Financial Math & Options

Cheatsheet Content

Nominal and Effective Rates Nominal Rate: Stated interest rate. Effective Rate: Actual annual rate of return, considering compounding frequency. Relationship: $1 + \text{effective rate} = (1 + \frac{\text{nominal rate}}{n})^n$ effective rate = $(1 + \frac{\text{nominal rate}}{n})^n - 1$ nominal rate = $((1 + \text{effective rate})^{1/n} - 1) \times n$ where $n$ = no. of compounding periods in a year. Daily Compounding Daily Equivalent Rate ($R_D$): For an annual rate of $R_A$, $(1 + R_D)^{365} = 1 + R_A$. Continuous Compounding Equivalent Rate ($r_c$): For an annual rate $R_A$, $e^{r_c} = 1 + R_A \implies r_c = \ln(1+R_A)$. Relationship to nominal rate $i$: $r_c = \ln(1+i) \implies i = (e^{r_c}-1)$. Example: Given an annualized compound rate of 4.04% for 146 days. Growth factor for 146 days: $(1.0404)^{146/365} \approx 1.01594$ Continuous rate for 146 days: $\ln((1.0404)^{146/365}) = \frac{146}{365} \ln(1.0404) \approx 0.01581$ Annualized continuous rate: $\frac{\ln(1.0404)}{1} \approx 0.0396$ or $3.96\%$ Geometric Series Sum of first $n$ terms: $S_n = a \frac{1-r^n}{1-r}$ Sum to infinity: $S_\infty = \frac{a}{1-r}$ (where $|r| Where: $a$ = first term $r$ = common ratio $n$ = number of terms Example: Solving for a Payment (pmt) in a Bond-like Structure Interpretation: You pay 100 at $t=0$ and receive a sequence of payments and the redemption of 100 at $t=100$. The payment at period $k$ is $1.01^k \cdot \text{pmt}$. The per-period discount rate is $5\% = 0.05$. Setting NPV = 0 gives: $$ -100 + \sum_{k=1}^{100} \frac{1.01^k \cdot \text{pmt}}{(1.05)^k} + \frac{100}{(1.05)^{100}} = 0 $$ Rearranging to solve for pmt: $$ \text{pmt} \sum_{k=1}^{100} \left(\frac{1.01}{1.05}\right)^k = 100 \left(1 - \frac{1}{1.05^{100}}\right) $$ The left sum is a geometric series with ratio $q = \frac{1.01}{1.05}$. So, $$ \sum_{k=1}^{100} q^k = q \frac{1-q^{100}}{1-q} $$ Thus, $$ \text{pmt} = \frac{100 \left(1 - 1/1.05^{100}\right)}{q \frac{1-q^{100}}{1-q}} \quad \text{with } q = \frac{1.01}{1.05} $$ So, $$ \text{pmt} \approx 4.0128178125 $$ Future Value (FV) and Present Value (PV) Simple Interest: $FV = PV \times (1 + i \times \frac{\text{days}}{\text{year}})$ Annually Compound Interest: $FV = PV \times (1 + i)^{\text{year}}$ Continuous Compound Interest: $FV = PV \times e^{r_c \times \frac{\text{days}}{\text{year}}}$ Yield/Rate of Return Simple Yield (Short-term): $\text{Simple Yield} = (\frac{\text{FV}}{\text{PV}} - 1) \times \frac{\text{year}}{\text{days}}$ Compound Yield (Short-term): $\text{Compound Yield} = (1 + \text{simple yield})^{\frac{\text{year}}{\text{days}}} - 1$ Compound Yield (Long-term): $\text{Compound Yield} = (\frac{\text{FV}}{\text{PV}})^{\frac{1}{\text{year}}} - 1$ Discount Factors Simple Interest: $\text{Discount Factor} = \frac{1}{1 + i \times \frac{\text{days}}{\text{year}}}$ Compound Interest: $\text{Discount Factor} = \frac{1}{(1+i)^{\text{year}}}$ Continuous Compound Interest: $\text{Discount Factor} = e^{-r_c \times \text{year}}$ Net Present Value (NPV) Definition: $NPV = \sum \frac{CF_t}{(1+r)^t}$ Mortgage Payment (PMT) Definition: The periodic payment required to amortize a loan over a set period, including both principal and interest. Formula: $$PMT = \frac{PV \times \text{rate}}{1 - (1 + \text{rate})^{-N}}$$ $PV$ = Present Value (loan amount) $\text{rate}$ = Periodic interest rate (e.g., annual rate / 12 for monthly payments) $N$ = Total number of payments (e.g., loan term in years $\times$ 12 for monthly payments) Time Deposit/Loan Term: 1 day to several years (usually < 1 year). Interest: Usually