^{Y} N(-a_3)\) Where:
Payoff: $P_T = M_T - S_T$ (Sell at max, buy at spot) \(P = M_t e^{-r\tau} N(-b_2) - S e^{-q\tau} N(-b_1) + S e^{-r\tau} \frac{\sigma^2}{2b} [ -(M_t/S)^{2b/\sigma^2} N(b_3) + e^{b\tau} N(b_3 + \frac{2b}{\sigma}\sqrt{\tau}) ]\) (Formula form varies by recursive $M_t$ logic, standard form above simplifies for $t=0, M_0=S_0$). Standard $t=0$ ($M_0=S_0$): \(P = S e^{-q\tau} [ \frac{\sigma^2}{2b} N(-x_1) - N(-x_2) ] + S e^{-r\tau} (1 + \frac{\sigma^2}{2b}) N(x_2 - \frac{2b}{\sigma}\sqrt{\tau})\)
Payoff: $C_T = \max(M_T - K, 0)$ \(C = S e^{-q\tau} N(d_1) - K e^{-r\tau} N(d_2) + S e^{-r\tau} \frac{\sigma^2}{2b} [ -(S/K)^{-2b/\sigma^2} N(d_1 - \frac{2b}{\sigma}\sqrt{\tau}) + e^{b\tau} N(d_1) ]\) (Note: $M_t$ term omitted for $t=0$ case where $M_0=S_0 < K$). If $M_t > K$, split into certainty part + option part.
Payoff: $P_T = \max(K - m_T, 0)$ \(P = K e^{-r\tau} N(-d_2) - S e^{-q\tau} N(-d_1) + S e^{-r\tau} \frac{\sigma^2}{2b} [ (S/K)^{-2b/\sigma^2} N(-d_1 + \frac{2b}{\sigma}\sqrt{\tau}) - e^{b\tau} N(-d_1) ]\) (For $t=0$ case where $m_0=S_0 > K$).
Payoff depends on the average price $A_T$.
Geometric Asian Option (Closed Form): Average defined as $G_T = \exp\left( \frac{1}{T} \int_0^T \ln S_t dt \right)$. $G_T$ follows a lognormal distribution. We can use the Black-Scholes formula with adjusted parameters:
Arithmetic Asian Option: Approximations (e.g., Edgeworth expansion, Moment matching) or Monte Carlo are required. No exact closed form.
Exercisable at possible any time $t \le T$.
Infinite maturity ($T \to \infty$). The solution becomes time-independent (ODE).
\[P(S) = \frac{K}{1 - \gamma} \left( \frac{(\gamma - 1)S}{\gamma K} \right)^\gamma\]Optimal Exercise Boundary: \(S^* = \frac{\gamma}{\gamma - 1} K\)
where $\gamma$ is the negative root of the characteristic equation (assuming $q=0$ for simplicity): \(\frac{1}{2}\sigma^2 \gamma(\gamma - 1) + r \gamma - r = 0 \quad \Rightarrow \quad \gamma = \frac{-(r - \frac{\sigma^2}{2}) - \sqrt{(r - \frac{\sigma^2}{2})^2 + 2\sigma^2 r}}{\sigma^2}\)
In the Black-Scholes framework, risk can be completely eliminated because the market is Complete. This allows the creation of a Replicating Portfolio that perfectly mimics the option’s behavior using the underlying asset $S$ and cash (bond).
For a Delta-Hedged portfolio (Short Option + Long Stock), the Total PnL at maturity $T$ is:
\[\text{Total PnL} = \underbrace{\text{Premium Received}}_{V_0} - \underbrace{\text{Option Payoff}}_{V_T} - \underbrace{\text{Total Rebalancing Cost}}_{\text{Hedge Cost}}\]If the hedge is perfect (and assumptions hold), $\text{Total PnL} \approx 0$ (ignoring interest rate for simplicity).
Let $h_t = \Delta_t$ be the number of shares held. The cost can be analyzed in two ways using integration by parts ($d(hS) = h dS + S dh$):
1. Cash Flow View ($\int S dh$) - Practical View Focuses on the cash spent/received to adjust the position size. \(RC = \int_0^T S_t dh_t \approx \sum_{i=1}^N S_i (h_i - h_{i-1})\)
2. Capital Gain View ($\int h dS$) - Theoretical View Focuses on the trading gains/losses from holding the stock. \(RC = h_T S_T - h_0 S_0 - \int_0^T h_t dS_t\)
Scenario: Short 1 Call Option ($V$), Hedged with Stock ($S$). Objective: Delta Neutral ($\Delta_{portfolio} = 0 \implies h = \Delta_{call}$).
| Time ($t$) | Stock ($S_t$) | Option ($V_t$) | Delta ($\Delta_t$) | Stock Pos ($h_t$) | Trade ($dh_t$) | Trade Cost ($S_t dh_t$) | Cash Balance ($B_t$) |
|---|---|---|---|---|---|---|---|
| 0 | $S_0$ | $V_0$ | $\Delta_0$ | $h_0 = \Delta_0$ | $+\Delta_0$ | $S_0 \Delta_0$ | $B_0 = V_0 - S_0 \Delta_0$ |
| 1 | $S_1$ | $V_1$ | $\Delta_1$ | $h_1 = \Delta_1$ | $\Delta_1 - \Delta_0$ | $S_1 (h_1 - h_0)$ | $B_1 = B_0 - S_1 dh_1$ |
| 2 | $S_2$ | $V_2$ | $\Delta_2$ | $h_2 = \Delta_2$ | $\Delta_2 - \Delta_1$ | $S_2 (h_2 - h_1)$ | $B_2 = B_1 - S_2 dh_2$ |
| … | … | … | … | … | … | … | … |
| k | $S_k$ | $V_k$ | $\Delta_k$ | $h_k$ | $h_k - h_{k-1}$ | $S_k dh_k$ | $B_k = B_{k-1} - S_k dh_k$ |
| … | … | … | … | … | … | … | … |
| T | $S_T$ | $V_T$ | $\Delta_T$ | $h_T$ | $B_T = B_{T-1} - S_T (h_T - h_{T-1})$ |
Final Portfolio Value: \(\Pi_T = \underbrace{B_T}_{\text{Cash}} + \underbrace{h_T S_T}_{\text{Stock liquidation}} - \underbrace{V_T}_{\text{Option Payoff}}\)
If hedging is continuous and parameters match ($r=0$ case for simplicity): \(\Pi_T \approx 0\) This demonstrates that the initial premium $V_0$ plus trading gains/losses exactly covers the final payoff $V_T$.
The PnL of a delta-hedged strategy depends on the difference between Realized Volatility ($\sigma_{real}$) and Implied Volatility ($\sigma_{imp}$).
\[\text{PnL} \approx \frac{1}{2} S^2 \Gamma (\sigma_{imp}^2 - \sigma_{real}^2) dt\]References