Implement lookback options (#52)

* Update readme example

* Implement floating strike lookback options

* Implement floating strike lookback options

* Implement floating strike lookback options for blackscholes

Author: carlobortolan

README.md

@@ -45,8 +45,8 @@ Quantrs supports options pricing with various models for both vanilla and exotic
 | ²Double Barrier             | ❌ (mod. BSM)   | ❌       | ⏳           | ⏳           | ❌ (complex)  | ⏳     |
 | ²Asian (fixed strike)       | ❌ (mod. BSM)   | ❌       | ❌           | ✅           | ⏳            | ⏳     |
 | ²Asian (floating strike)    | ❌ (mod. BSM)   | ❌       | ❌           | ✅           | ⏳            | ⏳     |
-| ²Lookback (fixed strike)    | ⏳              | ❌       | ❌           | ⏳           | ⏳            | ⏳     |
-| ²Lookback (floating strike) | ⏳              | ❌       | ❌           | ⏳           | ⏳            | ⏳     |
+| ²Lookback (fixed strike)    | ❌              | ❌       | ❌           | ✅           | ⏳            | ⏳     |
+| ²Lookback (floating strike) | ✅              | ❌       | ❌           | ✅           | ⏳            | ⏳     |
 | ²Binary Cash-or-Nothing     | ✅              | ❌       | ✅           | ✅           | ❌ (mod. PDE) | ⏳     |
 | ²Binary Asset-or-Nothing    | ✅              | ❌       | ✅           | ✅           | ❌ (mod. PDE) | ⏳     |
 | Greeks (Δ,ν,Θ,ρ,Γ)          | ✅              | ✅       | ⏳           | ❌           | ❌            | ❌     |
@@ -95,7 +95,7 @@ Add this to your `Cargo.toml`:
 quantrs = "0.1.6"
 ```
 
-Now if you want to e.g., model binary call options using the Black-Scholes model, you can:
+Now if you want to e.g., calculate the arbitrage-free price of a binary cash-or-nothing call using the Black-Scholes model, you can:
 
 ```rust
 use quantrs::options::*;
@@ -131,7 +131,7 @@ Greeks { delta: 0.013645840354947947, gamma: -0.0008813766475726433, theta: 0.17
 
 Quantrs also supports plotting option prices and strategies using the `plotters` backend.
 
-E.g., Plot the P/L of a slightly skewed Condor spread using the Monte-Carlo model:
+E.g., Plot the P/L of a slightly skewed Condor spread consisting of fixed-strike Asian calls using the Monte-Carlo model with the geometric average price path:
 
 <details>
 <summary><i>Click to see example code</i></summary>
@@ -142,20 +142,20 @@ use quantrs::options::*;
 // Create a new instrument with a spot price of 100 and a dividend yield of 2%
 let instrument = Instrument::new().with_spot(100.0).with_cont_yield(0.02);
 
-// Create a vector of European call options with different strike prices
+// Create a vector of fixed-strike Asian calls options with different strike prices
 let options = vec![
-    EuropeanOption::new(instrument.clone(), 85.0, 1.0, Call),
-    EuropeanOption::new(instrument.clone(), 95.0, 1.0, Call),
-    EuropeanOption::new(instrument.clone(), 102.0, 1.0, Call),
-    EuropeanOption::new(instrument.clone(), 115.0, 1.0, Call),
+    AsianOption::fixed(instrument.clone(), 85.0, 1.0, Call),
+    AsianOption::fixed(instrument.clone(), 95.0, 1.0, Call),
+    AsianOption::fixed(instrument.clone(), 102.0, 1.0, Call),
+    AsianOption::fixed(instrument.clone(), 115.0, 1.0, Call),
 ];
 
 // Create a new Monte-Carlo model with:
 // - Risk-free interest rate (r) = 5%
 // - Volatility (σ) = 20%
 // - Number of simulations = 10,000
-// - Number of time steps = 365
-let model = MonteCarloModel::geometric(0.05, 0.2, 10_000, 365);
+// - Number of time steps = 252
+let model = MonteCarloModel::geometric(0.05, 0.2, 10_000, 252);
 
 // Plot a breakdown of the Condor spread with a spot price range of [80,120]
 model.plot_strategy_breakdown(

examples/images/strategy.png

@@ -711,24 +711,24 @@ fn example_strategy() {
 }
 
 fn example_plots() {
-    // Create a new Monte-Carlo model with:
-    // - Risk-free interest rate (r) = 5%
-    // - Volatility (σ) = 20%
-    // - Number of simulations = 10,000
-    // - Number of time steps = 365
-    let model = MonteCarloModel::geometric(0.05, 0.2, 10_000, 365);
-
     // Create a new instrument with a spot price of 100 and a dividend yield of 2%
     let instrument = Instrument::new().with_spot(100.0).with_cont_yield(0.02);
 