paid at maturity. Certificate of Deposit (CD) Pricing: maturity proceeds = face value $\times (1 + \text{coupon rate} \times \frac{\text{coupon period (days)}}{\text{year}})$ Price = $\frac{\text{face value} \times (1 + \text{coupon rate} \times \frac{\text{coupon period (days)}}{\text{year}})}{1 + \text{interest rate} \times \frac{\text{days purchase to maturity}}{\text{year}}}$ Return (Yield): yield = $(\frac{\text{face value}}{\text{purchase price}} - 1) \times \frac{\text{year}}{\text{days}}$ yield = $(\frac{\text{sale price}}{\text{purchase price}} - 1) \times \frac{\text{year}}{\text{days held}}$ Treasury Bill (T-bill) Discount Rate Quote: Price = Face Value $\times (1 - \text{Discount Rate} \times \frac{\text{days to maturity}}{\text{year}})$ Discount to Yield Conversion: yield = $\frac{\text{discount rate}}{1 - \text{discount rate} \times \frac{\text{days to maturity}}{\text{year}}}$ discount rate = $\frac{\text{yield}}{1 + \text{yield} \times \frac{\text{days to maturity}}{\text{year}}}$ Commercial Paper (CP) Term: US: 1-270 days; ECP: 2-365 days. Interest: Non-interest bearing, issued at a discount. Bill of Exchange/Banker's Acceptance Term: 1 week to 1 year. Interest: Non-interest bearing, issued at a discount. Forward Rate Agreements (FRAs) Definition: An agreement to fix a future interest rate. Principal is not exchanged. The FRA simply compensates (or charges) the buyer as if they had borrowed at the contracted FRA rate, without requiring actual borrowing from the market. Money Market (MM) & FRA Valuation Money Market: $D = \frac{1}{1+rt}$ $r = (\frac{1}{D} - 1) \times \frac{1}{t}$ FRA: $D_S$: discount factor on forward start date $D_E$: discount factor on forward maturity date $t$: period of FRA $r$: FRA forward rate $r = \frac{D_S - D_E}{D_E \times t}$ $D_E = \frac{D_S}{1+r \times t}$ Forward-Forward Rate Calculation Formula: $\text{forward-forward rate} = (\frac{1 + i_L \times \frac{\text{days}_L}{\text{year}}}{1 + i_S \times \frac{\text{days}_S}{\text{year}}} - 1) \times \frac{\text{year}}{\text{days}_L - \text{days}_S}$ where $L$ and $S$ stand for longer and shorter period respectively. Example: Determining Rates for Floating Leg PV (Valuation Date = 14 Dec 2024) To calculate the PV of the floating leg, we need to know the interest rate for each future coupon period. These are determined based on whether the rate has already been fixed (known) or needs to be projected using forward rates. Coupon Period Known? Rate Used 23 Aug 2024 $\rightarrow$ 23 Feb 2025 Known Previous fixing (9.3%) 23 Feb 2025 $\rightarrow$ 23 Aug 2025 Unknown Forward rate (derived from current yield curve) 23 Aug 2025 $\rightarrow$ 23 Feb 2026 Unknown Forward rate (derived from current yield curve) ... and so on ... FRA Strip Example Given: 3M HIBOR: 4% (This is for a 3-month period starting now, assumes 92 days) FRA 3v6 months: 4.2% (This is for a 3-month period starting in 3 months, assumes 92 days) To calculate the effective rate for a 6-month period (e.g., 184 days), we "strip" these rates: Step 1: Future Value after 3 months (1M -> 3M) $1 + \text{current 3M rate} \times \frac{\text{days}}{365} = 1 + 0.04 \times \frac{92}{365} = 1.01008219$ Step 2: Future Value after another 3 months (3M -> 6M) Multiply the result from Step 1 by the forward rate: $1.01008219 \times (1 + 0.042 \times \frac{92}{365}) = 1.01008219 \times (1.01058082) = 1.020775032$ Step 3: Implied 6-month Fixed FRA Rate This compounded value represents the total return over $92+92=184$ days. To find the equivalent annual fixed rate for 6 months: Fixed FRA for 6M = $\frac{(1.020775032 - 