Understand Rho and interest rate impact — Hinglish notes

5.3.5 · Stock-Market › The Greeks

Rho KYA hai?

Call rho positive hota hai: higher rates → higher call value.
Put rho negative hota hai: higher rates → lower put value.
Rho short-dated options ke liye sabse zyada ignore kiya jaane wala Greek hai (effect chhota hota hai) lekin LEAPS (long-dated options) ke liye bahut matter karta hai, kyunki ye time ke saath multiply hota hai.

Interest rate price ko kyu move karta hai? (First principles)

Black–Scholes call price hai: $C = S_{0} N (d_{1}) - K e^{- r T} N (d_{2})$

Key player hai term $== K e^{- r T} ==$ , jo strike ka present value hai jo tum expiry par pay karoge.

Black–Scholes se Rho kaise derive karein

Call formula se shuru karo. Sirf $K e^{- r T} N (d_{2})$ aur $d_{1}, d_{2}$ (through $r$ ) $r$ par depend karte hain.

Step 1 — derivative set up karo. $\frac{\partial C}{\partial r} = \frac{\partial}{\partial r} [S_{0} N (d_{1}) - K e^{- r T} N (d_{2})]$ Ye step kyun? Rho defined hai $\partial C / \partial r$ ke roop mein; hum har $r$ -dependent piece ko differentiate karte hain.

Step 2 — discount term ko product rule se differentiate karo. $\frac{\partial}{\partial r} [K e^{- r T} N (d_{2})] = from e^{- r T} - T K e^{- r T} N (d_{2}) + from d_{2} K e^{- r T} N^{'} (d_{2}) \frac{\partial d _{2}}{\partial r}$ Ye step kyun? $e^{- r T}$ aur $d_{2}$ dono mein $r$ hai; product rule contribution ko split karta hai.

Step 3 — "hidden" terms cancel ho jaate hain. $\partial d_{1} / \partial r$ ke pieces (jo $S_{0} N (d_{1})$ mein hain) aur $\partial d_{2} / \partial r$ (upar wala doosra term) exactly cancel ho jaate hain, kyunki $S_{0} N^{'} (d_{1}) = K e^{- r T} N^{'} (d_{2})$ (ek standard Black–Scholes identity) aur $\partial d_{1} / \partial r = \partial d_{2} / \partial r$ . Ye step kyun? Ye famous simplification hai — $d_{1}, d_{2}$ ki $r$ ke saath sensitivity wash out ho jaati hai, sirf clean discount term bachta hai.

Step 4 — jo bachta hai use collect karo. $ρ_{call} = \frac{\partial C}{\partial r} = K T e^{- r T} N (d_{2})$

Put–call parity ( $C - P = S_{0} - K e^{- r T}$ ) se, $r$ ke w.r.t. differentiate karo: $\frac{\partial C}{\partial r} - \frac{\partial P}{\partial r} = K T e^{- r T}$ Toh: $ρ_{put} = \frac{\partial P}{\partial r} = - K T e^{- r T} N (- d_{2})$

Figure — Understand Rho and interest rate impact

Worked Examples

Steel-manned Mistakes

Rho kya measure karta hai?

Risk-free interest rate mein 1% (0.01) change hone par option ki price ka change,

ρ = \partial V / \partial r

Call rho vs put rho ka sign?

Call rho positive (rates up → call up); put rho negative (rates up → put down).

Call rho ka formula (Black–Scholes)?

ρ_{call} = K T e^{- r T} N (d_{2})

Put rho ka formula?

ρ_{put} = - K T e^{- r T} N (- d_{2})

Call price mein kaun sa term rho drive karta hai, aur kyun?

K e^{- r T}

, strike ka present value; zyada

r

ise shrink karta hai toh call (jo strike baad mein pay karta hai) zyada valuable ho jaata hai.

Short-dated options ke liye rho chhota kyun hota hai?

Rho

T

ke saath scale karta hai; chhota

T

K T e^{- r T} N (d_{2})

ko tiny bana deta hai.

Kaun se options sabse zyada rho-sensitive hote hain?

Long-dated options (LEAPS) — bada

T

rho ko magnify karta hai.

Put–call parity se rho ka relation?

ρ_{call} - ρ_{put} = K T e^{- r T}

(

C - P = S_{0} - K e^{- r T}

r

ke w.r.t. differentiate karo).

Rates badhne par put value kyun girta hai?

Put tumhe strike baad mein receive karne deta hai; zyada

r

us paison ke present value ko kam karta hai, toh put kam valuable ho jaata hai.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho tumhare paas ek coupon hai jisse tum agले saal ek video game $50 mein khareed sakte ho. Agar banks tumhe zyada interest dene lagte hain, toh $50 jo tumhare pocket mein rakhte ho agले saal tak zyada badh jaata hai — toh $50 abhi ki jagah baad mein pay karna ek meetha deal hai. Tumhara coupon (ek call) zyada valuable ho jaata hai. Rho sirf ek number hai jo tumhe bataata hai: "har 1% jab bank interest badhata hai, tumhare coupon ki value itni badhti (ya ghatti) hai." Jo coupons bahut door future mein expire hote hain (LEAPS) unhe bahut fark padta hai; jo agले hafte expire ho rahe hain unhe barely notice hota hai.

Connections

The Greeks — delta/gamma/theta/vega/rho ka overview.
Black-Scholes Model — $C = S_{0} N (d_{1}) - K e^{- r T} N (d_{2})$ ka source.
Present Value and Discounting — $e^{- r T}$ jo rho ko power karta hai.
Put-Call Parity — call aur put rho ko link karta hai.
Theta and Time Decay — doosra $T$ -driven Greek.
Cost of Carry and Dividends — $q$ rate sensitivity ko kaise modify karta hai.