-    // Create a vector of European call options with different strike prices
+    // Create a vector of fixed-strike Asian calls options with different strike prices
     let options = vec![
-        EuropeanOption::new(instrument.clone(), 85.0, 1.0, Call),
-        EuropeanOption::new(instrument.clone(), 95.0, 1.0, Call),
-        EuropeanOption::new(instrument.clone(), 102.0, 1.0, Call),
-        EuropeanOption::new(instrument.clone(), 115.0, 1.0, Call),
+        AsianOption::fixed(instrument.clone(), 85.0, 1.0, Call),
+        AsianOption::fixed(instrument.clone(), 95.0, 1.0, Call),
+        AsianOption::fixed(instrument.clone(), 102.0, 1.0, Call),
+        AsianOption::fixed(instrument.clone(), 115.0, 1.0, Call),
     ];
 
+    // Create a new Monte-Carlo model with:
+    // - Risk-free interest rate (r) = 5%
+    // - Volatility (σ) = 20%
+    // - Number of simulations = 10,000
+    // - Number of time steps = 252
+    let model = MonteCarloModel::geometric(0.05, 0.2, 10_000, 252);
+
     // Plot a breakdown of the Condor spread with a spot price range of [80,120]
     let _ = model.plot_strategy_breakdown(
         "Condor Example",

examples/options_pricing.rs

@@ -30,10 +30,6 @@ use rand_distr::{Distribution, Normal};
 pub struct Instrument {
     /// Current price of the underlying asset or future price at time 0.
     pub spot: Vec<f64>,
-    /// Maximum spot price of the underlying asset.
-    pub max_spot: f64,
-    /// Minimum spot price of the underlying asset.
-    pub min_spot: f64,
     /// Continuous dividend yield where the dividend amount is proportional to the level of the underlying asset (e.g., 0.02 for 2%).
     pub continuous_dividend_yield: f64,
     /// Discrete proportional dividend yield (e.g., 0.02 for 2%).
@@ -55,8 +51,6 @@ impl Instrument {
     pub fn new() -> Self {
         Self {
             spot: vec![0.0],
-            max_spot: 0.0,
-            min_spot: 0.0,
             continuous_dividend_yield: 0.0,
             discrete_dividend_yield: 0.0,
             dividend_times: Vec::new(),
@@ -93,32 +87,22 @@ impl Instrument {
         self
     }
 
-    /// Set the maximum spot price of the instrument.
+    /// Get the maximum spot price of the instrument.
     ///     
-    /// # Arguments
-    ///
-    /// * `max_spot` - The maximum spot price of the instrument.
-    ///
     /// # Returns
     ///
-    /// The instrument with the maximum spot price set.
-    pub fn with_max_spot(mut self, max_spot: f64) -> Self {
-        self.max_spot = max_spot;
-        self
+    /// The maximum spot price of the instrument.
+    pub fn max_spot(&self) -> f64 {
+        *self.spot.iter().max_by(|x, y| x.total_cmp(y)).unwrap()
     }
 
-    /// Set the minimum spot price of the instrument.
-    ///
-    /// # Arguments
-    ///
-    /// * `min_spot` - The minimum spot price of the instrument.
+    /// Get the minimum spot price of the instrument.
     ///
     /// # Returns
     ///
-    /// The instrument with the minimum spot price set.
-    pub fn with_min_spot(mut self, min_spot: f64) -> Self {
-        self.min_spot = min_spot;
-        self
+    /// The minimum spot price of the instrument.
+    pub fn min_spot(&self) -> f64 {
+        *self.spot.iter().min_by(|x, y| x.total_cmp(y)).unwrap()
     }
 
     /// Set the continuous dividend yield of the instrument.