1)}{(184/365)} = \frac{0.020775032}{0.50410959} \approx 0.041211 \text{ or } 4.1211\%$ Settlement Amount (for FRA period < 1 year) $\text{Buyer paid} = \text{notional} \times \frac{(\text{FRA rate} - \text{LIBOR/SOFR}) \times \frac{\text{days}}{\text{year}}}{1 + (\text{LIBOR/SOFR} \times \frac{\text{days}}{\text{year}})}$ If LIBOR/SOFR > FRA, buyer receives the net interest difference. If LIBOR/SOFR < FRA, buyer pays the net interest difference. Interest Rate Futures (IRF) A contract where the commodity is considered as being delivered at some future date. Exchange traded. Users of Futures: Hedgers: Use futures to cover risk. Speculators: Use futures for highly leveraged positions. Profit and Loss on Hedges: Net P/L: Calculated by $\text{FRA settlement P/L} - \text{Futures P/L}$ Futures P/L Example: Market maker shorted $N_f = 98.16$ futures contracts. Initial Futures Price ($P_{f, \text{initial}}$) = 92.50 Final Futures Price ($P_{f, \text{final}}$) = 92.38 Notional Unit ($N_{f, \text{Unit}}$) = €1,000,000 Multiplier for 1 basis point (0.01%) move on a 3M future: $€1,000,000 \times 0.0001 \times \frac{90}{360} = €25.00$ P/L per Contract (Based on Price Change): Price Change = $92.50 - 92.38 = 0.12$ Equivalent basis point change: $0.12 \times 100 = 12$ bps Profit per Contract = $12 \text{ bps} \times €25.00/\text{bp} = €300.00$ Total Futures P/L: $$ \text{P/L}_{Futures} = 98.16 \times €300.00 = €29,448.00 $$ The Futures results in a Profit of €29,448.00. Interest Rate Swaps (IRS) Definition: Exchange of one stream of cash flows against another. Hedging with an IRS Applied to a series of cashflows. Characteristics of IRS No exchange of principal. Only interest flows are exchanged and netted. Settlement amount paid at the end of the relevant period. Motivation for IRS (Win-Win Scenario) Leverage comparative advantages to achieve lower borrowing costs. Example: Comparative Advantage in Borrowing Company A: Floating-rate borrowing: LIBOR + 0.1% Fixed-rate borrowing: 8.0% Prefers floating-rate. Company B: Floating-rate borrowing: LIBOR + 0.8% Fixed-rate borrowing: 9.5% Prefers fixed-rate. Comparative Advantage Analysis: Company A has a 0.7% advantage in floating rates (LIBOR+0.8% - (LIBOR+0.1%)). Company A has a 1.5% advantage in fixed rates (9.5% - 8.0%). Company A has a greater comparative advantage in fixed-rate borrowing (1.5% vs 0.7%). Thus, Company A should borrow fixed, and Company B should borrow floating. IRS Transaction: Company A borrows fixed at 8.0% from the market. Company B borrows floating at LIBOR + 0.8% from the market. Company A enters an IRS to *receive* 8.4% fixed and *pay* LIBOR to Company B. Company B enters an IRS to *pay* 8.4% fixed and *receive* LIBOR from Company A. Valuation of Swap - Basic Concept Long Swap = + Long a Fixed rate bond - Floating rate borrowing. Swap Value = + PV of Fixed leg cash flow - PV of Floating leg cash flow. At inception, NPV of all cash flows = 0. Formula for Swap Valuation Where $P$ = hypothetical principal notional, $t_i$ = day count fraction of each interest payment period $i$. $C_i = \text{interest cashflow at time period } i = P \times r \times t_i$. $D_i$ = discount factor at time $i$. $D_n$ = discount factor at time $n$ (i.e. at maturity). $r$ = swap par rate (fixed leg). NPV: $NPV = -P + \sum_{i=1}^{n} C_i D_i + P D_n$ Present Value of Fixed Leg (Payer) Example of Fixed Leg PV Calculation: Notional: €10,000,000 Receive Rate: 7.4% Discount Factors: 23 Aug 2025: $D_1 = 0.9249$ 23 Feb 2026: $D_2 = 0.8825$ Day count for periods (assuming ACT/360): Period 1 (23 Aug 2024 $\rightarrow$ 23 Feb 2025): $\delta_1 \approx 184/360 \approx 0.5111$ Period 2 (23 Feb 2025 $\rightarrow$ 23 Aug 2025): $\delta_2 \approx 181/360 \approx 0.5027$ Fixed Coupon Payments: $CF_1 = €10,000,000 \times 0.074 \times \delta_1 \approx €377,999.99$ $CF_2 = €10,000,000 \times 0.074 \times \delta_2 \approx €372,016.67$ Present Value of Fixed Leg: $$ PV_{\text{fixed}} = (CF_1 \times D_1) + ((CF_2 + \text{Notional}) \times D_2) $$ Present Value of Floating Leg (Receiver) Example of Floating Leg PV Calculation (First Payment): Notional: €10,000,000 First future floating payment (23 Feb 2025) uses previous fixing = 9.3% Period: 23 Aug 2024 $\rightarrow$ 23 Feb 2025 = 184 days Day Count for this period (assuming ACT/360): $\frac{184}{360}$ Coupon for this period ($CF_1$): $$ CF_1 = 10,000,000 \times 0.093 \times \frac{184}{360} = 475,333 $$ Discount Factor for 23 Feb 2025: $D_1 = 0.9703$ (assuming this is the discount factor for the payment date) Present Value of this payment ($PV_1$): $$ PV_1 = 475,333 \times 0.9703 = 461,662 $$ Note: This example only calculates the PV of the *first* floating payment. A full floating leg PV calculation would involve summing the PVs of all future floating payments and the discounted notional principal at maturity. Par Swap Rate ($r$): $r = \frac{1 - D_n}{\sum_{i=1}^{n} t_i D_i}$ IRS Rate Calculation Example: Define Payment Dates and Day Count Fractions ($\delta_i$): Swap Term: 3 Jun 2020 to 3 Jun 2021 (1 year). Payment Frequency: Semi-annual (2 payments). Day Count Convention: ACT/365. Period (i) Accrual Dates Actual Days Discount Factor ($D_i$) 1 3 Jun 2020 to 3 Dec 2020 183 days $D_1 = 0.9632$ 2 ($n$) 3 Dec 2020 to 3 Jun 2021 182 days $D_2 = 0.9306$ The day count fraction ($\delta_i$) for each period is calculated as: $\delta_i = \frac{\text{Actual Days}}{365}$ Period 1: $\delta_1 = \frac{183}{365} \approx 0.501370$ Period 2: $\delta_2 = \frac{182}{365} \approx 0.498630$ Calculate the Numerator ($1 - D_n$): The numerator represents the present value of the final notional principal exchange. $$ \text{Numerator} = 1 - D_2 = 1 - 0.9306 = \mathbf{0.0694} $$ Calculate the Denominator ($\sum D_i \times \delta_i$): The denominator is the Present Value of the Annuity Factor (PVAF): $$ \text{Denominator} = (0.9632 \times 0.501370) + (0.9306 \times 0.498630) \approx \mathbf{0.946927} $$ Calculate the IRS Rate ($R$): $$ R = \frac{\text{Numerator}}{\text{Denominator}} = \frac{0.0694}{0.946927} \approx 0.073299 $$ Final Answer: Rounding the IRS rate to 4 decimal points: $$ \text{IRS Rate} \approx \mathbf{0.0733} $$ Pricing IRS from Futures or FRAs Use discount factors from futures to calculate par swap yield. Reversing a Swap Transact another swap in the opposite direction. Discount Factor & Forward Rate $\text{forward rate} = (\frac{D_1}{D_2} - 1) \times \frac{\text{year}}{\text{period}}$ $D_2 = \frac{D_1}{1 + \text{forward rate} \times \frac{\text{period}}{\text{year}}}$ Zero Coupon Rate $D_t$: discount factor at time $t$ $ZC_t$: zero coupon rate at time $t$ $D_t = \frac{1}{(1+ZC_t)^{\text{days/year}}}$ $ZC_t = (\frac{1}{D_t})^{1/\text{days/year}} - 1$ Day Count Basis HIBOR: 365 days LIBOR / USD Deposits: 360 days SOFR OIS (compounding): 365 days Important: Do NOT adjust the end date for weekends when counting days for interest. Yield Curve Shapes and Interpretation Yield Curve Shape Typical Market Interpretation (Expectations Hypothesis) What it really means in practice Steep upward Future short-term rates expected to rise Strong economic growth expected Flat Future short-term rates expected to stay about the same Uncertainty, transition phase, or neutral