src/options/instrument.rs

@@ -15,48 +15,14 @@
 //! including the current price of the underlying asset, the strike price of the option, the time to expiration, the risk-free interest rate,
 //! and the volatility of the underlying asset.
 //!
-//! ## Formula
-//!
-//! The price of an option using the Black-Scholes model is calculated as follows:
-//!
-//! ```text
-//! C = S * N(d1) - X * e^(-rT) * N(d2) for a call option
-//! P = X * e^(-rT) * N(-d2) - S * N(-d1) for a put option
-//! ```
-//!
-//! where:
-//! - `C` is the price of the call option.
-//! - `P` is the price of the put option.
-//! - `S` is the current price of the underlying asset.
-//! - `X` is the strike price of the option.
-//! - `r` is the risk-free interest rate.
-//! - `T` is the time to maturity.
-//! - `N` is the cumulative distribution function of the standard normal distribution.
-//! - `d1` and `d2` are calculated as follows:
-//!     ```text
-//!     d1 = (ln(S / X) + (r + 0.5 * σ^2) * T) / (σ * sqrt(T))
-//!     d2 = d1 - σ * sqrt(T)
-//!     ```
-//! - `σ` is the volatility of the underlying asset.
-//!
-//! The payoff of the option is calculated as:
-//!
-//! ```text
-//! payoff = max(ST - K, 0) for a call option
-//! payoff = max(K - ST, 0) for a put option
-//! ```
-//!
-//! where:
-//! - `ST` is the price of the underlying asset at maturity.
-//! - `K` is the strike price of the option.
-//! - `max` is the maximum function.
-//!
 //! ## References
 //!
 //! - [Wikipedia - Black-Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model)
 //! - [Black-Scholes Calculator](https://www.math.drexel.edu/~pg/fin/VanillaCalculator.html)
 //! - [Cash or Nothing Options' Greeks](https://quantpie.co.uk/bsm_bin_c_formula/bs_bin_c_summary.php)
 //! - [Asset or Nothing Options' Greeks](https://quantpie.co.uk/bsm_bin_a_formula/bs_bin_a_summary.php)
+//! - Musiela, M., Rutkowski, M. Martingale Methods in Financial Modelling, 2nd Ed Springer, 2007
+//! - Joshi, M. The Concepts and Practice of Mathematical Finance, 2nd Ed Cambridge University Press, 2008
 //!
 //! ## Example
 //!
@@ -74,8 +40,9 @@
 use crate::options::{
     types::BinaryType::{AssetOrNothing, CashOrNothing},
     Instrument, Option, OptionGreeks, OptionPricing, OptionStrategy, OptionStyle, OptionType,
-    RainbowType,
+    Permutation, RainbowType,
 };
+use rand_distr::num_traits::Pow;
 use statrs::distribution::{Continuous, ContinuousCDF, Normal};
 
 /// A struct representing a Black-Scholes model.
@@ -297,6 +264,57 @@ impl BlackScholesModel {
         )
     }
 
+    /// Calculate the price of a lookback option using the Black-Scholes formula.
+    ///
+    /// # Arguments
+    ///
+    /// * `option` - The option to price.
+    /// * `normal` - The standard normal distribution.
+    ///
+    /// # Returns
+    ///
+    /// The price of the option.
+    pub fn price_lookback<T: Option>(&self, option: &T, normal: &Normal) -> f64 {
+        let max = option.instrument().max_spot();
+        let min = option.instrument().min_spot();
+        let sqrt_t = option.time_to_maturity().sqrt();
+        let s = option.instrument().spot();
+        let r = self.risk_free_rate;
+        let t = option.time_to_maturity();
+        let vola = self.volatility;
+
+        assert!(s > 0.0 && max > 0.0 && min > 0.0, "Spot prices must be > 0");
+
+        println!("max: {}, min: {}", max, min);
+
+        let a1 = |s: f64, h: f64| ((s / h).ln() + (r + 0.5 * vola.powi(2)) * t) / (vola * sqrt_t);
+        let a2 = |s: f64, h: f64| a1(s, h) - vola * sqrt_t;
+        let a3 = |s: f64, h: f64| a1(s, h) - 2.0 * r * sqrt_t / vola;
+
+        let phi = |x: f64| normal.cdf(x);
+
+        match option.option_type() {
+            OptionType::Call => {
+                s * phi(a1(s, min))
+                    - min * (-r * t).exp() * phi(a2(s, min))
+                    - (0.5 * s * vola.powi(2)) / (r)
+                        * (phi(-a1(s, min))
+                            - (-r * t).exp()
+                                * (min / s).pow((2f64 * r) / (vola.powi(2)))
+                                * phi(-a3(s, min)))
+            }
+            OptionType::Put => {
+                -s * phi(-a1(s, max))
+                    + max * (-r * t).exp() * phi(-a2(s, max))
+                    + (0.5 * s * vola.powi(2)) / (r)
+                        * (phi(a1(s, max))
+                            - (-r * t).exp()
+                                * (max / s).pow((2f64 * r) / (vola.powi(2)))
+                                * phi(a3(s, max)))
+            }
+        }
+    }
+
     /// Calculate the option price using the Black-Scholes formula with a given volatility.
     ///
     /// # Arguments
@@ -369,6 +387,7 @@ impl OptionPricing for BlackScholesModel {
             (_, OptionStyle::Binary(AssetOrNothing)) => self.price_asset_or_nothing(option, &normal),
             (OptionType::Call, OptionStyle::Rainbow(_)) => self.price_rainbow_call(option, &normal),
             (OptionType::Put, OptionStyle::Rainbow(_)) => self.price_rainbow_put(option, &normal),
+            (_, OptionStyle::Lookback(Permutation::Floating)) => self.price_lookback(option, &normal),
             _ => panic!("BlackScholesModel does not support this option type or style"),
         }
     }