expectations Inverted Future short-term rates expected to fall Recession often anticipated Linear Interpolation Formula (for Discount Factors): $DF_{interp} = DF_1 + \frac{t_{interp} - t_1}{t_2 - t_1} \times (DF_2 - DF_1)$ Where $DF_1$ and $DF_2$ are known discount factors at times $t_1$ and $t_2$. $t_{interp}$ is the time for which the discount factor is being interpolated. Option Price What is option price? Option price (Premium) = Intrinsic value + Time value Intrinsic Value Call option: Intrinsic value = Underlying price - Strike price Put option: Intrinsic value = Strike price - Underlying price Time Value Time value = option price - Intrinsic value Moneyness Table Moneyness Call Put Intrinsic Value Remarks In-the-money $S > X$ $X > S$ Max(0, $S-X$) / Max(0, $X-S$) Intrinsic Value $> 0$ At-the-money $S = X$ $S = X$ 0 Strike price = Underlying security price Out-of-the-money $X > S$ $S > X$ 0 Intrinsic Value $= 0$ Where $X$ = Strike / Exercise price, $S$ = Underlying asset price Factors Affecting Option Pricing Factors Call Price Impact Put Price Impact Underlying Asset Price $\uparrow$ $\downarrow$ Strike Price $\downarrow$ $\uparrow$ Time to Expiry $\uparrow$ $\uparrow$ Volatility of Underlying $\uparrow$ $\uparrow$ Interest Rate $\uparrow$ $\downarrow$ Dividend Income $\downarrow$ $\uparrow$ Basic Vanilla Option Strategies Option Premium: $6 Strike Price: $100 Current Stock Price: $105 Long Stock Short Stock Long Call Short Call Long Put Short Put Strike (X) 100 100 100 100 Premium (P) -6 6 -6 6 Max Loss -105 unlimited -6 unlimited -94 -94 Max Gain unlimited 105 unlimited 6 94 6 Breakeven 105 105 106 106 94 94 Complex Strategies Straddles Call Premium: $15 Put Premium: $6 Strike Price: $100 Current Stock Price: $105 Long Call Long Put Long Straddle Short Call Short Put Short Straddle Strike (X) 100 100 100 100 100 100 Premium (P) -15 -6 -21 15 6 21 Max Loss -15 -6 -21 unlimited -94 unlimited Max Gain unlimited 94 unlimited 15 -6 15 Breakeven 115 94 115 94 Breakeven (upside) 121 79 Breakeven (downside) 79 121 Strangles Call Strike: $110 Call Premium: $6 Put Strike: $100 Put Premium: $10 Current Stock Price: $105 Long Call Long Put Long Strangle Short Call Short Put Short Strangle Strike (X) 110 100 C110/P100 110 100 C110/P110 Premium (P) -6 -10 -16 6 10 16 Max Loss -6 -10 -16 unlimited -90 -16 Max Gain unlimited 90 unlimited 6 90 16 Breakeven 116 90 116 90 Breakeven (upside) 126 110 Breakeven (downside) 84 100 Option Pricing Put-Call Parity Relationship $C + Xe^{-rT} = P + S_0$ Where: $C$ = Call price $S_0$ = Current stock price $X$ = Strike price $r$ = Continuous compounding risk-free rate annually $T$ = Option time to expiry $P$ = Put price Discrete-Time Option Pricing: Binomial Model Intermediate Parameters: $\Delta t = \text{option time to expiry} / \text{periods}$ $DF = e^{-r \times \Delta t}$ (where $r$ is the annual risk-free rate) $u = e^{\sigma \sqrt{\Delta t}}$ (where $\sigma$ is the volatility) $d = 1/u$ $p = \frac{e^{r \Delta t} - d}{u - d}$ European Call Option Price: Option Price at node = $(p \times \text{Option Price Up Node} + (1 - p) \times \text{Option Price Down Node}) \times DF$ Differences for American Call Option: American call options can be exercised at any time. At each node, compare the calculated option price with the intrinsic value ($\max(0, \text{Stock Price} - \text{Strike Price})$). The option price at that node is the larger of these two values. Continuous-time Option Pricing: Black-Scholes Model (No Dividend) European call price: $C = S_0 N(d_1) - X e^{-rT} N(d_2)$ European put price: $P = X e^{-rT} N(-d_2) - S_0 N(-d_1)$ Where: $d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}$ $d_2 = d_1 - \sigma \sqrt{T}$ $N(d)$ = standard normal cumulative probability distribution function. Volatility Measures price fluctuation; represents future price uncertainty. Only variable not directly observable. Example: Estimating Historical Volatility Volatility $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ Yearly variance: $\sigma^2 = \text{monthly variance} \times 12$ Option Risks Parameters - The Greeks Delta ($\Delta$) Sensitivity of option price to a $1 change in the underlying asset's price. Interpretation: A Delta of 0.60 means the option price changes by $0.60 for every $1 change in the underlying. Hedge Ratio: $\Delta = \frac{\text{change in option price}}{\text{change in underlying asset price}}$. For a call option in a one-step binomial model, $\Delta = \frac{C_u - C_d}{S_u - S_d}$. For Long Call: Deep ITM: $\Delta \rightarrow +1$ (highest) At-the-Money: $\Delta \approx 0.5$ Deep OTM: $\Delta \rightarrow 0$ (lowest) For Long Put: $-1 Delta hedge: Create a portfolio whose value is insensitive to small changes in the underlying asset's price. Delta call: $\frac{\partial C}{\partial S} = N(d_1)$ Delta put: $\frac{\partial P}{\partial S} = N(d_1) - 1$ Gamma ($\Gamma$) Sensitivity of Delta to a $1 change in the underlying asset's price (rate of change of Delta). Interpretation: A Gamma of 0.10 means Delta changes by 0.10 for every $1 change in the underlying. E.g., if Delta is 0.50 and stock price increases by $1, new Delta is $0.50 + 0.10 = 0.60$. Highest Gamma: When Stock Price $\approx$ Strike (At-the-Money, ATM). This is where small price moves cause the biggest change in Delta. Lowest Gamma: When the Stock is far from the Strike (deep In-the-Money or Out-of-the-Money). This means Delta is relatively stable. Gamma for both calls and puts is positive. $\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T}}$ Delta-Gamma Adjusted Hedge A strategy to make a portfolio immune to small changes in the underlying asset's price and to changes in Delta itself. This involves adjusting the portfolio to have both zero Delta and zero Gamma. Requires trading in the underlying asset and at least two different options. Vega ($v$) Sensitivity of the option price to a 1% change in the underlying asset's volatility. Interpretation: A Vega of 0.15 means the option price changes by $0.15 for every 1% increase in implied volatility. Vega peaks ATM and declines as the option becomes deep ITM/OTM. Vega for both calls and puts is positive. $v = S N'(d_1) \sqrt{T}$ Theta ($\Theta$) Sensitivity of the option price to the passage of one day (time decay). Interpretation: A Theta of -0.05 means the option price decreases by $0.05 each day, assuming all other factors remain constant. Theta is typically negative for long options (calls and puts). Time decay is strongest ATM; deep ITM/OTM options lose less time value per unit time. $\Theta = -\frac{S N'(d_1) \sigma}{2\sqrt{T}} - r X e^{-rT} N(d_2)$ (for call) Rho ($\rho$) Sensitivity of the option price to a 1% change in the risk-free interest rate. Interpretation: For a call option, a Rho of 0.02 means its price increases by $0.02 for every 1% increase in the risk-free rate. For a put option, a Rho of -0.01 means its price decreases by $0.01 for every 1% increase. Rho is generally positive for calls and negative for puts. For puts, its magnitude is largest when the option is ITM; for OTM options, Rho approaches 0. Rho increases with longer time to expiry. $\rho_c = X T e^{-rT} N(d_2)$ (for call) $\rho_p = -X T e^{-rT} N(-d_2)$ (for put)